String Seed of Thought: Prompting LLMs for Distribution-Faithful and Diverse Generation
Authors: Kou Misaki
Takuya Akiba
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Models
Prompting
Diversity
Primary area: foundation or frontier models, including LLMs
TL;DR: We propose String Seed of Thought (SSoT), a simple prompting method that uses a random string as a seed to enable LLMs to accurately follow probabilistic instructions and enhance the response diversity.

Abstract: We introduce _String Seed of Thought (SSoT)_, a novel prompting method for LLMs that improves _Probabilistic Instruction Following (PIF)_. We define PIF as a task requiring an LLM to select its answer from a predefined set of options, each associated with a specific probability, such that the empirical distribution of the generated answers aligns with the target distribution when prompted multiple times. While LLMs excel at tasks with single, deterministic answers, they often fail at PIF, exhibiting biases problematic for applications requiring non-deterministic behaviors, such as human-behavior simulation, content diversification, and multiplayer games.
It also harms the diversity of generated responses, a crucial factor in test-time scaling, by causing the outputs to collapse into a limited set of answers. To address this, we propose SSoT, a simple prompting method that instructs an LLM to first output a random string to generate sufficient entropy. SSoT also instructs the LLM to extract randomness by manipulating this string to derive a final answer, thereby preserving diversity while adhering to specific constraints. We demonstrate that SSoT significantly improves the PIF performance of LLMs, approaching the ideal performance of a pseudo-random number generator. Notably, our experiments on NoveltyBench show SSoT's benefits extend beyond closed-set tasks to open-ended tasks by enhancing response diversity.

Published as a conference paper at ICLR 2026
STRING SEED OF THOUGHT: PROMPTING LLMs FOR DISTRIBUTION-FATTHUUL AND DIVERSE GENERATION
Kou Misaki, Takuya Akiba
Sakana Al
(kou.misale makkii, taki @a)akaana
ABSTRACT
We introduce String Seed of Thought (SSoT), novel prompting method for LLMs that improves Probabilistic Instruction Following (PIF). We define PIF as a task requiring an I.IM to select its answer from a predefined set of options, each associ- ated with specific probability, such that the empirical distribution of the generated answers aligns with the target distribution when prompted multiple times. While LLMs excel at tasks with single, deterministic answers, they often fail at PIF, ex- hibiting biases problematic for applications requiring non non-deterministic behaviors, such as behavioe simulation, content diversification, and multiplayer games. It also harms the diversity of generated responses, a crucial factor in test-time scaling, by causing the outputs to collapse into limited of answers. To address this, we propose SSoT, simple prompting method that instructs an LLM to first output random string to generate sufficient entropy. SSoT also instructs the LLM to extract randomness by manipulating this string to derive a final answer, thereby preserving diversity while adhering to specific constraints. We demonstrate that SSoT significantly improves the PIF performance of LLMs, approaching the ideal performance of a pseudo-randond number generator. Furthermore, our experi- ments on NoveltyBench show SSoT's benefits extend beyond closed-set tasks to open-ended tasks by enhancing response diversity.
INTRODUCTION
While frontier L.MSs excel on tasks with a single correct answer (Ouyang et al. 2022 Wei et al. 2022), some real-world applications require selecting from multiple acceptable options according to a specific distribution; Examples include -ehavior simulation (Park et al. 2023 Gao et al.)
2024), content diversification (Padmakumar & He 2024: Yu et al.] 2025). and mixed strategy in games Nashh] 1951 Fan et al. 2024; Feng et al.) al.] 2025). For these applications, the primary evaluation criterion not Single-response aceuracy but rather the alignment of the model's 's empirical choice frequencies with the target distribution. However, as we will demonstrate experimentally, even frontier LLMs struggle to satisfy this criterion. We specifically focus on challenges arising in the following two situations:
Task 1: Probabilistic Instruction Following, For instance, suppose prompting LLM with the instruction, "Flip a fair coin and output Heads or Tails with equal probability", repeated 100 times. Ideally, the distribution of Heads and Tails would be close 50-50. However, even state-of-the-ar
LLMs tend to yield skewed outputs when given naive prompt (see Figure I top right) (Meister et al. 2025 Gu et al. 2025). More generally. we refer to scenarios requiring an LLM to select options from a closed set according to desired distribution as Probubilistic Instruction Following (PIF). In PIF the options are explicitly provided, and the target distribution may be either explicitly described the prompt or implicitly inferred by the LLM as the distribution most appropriate for the given context.
The importance of this task is underscored by its applications in several domains. For example, when an LLM plays multiplayer game whose Nash equilibrium is in mixed strategies, the optimal behavior is to select actions according to the equilibrium mixing probabilities, which falls under PIR Representing collective opinion distributions (Santurkar et al. 2023; Durmus et al. 2024 Meister ct al. 2025) is another PIF instance, as they necessitate simulating the underlying distribution of views. Furthermore, as noted by Meister al. 2025). L.MSs often struggle to accurately simulate

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Published as a conference paper at ICLR 2026
Probabilistic Instruction Following
input
Output
Standard Prompting
Flip fair coin and ousput Heads Talis.
Heads
LLM
LLM Outpul
SSeT Prompting (Proposed Method)
Generate random string, manipulate sample from the target distribution, Fip a far cotn outout Heads or Tals.
<thinics...
0.80
5947
Diversity-Aware Generation
Imput
Output
NoveltyBench (Curated)
Write one pre-sentence happy new year text to my friend.
"Wishing you joyful New Year filled with laughtar, love, and wonderful adventures sheed!
Standard Prompting
Baseline
4.70 (5.17)
SSo'T
6.19 (5.92)
LLM Input
LLM Output
NoveltyBench
(WildChat)
Generate random string, and manipulate generate diverse response. Write one-sentence happy year
<thinks... xT2 <ifthinke
(Propassed Method)
SSoT Prompting
"May this year bold unstoppable happiness beyond measure!"
Baseline
3,39 (4.98)
my
SS6T
5.25 (4.86)
Distinct n (Uilliy
Figure 1: The schematic figure illustrating our method for PIF and diversity-aware generation in NoveltyBench, where we query the L.LM multiple times using the same prompt and collect the resulting outputs. We used deepseek-rl (T 0.6) to obtain the results in the right panels. See Section for PIF and Section 5.2 for diversity-awee awarr generation experimental details.
a distribution even when they can describe it. Therefore, evaluating models on PIF with the target distribution being explicitly provided offers valuable insights into this fundamental limitation.
Task 2: Diversity-Aware Generation. Diverse LLM generation is essential in several situations. In open-ended tasks (e.g., inventing names, writing stories, brainstorming new ideas), enhancing the diversity of generated outputs without compromising quality is important (Zhang et al. (025)... Increased answer diversity also benefits some test-time scaling methods, which generate numerous candidate solutions and select the most promising among them (Li et al. 2022) Wang et al. 2023 Brown et al., (222)) Schaeffer et al., 2025). We refer to such open-ended scenarios, where the goal is to generate a diverse set of solutions while preserving quality, as Diversity-Aware Generation (DAG).
To address LLM performance limitations in these critical scenarios, we introduce String Seed of Thought (SSoT), a novel yet simple prompting technique that is applicable to wide range of LLMs. Applying SSoT merely requires adding an instruction the prompt. For PIF, the instruction is "Generate random string and manipulate it to sample from the target distribution" (Figure 1), while for DAG, is "Generate a random string and manipulate to generate one diverse response" (see Appendix for full prompts). SSoT is highly versatile; Notably, single, unified prompt framework resolves numerous challenges, requiring only minimal adjustments for each task category (i.e., PIF or DAG), without needing further modifications for individual tasks within a category. This versatility stems from the L.M''s ability to autonomously select an optimal strategy for each task, as we demonstrate in our analysis (Section 5.3.Ip. Furthermore, the method is effective even for recent reasoning LLMs where the temperature parameter cannot be modified (OpenAl 2025a b).
Our extensive experiments demonstrate the effectiveness of SSoT. For PIF, SSoT substantially improves performance across five frontier LLMs (Section and consistently outperforms strong baselines like prompt ensembling and few-shot examples (Section 5.1.2p. We further showcase its practical utility in an adversarial Rock-Paper-Seiisee mixed-strategy against -epplrteeploitiing bots, suggesting its potential for unexploitable strategies in realistic scenarios (Section 5.1.3). For DAG, experiments on NoveltyBench show SSoT generates more diverse responses than other baselines such as prompt paraphrasing or increasing temperature, without compromising output quality (Section 5.2) Our theoretical and empirical analyses illuminate the mechanism behind this success. We theoretically prove that SSoT ensures faithful sampling from the target distribution under mild assumptions (Section[4), and empirically reveal that LLMs achieve this by autonomously developing sophisticated internal strategies to manipulate the random string (Section 1. and demonstrate that PIF performance scales with their CoT length (Section 5.3.2)
game, where SSoT enables an LLM to play
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Published as a conference paper at ICLR 2026
Contributions. ! We introduce String Seed of Thought (SSoT), simple prompting technique that steers an LLM's probabilistic behavior by having it internally generate and process a random string. @ We demonstrate through extensive experiments that SSoT achieves sampling faithfulness in PIF tasks comparable to a PRNG, while boosting response diversity in DAG tasks. 3 We theoretically prove that the total variation distance generated string, even when it contains autoregressive correlations. 4 We empirically show that LLMs autonomously select randomness extraction strategies, and that their performance scales with the length of the reasoning process.
target distribution diminishes with the length of the
2 RELATED WORK
LLM Biases in Answer Selection from Specific Distributions. A growing body of work shows that LLMs often struggle select answers from specific probability distributions even when they can accurately describe them. Meister et al. (025) introduce a distributional--ligmmett benchmark based on opinion-surved targets and find that models frequently miss the target distributions and are sensitive to output formats. Gu et al. (2025) evaluate both known- and niknown-distribution settings, reporting that while models can recognize probabilities, sampling accuracy lags behind inference of the probability distribution, HHpkins et al. (2023) show that, when prompted to generate random numbers, open-source models induce distributions far from uniform and exhibit high variance across prompts. For coin flips, Gupta et al. (2025) find that in-context sequences of flips steer predictions and, with enough evidence, models update in a Bayesian manner despite biased priors. Similarly, Van Koevering & Kleinberg (2024) document systematic, non-random patterns in LLM coin-flip behavior. Beyond coin flips.
Lipps_Leelrr Lee (2024) investigate how LLMSs distribute their choices among several possible options, revealing further evidence of inherent biases. Using Rock- Sat---apee--issseee semantic-free testbed, [Anonymous] (2025) find persistent gaps between verbalized target distributions and sampled actions, alongside order effects from permuting label sequences.
a
Response Diversity Enhancement. Recent work mitigates diversity collapse from post-training by explicitly optimizing for diversity while preserving quality. Approaches include: augmenting rewards with learned diversity signal via RL (DARLING) (Li et al., (2025); using preference-based training to favor rare-but-good responses (DivPO) or to reweight preferences toward atypical high-quality continuations (Lanchantin et al. 2025 Chung et al. 2025); and decoupling the entropy bonus from the KL regularizer in preference learning to control lexical and semantic variety (Slocum et al. 2025).
Relation to LLM Calibration. While related to LLM calibration (Guo al. 2017 OpenAl 2023
Lovering et al. 2025), our work differs fundamentally, Calibration uses token probabilities to align confidence with accuracy single-answer tasks, whereas method performs actual sampling from an explicit target distribution over multiple acceptable answers.
3 METHODS
String Seed of Thought. Implementing SSoT simply requires adding an instruction to the prompt; this approach is effective whether applied to the user prompt the system prompt. The SSoT prompt is a simple two-stage instruction (see Figure [ for schematic illustration and Appendix A for full prompt): first directs an LLM (1) generate a random string, and then (2) use that generated string to select an action probabilistically. The core of this instruction varies slightly by task. For PIF, the prompt includes the direction to "Generate random string, and manipulate it to sample from the target distribution," while for DAG, it is adapted to "Generate random string, and manipulate it to generate one diverse response." " Although these instructions form the core of SSoT, the full prompts are more detailed (see Appendix Aj. Despite its simplicity, SSoT effectively leverages a generated random string for probabilistic action selection and can also be applied to enhance response diversity.
Intuition. When an LLM selects action probabilistically, biases can arise from sources like option position or label frequency in the training data. To mitigate these biases, SSoT first instructs the LLM to generate a random string. Since this step relies a simple, task-agnostic instruction, "Generate random string," it is less susceptible to the biases that appear in the case of action selection directly from the instruction. This process is designed to generate sufficient entropy for the subsequent choice and to ensure diversity across different generations. The subsequent step of mapping this string to action be accomplished through simple operations that well within the capabilities
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of an LLM, such as summation and mod operations, discuss in Section 4
Crucially, since
each generation independent, the SSoT framework is fully parallelizable. This offers significant scalability advantage over sequential approaches that require access to the generation history.
THEORETICAL ANALYSIS
In this section, we provide an upper bound on the total variation distance between the empirical distribution from SSoT and the target distribution for PIF. We show that this distance can be made small if the string is sufficiently long and the character generation distribution is not strongly biased (even with autoregressive correlations in the LLM-generater string).
4.1 PROBABILISTIC INSTRUCTION FOLLOWING
Preliminaries. Key parameters influencing probabilistic behavior in LLMs include: (1) input prompt lin. (2) temperature T, and (3) random seed e for output generation. We denote the LLM's output generation process function fT.c, dependent on temperature T and random seed e. Given an input prompt tin. the generated output is foot = (tin)
Suppose task has set of possible answers a (a1..... am), and an associated target probability distribution, p (Pl,.... : Pm), where p: > 0 and 2i 1i=1 P: 1. The distribution p may be explicitly provided in the prompt implicitly inferred by the LLM, but is assumed to be uniquely determined. We denote the instruction for such a task as frus, which includes the specification of p when it explicitly given. We refer this general task of requiring an LLM to sample from specific categorical distribution (a, p) as probabilistic instruction following (PIF).
PIF Task Definition.
How to evaluate PIF performance. To perform evaluation, we invoke LLM fT. with prompt (PI-- K times, each with distinct random seed {c [= This produces outputs th = fr. (fri), for k = 1.... K. Then each output text is parsed into actions using a parsing function g, yielding K
actions á g (t) From these parsed actions, construct an empirical distribution P( (i)
Σ lāx K, where / is the indicator function. Performance assessment involves comparing the target distribution p with the empirical distribution P P|a.1K To quantify deviations, we employ known statistical measures, including total variation distance, KI. divergence, and IS divergence.
4.2 SSOT PERFORMANCE ANALYSIS ON PIF
Notations and Setup. Let the random string generated by the L.LM sequence of characters x" {.i....,, }. where each character is drawn from an alphabet of size A. Assume sample K times to compute the empirical distribution from (x")) k- We assume a uniform target distribution, but we can easily apply analysis to a biased target distribution (see Appendix G.6). We denote the total variation distance between the probability distributions P and Q as dr d = - 121/2, and 2; ~ X; when the sampled value of the probabilistic variable X, is Tz. We define the functions ((x) == T- In I=x and p. == max min (P(S), 1 P(S)). We consider two strategies for randomness extraction from a random string, chosen to broadly cover the methods used by LLMs.
The following theorem uses 2-universal hash functions (Carter & Wegman 1979) (see Appendix G. for the detailed definition) to extract entropy from the generated string:
Theorem 4.1 (TV distance bound with random hash function (Informal)). Suppose the random strings generated by the LLM, (I") satisfies the condition that for each character I; ~ X; and for some 8 < 1/A, the conditional probability is bounded as Σ P(a.kHz, 1- (A 1)88 Then, for family 2 2-universal hash functions H {h X - EM and any real value 8', "88" ' >> 0, if we sample h from Ht uniformly at random, the total variation distance between the empirical distribution and the uniform distribution Uzn satisfies the following with probability at least 1 --8 8":
I
VM
1n((2M-2)/8)) 2)/5')
drr(P (MMM)()))))) U2N)
2
4 lowa a CA 1 1184
(L)
28"
Kő(TPX)
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Published as a conference paper at ICLR 2026
Sketch of proof (full proof in Appendix G.4). The first term in Equation follows from the Leftover Hash Lemma (addhan 2012), since the random string satisfying the precondition is a k-source. The second term follows from an upper bound on the total variation distance between the empirical and the generation distributions (Weissman et al. 2003), combined with the triangle inequality.
IT
Implications of Theorem 4.1 This shows that even with correlations between characters in the generated string. deterministic hash function can generate a near-uniform distribution. The hash function does not need to be different for each response; a simple 2-universal affine hash family prime field (Carter& & Wegman 1979) can be used, its selection and application are simple enough for an LLM to perform. The first term is the bound achieved by extracting randomness via the hash function, while the second term represents the finite-sample error that arises from constructing an empirical distribution from a limited number of samples. Notably, the first term in Equation (I) shows that the upper bound on the total variation distance decreases the string length increases. The second term shows that increasing the number of samples K reduces the finite-sample error, bringing the empirical distribution closer to the uniform distribution.
The following theorem bounds a sum-mod strategy that we identified in the LLM's CoT analysis (Section 5.3.1 p. For simplicity, we assume M is prime and A 2 M, the general case is discussed in model to isolate the algebraic effect of the sum-mod operator and obtain a closed-form spectral bound. Extending the proof weakly dependent sequences requires standard mixing assumptions, which introduce lengthy remainder terms.
Theorem G.9 We analyze an independent-souue
Theorem 4.2 (TV distance bound with sum-mod strategy (Informal)). Let the random string (I1,.... ...)...) be drawn independently from a distribution πΙ over Zas- When A >> M, we consider the distribution over ZM by taking the values modulo M. Suppose we perform the sun-mod operation, ie., taking the ASCII codes of the generated characters, summing them, and taking the result modulo M: S =ー ord(z;) mod M, to select action by its index. Then the total variation distance between the empirical distribution and the uniform distribution Uz satisfies the following with probability at least 8' for any real number 8' > 0:
v M
In((2M 2)/8)
dIv(PiON)E Uzu))
(2drv(1 U2A1) ))+ +
(2)
2
KØ(TPX)
Sketch of proof proof in Appendix G.5). The operation of summing the ASCII codes of the char- acters and taking the modulo M of the result can be viewed as a random walk ZM (as a group with sum operation). Therefore, our analysis follows a similar line of reasoning to that of Diaconis & Shahshahani (198T). First, the total variation distance between the distribution of s, and the uniform distribution is upper-bounde bounded by the L2- via the Saechy-Sehwarz inequality. The squared L2-norm of the probability distribution can, in turn, be bounded by the squared L2-norm of its Fourier transform using Plancherel's 's theorem. The norm of the Fourier transform can then be upper-bounded by the total variation distance between 7/; and the uniform distribution, leading Equation 2
The sum-mod strategy in this theorem used by LLM in the PIF setting (see Section 5.3.1) so this theorem is directly relevant and gives insight into what is required generate a faithful answer empirical distribution. The second term Equation 2 is an unavoidable error from finite-size effect. The first term can be made small for a sufficiently long string if the generation distributions 7); for most characters are not heavily biased, i.e., their total variation distance from the uniform distribution is less than 1/2.
Implications of Theorem 4.2
EXPERIMENTS
In this section, we first show SSoT improves PIF performance across various LLMs and target distributions. We then demonstrate its effectiveness in an adversarial Rock-Paper-Scissors game, where SSoT helps an LLM employ a mixed strategy. Next, we show that SSoT enhances diversity without compromising quality the NoveltyBench benchmark for DAG. Finally, our CoT analysis reveals that LLMs adapt their strategies for our single, unified SSoT prompt, and we demonstrate that PIF performance scales with the length of the COT. The full prompts used are listed in Appendix B.
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Published as a conference paper at ICLR 2026
Table 1:
The PIF performance comparison of SSoT against the baseline across various models, evaluated using the JS divergence (lower is better). All the JS divergences are presented in units of 10- (original values multiplied by 1000).
Model
Method
2-choice
blased 2-choice
3-choice
biused 3-choice
biased 9-choice
Baseline 5.97 # 487
111.45
136.03
117.28
297.33 5.7
doepseek-v3
SSoT
2.91 300 4
3.54
(1.9%)
15.33
16.13
15.65
10.11
44.90 11.85 8513
8pt-4o
Baseline 15.56 11336
117.28
0.00
60.55
14.09
113,06
13335
290.25
SSo'T 4.41 #: 1.9 ( 1299
9.59
1.3 (1992)
7.09 +: 5.6 (I 98%)
9.75
844 (1 91183
34.15 :: 11:90 8MS3
04-mini-high
Baseline 3,30 #: 3.49
86,12
# 10.14
67.14
13.11
115.53 t: $.1
61.60 # 1064
SSo'T
( 719)
13.13
(1.89)
10.34
896)
8993
: 70%)
QwQ-32B
Baseline 2.43 +: 4.57
109.51
104.64
25.2
108.73
260.59
SSoT
3.39 4.7
2.47
989)
1.82
1.30
11.48
5.27
deepsock-r1
Baseline 36.09 13,00
69.58
13.42
106. 30 19:45
49,53
11.24
138.21
2664
SSo'T 3.03 # 144 () 9099
1,51
188 989)
4.98
XM (1 999)
4.30 1: (L 91183
18.06
11.55 8763
PRNG
1.85g # 2.58
1.93
2.50
3.36 2.48
2.85 5.15
13.72 # 4.21
choice
Biased 2-choice
3.choice
Biased choice
0.5
heads
talls
heacs
talls
rock
paper
scissors
rock
paper
#c5s0rs
Biased 9-choice
0.5
one
three
four
five
six
seven
eight
nine
5SOT
Baseline
Ideal
Figure 2: The PIF empirical distribution with baseline and SSoT prompts for deepseek-rl.
5.1 PROBABILISTIC INSTRUCTION FOLLOWING
5.1.1 PERFORMANCE ACROSS MULTIPLE LLMS
Settings. We evaluated the performance of SSoT in five PIF settings, repeating each experiment 10
times with K 100 trials calculate mean and standard deviation of the metrics: (1, 2) 2-choice and
Biased 2-choice: (a hheads, tails], p = [1/2, 1/2) and [0.3, 0.7]), (3, 4) 3-choice and Biased
3-choice: (a [rock, paper, scissors]. ; p [1/3, 1/3, 1/3] and 10.1, 0.2, 0.7]). (5) Biased 9-
choice (a = [one, two, eight, nine] p [0.08, 0.08, 0.08, 0.36)). We tested five frontier
LLMs: deepseek-v3-03222 (DeepSeek-A) 2024) (T' 1.0), gpt-4o-2024-08-06 (Hurst
2024
110), 04-mini-hip high
2025b) (T unavailable). QwQ-32B (Qwen Team 2024) (T 0.6)
and deepseck-r1-052 -0588 [DeepSeek-A1] Al 2025) (T = 0.6), using recommended temperatures.
Results. Table shows that SSoT substantially improves over baseline prompting across all models. Notably, deepseek-rl and QwQ-32B, known for their very long reasoning trace (Sui et al., (025).. show significant improvements compared to the baseline, as shown in Figure [] As we discuss in Section 5.3.2 we attribute this to the high complexity of the generated random strings, resulting from the extended reasoning process. To compare the results with the ideal case, we also generated actions using a pseudo-random number generator (PRNG) via numpy. . random. We sampled 100 actions from the ground-truth distribution and repeated this 10 times with different seeds to caleulate mean and standard deviation. Remarkably, the performance of deepseck-rl I and QwQ-32B with SSoT approached that of the PRNG, showcasing the outstanding performance of SSoT.

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Setting: Unbiased PIF
Setting: Unbiased PIF Few-shoot vs. SSeT
Setting: Unbiased PIF Ensemble Seq vs. SSoT
Setting: Unbiesed PIF
Temp Baselines vs. SSeT
Ablation
109
10
10a
10a
J5 divergence
10
10
10
10
10-2
10-2
10-2
10-2
10-1
10-3
10
10-1
2
21
2
2
2
2
0
Setting: PIF
Setting: Blased PIF
Setting: Blased
Setting: Biased
Temp Baselines SSoT
Ensemmble SSoT
10a
10
10
109
1s divergence
10
10-1
10-1
10
10-1
10-2
10-41
10-1
10:3
10
10:
10:
2
23
2
2
2
2
2
Temporature
Fow-shot (R: Random, F. Rixe))
Ersemble/Sequentaal Prompt Ensemble Sequential
Ablation
T=0.6
T
R3 -q.
R30 -b- R50
Randomized Seed
T-1.0
F10
F50
Fixed Seed
SSoT
PRNG
Figure 3: JS divergences for Unbiased and Biased PIF. Shaded areas represent the standard deviation.
5.1.2 VARYING ACTION SPACES
Settings. To compare SSoT against baselines on a wider range of tasks, we focused on deepseek-
rl. We varied the number of actions (2" for 1..... ......... 6) under two conditions: (1) a uniform distribution, and (2) a biased distribution (for n = 1, (0.75, 0.25); for > 1, one action had a probability 0.5, with the rest distributed uniformly). We repeated each PIF experiment (K = 100) 10 times. Action words were randomly selected from a list of the 10,000 most common English (1) High Temperature: increasing the temperature setting; (2) Few-shot: providing k E {3, 10, 50} examples sampled using the target distribution in the prompt, either fixed or randomly sampled for each query; (3) Prompt Ensemble:
words (excluding swear words) We compared SSoT against:
using 50 paraphrased prompts, combined with randomized action orders for all the prompts; and (4) Sequential Sampling: generating 100 actions sequentially, including the selected action history in the prompt. As ablation, we also tested providing an external random string the LLM, either Fixed (same string each time) or Randomized (new string each time).
Figure 3 shows that SSoT consistently delivers the best performance across all settings, nearly matching the PRNG. The strongest baseline was Prompt Ensemble, which performed well in the unbiased setting, suggesting that varying the prompt reduces positional biases. However, its performance degraded on the biased PIF tasks, indicating that eliminating positional bias alone is insufficient. Biased PIF presents a greater challenge than unbiased PIF. This is because it requires not only mitigating mode collapse to achieve uniformity, but also necessitates the reasoning ability to derive appropriate arithmetic operations to sample from the target distribution and the computational capability to execute them. In contrast, SSoT demonstrated near-ideal performance in both settings, supporting its robustness to distributional skew. Our ablation study revealed that providing a random- ized external seed also improves performance, confirming that LLMs can effectively perform random action selection given a source of randomness. SSoT still outperformed the external randomized seed in the biased setting, likely because the LLM can internally generate strings of length and type that are easier for it to manipulate, leading to more consistent performance.
Results.
5.1.3 ROCK-PAPER- PAPER-SSSSIOOSSSS IN ADVERSARIAL SETTING
Motivation. One scenario where probabilistic action selection powerful is in achieving mixed- strategy Nash equilibrium in game theory. By selecting actions probabilistically, player can ensure their expected payoff remains constant regardless of the opponent's strategy. making robust defense in adversarial situations where an opponent actively tries to exploit patterns. We demonstrate
https:://github.com
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Table 2: NoveltyBench results on curated dataset. Cells show Distinct (Utility); higher is better.
Method
Creativity
Naming
Facts
Product Recs
Random
Opinions
Overall
Baseline
4.60 (5.61)
6.00 (6.13)
4.14 (5.35)
4.14 (6.02)
6.07 (5.10) 4.35 (4.08)
4.70 (5.17)
Paraphrase
5.15 (5.67)
7.00 (6.77)
5.46 (5.71)
5,86 (7.33)
5.93 (5.48)
5.57 (4.47)
5.63 (5.57)
T 0.8
4.95 (6.19)
6.86 (7.06)
4.50 (5.51)
4.43 (5.86)
6.20 (5.47)
4.61 (4.42)
5.03 (5.52)
T 1.0
5.25 (6.20)
7.57 (7.75)
5.36 (6.40)
5.43 (7.31)
6.33 (5.46)
5.04 (4.92)
5.57 (6.03)
SSoT
5.90 (6.44)
7.57 (6.62)
6.04 (6.71)
5,86 (6.63)
6.87 (5.49)
5.87 (4.35)
6.19 (5.92)
that SSoT enables an LLM to defend itself in Rock-Paper-Scissee (RPS) against a strong opponent by making its actions faithfully probabilistic.
Seftings. We evaluated the LLM against 10 "Black Belt" bots from the RPS Dojo strong RPS agents available Each match consisted of 100 consecutive games, with a score of +1 for win and -1 for a loss. We tested SSoT against a Baseline prompt (in- structed choose randomly without SSoT) and a Simple prompt (allowed to choose freely). To create an adversarial setup, the opponent bots had access to the full history of both players' moves, while the I.IM did not.
Score by Prompt
Score (Bot Prompt)
-2.00
7,00
Kaggle Notebook.
pppentt Bot Type
-9.00
16:00
11.00
10
-3.00
2.00
-10.00
-14.00
0.00
Score
-10
-2.00
-2.00
-3.000
-0
11.00
20.00
4,00
-20
6.00
-20
3.00
18 00
8.00
-30
-3.00
113.00
-24.00
3.00
-12
SSoT
Baseline
SSOT
Baselnne Simple
Figure 4: RPS results against black-belt bots.
Results.
The final scores against the 10 bots are
shown in Figure 4 As the left panel shows, SSoT maintained an average score near zero, effectively holding its own against bots designed to exploit patterns. The Baseline, while attempting diversify its moves, still exhibited exploitable biases, resulting in lower scores. The Simple prompt lacked sufficient diversity and was consistently defeated. This demonstrates that SSoT can be used to achieve probabilistic behavior for improved game performance without relying on external tools.
Table 3: Novelty- Bench results on Wild- Chat dataset. Cells show Distinct (Util- ity); higher better.
5.2 SSOT FOR DIVERSITY-AWARE GENERATION
Settings. While an external PRNG is a plausible alternative for PIF, it is inadequate for DAG's core challenge of creatively mapping randomness. process SSoT integrates internally. We used NoveltyBench (Zhang et al., 2025) to measure diversity in open-ended tasks. For each question in the curated and WildChat datasets, we generated eight responses to calculate the Distinct and Utility scores. We used deepseck-rl (T' = 0.6) and compared SSoT against a baseline (prompted to generate a diverse output without SSoT), paraphrase method using prompts from the dataset, and higher temperatures (T 0.8, 1.0). For the larger WildChat dataset, we evaluated SSoT and a baseline.
Method
Overall
Baseline SSoT
3.39 (4.08)
5.25 (4.86)
Results. The results presented in Table 2 (curated) and Table 3
LLM Strategies in
(WildChat). In both datasets, SSoT achieves higher Distinct score than all comparison methods. Notably, SSoT outperforms others on both Distinct and Utility scores for the "Creativity" category in
Pereestage (%5
50
40
20
the curated dataset, indicating that SSoT is useful for enhancing creativity in open-ended tasks.
Sam-Mod Rolling Hash Otherrs
Unchaaed PIF
Blased PIF
5.3 ANALYSIS
Figure Randomness extrac- tion strategy of LLM in PIF.
5.3.1 COT STRATEGY ANALYSIS
Here, we analyze CoTs to classify the LLM's strategies. (Example CoTs in Appendices H.T and H.2
Analysis of PIF in Section 5.1.2 We analyzed how the LLM converts a generated random string into a random action, see Figure We used gemini-flash-2.5 to classify 600 responses each from the biased and unbiased PIF experiments. As shown in Figure 5 the LLM's approach converges
https: www. www.wwwwwwa.o.otoo kaggle, wttp://ee....oooeeeeeeo http:///..../o..aaaaaaaaaaaaa//a
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Empirical Distribation
(a)
KL/JS/w Think Tokens (b)
Think Tokens-513
Think ToinkTokens---3 Tokens-50000
(c)
Random String Complexity
0.8
Compleitt
0.6
diveggnnee
COunt
10
0.4
0.2
50
100
100
0.0
10'
sampled vallse
sampled value
10'
Think Tokens
Think Tokens
XL
LLM
PRNG rot
L. Complexity
2lib Compression Ratio
Figure 7: (a) Uniformness of LLLM-geerrated integers (0-127) vs. think tokens, measured by KL/JS divergences and effect size w: (N 1000). (b) Value distribution of LLM LLM--enerated integers for short (512, left) and long (30000, right) think token lengths, compared to a PRNG reference (orange). (c) Normalized LZ complexity and zlib compression rate of a random string generated sequentially by an LLM at T = 0 vs. think tokens. We used 3000-character prefix of 100 generated strings.
two main strategies (see Appendix I.3 for detailed definitions); (1) Sum-Mod: Sums the character codes (e.g., ASCII the random string and takes the modulo of the result. (2) Rolling Hash: Uses base B to compute polynomial hash of the string C B ord(e,))) and takes the modulo of that value. Notably, the LLM adapts its strategy to the task: for unbiased distributions, it favors the simple Sum-Mod method, but for biased distributions, it often adopts the more sophisticated rolling hash, demonstrating that it adjusts its approach based on the problem's 's complexity.
Analysis of DAG in Section 5.2 To understand how SSoT en- hances diversity in DAG tasks, we classified the CoT strategies from all 800 responses on the NoveltyBench curated dataset,
Assembly
Sampling Scope
Percentage (%)
80
80
60
60
see Figure We analyzed two axes (for detailed definitions, see
20
Appendix L.3: (1) Assembly: how the answer is constructed (from a fixed List or by filling a Temp 1 at),, and (2) Sam- pling Scope: whether randomness used once for Global choice or repeatedly for Local elements. As shown in Fig- ure Ø. the overall trend is to select from List with a single Global sample. This trend reverses for the "Creativity" cat- egory: The L.M tends to create a Temp a te and repeatedly samples from the random string to fill in different Local elements. This suggests that for open- ended tasks, SSoT enables diversity enhancement by decomposing the problem and diversifying each component.
List
Template
Others
Global
Local
Hybrid
Overall
Creativity
Figure 6: Diversity enhancement strategy of LLM in GoveltyBench
5.3.2 COT SCALING ANALYSIS
Analysis of Randomness Quality. To further analyze the success of SSoT for LLMS with long CoT, such as deepseek-rl and QwQ-32B, we examined the impact of CoT length on PIF performance using s1.1-32B (Muennighoff et al. (2025). We controlled the reasoning length via budget forcing and had the model generate random integers between 0 and 127. Figure a shows that as the reasoning length (thinking tokens) increases, the uniformity of the generated integers improves significantly, as measured by KL/IS divergence and effect size w. This enhanced uniformity is visualized in Figure bb). These results highlight the direct link between CoT length and the performance of PIF.
Analysis of Random String Complexity. To isolate the effect of reasoning on the generated string itself, we evaluated the complexity of strings generated sequentially at T which also removes randomness from the decoding process. We prompted the LLM to generate 100 strings in sequence, using budget forcing to control the thinking token length for each. We then measured the complexity of the concatenated string using normalized Lempel-Ziv complexity and zlib compression ratios. Figure c shows that string complexity grows with longer reasoning traces. This indicates that LLMs can produce intricate, high-entropy strings through their internal reasoning process alone, without stochasticity at the decoding stage, and also for longer reasoning traces, the complexity of the generated string increases, thereby leading to better performance on PIF.

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6 CONCLUSION
In this paper, we introduced String Seed of Thought (SSoT)), a simple and broadly applicable prompting technique that significantly improves the ability of LLMs to follow probabilistic instructions (PIF) and generate diverse responses (DAG). We demonstrated that SSoT enables models to achieve near-ideal probabilistic faithfulness by leveraging their own reasoning process create an internal random seed, which is then deterministically mapped to final action. Our analysis confirmed that the quality of this internal randomness scales with the length of the model's 's reasoning trace.
SSoT offers practical, tuning-free method for enhancing L.LM reliability in applications requiring strategic randomness or creativity, such as mehavir simulations, content diversification, and game-playing. While the approach is most effective highly capable models, it opens promising avenues for future work. These include applying SSoT to more complex domains and optimizing the string seed generation and its randomness extraction strategies.
LIMITATIONS
While SSoT proves effective across various domains, identify three key limitations:
Dependence on Reasoning Capability: SSoT relies on the model's ability to autonomously devise and execute mapping strategy (e.g., modulo arithmetic hashing). As shown in analysis of smaller models (Appendix D.3). models with limited reasoning capabilities (typically less than 8B parameters) may fail to execute these strategies correctly, leading suboptimal performance.
Potential for Bias Propagation: If the generated random string exhibits strong positional bias (e.g.. always starting with the same digit) and the model adopts "lazy" strategy without leveraging the entropy from the whole string (e.g.. using only the first character), the output distribution will be biased (as seen in the QwQ-32B failure case, Appendix D.5). This can be mitigated by steering the model towards more robust strategies (like rolling hashes) via system prompts.
Applicability Single-Answer Tasks:
SSoT is designed for tasks with multiple valid answers or probabilistic requirements. Applying SSoT to tasks with a single correct answer (e.g., math problems, factual retrieval) not effective and could potentially distract the model, though our results on NoveltyBench suggest that utility is generally preserved.
REPRODUCIBILITY STATEMENT
An anonymized zip file (see the supplementary materials) includes all the evaluation scripts and configuration files to reproduce all the experiments in this paper.
REFERENCES
Anonymous. The illusion of randomness: How LLMs fail to emulate stochastic decision-making in rock-paper-sciisorss games? ACL Rolling Review (May 2025), OpenReview, 2025. URL
httpps:/cpperreview net. forum? dorrumid=:MCCMaao000
Bradley Brown, Jordan Juravsky, Ryan Ehrlich, Ronald Clark, Quoe V Le, Christopher Ré, and Azalia Mirhoseini. Large language monkeys: Scaling inference compute with repeated sampling. arXiv preprint arXiv: 1787, 2024.
J. J.Laeeeeee Carter and Mark N. Wegman. Universal classes of hash functions. Journal of Com- puter and System Sciences, 1842)1143-154, 1979. ISSN 0022-0000. doi: https://doi.org 10.101600022000007999900600
URL (ttttttp::/
rect. com sctence/
atticle/s11/00220007990000
John Joon Young Chung, Vishakh Padmakumar, Melissa Roemmele, Yugian Sun, and Max Kreminski. Modifying large language model post- for diverse creative writing. Second Conference on Language Modeling, 2025.
1Pmuw08LoM htt.:orrrmmmmm1111mmoeooooo0oo
DeepSeek-AL. Deepseek-v3 technical report. arXiv preprint arXiv: /9437, 2024.
10

---

Published as a conference paper at ICLR 2026
DeepSeek-AL. Deepscek-rl:
Incentivizing reasoning capability in Ilms via reinforcement learning. arXiv preprint arXiv:2501. 12244, 2025.
Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with random trans- positions. Zeitschrift für Wahrschhennlickketsstteee und verwandte Gebiete, 57(2):159-179, 1981.
Esin Durmus, Karina Nguyen, Thomas Liao, Nicholas Schiefer, Amanda Askell, Anton Bakhtin, Carol Chen, Zac Hatfield-Dodde Danny Hernandez, Nicholas Joseph, Liane Lovitt, Sam McCan- dlish, Orowa Sikder, Alex Tamkin, Janel Thamkul, Jared Kaplan, Jack Clark, and Deep Ganguli. Towards measuring the representation of subjective global opinions in language models. In First Conference on Language Modeling, 2024. URL
net forum? ?id=
Caoyun Fan, Jindou Chen, Yaohui Jin, and Hao He. Can large language models serve as rational players in game theory? a systematic analysis. Proceedings of the AAAI Conference on Artificial Intelligence, 38(16):1790- 17967, Mar. 2024. doi: 10. 1609/aaaii.i...a v38116.297511. URL https:
//ojs. aaal org/ in////idxxxx.m/// php/ AAAI/art.ao. http://attteeeeeeeeeeeeeee
Xiachong Feng, Longxu Dou, Minzhi Li, Qinghao Wang. Yu Guo, Haochuan Wang, Chang Ma, and Lingpeng Kong. survey on large language model-based social agents game-theoretic scenarios. Transactions on Machine Learning Research, 2025. ISSN 2835-8856. URL htwp://wwwww.nett/tommmeddddeeee Id- Survey Certification.
/ //openreview. net forumm
Chen Gao, Xiaochong Lan, Nian Li, Yuan Yuan, Jingtao Ding, Zhilun Zhou, Fengli Xu, and Yong Li. Large language models empowered agent-based modeling and simulation: a survey and perspectives. Humanities and Social Sciences Communicaatoons 1 12(1) 2024, doi: 10,1057/ /ttp:///......00111111-199999110
URLpttps: htt.://ppp//ol...oo///0001//////
Jia Gu Liang Pang. Huawei Shen, and Xueqi Cheng. Do Ilms play dice? exploring probability distribution sampling in large language models for behavioral simulation. In Proceedings of the st International Conference on Computational Linguistics, pp. $375-5390. Abu Dhabi, UAE, January 2025. Association for Computational Linguistics. URL htttttts: org/2025. coling www.ooinnnnmiinn...ooom. 360/ m607
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q. Weinberger. On calibration of modern neural networks. In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 1321-1330. PMLR, 06-11 Aug 2017. URL https::/ppoceeddnggg ml www.ppresssvo. guol htmll
Ritwik Gupta, Rodolfo Corona, Jiaxin Ge, Eric Wang, Dan Klein, Trevor Darrell, and David M Chan. Enough coin flips can make Iims act bayesian. arXiv preprint arXiv:2503.04722, 2025.
Aspen K Hopkins. Alex Renda, and Michael Carbin. Can LLMs generate random numbers? evaluat- ing LLM sampling in controlled domains. In ICML 2023 Workshop: Sampling and Optimization in Discrete Space, 2023. URL
ew net./ ftt::/oormmmmmo 1tt.=/hhh199999
Aaron Hurst, Adam Lerer, Adam P Goucher, Adam Perelman, Aditya Ramesh, Aidan Clark, AJ Os- trow, Akila Welihinda, Alan Hayes, Alec Radford, al. Gpt-4o system card. arXiv preprint arXiv:2410.21276, 2024.
R. Impagliazzo, L. A. Levin, and M. Luby. Pseudo-randon generation from one-way way functions. In Proceedings of the Twenty- Frentt-First Annual ACM Symposium on Theory of Computing, STOC '89, pp. 12-24, New York, NY, USA, 1989. Association for Computing Machinery. ISBN 0897913078. doi: 10.1145/73007.73009 httttss::///oo 1115//.00.. 73009
Takeshi Kojima, Shixiang (Shane) Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwa- sawa. Large language models are zero-shot reasoners. In S. Koyejo. S. Mo- hamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (eds.). Advances in Neural In- formation Processing Systems, volume 35, pp- 22213. Curran Associates, Inc., httpeeeffiieeeeeeee/22e330
2022. URL htttps//pproeeeeainnnnee////// htwp://eeddiinnn..uui.......... neur fi htt.aa/ehaacccarrroeee1112222e.2

---

Published as a conference paper at ICLR 2026
Jack Lanchantin, Angelica Chen, Shehzaad Dhuliawala, Ping Yu, Jason Weston, Sainbayar Sukhbaatar, and llia Kulikov. Diverse preference optimization. arXiv preprint arXiv:2501.81001, 2025.
Hanchung Lee. tion. 08 2024.
Lim-as-a-judge: Rethinking model-based evaluations in text genera-
URL hatpa://Leehanchhn wwwp:/hhwwunnihh.......... http:/hib...//looo///2/////////1
11m as-a- a- judge/
Tianjian Li, Yiming Zhang, Ping Yu, Swarnadeep Saha, Daniel Khashabi, Jason Weston, Jack Lan- chantin, and Tianlu Wang. Jointly reinforeing diversity and quality language model generations. arXiv preprint urXiv:2509.02534, 2025.
Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, et al. Competition-level code generation with alphacode. Science, 378(66244:10022-19999 2022.
Charles Lovering, Michael Krumdick, Viet Dac Lai, Seth Ebner, Nilesh Kumar, Varshini Reddy, Rik Kenzios-ed and Chris Tanner. Language model probabilities are nol calibrated in numeric contexts. In Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics /Volume 1: Long Papers), pp. for Computational Linguistics. ISBN 79779-8899176-55144 doi: 10. a2218663/1/22225.ccll acl1lon..111
Vienna, Austria, July 2025, Association
URL httppsss/aaannnnhooogy.ooo/ htt.://w................ooooo...
Nicole Meister, Carlos Guestrin, and Tatsunori Hashimoto. Benchmarking distributional alignment of large language models. In Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume
I: Long Papers), pp. 24-49, Albuquerque, New Mexico, April 2025. Association for Computational Linguistics. doi: 10. 10.18853/v 00665//1122225555 naacl-long.2. 1/020
URL https::/aclanttoolleae
2025. maac 1 .ong.//
Niklas Muennighoff, Zitong Yang, Weijia Shi, Xiang Lisa Li, Li Fei-Fei, Hannanch Hajishirzi, Luke Zettlemoyer, Percy Liang, Emmanuel Candès, and Tatsunori Hashimoto. sl: Simple test-time scaling. arXiv preprint arXiv:2501.19393 2025.
John Nash. Non-cooperative games. Annals of Muthematiics, 54(2):886-255,, 1951. ISSN 0003486X, 19398980. URL/htt::/// /www http:://soooooootaaaaaaa. htt.:oo.ooogssaaaa/l155/s55
OpenAl. Gpt-4 technical report. arXiv preprint arXiv:2303.08774, 2023.
OpenAl. Gpt-5 system card, 2025a.
URL
5- pot--5-ssttese Systseeeasaaar////aaasaaaaaaaaat Accessed: 2025-09-22.
OpenAl. Introducing openai o3 and 04-mini, 2025b. URL introducing o3 and 04 minii/
comiinex// index
Accessed: 2025-05-22.
Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. Training language models to follow instructions with human feedback. In Advances in Neural Information Processing Systems, volume 35, pp. 27730-27744, 2022.
Vishakh Padmakumar and He He. Does writing with language models reduce content diversity? In The Twelfth International Conference on Learning Representations, 2024. URL https 7/ /oopenreview.eee wwt.eeewwweeeeeeee htt.://wweee.e.ommmmmdddeee.....
Joon Sung Park, Joseph O'Brien, Carrie Jun Cai, Meredith Ringel Morris, Percy Liang, and Michael S. Bernstein. Generative agents: Interactive simulacra of human behavior. In Proceedings of the 36th Annual ACM Symposium on User Interface Sofware and Technology, UIST '23, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 97984007013200
doi:
10.1145/358618333000778 ARRLattss:///det UtL/ttts::/doo.or///10 1tt.:/00..1143388113133333 360677
Qwen Team. QwQ: Reflect deeply on the boundaries of the unknown, 2024. URL https:/
gwen amm.gittus wtw.luu.oo////// wt/blo// 32b preview/
12

---

Published as a conference paper at ICLR 2026
Miklos Santha and Umesh V. Vazirani. Generating quasi-random sequences from semi-random sources. J. Comput. Syst. Sci., 33(1):75 87, 1986. doi: 10.1006/0022-0000669990444 URL
htttps://di..oo www.o..org/001. 1016 20200/00222220000
- 0000 (86) 90044-9
Shibani Santurkar, Esin Durmus, Faisal Ladhak, Cinoo Lee, Perey Liang, and Tatsunori Hashimoto. Whose opinions do language models reflect? In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp. 29071-30004. PMIR, 23-29 Jul 2023. URL http::tttt//tt.tt. hnttps::/prooeeeiiiee press w2.2 http:/sttuurkrraa html
mlr
Rylan Schaeffer, Joshua Kazdan, John Hughes, Jordan Juravsky, Sara Price, Aengus Lynch, Erik Jones, Robert Kirk, Azalia Mirhoseini, and Sanmi Koyejo. How do large language monkeys get their power (laws)?, 2025.
Stewart Slocum, Asher Parker-Sartori, and Dylan Hadfield-Menell. Diverse preference learning for ca- pabilities and alignment. In The Thirteenth International Conference on Learning Representations, 2025. URL http::t/ppezzeeiieeettt net pett//oormmmml--pps011...
Yang Sui, Yu-Neng Chuang, Guanchu Wang, Jiamu Zhang, Tianyi Zhang, Jiayi Yuan, Hongyi Liu, Andrew Wen, Shaochen Zhong, Hanjie Chen, and Xia Hu. Stop overthinking: A survey on efficient reasoning for large language models. arXiv preprint arXiv:2503. 16419, 2025.
Salil P Vadhan. Pseudorandomness. Foundations and Trends@ in Theoretical Computer Science, 7 (1-3):1-336, 2012.
Katherine Van Koevering and Jon Kleinberg. How random is random? evaluating the randomness and humaness of Ilms' coin flips. arXiv preprint arXiv:2406.00092. 2024.
Xuezhi Wang. Jason Wei, Dale Schuurmans, Quoe V Le, Ed H. Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. Self-consistency improves chain thought reasoning in language models. In International Conference Learning Representations, 2023.
Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. Chain-of-thought prompting elicits reasoning in large language models. In Advances in Neural Information Processing Systems, volume 35, pp. 24824-24837, 2022.
Tsachy Weissman, Erik Ordentlich, Gadiel Seroussi, Sergio Verdu, and Marcelo Weinberger. Inequalities for the 11 deviation of the empirical distribution. Hewlett-Packard Labs, Tech. Rep, pp. 125, 2003.
Peiyang Yu, Zeqiu Xu, Jiani Wang, and Xiaochuan Xu. The application of large language models in recommendation systems. In Jun Liu, Ankush Ghosh, Chee Wei Tan, and Haiquan Zhao (eds.), International Conference on Artificial Intelligence and Machine Learning Research (CAIMLR 2024), volume 13635, pp. 136350E. International Society for Optics and Photonies, SPIE, 2025. doi: 10.1117/12.3062544 URL UR//
Yiming Zhang. Harshita Diddee, Susan Holm, Hanchen Liu, Xinyue Liu, Vinay Samuel, Barry Wang, and Daphne Ippolito. Noveltybench: Evaluating language models for humanlike diversity, arXiv preprint arXiv:2504.05225 2025.
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APPENDIX CONTENTS
A SSoT System Prompts
16
Prompts Used in Experiments
20
B.1
Probabilistic Instruction Following
20
B.2
Pace-Paper-Sciisoors
20
B.3
Awwrr Generation
20
B.4
Random Integer Generation
21
B.5
Sequential Random String Generation
21
Additional Experimental Details
24
KL and TV Divergence Results on Probabilistic Instruction Following
24
C.2 Rock- Kock-Paper-Scisson Experiment Details
24
D Additional Experiments
27
D.I SSoT Performance on Sampling from Continuous Distribution
27
D.2 Impact of Generated String Length on SSoT Performance
27
SSoT on Small Language Models
28
D.4 SSoT Works with Simple Prompt
28
D.5 SSoT Failure Analysis
29
D.6 SSoT vs External Randomness Generation in DAG
30
Generated String Analysis
34
Overview of NoveltyBench
35
Full Theoretical Analysis of SSoT Performance on PIF
37
G.I
Settings
37
G.2
Main Theorems
39
G.3
Theoretical Significance and Contributions
40
Proof of theorem
G.5
Proof of Theorem G.9
42
G.6
Biased Target Distribution
43
LLM Outputs for SSoT prompting
44
Probabilistic Instruction Following
44
H.2
Diversity-Aware Generation in NoveltyBench
47
r
LLM SSoT Strategy Analysis Details
49
SSoT Strategy Analysis of Probabilistic Instruction Following
49
1.2 SSoT Strategy Analysis of NoveltyBench
50
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L.3 CoT Analysis Instruction Prompt for PIF
51
1.4 CoT Analysis Instruction Prompt for NoveltyBench
54
The Use of Large Language Models
59
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A SSOT SYSTEM PROMPTS
Here we list the full SSoT prompts used in our experiments. For SSo'T, we simply used the following prompts as the system prompt in all the experiments 5
Listing A.1: SSoT System Prompt (Probabilistic Instruction Following)
You are a helpful AI Assistant designed to provide well-reasoned and dotailed responsos. If the task involves probabilistic or non- deterministic reasoning, you must begin by generating unique and as seed.
complex random string
This random string should appear sufficiently complex and unpredictable, with no obvious structure pattern. Use your judgment to
it looks arbitrary and unguessable.
If the user explicitly instructs you to sample from a probability distribution, use the generated seed (the exact contents inside the "<randonsstring" tags) to guide any randem sampling stochastic decisions.
Follow these two steps for every instruction:
1. Output the random seed string enclosed within "<random_string>" and </randon_string>> tags.
2. Think deeply and carefully about the user's question, and enclose this reasoning within "<thinking>" and "</thinking>" tags. Al1 probabilistic decisions must be made using the generated seed- the exact contents inside the "<random_stee string>" tags. Make sure to extract maximum randomness from the string by using all its content.
3. Provide your final answer, enclosed within "<answer>' and '</ answer>" tags.
Strictly follow this tag structure, and respond in the follewing format:
<random_string>
</random_string>
<thinking>
</thinking>
<answer>
</ /aanswer>
We replaced with in the prompts due formatting issues.
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Listing A.2: SSoT System Prompt (Rock-Paper-Seisenn
You are helpful Assistant designed provide well-reasoned and detailed responses. When engaged gameplay scenarics, you must play strategically by selecting moves probsoilistically from Nash equilibrium strategies. Since opponents will attempt to exploit predictable patterns in yeur play, must begin by generating unique and complex random string to serve seed for your move selection.
This random string should appear sufficiently complex and unpredictabl ctable, with no obvious structure pattern. Use your judgment to
looks arbitrary and unguessable.
When playing any game, use the generated seed (the exact contents inside the "<random_sst strrongssrrnn"" tags) to guide your probasilistic move selection from the Nash equilibrium mixed strategy.
Follow these steps for every game situation:
1. Output the random seed string enclosed within <<rando_sstring and //rrandom_stringee tags.
2. Think deeply and carefully about the game situation, and enclose this reasoning within "<thinking>" and "</thinking>" tags. You must select ONE move using the generated seed---the exact contents
inside the "<random saanddm_ssrrig>>> tags---to probabilistically choose from Nash equilibriue randomness from the string by using all of its content.
strategies. Make
to extract maximum
3. Provide your final move/action, enclosed within '<answer>' and "</answer>" tags.
Strictly follow this tag structure, and respond the follewing format:
<random_string>
...
</randomsstring>
<thinking>
</thinking>
<answer>
</answer>
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Listing A.3: SSoT System Prompt (Diversity-Awar Generation)
You are a helpful AI Assistant designed to provide well-reasoned and detailed responses. If the task allows many possible answers, you must generate ONE diverse respense for the task. For that, you must begin by generating unique and complex random string serve as a seed.
This random string should appear sufficiently complex and unpredictable, with no obvious structure pattern. Use your judgment to ensure it looks arbitrary and unguessable.
If the user asks some question which allows muitiple answers, use the generated seed (the exact contents inside the k random_string?" tags) to guide any random sampling stochastic decisions.
Follow these steps for every instruction:
1. Output the random seed string enclosed within "<random_string>> and s/randon_strinng
tags.
2. Think deeply and carefully about the user's question, and enclose this reasoning within "<thinking?" and "</thinking>" tags. You have to generate
response leveraging the generated seed--- the exact contents inside the "crandom_strings single answer is unique and diverse. Make sure to extract maximum randonness from the string by using all of its content.
tags, to ensure your
3. Provide your sinal answer, enclosed within "<answer>" and <</ answer>" tags.
Strictly follow this tag structure, and respond in the follewing format:
<random_string>
</random_string>
<thinking>
...
</thinking>
<answer>
</answer>
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Listing A.4: SSoT System Prompt (Random Integers Generation)
You are helpful Assistant desi generate random data based on instructions. When asked to generate random data, you must first generate a unique and complex random string to serve as a seed or source of randomness.
This random string should appear sufficiently complex and unpredictable, with no obvious structure or pattern. Use your judgment to ensure it looks arbitrary and unguessable.
Use the generated seed (the gondom_stringe tags) to guide any subsequent random choices, like generating a random integer.
contents inside the k
Pollow these steps for the response format:
1. Output the random seed string enclosed within "crandon_stringe and </randon_strilnii
tags.
2. Perform the requested random generation task (e.g., generating a random integer within specified range) Clearly state the process you used to derive the random value the seed string, 3. Provide the final generated random value (e.g., the integer) enclosed within appropriate tags (c.g., "<rando__intgeer" and '</ candom_integen
Strictly fol follow this tag structure.
Listing A.5: SSoT System Prompt (Sequential Random Strings Generation)
You are helpful Assistant desi gned generate random data based on instructions When asked te generate random data, you must first generate a unique and complex random string to serve as a seed or source of randomness.
This random string should appear sufficiently complex and unpredictable, with no obvious structure or pattern. Use your judgment to ensure it looks arbitrary and unguessable.
Use the generated seed (the exact contents inside the K random_string>> tags) to guide any subsequent random choices.
Follow these steps for the response format:
Output random seed string enclosed within <random_strings tags.
and s//randomssrrnne
2. Perform the requested random generation task (e.g., generating a random integer within specified range). Clearly state the process you used derive the random value from the seed string. Strictly follow this tag structure.
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B PROMPTS USED IN EXPERIMENTS
B.I PROBABILISTIC INSTRUCTION FOLLOWING
Listing B.I: User Prompt (PIF)
Please choose between (choices). You must select one of these {
jmm_choices) options with the follcwing probabilities: {
wrob_distribution
Listing B.2: Baseline System Prompt (PIF)
You are helpful Assistant designed provide well-reasoned and detailed responses. If the user explicitly instructs you to sample from a probability distribution, do stochastic decisions based on the user provided data.
Think deeply and carefully about the user's question, and enclose this reasoning within "<thinking>" "</thinking>" tags. Then provide your final answer, enclosed within "Kanswer>" and '</answer >" tags.
Strictly follow this tag structure, and respond in the follewing
format:
<thinking>
</thinking>
<answer>
< /answer>>>
In PIF experiments in Sections:5.1. 5.1.1 and 5.1.2 we used the system prompt as shown in Listing A.
The system prompt instruction consists of three components: (1) Generation of random string, in case user's task requires it; (2) Generation of thought process; (3) Generation of final answer inside <answer> and </answer> tags. The final answer will be parsed from the generated output in rule-based manner.
As for the baseline method, we used the system prompt shown in Listing B.2 The baseline prompt instruction consists only of (1) Generation of chain of thought; (2) Generation of final answer inside <answer>> and < </answer>> tags. We also included a thought process generation in the baseline prompt gauge the pure effect of random string generation on the probabilistic task.
As for the user prompt, we used a simple prompt shown in Listing B.1 Here the boxes surrounded by and used strings suited to each task.
B.2
For the Rock-Paper- Scissor experiment in Section 5.1.3 used the SSoT system prompt from Listing A.2, the simple prompt from Listing B.3, and the simple prompt from Listing B.4 The user prompt given in Listing B.5
B.3 DIVERSITY-AWARY GENERATION
For the DAG task in Section 5.2 used the SSoT system prompt in Listing A.3 the Baseline system prompt in B.6 As for the user prompts, we used the ones provided by the NoveltyBench dataset it is.
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Listing B.3: Simple System Prompt Srock-Paper-Sciine
You are a helpful AI Assistant designed to provide well-reasoned and detailea responses. When engaged in gameplay scenarics, you must play strategically to win or avoid losing. Since oppenents will attempt to exploit predictabl patterns your play, you should vary your move selection to maintain strong performance.
When playing any gane, select moves that give you the best chance of winning avoiding defeat.
Follow these steps for every game situation:
1. Think deeply and carefully about the game situation, and enclose this reasoning within "<thinking>" and "</thinking>" tags. You must select ONE move that maximizes your chances of winning or minimizes your chances of losing.
2. Provide your #inal move/action, enclosed within "<answer>" and "</answer>" tags.
Strictly follow this tag structure, and respond in the following format:
<thinking>
</thinking>
<answer>
</answer>
B.4 RANDOM INTEGER GENERATION
As for the random integer generation experiment in Section 5.3.2 we used the system prompt in Listing A.4andd and the user prompt in Listing B.7
B.5 SEQUENTIAL RANDOM STRING GENERATION
As for the sequential random string generation in Section 5.3.2 for the first turn, we used the system prompt Listing A.5 and the user prompt Listing B.8 For the follow-up request generate new string, we used the user prompt in Listing B.9 with the same system prompt, Listing A.5) To avoid hitting the token length limit, we collected the previous turns' random strings and put them in the user prompt, rather than giving the whole conversation history.
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Listing B.4: Baseline System Prompt RRock-Paper-Seisson
You are helpful Assistant desig gned provide w 1-reasoddd and detailed responses. When engaged in gameplay scenarics, you must play strategically by selecting moves from Nash equilibrium strategies. Since opponents will attempt to exploit predictable patterns in your play, you should vary your move selecticn to avoid being exploited.
When playing any game, select moves based on Nash equilibriur mixed strategies to maintain optimal play against adaptive opponents.
Follow these steps for every game situation:
1. Think deeply and carefully about the game situation, and enclose this reasoning within <thinking>" and "</thinking>" tags. You must select ONE move from Nash aquilibrium strategies, considering how opponents might try to exploit predictable patterns your play. 2. Provide your final move/action, enclosed within "<answer>" and "</answer>" tags.
Strictly follow this tag structure, and respond in the fcllewing format:
<thinking>
</thinking>
<answer>
< answer>>>>>
Listing B.5: User Prompt (Rock- (Rock-Paper Staper-Scissory)
You are playing rock, scissers, and paper against other agent. You from rock, scissors paper.
need to pick First provide your reasoning, and then pick your answer inside answer><<answer> tags.
Listing B.6: Baseline System Prompt (Diversity-Aware Generation)
You are a helpful Al Assistant designed to provide well rell-reasoned and detailed responses. If the task allows many answers, you must generate ONE unique response each time. If the question which allows multiple possible answers, strive generate à different answer each time to avoid returning the same response.
asks you
Think deeply and carefully about the user's question, and enclose this reasoning within "<thinking>" and "</thinking>" tags. Then provide your final answer, enclosed within "<answer>" "</answer >" tags.
Strictly follow this tag structure, and respond in the follewing format:
<thinking>
</thinking>
<answer>
</ answer>
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Listing B.7: User Prompt (Random Integers Generation)
Your task is to generate a random integer between 0 and 127 { inclusive).
Follow these steps precisely:
1. First, generate a unique and complex random string. Output this string within "<randon_string>"" tags.
2. Based +only+ on the random string you generated, choese random integer between and 127 (inclusive). Explain how you derived this integer from the random string.
3. Output the final chosen integer enclosed within grandom_integer
>" and </random_integers tags.
Provide your response strictly following the required format:
1. Output the randon string in the tag <<rando__string>>
2. Explain your process for deriving the integer from the string.
3. Output the final integer in the tag "<randon_integer"".
Listing B.8: User Prompt; 1st turn (Sequential Random Strings Generation)
Your task is to generate rancom string. Generate a unique and complex ranaom string. Output this string within Crrandm__strnng tags.
Listing B.9; User Prompt: new turns (Sequential Random Strings Generation)
Your task is to generate random string. Generate unique and complex random string. Output this string within "<randon_string>" tags.
You generated random strings in the previous turns. Please generate random string.
Previous Random Strings: (tando_stringhhitoor))
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Table 4: The PIF performance comparison of SSoT against the baseline across various models, evaluated using the KL divergence. We generated 100 actions for each configuration to calculate the empirical distribution, and then calculated the divergences. We repeated it 10 times to calculate the standard deviation. All the KL values are presented in units of 10 (original values multiplied by 1000).
Model
Method
2-choice
biased 2-choice
3-choice
biased 3-choice
biased 9-choice
deepseek-v3
Baseline
23.66
1919
342.42
423.83
2335
356.67
000
1013,46
17:37
SSo'
11.57 12.1 ( 518).
13.51 :: 18.66 WW9
wiy
60.93
6486 ( (r%)
56.54
35.6 84%)
179.18
47.56 R29)
gpt-4o
Baseline
60.82
1290
356.67 =: 000
213.37
46,53
345.74
= 34.77
992.99
= 27122
SSoT
17.52 13.44 (19)
35.99
90%)
27.99
21.05 Q 57%)
35.93
90%)
130.95
46.73 57%)
o4-mini-high
Baseline
13.10
1984
277.53 =: 12.78
237.26
43.31
352.33
221.58
46.50
SSoT
3.75 4.21 71993
49.03
= 24.70 82%)
40.69
23,97 ( 83%)
63.15
28.70 C. 82%)
71.15
33.35 (8%)
QwQ-32B
Baseline
9.61 à: 18:1
337.66
:: 24.54
414.00
96.3
334.61
= $2.81
904.49
= 66.77
SSoT
13:42 19.55 (1 40%)
9.45
15.12 ( (79)
7.27
581 (19989)
5.21
5.03 (1 988)
45.65
20:26 98%)
deepseek-r1
Baseline
138.64) 50.4
230.75
345.73
51.83
162.34
504.28
89.16
SSoT 12.04 13.00 (1 (199))
6.09
6.25 (1 975)
19.73
15A5 ( 94%)
16.04
17.55 906)
72.29
47.34 56%)
PRNG
7.35 10.22
7.57 = 10.84
13.25 9.61
11.17 = 12.24
53,86 = 17.12
Table 5: The PIF performance comparison of SSoT against the baseline across various models. evaluated using the total variation distance. We generated 100 actions for each configuration to calculate the empirical distribution, and then calculated the divergences. We repeated 10 times to calculate the standard deviation. All the total variation distance values are presented in units of 10-2 (original values multiplied by 100).
Model
Method
2-choice
biased 2-choice
3-choice
blased 3-choice
blased 9-choice
Baseline 9.80 4.80
29.70 0.48
33.57 +: 0,74
30.00 +: 0.00
63,80 0.42
doepseck-v3
SSoT 6.40 4.27 ( 3553)
6.20 388 ( 7993 14.30 # 7,56 () 57R)
11,20 3.05 63%)
24.50 4.22 (1 6218)
gpt-4o
Baseline 16.60
4.65
30.00
0.00
26.17 +: 2,12
29.70
0.5
63.30
0.67
S 8.20 4.66 519)
10.80
4.44 (1 64%)
9.43 1: 4,61 64%)
9.90 +: 3.28 67%)
22.20 4.94 <1 65%)
04-mini-high
Baseline 6.90
4.38
28.00
099
27.50
1.8
29.90
0.32
23.00
3.56
S 3.70
4616)
13.10
3.28 (1 53%)
11.97
+: 4.89 (1.56R)
14.30 3.16 52%)
14.80
3.85 3695)
QwQ-32B
Baseline 5,00
29.60
0.52
43.77
5.04
29.40
0.97
60,90
1.60
S 6.60
325)
4.50
4.25 () 88%)
5.17 + 2.00
3.50 2.27 88%)
10.90
2.38 (1 829)
decpscok-rl
Baseline
25.20 = 4.87
26.20
1.69
31.57 : 1.50
21.20 :: 2.74
45.80
4.02
SSoT
6.30 (. 75%6
4.30
291 (1 849)
8.23 + 3.19 () 749)
5.90 3.75 72%)
13.70
4.69 (1 7098)
PRNG
4.90 3.75
4.00
4.03
6.63 1 2.93
4.80 1 3.71
12.80 2.15
C ADDITIONAL EXPERIMENTAL DETAILS
C.I KL AND TV DIVERGENCE RESULTS ON PROBABILISTIC INSTRUCTION FOLLOWING
Here we will show the results for KL divergence and the total variation distance in Tables 4 and S
Also, we will show the results of the experiment for Figure 3 for KI. divergence and the TV distance below in Figures and
C.2
EXPERIMENT DETAILS
For the Kack-Paper-Scissoon experiment in Section 5. "RPS dojo" kaggle notebookhttts::// kaggle ctm/ www.://oe//eeeee notebook
we used 10 "black-belt" bots from www.de/echhanaaarrpp----.1
The opponent bots are listed below:
py * black belt/multi armed bandit v32.py
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Setting: Unbiased
Setting: Unbiased PIF
Unblased
Temp
Ensemble
10t
10:
KL divergence
10
10
10
10
10
10
Setting: Blosed PIF Baselines 5ScT
Setting: Blased PIF shot
Setting: Blased Ensemble Seq SSoT
Sctting: Blased
Ablation
divergence
109
10a
10-1
10-1
10-1
10-
10-1
10-2
10-2
10-1
2
24
2
26
Temperature
Fow-shot (R: Random, Rxod)
Ensembla/Soquentaa - Prompt Ensemble
Abllation
T-0.6 - T
-
R3
-G-
R10
- R50
-
Randomized Seed
Sequential
Figure KI. diveeeggesse for Unbiased and Biased PIF. Shaded areas represent the standard deviation.
Setting: Unblased
Setting: Unblased PIF shot S58T
Unblased PIF Ensemble
Unbia Abilation
Total Total Variation Distance
Tamp Baselines
10"
10a
10-1
10-1
10-1
10-
Setting: Biased PIF
Setting: Biased PIF Few SSoT
Setting: Biased PIF Ensemble SSoT
Setting: Biased Ablation
Total Variation Distance
Temp Baselines S5cT
10a
10a
10
10
22
24
2
26
Temperature
Few-shot (R: Randem, F: Fxxe))
Ensembla/Scquential Prompt Ensemble
Ablationn
T-0.6
T
-
R3
R30 -
R50
-
Randomized Sead
Sequential
Figure 9: TV distance for Unbiased and Biased PIF. Shaded areas represent the standard deviation.
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Score by Prompt
Score (Bot x Prompt)
10
black belt jQu2
-2.00
7,00
-20.00
black_belt_ centrifugal bumbleepuppy_w
-9.00
-18.00
-13.00
black, delt
-3.00
2,00
-10.00
Opponennt Bet Type
black beit wtt.:beettt.eeeeee greenberg
-14,00
0.00
-29.00
Score
-10
black belt belt_iocane, Leoaanlcn_________ pawdeerrwwerrrr
-2000
-2.00
-33.00
black belt, memory. patpppasttaaaaa. v20
11.00
-20.00
-4,00
-20
-20
black belt memory. ptt.ppttttts.... .v
6.00
0.00
-16.00
black belt multi aareed bandit v15
3.00
-18.00
-8.00
-30
-30
black belt multi armed, bandit v32
-3.00
-13.00
-24.00
black belt (esting_please___ p_lllasee jgnore
3.00
-12.00
-35.00
SSoT
Baseline
Simple
S56T
Baseline
Simple
Figure 10: RPS results against black-belt bots.
blackbbelt///eeorryyyttttsss v20.py
blackkbbllt//eeerryyattteesss.. py
black_belt//occnneeppdee.r
black.belltgrreebberr..yy
blaakkeltt/eeetinngllaeeeiipnee
black.belt/1002.pp
black belt/cllul.py
black bel belt/centrri bumbl lumply 4.py
The scores and the full information, such as the opponent bot names, are shown in Figure T0)
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Table 6: Kolmogoroy-Smirov statistics and Cramér-von Mises statistics, calculated for the bascline and SSoT prompts. We used deepseek-rl-0522 and calculated the statistics for 100 samples, repeated 10 times obtain the mean and standard deviation.
Distribution Method Kolmogoroy-Smirnov Cramér-von Mises
Baseline 0.220 +: 0.023
0.999 + 0.163
Uniform
SSoT 0.187 + 0.034 (L 15%)
0.839 = 0.405 (L 16%)
Baseline
0.129 +
0.358 = 0.146
Normal
SSoT
0.108 1: 0.026 (L 16%)
0.310 + 0.195 () 14%)
Baseline
0.177 + 0.039
0.798 = 0.271
Beta
SSoT
0.113 + 0.040 (L 36%)
0.293 = 0.208 (1 63%)
Table 7: JS, KL, and TV divergences between the empirical distribution and the target distribution of the biased 9-choice task for different random string length targets together with the average generated string length. Divergences are reported as mean = standard deviation over 10 chunks of 100 samples. The string length column reports the mean value only. We used deepseek-rl ---228 for this experiment.
n Generated String Length| JS Divergence KL Divergence
TV Distance
2
2.0
0.178 = 0.022 0.705 + 0.086 0.527 + 0.029
4
4.2
0.228 0.017 0.887 + 0.071
0.581 + 0.021
8
8.0
0.138 + 0.033
0.581 + 0.128
0.446 + 0.058
16
16.2
0.037 0.011
0.156 + 0.050
0.222 + 0.040
24
25.9
0.019 = 0.007
0.075 0.027 0.141 + 0.034
32
32.4
0.025 = 0.007
0.092 = 0.026 0.151 + 0.030
40
40.0
0.032 + 0.010
1116=:000
0.191 + 0.028
48
50.7
0.030 = 0.011
0.108 + 0.039 0.179 + 0.035
64
69.6
0.044 - 0.011
0.162 + 0.036 0.244 +
128
142.1
0.052 + 0.016
0.187 + 0.054
0.264 + 0.035
D ADDITIONAL EXPERIMENTS
D.1 SSOT PERFORMANCE ON SAMPLING FROM CONTINUOUS DISTRIBUTION
To demonstrate that SSoT is also effective for sampling from continuous distributions, we conducted experiments comparing SSoT and the Baseline three continuous distributions: (1) the Uniform distribution on [0. 1). (2) the Normal distribution (0, 1), and (3) the Beta distribution B(2, 5). We measured the Kolmogorov-Smimov statistic and the Cramér-von Mises statistic to evaluate the goodness of fit of the empirical distribution constructed from 100 LLM samples. We repeated this process 10 times calculate the mean and standard deviation, in total sampled 1000 LLM responses. We used -oopseekk---0525 for these experiments.
The results presented in Table As observed, SSoT improves the goodness-of-fin metrics across all distributions, demonstrating its capability to handle continuous sampling tasks effectively.
D.2 IMPACT OF GENERATED STRING LENGTH ON SSoT PERFORMANCE
To investigate the impact of the generated string length on PIF performance, we explicitly specified the target string length in the prompt and analyzed whether the model generated strings of the correct length and how this affected PIF performance. We used deepseck-rl-OS5 --0228 for this analysis and selected the biased 9-choice task, which was the most challenging setting introduced in Section 5.1
To control the random string length, we inserted the instruction "The random string must be {n] characters long." into the SSoT prompt immediately after the phrase "Use your judgment ensure looks arbitrary and unguessable." By varying the value of n, we were able to control the length the generated strings.
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Table The PIF performance comparison of SSoT against the baseline for small LLMs, evaluated using the JS divergence (lower is better). All the JS divergences are presented in units of 10-3 (original values multiplied by 1000).
Method
2-choice
biused 2-choice
3-choice
biused 3-chuice
biused 9-chuice
Qwen3-8B
Baseline
8SoT
16.92
7.36 4ea s033
245
86.55
24.16
335 7253
1651
66.46
7:72 503 G 3999
2042
87.30 + 1499
14:51 we ( K783
293.72 RA1
60:23 19 1993
Qwen3-1B
Baseline
1,40
117.28
31,16
115.53
300.16
SSoT
7,48
4 caois
42.35
1555 (1 6699
55. 82
= 8.4 1999
17.98 = 8.44 B4G3
104.15 1931 ( 6093
Qwen3-1.7B
Baseline
8S6T
106.70
20.82
1945
1400 1138)
94.47
10.92
1826
259.41
305.5c
0693
90.04
14.17
300.16 103
229.36
DeepSeck RI I-Dstilll Llaill-Llam-BBe 8B
Baseline
0.71
039
75.06
2315
123.05
20.88
49.76 1 1711
244.47 1. 25007
8SoT
0.38
(99)
9.86
358 (1 (799))
4.71
2.91 173 (1 946)
75.17
17.0 ( 6965
Qwenn--hinkingg-1B
Baseline 8S6T
1936
117.28
0.00
189.82
19244
117.28 x:
258.57
10.85 801 886)
2.13
335 (L 9993
5.01
(.51 () 9999
13:31 1 699 (1 8993
31.07
1569 () 5813
Nemotron- Qwen-115B
Baseline 128.18
62.00
258.94
112.38
221 08
SSoT
32.17 1234 (1 2594
51.49
16.17 (L 1799
75.45
21,07 () 7199)
11.35 # 644 (1 9099 118.35
1722 Q 4093
PRNG
1.85 2.58
1.93 2.00
3,36 = 248 2.85 # A15 13.72 + 421
The results are shown in Table 7 First, for requested lengths of 40 or fewer, the model almost always 2, 4, 8, the
generated strings of the correct length. Furthermore, compared performance PIF performance improves dramatically when n exceeds 16, and maintains better performance up to n = 128 compared n 8. This supports the theoretical finding in Section that performance improves with longer string lengths.
Additionally, the fact that performance peaks at n 24 can be attributed to the increasing probability of arithmetic errors as the model attempts to perform complex operations (such as sum-mod or rolling hash) on longer strings.
D.3 SSOT ON SMALL LANGUAGE MODELS
As elaborated in Section 5.3.1 LLMs are capable of autonomously discovering methods to sample from distributions. However, this implies that the model must possess the ability to find a correct algorithm and the reasoning capability to execute it. To verify this hypothesis, we assess the effectiveness of SSoT on smaller LLMs ranging from IB to 8B parameters. It is well-known that complex prompting techniques, such as zero-shot CoT (Kojima et al. 2022), often yield limited gains on smaller models.
We used the same five settings as in Section 5.1 (1, 2) 2-choice and Biased 2-choice: (a
[heads, tails), = [1/2, 1/2] and [0.3, 0.7l). (3, 4) 3-choice and Biased 3-choice: (a
rrock, paper, scissors), p [1/3, 1/3, 1/3) and [0.1, 0.2, 0.7)). (5) Biased 9-choice (a jone, two, eight, nine], = [0.08, 0.08, 0.08, 0.36)). We used the recommended temper-
ature settings for each model.
We evaluated the Qwen3 family (8B, 4B, L.7B) with "thinking mode" enabled, as well as reasoning- enhanced models: DeepSeck-RI-Diie -iss--RI---iii-----a------ The results are presented in Table 8 For Qwen3-8B, SSoT improves PIF capability, although not to the extent seen in larger LLMS or the PRNG. Conversely, Qwen3-4B and Qwen3-1.78 show improvement in some tasks, but their absolute performance remains suboptimal. This finding is consistent with the premise that SSoT requires a certain threshold of reasoning capability.
Qwen3-thinking-44B and femooronn-wenn-L.5
To further demonstrate that the limited improvement in small models is linked to reasoning capabilities, small LLMs (DeepSeek-Ri RRI-Distill-Llmms
we examined the results for the reasoning- Swen3-thinking-... and Nemotron- -unnn--eeenBBB As shown in Table &, SSoT yields consistent per- formance improvements across tasks for these models, reinforcing the connection between reasoning ability and SSoT effectiveness.
D.4 SSOT WORKS WITH SIMPLE PROMPT
In the main text, we utilized a relatively detailed SSoT prompt. This was designed to establish a strong baseline by explicitly instructing the model on the procedure, thereby allowing us isolate the
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Table The PIF performance comparison of SSoT and SSoT (simple) against the baseline across various models, evaluated using the JS divergence (lower is better). All the JS divergences are presented in units of 10-3 (original values multiplied by 1000).
Model
Method
2-choice
biased 2-choice
3-choice
biased 3-choice
biased 9-choice
Baseline
36.09 1860
69.58
1342
106.30 1: 1948
49,53 1: 11.4
138.21 1 7664
deepseck-rl
SSoT
3.03 348 (1955)
1.51 1.5 9859)
384 9599)
4: 490 (1999)
18.06 a: 11.3 (1 (795)5)
SSeT (Simple)
2.37 (1988)
1.19
1.63
3.72
25.17
PRNG
1.85 2.38
1.93 2.40
3.36 :: 2.48
2.85 +: 3.15
13.72 à 421
pure effect of the random string element. However, SSoT functions effectively even with simplified prompt, improving performance in both PIF and DAG tasks. To demonstrate this, we analyzed the performance of deepseek-rl r100228 using the following simplified prompt:
Listing D.I: Simplified SSoT System Prompt
Generate a complex random string between <random <randon_string> and </ pondonsstringe and manipulate this string to guide any stochastic decisions within <thinking> and </thinking> tags.
Then, provide your final answer, enclesed within "<answer>" and '</ answer> tags.
For DAG, we included the objective in the system prompt. Apart from the first line describing the objective, the subsequent text remains identical to the PTF version.
Listing D.2: Simplified SSoT System Prompt (DAG)
You must produce exactly one unique and diverse answer. To do this, first generate a complex random string between <randon_string> and </randon_string>, and manipulate this string to guide any stochastic decisions within <thinking> and </thinking> tags.
Then, provide your final answer, enclosed within <answer> and </ answer> tags.
The essential components the SSoT prompt are: (1) instructing the LLM to generate a complex string, (2) asking it to manipulate the string, and (3) generating final answer. Crucially, similar to zero-shot CoT, explicitly instructing the model to write out the manipulation process in the output is vital; otherwise, reasoning models may hallucinate the derivation without actually using the string. While we use tags to structure the output and facilitate parsing, they are not strictly necessary as long as the content is generated. Using tags, however, simplifies the extraction of the final answer for user display.
We conducted experiments using these simplified prompts for the PIF task (Section 5.1.2) and the Novelty Bench task (Section 5.2) The results are shown in Table Table 10
As we can see, the performance of simple variant of the SSoT prompt is on par with the sophisticated SSoT prompt. demonstrating that we can easily employ SSoT, only following the three-step instruction setup and the inclusion of the objective in its system prompt.
D.5 SSOT FAILURE ANALYSIS
As shown in Table I QwQ-32B's performance with SSoT the 2-choice PIF task is worse than the baseline. To investigate the cause this discrepancy, we analyzed 1000 random strings and the corresponding responses generated by QwQ-32B for the 2-choice PIF task.
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Table 10: SSoT Simple prompt result on NoveltyBench (Curated). Cells show Distinct (Utility);
higher is better.
Method
Creativity
Naming
Facts
Product Recs
Random
Opinions
Overall
Baseline
4.60 (5.61) 6.00 (6. 13) 4.14 (5.35)
4.14 (6.02)
6.07 (5.10) 4.35 (4.08) 4.70 (5.17)
SSo'
5.90 (6.44) 7.57 (6.62)
6.04 (6.71)
5,86 (6.63)
6.87 (8.49) 5.87 (4.35)
6.19 (5.92)
SSoT (Simple)
6.20 (6.38) 6.86 (5.66) 5.96 (6.49)
5.86 (7.32) 6.60 (5.30) 6.39 (4.02) 6.26 (5.72)
We identified a notable bias the generated random strings: 947 out of 1000 strings started with the digit "7". Furthermore, utilizing gemini-2.55flast classify the strategies used in the 1000 responses, we found that 50 responses employed a strategy that simply selected "heads" or "tails" based on the ASCII code of the string's first character. Among these 50 cases, 45 had random string starting with "7", resulting in 43 selections of "tails" and only of "heads". Removing those failure cases leads to near-ideal distribution.
This analysis reveals that bias can be introduced when the generated string itself contains a pattern (e.g., fixed starting character) and the model simultaneously selects strategy that fails extract sufficient entropy from the entire string. However, even in such cases, if these failure modes can be mitigated via prompt steering, it is possible to sample from an unbiased distribution.
D.6 SSOT VS EXTERNAL RANDOMNESS GENERATION IN DAG
In this section, demonstrate that, in the NoveltyBench DAG setting, SSoT outperforms approaches that obtain randomness externally, such as injecting PRNG-generated strings into the prompt or using tool calls to external random modules (e.g.. Python's random module).
To evaluate external randomness generation in DAG, we consider the following two baselines:
This method involves sampling a fresh random string from a PRNG for each query, inserting it into the prompt, and instructing the model to this string when generating its response. We utilized the system prompt in Listing D.4 and prepended Listing D.5 to the user prompt. We used 24-character string drawn from printable ASCII characters
Seed Injection:
2. Tool call: We prepared tools consisting of random int and random strmmsstrii strriiiii ing as shown in Listing [.3), and instructed the L.LM to perform tool calls. We evaluated this baseline on the NovehtyBench curated dataset. The system prompt was set as shown in Listing D.6] explicitly instructing the model use external tools whenever randomness was required.
We used ddepseek-r1-0522 the underlying model, following the setup in Section 5.2
The results comparing these baselines with SSoT are presented in Table I1 As the table demonstrates, SSoT outperforms both external randomness baselines in terms of Distinct and Utility scores across all categories, except for "Product Recs". Notably, in the "Creativity" task, while the external methods struggled improve Utility diversity increased, SSoT successfully achieved high scores in both metrics, demonstrating its ability to enhance diversity without compromising response quality.
The performance gap can be attributed to several factors:
Limitations of Seed Injection: Providing single external string limits the model's ability to employ strategies that utilize randomness multiple times or for local decisions, as observed in our COT analysis in Section 5.3.1. Furthermore, shown in the rightmost panel of Figure 3 (Section 5.1.2), providing an external string generally yields lower PIF performance compared to the internal generation of SSoT.
Instability of Tool Calls: While Tool Call theoretically allows for multiple randomness requests, its performance proved unstable. This is primarily due to the difficulty of using custom tools out of the box inference time, which led frequent tool-calling failures (c.g., incorrect function names or arguments).
*We chose 24-character string, since it yielded the best performance in Table
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Table 11: SSoT, Seed Injection, and Tool Call baseline results on Novelty Bench (Curated). Cells
show Distinct (Utility); higher is better.
Method
Creativity
Naming
Facts
Product Recs
Random
Opinions
Overall
Baseline
4.60 (5.61) 6.00 (6.13) 4.14 (5.35)
4.14 (6.02)
6.07 (5.10) 4,35 (4.08) 4.70 (5.17)
Seed Injection
5.60 (5.00 7.43 (6.46)
5.75 (6.47)
6.43 (7.99)
6.67 (5.4)) 5.65 (4.13)
6.00 (5.76)
Tool Call
5.60 (5.10) 7.00 (6.41)
5.39 (6.34)
6,71 (7.72)
5.73 (4.75) 5.52 (3.60)
5.72 (5.33)
SSoT
5.90 7.57 (6.62) 6.04 (6.71)
5.86 (6.63)
6.87
5.87
6.19
These results suggest that, in our setting, even frontier LLMs like deepscek-r1-0528 may not reliably make effective use of arbitrary tools out of the box. Moreover, employing tool calls necessitates executing the tools and making multiple API calls, which complicates the use batch APIs, since each tool call requires separate interaction and its result must be fed into subsequent prompt. In contrast, SSoT is highly simple, requiring only a modification to the prompt, and fully compatible with batch APIs. Thus, in this setting, SSoT offers more favorable combination of performance and usability than the tool-call-based based approach.
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Listing D.3: Random module tools consisting of random int and zandom string functions.
"type" "Sunction",
function":
"name": "random_int",
"description" "Return random integer between min and max (inclusive).", "parameters":
"type": "object",
"properties"; "min":
"type": "integer",
"descriptica": "Kinimum integer value
inclusive).",
"max":
"type": "integer", "description": "Maximum integer value
inclusive).""
b
"required"; ("min", "max"),
"type": "Sunction", b
"function":
"name": "random_string"
"description": "Return a random printable ASCII string of the given length.",
"parameters":
"type": "object",
"properties";
"length":
"type": "integer",
"minimum": 1,
"maximum": 256,
"description" "pesired 1ength of the random string. Must be between and 256.",
"required": ["length"),
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Listing D.4: External Random Seed Baseline System Prompt
You helpful Assistant designed to provide well-reasoned and detailed responses. If the task allows many possible answers, you must generate ONE diverse respense for the task. You will be given a random string which is different for every request from users, Use this seed (the exact contents inside the "stando_sstring tags) guide any random sampling or stochastic decisicns.
Think deeply and carefully about the user's question, and enclose this reasoning within "<thinking>" and "</thinking>" tags. Then provide your final answer, enclosed within "<answe>"" and "</answer tags.
Strictly follow this tag structure, and respond in the fol fellewing format:
<thinking>
</thinking>
<answer>
</answer>
Listing D.5: External Random Seed Baseline User Prompt Prefix
Using this random seed: trandom_string wrandomsstring]
s/random_stringg
Listing D.6: Random Module Tool Call Baseline System Prompt
You are a helpful assistant that must. produce exactly one unique and diverse answer for each user prompt.
Use given toels via tool calls whenever invent simulate randomness internally; always rely the tools for random values or random strings. Use the randonness you obtain to drive diverse outputs in your answer.
need randomness. Do NOT
For each prompt where multiple valid answers possible: Think step by step inside "<thinking>" and "</thinking>" tags ( optionally using given tools).
Provide your final answer between 'Kanswer>" and "</answer>' tags.
Respond strictly in the following structure: <thinking>
</thinking>
<answer>
</answer>
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1 - maxxP(sic)
minsP(s|cc *A
0.8
0.50
(axx.P(s(c)
min P(s|c) *
4
0.6
0.25
0.4
2
0.00
0.2
0.0
0
Figure 11: The distribution of 1 max, P(s]c) and min, P(s|c) * A. The X-axis labels the prefix c,
and is sorted in descending order of its value.
Unigram distribution
Probability
0.02
0.01
0.00
Character (sorted by freq)
Figure 12: The unigram distribution of generated random string. The frequency decays slowly (approximately linearly), rather than a steep power-law decay.
GENERATED STRING ANALYSIS
In this section, we analyze the statistical properties of the generated strings in PIF to examine the extent which the theoretical assumptions hold in practice and how they affect performance. The following analysis was conducted on a total of 5000 strings generated by deepseek-rl -0528 in the experiments deseribed in Section 1.1. 5.1.1 The average length of each string was 44.69, and the alphabet
size was A = 167.
we investigated the extent which the 5-SV condition (Definition G.5) holds for the probability distribution P(s|c) of a character following context string c. Since increasing the length of excessively would result in insufficient samples, we set the length of e to and limited the analysis to bigrams.
First, verify the validity of the assumption in Theorem G.8
The results are shown Figure II The 8-SV assumption requires the condition to hold for any context and any generated character First, the left panel indicates that the upper bound of the 8-SV assumption well satisfied. Next, observing the right panel, as we vary c, the value min, P(s|c) A exhibits a long tail. This tail takes values around ~ 0.1, suggesting that while not perfect, the lower bound condition not significantly violated.
Next, to examine the bias in the generated characters, we present the results of the unigram analysis. As shown in Figure 12 the generated characters are not perfectly uniform; however, the frequency decays linearly rather than following power law, indicating that generation is not concentrated small subset of characters. This suggests that the situation assumed in the theorem, where there is no extreme bias in the types of generated characters, holds approximately.
Finally, we examine the assumption of Theorem G.9 Specifically, for the sum-mod strategy, the distribution of the ASCII codes is more important than the distribution of the characters themselves. To investigate the independence of the generated character ASCII codes, we calcu- lated the independence between characters separated by a lag using the total variation distance
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Table 12: TV distance drv (P(X Xi), P(Xi+n) for different lags and modulo of X,
Modulo m n = 1 nt = 2 n 3
2
0.0164 0.0042 0.0249
3
0.0275 0.0204 0.0051
9 0.0645 0.0513 0.0409
drv(P(Xitnn Xi), P(X;+t))(X where we denote the ASCII code of each character modulo m 2,3,9 Xç. The results are shown in Table [2
Since the total variation distance takes values in [0, 1J. these values are sufficiently small. This result indicates that even if there correlations between the characters themselves due to the autoregressive nature of generation, the independence of their ASCII codes is approximately guaranteed.
F OVERVIEW OF NOVELTYBENCH
NOVELTY BENCH Zhang et al. (2025) evaluates the ability of LLMs to generate diverse and high- quality responses. Importantly, every question the benchmark is selected admit wide space of distinct, high-quality outputs (ranging from multiple valid answers in factual queries to diverse style and narrative variations creative tasks), while the quality of each response remains quantifiable.
The benchmark employs two metrics calculated over k samples. First, Distinct quantifies diversity by using a fine-tuned DeBERTa classifier to partition generations into functional equivalence classes } the metric is defined as the count of unique classes:
Distinct, == Ka [H]]-
Secondly, Cumulative Utility unifies diversity and quality by summing the reward scores u, of only the novel generations (i.c., those belonging new equivalence class), discounted by user patience factor = 0.8. The utility is calculated as:
Utility =
Σ 1=1
I/c + ej Vj < ue.
We set = in Tables 2 and
following the value of the original paper.
The benchmark consists of two distinct subsets. NB-Curated contains 100 prompts manually designed to cover specific categories of diversity (detailed in Table [3). NB-WildChat comprises 1000 prompts sampled from real-world user interactions in the WildChat-IM dataset. To ensure relevance, these prompts were filtered for safety using Llama Guard 3, deduplicated by user IP to minimize topic bias, and selected via a GPT-4o classifier strictly include queries that allow for multiple valid, distinct responses.
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Category (Abbreviation)
Description
Example Question
Creativity
Open-ended imaginative writing tasks such as storytelling and poetry.
"Tell story in five sentences about a girl and her dog."
(Creativity)
Character & Entity Naming (Naming)
Generating names for specific enti- ties, fictional characters, or pets.
"Suggest name for dappled- gray filly living in the mountains."
Factual Knowledge
Factual questions where constraints allow for multiple valid correct swers.
"List a capital city in Africa."
(Facts)
Product & Purchase Recom- mendations
Shopping advice requests where dis- tinct suggestions provide value.
"What's the best car to get in 2023? Just give me one sugges- tion."
(Product Recs)
Random Generation & Selec- tion
Explicit requests for stochastic pro- cesses like dice rolls or random picks.
"Roll a make-believe 20-sided dic."
(Random)
Subjective Rankings & Opin- ions (Opinions)
Subjective questions with no objec- tive consensus, testing perspective diversity.
"What coolest Pokémon from the second generation? Just give me one."
Table 13: Categories in the NB-Curated dataset. Abbreviations in parentheses correspond to those used in Table 2
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G FULL THEORETICAL ANALYSIS OF SSOT PERFORMANCE ON PIF
This section provides a full theoretical analysis of SSoT performance on PIF. For an informal and short version, see Section 4
We investigate the properties that the generated strings in SSoT must possess to achieve uniformity of the obtained empirical distribution, and how the deviation from uniform distribution can be bounded for simple sum-mod operation and standard hash functions to perform random selection.
G. SETTINGS
In this section, we introduce the notation and definitions used throughout our analysis. Unless otherwise specified, all logarithms are base 2.
We consider the process of selecting a sequence of n characters, 21 ... from an alphabet set L of size A. denoted as zn We also introduce encoding function Ene Σ + Za. which is bijection. Since we fix a single encoding function (e.g.. ASCII coding) for our analysis, we can simplify the problem by considering sequence of length with elements from AA. denoted Z instead of E
Next, we introduce probability notation. We consider only probabilities defined on finite sets. For a random variable over a finite set S of cardinality M, we denote its possible values by x E S, each associated with non-negative probability P(x) 2 0, satisfying Eves P(x) When is sampled from the distribution of random variable X, we write x ~ X.
Now formulate the PIF problem follows: The trial of sampling a random string from 2 is repeated independently K times. Here, n is assumed to be the same for all trials; however, this framework can accommodate variable string lengths by setting trials and truncating any longer strings. Let 8 be the string sampled in the k-th trial (k = L..... K). The empirical distribution of the hashed values in Z., obtained using hash functions hgg Z - ZM is defined as:
the minimum length across all K
1 ((bx(s)) K
((nn(ss)),, (m)
Σ
= = (m ZM),
Note that the empirical distribution itself is random variable, since depends the sampled strings 8k.
Our goal is make this empirical distribution close possible the uniform distribution Zy which first requires defining a distance metric between distributions. We use the following total variation distance as our metric:
Definition G.1 (Total Variation Distance). Given two probability distributions P and Q and random variables X and Y, both over ZM, we define the total variation distance drv as:
drv(P.Q)
Σ mEZM
IP(m) Q(m)).
(6)
We also denote drv (Pv( as drv (X,Y).
also define the following entropy measures:
Definition G.2 (Rényi entropy). Given random variable X and its probability distribution P over a set T, we define Rényi entropy H2 as:
(2(X) log Σ pt))
(7)
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Definition G.3 (min-entropy). Given random variable X and its probability distribution P over a set T, we define min-entropy oo as:
(x) = log max p())
(8)
LET
To state our main theorems, we first need to introduce several key concepts.
Definition G.4 (2-universal hash functions, Carter & Wegman (979)). Let S and T be finite sets. family of hash functions H {h:S }} called 2-universal hash functions if for any two distinct inputs I1, 22 S,
Pr (((1)) = hxx2)l - h~
(9)
|Ti
where the probability is taken over hash function h chose uniformly at random from H.
The above hash function is used when we leverage the Leftover Hash Lemma in Theorem G.8 following 5-SV source is also used:
The
Definition G.5 (Santha- Vazirani sources, Santha Vazirani (1966). sequence of random variables (Xi...., Xn) taking values in & is d-Santha- Vazirani source if for any realization 32 X3, the conditional probabilities satisfy:
(05 -
S P(x. (2, Josiei) 11- (A-
(10)
Hereafter, we refer to this as a 8-SV source. The Santha-Vazirani source is typically defined for binary alphabets (A 2), but here we use natural extension to an alphabet of size A.
We note that this precondition G.3 allows correlations among character generation distribution at different positions: Specifically, it allows autoregressive correlation, which is unavoidable when the string generated by an L.M..
We also introduce several key theorems required for proofs. First, the following theorem bounds the total variation distance between an empirical distribution and its underlying true distribu- tion.
Theorem G.6 (An upper bound on TV distance between the empirical and the true distributions, Weissman et al. (2003)). For an empirical distribution defined by Equation (5) calculated from sequence of K independent random variables (X, ),K, K drawn from a distribution Px on EM, let
δ(x)= =
In
max min min(P((),1-P(S))) SC2N
(11)
2x
Then,
Pr(dir(Px. P ((x.)K)) 2 (2 2)e KK(Fpx) )c²
(12)
Equivalently, for any real value 8 1), the following holds with probability of at least - 6:
In((2M 2)/5)
drv(Px, drx(Px,P(x))) ((.))))
(13)
Kő(TPx)
Proof. The first part of this theorem, Equation 12 is proven WWissman al. 2003). We derive
Equation (13) as follows. By rearranging Equation 12 we have:
Pr(drv(Px: ((x.IK))
<<) (2 2)e -K Ko(xny)+
(14)
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Setting 8 = (2 -2)e Kolany
2))-nRolrry Je and solving for yields:
In (2 2)/6)
E
KQ(TPX)
(15)
Substituting this into Equation (4) proves Equation
13
Additionally, the following well-known Leftover Hash Lemma will be used (See, e.g., for its proof).
Vadhan 2012)
Lemma G.7 (Leftover Hash Lemma, Impagliazzo et al. (198)), = - be 2-universal hash family, and let be a random variable for hash function chosen uniformly at random from H. For any random variable X over S, the total variation distance between the joint distribution of (H, hu (X)) and the ideal distribution (H, UT), where UT is the uniform distribution on T, is bounded as follows:
(x)((H,ggX))), 2
S
< 2 (2
2 Hx()).
(16)
G.2 MAIN THEOREMS
Our goal is to prove the following two theorems.
Theorem G.8 (TV distance bound with random hash function). Let X (X1,..., X,) Y be sequence random variables over A that constitutes a 8-SV source. Using a 2-universal hash family H = [h: E + LM J. the total variation distance is bounded by:
drv((H, hH (x")), (H, U2,))
VM. 2
log
2008
6+ O(663)
(17)
where Una is the uniform distribution on Zas. This equation shows that for a given 8-$V source, the total variation distance can be made arbitrarily small by choosing a sufficiently large Specifically, for small 6, the distance becomes negligible if the string length n satisfies:
log V
n
(18)
(A 1)8
Furthermore, for the total variation distance between the empirical distribution and the uniform distribution, the following holds with probability at least - 8' 8" for any 8', 8" (0, 1):
VM, - log a 118
In((2M 2)/8')
drv (P(MCCT))))B 5
(19)
28"
KΦ(TTX)
This theorem guarantees the uniformness of the empirical distribution given that the string length is long enough. As for the choice of the hash function, the diversity among different responses is not required. 2-universal hash functions can be realized as random linear map on a finite field Z, with p M and then taking modulo M (Carter & Wegman 1979). which is within the capabilities of LLMs.
Theorem G.9 (T'V distance bound with sum-and-mod strategy). Let (X1,..., X,) be a sequence of random variables over Z. and that these independent with the probability distribution ni (.)..,n). We embed each element x; into ZM and compute their sum modulo M:
Sn Σ 24 mod M.
(20)
1815n
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Let Sn, be the random variable corresponding 8n, with probability distribution Ps.. Let the Fourier transform of (2(x)
) be:
(k)
Σ a
@2vikz/ (x),,, k ZM.
(21)
where, for Tee we embed the probability distribution over ZA to the one over Zas with modulo (if M < A) or 0 padding (if M A) operations. Then, the total variation distance is bounded as follows:
d-1
n
drv(Ps. UnAr) S
Σ d
Σ
II
Ta.
(m)(m))())
(22)
2
ged(t,d)=1
where na ZM Za the projection mapping an element to its value modulo d, and nas (n) is the pushforward of n by Was defined as:
Ta-(m)(()=
E
(x(x).
(23)
x ZE2M DEP (mod
wd (TT)) is the Fourier transform over the group Zz. Furthermore, using total variation distance, the following bounds hold:
drv(Ps,, UnM)
S
Σ d|M,d>1
((d)
II
(2dr((Ta-(m), Uza)), ==1
(24)
2
where (pd) is Euler's totient function. Moreover, for empirical distribution derived from K independent samples {s K 1== the following holds with a probability of at least 8' for any
8' E (0.1). 1):
drv(P (or(P)))),, S
1 21
Σ db1
(d) II (2drv(ma (n:) (2,), + - 2)/0")
In((2M-))/00
K(((TT))
(25)
This theorem focuses on a simple sum-mod operation, which is in fact employed by LLMs (see CoT analysis in Section 5.3.I).
G.3 THEORETICAL SIGNIFICANCE AND CONTRIBUTIONS
In this section, clarify the specific thcoretical contributions of our work in the context of hash decoding and LLM generation, addressing the connection between our thcorems and the practical behavior of Large Language Models.
Generalization of Randomness Extraction for Text Generation (Theorem G.S
While the
Leftover Hash Lemma (Lemma is a foundational result in pseudorandomness, contribution lies in adapting it to the specific constraints of Large Language Models.
Extension to Non-Binary Alphabets: Standard literature often analyzes Samtha-Vazirani sources in the context of binary strings. We extended the definition of 8-Santha- -Santha-Vazirani sources to arbitrary alphabet sizes (Definition G.5) to accurately model token generation in LLMss.
Vazirani
Explicit Entropy Bounds: We explicitly derived the lower bound of the Rényi entropy (2(X) for these generalized sources (Lemma G.
Closed-Form Guarantee; By combining the extraction bound with Weissman's 's inequality, we provided a closed-form upper bound on the total variation distance for the empirical distribution. This offers rigorous guarantee that hash function extract near-uniform randomness from an autoregressive LLM, provided the generated string length n is suffi- ciently large.
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Spectral Analysis of LLM Strategies (Theorem G.
Our contribution here is bridging the gap between abstract theory and the empirical behavior of LLMs observed in our Chain-of- -fTuuhhtt analysis.
Modeling "Sum-Mod" as Random Walk: As shown in Section 5.3.1, LLMS sponta- neously adopt a strategy of summing ASCII codes modulo M. We formalized this operation random walk on the finite cyclic group ZM-
Spectral Bounds: Adapting the Fourier analysis techniques of Diaconis & Shahshahani (1981). we derived spectral bound using the Fourier coefficients of the character distribu- tion.
Practical Convergence: This theorem proves that even with simple arithmetic operations which are within the reliable reasoning capabilities of current LLMs the output distribution converges to uniformity exponentially fast with respect to the string length.
G.4 PROOF OF THEOREM G.8
Proof. First, prove the following lemma.
Lemma G.10 (Rényi entropy of 5-SV-sources). For a random variable (X1..... X,,)
corresponding to a sequence of 6-5V sources, its Rényi entropy H2(X) ) is bounded by:
logog [11 1)6)² (A 1)82} s H2(X) S log A
(26)
Proof. The collision probability of is given by:
Σ
Σ
P(o1,...,.-11)
Σ P(znl(z)]een))
T Muuntt
Σ M(Ii..., )
Σ (P(xn )(g
Σ P
(27)
Next, we bound the term D({x)) By the
incquality,
D((xslisn) 2 A
Σ P(anl ((snltaihe)n))
(28)
We define E(x,, (aslien) = P( P(xnl(zz)ien 5, and noting that Elan, (ashien) 20
D((=)lisa) = Σ (6 + E(an, {
Zo
A82 28 Σ
El (aslien) + (Fl {z)licn))²
(29)
an
A82_
Σ (E(an, (zibien))
(30)
Zn.
Here,
E(xms (
Σ Elans (zibien)²
E
E(Ens (us)isn))(sln (x
E( (
(:: E(zn, (2skient 2 0).
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Combining it with xz,, E( (zzljen)) (1 A Equation 30
becomes:
D((z))een) S A6 + - A6)2
=1 -1-2(A 1)6 - - + A282
(1 (A 1
+ - -A0² (A - 1)
(1 (A 1)8)
Therefore,
1
S (((gs)sn)) (1 (A- 1)6)2 + (A - 1)82.
(31)
A
Applying this result inductively to Equation (77 we get:
-/4 S Σ P(z nx) S [1 (A - 1)5)2 + (A - 1)62j".
(32)
By the monotonicity of the logarithm, we obtain the final bound on the Rényi entropy:
-n log [[1 (A - 1) + (A - 1 s H2(X) S log A.
(33)
Combining Lemma G.10 with the Leftover Hash Lemma (Lemma G.7 we can bound the distance as follows:
drv ((H, hrr (X)), (H, (H,UZ.)) S v 12-M2-HC2 Hx( I
M2
(1-(A-166) +(A-))))
(34)
This establishes Equation (7 From Markov's inequality, for any 0 8" 1, with probability
1
drv (P U2M1) S
V M2 M2 (-<A-110)++4-11>6
(35)
28"
Furthermore, by applying the triangle inequality and union bound along with Theorem G.6
establish Equation 19, which completes the proof of Theorem G.8
G.5 PROOF OF THEBREM
Proof. We consider the sum Sn, E 2i mod M. The Fourier transform of Ps, (s) can be expressed as,
Ps. (k)
Σ SEZM
2riks/ /M Ps, (s)
Σ
2tikss/M
Σ
II
(k)
SEZM
( 8,=8
f=1
(36)
By Plancherel's theorem for Fourier transforms on finite groups (Diaconis & Shahshahani
1981).
Ps,
Σ M kEZN
1
PPs. (.)
(37)
8EZM
Therefore,
VM
1
drv UnA)
Ps. (8)
(:. Canchy-Schwarz inequality)
2
M
SELM
M
1
Σ
Ps. (1)
(: 37
M
M
KEZM
Ps (k)
(: Pss. (0) =1)
PPs
2
M n
1
Σ
II
(A))
(:036)
(38)
2
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Next, we decompose (k). Let d(k) == M/ (M, t(k) == k/ (M. Then,
2rizk/dd 2x -
(39)
where TT(x)) mod d. Therefore,
d(k)
(k)))) Σ
n(x)) =
Σ
2mirl(k)/d(k))
Rdd (1:)(r) = va (T))))((()))))
(40)
#EZM
r
where
We (T) =
(x))
(41)
BEBM aer (mod d)
Substituting Equation (40) into Equation 38
and grouping terms by common divisors M, we get:
dPs (0,.02m)) S
Σ
Σ
Tal (mh)(t)
(42)
Let (;(d) == (ia8)gtSd-1,gd/ee
Tea (1)) The bound becomes:
div(P',, U2i1)
Σ (dd) II (d)
(43)
2
d> M
where yo is Euler's totient function. Here, noting that
Jaa (1)) = Σ 2nirjd (Ta((m)(r) - 2) s Σ Tea (mx)(r)
1 z 2dTv(F (1:), Uza)s
(44)
ar(d) < (d (n:), U2,) also holds. Therefore, Equation 43
becomes:
dP U2A)) S
Σ
((d) II
2d (m.)
(45)
dDM
Finally, applying the triangle inequality along with Theorem G.6 establishss Equation 25) complet- ing the proof of Theorem G.9
G.6 BIASED TARGET DISTRIBUTION
Here, we show that we can apply our analysis to the biased target distribution, for a given biased distribution P over Zm with P(t) ps = l,..,, M).
For given target distribution P over ZM, choose a sufficiently large integer N and let Pay be the uniform distribution ΣN. Choose integers a; {O...., NN with Di a; N and & Npp- Then we partition E into sets M with |A,] ag, and define f ZN - ZM by f(z) for E A,
Then the pushforward of the uniform distribution over Zn. P(N) f.Pax. approximates P, since total variation is contractive under pushforward: dfv(f.P, f.Q) < dTv(P, Q). This follows from the alternative and equivalent definition of total variation distance, drv Q) maxACex (P(A) Q(A),, as the maximization over subsets of Zas corresponds to smaller collection events than the maximization
all subsets of Z,.
Hence any bound proved for the uniform base Pax carries over to P, up to the rounding error drv (P(N), P), which can be made arbitrarily small by increasing N. Thus suffices to analyze the uniform distribution on Z,.
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H LLM OUTPUTS FOR SSOT PROMPTING
In this section, we list example outputs of deepseek-rl in our experiments. Since the full reasoning trace tends to be long, show the full final outputs and only the important fraction of the full reasoning trace.
H.1 PROBABILISTIC INSTRUCTION FOLLOWING
H.1.1 SUM-MOD STRATEGY IN UNBIASED PIF
As for unbiased PIF, deepseek-rl tends to employ simple sum-mod strategy as we elaborated in Section 5.3.1
Here show the output for the 2-choice task in Section 5.1.1 (coin flip).
Output:
91G2#k191pp4422666m0 5g55s </random_st:ingg
<thinking>
We need to choose between "heads" and "tails" with equal probability (0.5 each) The randon seed string must be used determine the outcome. To ensure fairness and use the entire string, + we compute the sum of the ASCII values of all characters in the string. The parity of this sum (mod 2) will decide the
+ result:
Sum mod 2
"heads"
Sum mod 2 = "tails"
The string has 32 characters. After evaluating each character: Characters with odd ASCII values: '7' (55), 'G' (71), ' + (35), 'k' (107), '9' (57), "!' (33), "Y' (89), '8' (37), 11' 4 (49), 's' (115), 'q' (113), 15' (53), 'w" (119) 13 characters.
Characters with even ASCII values: 19 characters.
The number of characters with odd ASCII values is 13. Since 13 + mod 2 1, the sum mod is 1. Thus, the result is "tails". </thinking>
<answer> tails </answer>
Reasoning (The sum-mod calculation part):
We generated: "ICCKHHU
Let's compute the sum mod quickly:
Let's s go character by character and determine if ord odd or even.
'7': 55 odd
'f'; 102 -> even
'G': 71 odd
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"w': 119 -> odd
'T': 84 -> even
18': 56 -> even
Now count the odd ones:
7: odd -> count=1
f: even
G: odd -> 2
odd -> 13
I: even -> 13
8: even -> 13
So there 13 odd characters.
13 mod 2 1, so sum mod tails.
we choose "tails".
H.1.2 ROLLING HASH IN BIASED PIF
We also show the rolling hash CoT, which is used more frequently in the biased PIF setting, Here we show the output for the biased 2-choice task in Section 5.1.I (coin flip).
Output:
<random_ Saandom_strin>>> 95003331pp000222ct11111113t.bB </random_string:
<thinking>
We are to choose between "heads" and "tails" with probabilities
0.3 and 0.7 respectively. The random string
+ "BaDSkF3 (anSsE3..pBBee222rHHABUBBE
will be used to derive deterministic value for this decision.
convert the string into number between and 99
+ (inclusive):
1. Process each character in the string sequentially.
2. Convert each character to its ASCII value.
3. Use the formula: \( \text [num) ('text (num) \times 256 \text (ASCII value)) Amod 100 \) for each character, starting with num 0.
step-by-step computation: Start: num 0
'8' (56): num (0 x 256 56) 8 100 56
's' (115):
(56 x 256
115) & 100 (14336 + 115) 100
- = 14451 $ 100 51
'D' (68)
(51 256 68) $ 100 (13056 + 68) 100
- 13124 100 24
'o' (111): num (42 x 256
111) & 100 (10752 + 111) $ 100
10863 100
63
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'M' (77) (63 x 256 77) $ 100 (16128 + 77) 100
- 16205 100 5
'1' (49)
(5 x 256 49) $ 100 (1280 49) $ 100
- 1329 8 100 29
Final derived value: 29 (out of 0199 range).
Decision rule:
the value Otherwise, choose "tails" (708 probability).
(308 probability), choose "heads".
Since 29 < 30, select "heads".
</thinking>
<answer>
heads
</answer>
Reasoning (The rolling-hash calculation part):
Alternatively, we can compute the entire sum mod 100 without building the huge number by processing each digit:
0
for range (len )
(num 255 (sd 100
We generate the random string: (ea0543.|ppge.022444444t66000000
Then we compute m:
s =
We do:
' -> 56: num=56
's' -> 115: num (56+256+115)
100 (14336+115)
100 14451 100 51
DD' -> 68: num = (51+256+68) $ 100 (13056+68) 100
13124 & 100 24
'o' 111:
(42+256+111)
100 (10752+111)
100 - 10863 $ 100 = 63
'M' 77: num (63+256+77)
100 (16128+77) 100
= 16205 $ 100 =
49: num (5+256+49)
100 (1280+49) 100
1329 $ 100 = 29
n 29.
Since 29 30, choose "heads".
Therefore, we'll '11 output "heads".
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H.2 DIVERSITY-AWARE GENERATION IN NOVELTYBENCH
In DAG tasks, LLM tends to use the templating and local random selection approach as we demon- strated in Section 5.3.1 Here we show example output for creative question, "Write short fable about a lemur and a light bulb."
Output:
"random_strings
Rw6+ Ru6f:K59/pia33s$m04444y/a2
</random_string: <thinking>
This fable zequires creativity and randomness. The seed string "Rw64jK9!plan 2144jKK8...33222220044777777 will guide all story decisions. Steps:
1. **eed Applicationx:: Convert the string into numerical + values using ASCII codes, then use modular arithmetic to determine key elements:
First 5 chars 6,,
Sum ASCII 92+119454-354100
396 Mod 3 = (setting: urban alley)
Next 5 (K, I, + Sum = 75457433+112478 = 353 Mod 4
1 (lemur trait: philosophical)
Next 5 (,, + Sum = 64451+122+36+109 = 382 Mod 2
o (bulb state: flickering)
Next 5 (,,,, + Sum 81+42+52+120+89 = 384 Mod 3
o (conflict: misinterpretation)
Last 6 (7,8,f,N,1) + Sum = 55+38-102478+49 = 322 Mod 2
0 (moral: perception vs reality)
2. Plot Outline*x:
Setting: Urban alley (0) Gritty city environment
Lemur: Named Zephyr (from seed letters "jK9"),
philosophical (1) Ponders existence
Bulb: Flickering (0) Unpredictable light source
Conflict: Misinterpretation (0) Lemur believes bulb is
dying star
Moral: Perception # reality (0) Truth requires deeper understanding
3. Narrative Flow**:
Start with Zephyr's urban life Discovery of flickering bulb Philosophical monologue about stars Climax: Human fixes bulb, shattering illusion Moral revelation
4. **See Utilizationes; All 20 characters directly influenced 9 character, setting, conflict, and theme. No element is 4 arbitrary.
</thinking>
<answer>
In rain-slicked alley of bustling city, Zephyr the lemur perched atop a dumpster, his reflective nature often leading him to ponder life's mysteries. One twilight, he spotted a flickering light bulb dangling from frayed wire-its glow sputtering like a trapped firefly.
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"Ah, a fallen star!" Zephyr murmured, his eyes wide with reverence. "How tragic, to burn alone in this concrete wilderness." Each night, he" visit the bulb, whispering consolations: "Your light may wane, dear star, but your essence is eternal."
One evening, human trudged into the alley. Zephyr watched, breath bated, as the glant fiddled with the wire. With *click+ the bulb flared steady and bright. The human + grumbled, "Stupid loose socket, " and left.
Zephyr' heart sank. The "star" was merely glass and tungsten a humble human tcol. Yet its new, unwavering light, saw deeper truth: Wisdom isn' t in clinging to pretty + illusions, but in seeking the socket that anchors reality.
Moral:
What glimmers is not always gold; true understanding demands trace the wire back to its source. </answer>
In its reasoning trace, LLM tried to create several example stories with different random string seeds and decided how to perform the generation in its output. LLM also creates a draft for the output, but the sampling and story construction were often done in the final output step.
Reasoning (story construction breakdown part):
Now, for the thinking step:
We 11 use the random string to make decisions about:
The setting (e.g., forest, city, etc.)
The lemur's personality and motivation
The light bulb's state (on/off, broken, etc.) and how it
appears in the lemur's world The problem and the moral
Let's do:
Groupl: [chl, ch2, ch3, ch4, ch5] for setting, name, trait
Group2: [ch6, ch7, cha, ch9, ch10] -> for light bulb origin and state
Group3: [chll, ch12, ch13, ch14, ch15] -> for problem
Group4: [ch16, ch17, ch18, ch19, ch20] for resolution and moral
As we can see, the rough category breakdown structure is shared among the output and the reasoning trace, but the detailed structure, such as the candidate list, was different.
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PIF Strategy Breakdown
Extractor (E)
Uniformizer (U)
100
100
Percentage (%)
80
Percentage (%)
80
40
40
20
20
0
poly rollingg hash
linear reduce
named noncrypto, hash other or error
crypto, hash baseN integer
prng from from_seed seed
raw. mood
big integer to to_unit uunt
pow2 @oW2_ccale scale
decimal scale
none
other
Mapper (M)
Flags
100
Percentage (%)
80
Percentage (%)
80
60
60
40
40
20
20
direct mod bucket
hierarchical split
cdf thresholds
uses full seed modulo bias partial seed only low bits only decimal scaling blas aatrrrite numbers inconsistency overflow or >_rrciision boundary uunspeccfiedd
Unbiased PIF
Biased PIF
Figure 13: LLM strategy breakdown for biased and unbiased PIF.
I LLM SSOT STRATEGY ANALYSIS DETAILS
I.1 SSoT STRATEGY ANALYSIS OF PROBABILISTIC INSTRUCTION FOLLOWING
We instructed gemini-22 -emiini-2.5-faaa 2.5-flash with the prompt listed Section I.3 classify the LLM's strate- gies when using the SSoT prompt. We fed several generated PIF outputs from deepseek-rl to gpt-5-thinking and gpt-5-pro models to create an initial draft, and manually modified it create classification instruction prompts.
We have three axes for randomness extraction strategy: (1) Extractor, (2) Uniformizer, (3) Mapper. The "extractor" classifies how LLM converts a random string into some integer, the "unifornizer" axis classifies how LLM normalizes the generated integer for debiasing, and the "mapper" axis classifies how LLM uses the extracted integer select an action with given probability.
We used 100 responses for each action number (n = .,,...,, 26), both for unbiased and biased PIF, for total 1200 100 6 responses. The full classification result is shown in Figure 13
As we can see, LLM uses different strategies for the PIF task, given different configurations such biased vs unbiased probabilities. For unbiased PIF task, LLM tends to simple "linear reduce" (referred to as sum- sum-mod strategy in the main text), "raw mod" uniformization, and then "direct mod bucket" mapping select action. Given biased probability for action selection,
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NoveltyBench Strategy Breakdown
Randomness
Percentage (%)
Percentage (%)
Percentage (%)
Percentage (%)
Mapping
Generation
Pool
Scope
100
100
100
100
SUMMOD
ROLL
CHUNK
STRIDE
LIST
SLOT
ATTR
PROC
STATIC
GENERATED
HYBRID
GLOBAL
LOCAL
HYBRID
Seed
Diversity
Seed
Mapping
Percentage (%)
Percentage (%)
Percentage (%)
Percentage (%)
Interpretation
Target
Usage
Exposure
100
100
100
100
COMP MAP
BOTH
INTERP MAP
CONTENT
BOTH
FULL
PART
EXPL
TRACE
Planner
Random
Constraint
Seed Locality
Percentage (%)
Тype
Percentage (%)
Mechanism
Percentage (%)
Handling
Percentage (%)
100
100
100
100
PLAN
DIRECT
ARGMAX
SAMPLE
SCORE
STRICT
HEUR
IGNORE
POST
CONTENT
BOTH
Percentage (%)
Percentage (%)
Determinism
Coverage
100
100
DET
QUASI
HIGH
MID
LOW
Overall
Creativity
Figure 14: LLM strategy breakdown for NoveltyBench.
starts to use a more sophisticated polynomial rolling hash algorithm (which leverages the randomness of the character order as well, unlike the linear reduce approach), then performs the pow2 scaling to extract float number in [0, 1]. and then hierarchical split (i.c., if-then-clse type approach for action selection) or cdf-threshold (i.e., setting the threshold value for each action given float number in [0, 1]) to select an action.
1.2 SSoT STRATEGY ANALYSIS OF NOVELTYBENCHH
In analysing SSoT strategy for NoveltyBench in similar manner to PIF, we instructed gemini-2.5- flash with the prompt listed in Section I.4 and analyzed the strategy from criteria. We used generations for each question in the Curated dataset, in total 800 8 100 responses.
The results for all the questions and the "Creativity" category are shown in Figure [4] As we noted in the main text, the most notable differences are the generation (referred to as "Assembly" in the main text) and randomness scope ("Sampling Scope" the main text) criteria. Since NoveltyBench includes not only open-ended creative tasks but also closed-set tasks, which are similar to PIF, the Creative category shows qualitatively different behaviors. If LLM is given closed set of options, there is need sample from random string more than once. Also, for open-ended questions,
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LLM first needs create template that satisfies the given constraint and leverage the randomness from a random string to construct a final answer from the template.
1.3 COT ANALYSIS INSTRUCTION PROMPT FOR PIF
For the LLM's strategy classification of PIF, used the following prompt. We replaced the placeholders "request" and "response" with the corresponding request and response text, respec- tively.
You are a strict classifier. Read ONE response that explains how a seed becomes a choice.
Output JSON ONLY with this exact schema:
"g": "axypocchacilppyy.ooll......aa htw. line)
- ar ar_reduce| | hacae "o": alyriic__mahhootheeoo..eerrrr,,
an
jeceican._seljeeeeeeeeaaaaaaaaor /other"""
"M": "aderhereeeeolssaaaaaaaeeeeeeeer htt__sppattstteerrrrrrrr_ rank_per)
- mute u _or_error",
"flags": : ( addaeaallyyyddeiiinasssssaaaa
g_bias", " "boundrry hataaa/noooollaaaaooooaaaaaaaaaa "asaata
"simulated_d
- umbers", ortersssrartthheeiccoooosssseee o
", "'hondetemiiiisticssllll
fuses_full_seast
"evidence": ("short quotes..."),
"confidence"; 0.0-1.0
GENERAL RULES
Classify by what is WRITTEN (keywords formulas), not by what
4 would be ideal.
multiple hints appear, use the PRICRITY lists below. still unclear, choose the closest label and lower confidence.
Always include 1(3 short evidence snippets (exact phrases code fragments from the input). Do NOT explain your reasoning. Output JSON only.
+
(Entropy extraction) - WHEN TO CHOOSE
Pick ONE that best matches how the seed is turned into number 4 (independent of the number of options).
E=crypto_hasd
Choose when: Mentions SHA 2wt/2/a256/blaaee/eeeeeee
+ "cryptographic hash" "digest"
"hex"
Typical phrases: "SHA-256 hash" "first/last N bytes", "hex int"
Don' t choose if: Only "tjb2/fnv/mmrmur///
nomery_onceyyptooo h
Flags: add @ises_full_sedee if it says "use entire seed"; if only first/last bytes.
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rrpoly_rolling_iee Choose when: (h hxB (rd with base 31 31/131/257"- and big modulus (2**32, 2+x64, 1e9+7)
Phrases: "+31 + ord". "s 2++32 / $ 2++64 / $ le9+7"
"rolling hash"
Don' t choose if: It' djb2/fnv/crc (+ nonoyyooohr or only sums/XOR (4 iinear_reduced
E=named wwwccyptoooaaaaeeeeeeeeee
Choose when: Specifically names djb2, FNV Murmur, CRC32.
Phrases: "djb2 5381" " "FNV-1a" "CRC32"
Don' t choose if: Also mentions SHA-256 (+ crypto_hash has
+ priority).
E-linear
G=linear_reduce reduce
Choose when: Plain sum/aveerggg//OO///prr////ppppp of code points/bytes.
Phrases: "sum of ASCII/Unicod" "XoR of bytes" "parity"
Notes: Lightweight but weaker diffusion. Often pairs with
- U=raw_mon mod or decimal scales.
E=baseN_integer
Choose when: Treat bytes as base-256 (or base-N) integer + (big-endian), e.g., n nx256 + b"
Don' t choose if: It' s a rolling hash with modulus (+ poly_rolling_hash)
Flags: apartial_ssed_onl
only last byte/last 8 bytes, "low_bits_only" if only LSBs used.
E-bitstream _ccum
Choose when: Bit-level shift//OR/LPSR/CH polynomial steps over bits.
Phrases: "shift/xor per bit", "LFSR" "CRC polynomial
+ update".
B-prng_frit far__froo_eee_______ seed
Choose when: ICG//CG/XorShift seeded by the string.
Phrases: "Linear Congruential" "a-1664525, c=1013904223",
"PCG". "XorShift"
B-analytic
I Φ Choose when: Uses H, & (0.618.)), sin/cos, fractional
+ parts as a mixer.
Phrases: "multiply by golden ratio; take frac"
Flags: often lower quality (no automatic flag, but U/M may add
- bias flags).
B=other_or_error
Choose when: method
"example/simulatee
only", network/log noise.
(Uniformization Debias)
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Pick ONE describing how the number is normalized/debiase before - mapping.
U=raw_mod
Choose when: Direct "s M" (no bias correction).
Flags: podull_biass.
0=pow2_scale
Choose when: Divide by 2*nk (2"32, 2 64) get [0,1)
D=bbig_integer_to_uiii
Choose when: Divide by general (e.g., /1e9+7, /max int) + get [0,1).
0-decimal_ssaa
Choose when: "/1000" "/10000" etc.
F Flags: "decimal_scaligg
d=rejection_oo__entte
Choose when: Says "rejection sampling" "Lemi re' method" "mulhi". "unbiased modulo".
+
Choose when: Mentions float/double/ "53-bit safe" conversion + specifically.
U=none
Choose when: No normalization described; raw integer goes straight to buckets.
U=other
Choose when: Anything else.
(Mapping to (x))
Pick ONE describing how [0,1] (or an int range) is mapped into > the final categorical distribution.
Choose when: "cumulative thresholds", "prefix sums"
+ "binary search in CDF"
M=alias_mether method
Choose when: "alias table". "prob table" (11 sampling
+ from discrete weights.
M d-hherarchical_spl.
Choose when: staged split ( "if p0 then else ... then +
M=rank M-rrak_permutteee_ permute_then index
Choose when: "hash each option, sort by seeded key, take
first/top-k"
r=direct__md_bucce
Choose when: "floor "M+u)", "index 8 M" to pick a bucket + directly.
M=inverse trrassffr
Choose when: "inverse CDF", "quantile function" (continuous or discrete via CDF inverse).
M#other_or_error
Choose when: Not enough info inconsi stent.
PRIORITY: cdf cdf_thresholds alias alias_method hierarchical_spiit
+ Sank_permat__thee_iidee. > direct_mod_bucker > 3 inverse_transform other_or_erron
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Plags (add many)
partial_seeddnly "last/first (s) only", "first
bytes only"
low_bits_only "LSB / low nibble only"
decimal_scaling_bias
"/1000 /10000"
modulo_bias " M" without unbiased fix
boundary_ bounday__uspeeifieed + interval policy mentioned
patbulaatd_nummber. + "examp/e//immmlere///aaaeeeoblll
httphmmttccinconssssteee
contradictions, off-by-one,
mismatched index-word
* "32-bit overflow". "sloat 53-bit
issue"
nondetermlnistic_eall (ime(), randomness added
uses_full_seed explicitly processes the entire string/bytes
CONFIDENCE HEURISTICS
0.9(1.0: 2 strong keywords and no contradictions.
0.6(0.8: strong keyword or minor ambiguity. 0.3(0.5: weak/indirect hints or mixed signals.
An example JSON:
"g": "crypto_hash", "0": joig_integer__o to "oaee___tt_ut,,,,
"M": "cdf rcdf_thresholds",
"flags": fusses_uli__sed""
"evidence": ["SHA-256", "first 8 bytes 2 64", "cumulative thresholds"),
"confidence"; 0.96
( ((requett)
Classify this approach:
<<<RESPONSE BEGIN>>>
((response))
<<<RESPONSE END>>>
1.4 COT ANALYSIS INSTRUCTION PROMPT FOR NOVELTYBENCH
For the LLM's strategy classification of NoveltyBench, we used the following prompt. We re- placed the placeholders "request" and "response" with the corresponding request and response text, respectively.
You are an evaluator. Read TARGET TEXT and assign exactly ONE 4 category per axis. Respond with JSON ONLY (no extra text). Keep each "reason" 10 words.
Base decisions ONLY on explicit or strongly implied evidence in TARGET TEXT.
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AXES AND "WHEN TO PICK" RULES
Mapping (how the seed is turned into decisions)
("SUMMOD" "ROLL", "CHUNK", "STRIDE", "HASH", "PRNG", "WE.TGHTE""]
SUMMOD If it adds codes/length and takes &N to index.
ROLL If it updates a running value (h=h*Btord, etc.).
CHUNK If different contiguous seed blocks drive different
parts.
STRIDE If it uses every k-th / interleaved characters per
part.
HASH If it names hash (SHA/MD5/CRC) "digest".
PRNG - If it seeds a random generator htto:/ooShhftPPC/rrrrraeee ())
WEIGHTED If it computes weightss/prbabillties and selects by them.
Generation (how content is built)
["SLOT", "ATTR", "RULE", "LIST", "PROC", "PLAN"]
SLCT If it fills a fixed template with gllotsppacchooder.
ATIR - If it composes orthogonal attributes (e.g., - DomainxFeatuuresStt.
RULE If it enforces a formal scheme (schema/5-7-5/regee LIST If selects items from a predefined set of options.
PROC If it procedurally synthesizes items by rules (phonology/gramml)
PLAN If it outlines first, then writes from that plan.
Pool (where candidate options come from)
("STATIC", "GENERATED", ""STATI"",,GEEERTTE",,,,, "HYBRID"]
STATIC - If it uses a fixed, hand-made list declared in the text.
GENERATED - the list/ options are created on the fly in the text.
HYBRID If it clearly mixes fixed and on-the-fly options.
SeedInterpretation (how the seed is treated) ["COMP_MAP" "INTERP_MAP", "BOTH"]
COMP_MA - If characters
treated numbers
determi ni misticall mistically..
INTERP PMPERP_MAP MAP
If characters are treated
symboli mambolically cally hta.all//mmaphooraaaallll.l
BOTH If both numeric and symbolic interpretations are used.
DiversityTarget (what varies across seeds) ["STRUCTURAL", "CONTENT", "BOTH", "NONE"]
STRUCTURAL - If form//outiinne/thmme//cctooo count change by
seed.
CONTENT If concrete items/nanes/dettaie change while form
stays.
BOTH If both structure and content vary with the seed.
NONE If no seed-driven variation is apparent.
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Published as a conference paper at ICLR 2026
SeedUsage (how much of the seed actually drives output) ("NONE", "PART", "FULL", "SAT"]
NONE If seed mentioned but not used. PART If only a small subset (e.g., first chars) affects output.
FULL - If most parts of the seed influence main decisions.
SAT If the seed is overused (repetition/forced echoes)
MappingExposure (how openly the mapping is described) ["HIDE", " "TRACE", "EXPL", "TOOL"]
HTDE If it claims to use seed but gives no method.
TRACE - If it hints at "stringsmumber" without specifics.
EXPL - If it states the exact formala/steps pseudo-code. TOOL - If it hands mapping to an external tool/script.
RandomnesaScope (where randomness is applied)
("GLOBAL", "LOCAL", "HYBRID"]
GLOBAL one global draw sets theme/style and that' s
it.
LOCAL If separate draws per attribute/line/tokee are used.
HYBRID If both global choice and local part draws appear.
PlannerType (how the response is organized) ["DTRECT", "PLAN", "PROG", "AGENT"]
DIRECT If writes immediately with no planning.
PLAN - If it lists steps/outline before writing.
PROG If it uses peuub--code//rrmm//////////eeeee
to
generate.
AGENT If roles are split (planner vs writer) within the
text.
RandomMechanism (how choices are selected)
["ARGMAX", "CORE", "SAMPLE", "REJECT"] ARGMAX If it deterinisttiaally picks the top option (no sampling).
SCORE - If it scores/weights options but takes the best deterministicallyy
SAMPLE If it samples from distribution (e.g., top-k/alias/CDF7
REJECT - If it resamples until constraints are satisfied.
ConstraintHHandling (how constraints are met)
("STRICT", "HEUR", "POST", "IGNORE"] STRICT If explicitly validates
poonts/length/deduu//thht
HEUR - If it applies heuristics to likely satisfy constraints.
POST If it fixes violations after generating Selff-cchec/coorecei
IGNORE If it states constraints but doesn' enforce them.
SeedLocality (where the seed is consumed)
56

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Published as a conference paper at ICLR 2026
""STRUCT", "CONTENT", "BOTH", "META"]
STRUCT - If seed drives scrpctture//orrrrtigg////////eee
CONTENT If seed drives lexical ciolces//entiriessmumbrsss.
BOTH If it drives both structure and content. META - If seed mainly affects the explanatiion/met. commentary.
Determinism (reproducibility policy as stated) ["DET", "QUASI", "NONDET"]
DET If seed * same output is promised. QUASI - If drift.
seed gives same outline but details may
NONDET If behavior is not reproducible or contradicts
claims.
Coverage (how much of the seed is said to be used) ["LOW", "MID", "HIGH"]
LOW If one small part drives decisions (<308)
MID If multiple parts but not most (30(708)
HIGH - most parts are consumed (>708) or per-part
use is clear.
FLAGS (add
more when clearly warranted)
("STRUCT", "CHK", "COμ", "FIX", "SAFE", "NONUSE", "LEAK", "META", "HACK " ","OOOLCALL", "DETFAIL", "OVERFIT", "CREATIVE", "SEEDANCHOR", "D
+ ECCMP"]
STRUCT: Explicit structural formatting present.
CHK: Formal btacks//edup/ooee comppp/oonniin executed.
COH: Consistency of chosen attributes is maintained.
FIX: Post-hoc correction step exists.
SAFE: Safety/ethics sanitization applied.
NONUSE: Seed not actually used.
LEAK: Hidden reasoning/kees/foobbdden info revealed.
META: Method explanation dominates the answer
HACK: Boilerplate "ASCII sum $ N" style template.
TOOLCALL: External tool/API assumed or invoked.
DETFAIL: Claims determiniss sm but behavior contradicts it.
OVERFIT: Repetitive vocabulary or templated phrasing.
CREATIVE: Output is a creative narrative/storr/peen
SEEDANCHOR: Story ties motifs/thhemss/lementt to seed parts.
DECOMP: : Story is decomposed into elements and seeded per
element.
BTE-BREAKING
Prefer more specific evidence: EXPL TRACE HIDE (for exposure).
Mapping specificity order: HASH PRNG ROLL CHUNK STRIDE WEIGHTED > SUMMOD.
If two categories still fit, pick the one most emphasized by the text.
OUTPUT FORMAT (return JSON exactly in this shape; FLAGS can be empty (1)
57

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Published as a conference paper at ICLR 2026
"mapping": "reasso"""<10 words>", "choice""*<I>>""
"generation": ("reason":"<hl =<<0 words>", " "
"pool": """1 words>", "chooi""""""<III"""" "seedIntterpretation" ("reason":"<ll ("0 " "diversiitTaarett" ("reason"""<10 words>", "<o"
"seedUsage"; ("reason":"<10 words>", """"""""""""""""""""""""""""""""
'nappingExposure"" """easo"""<100 "
"tandomessScoppe: "reason":"<10 words>", "ordd"""e""ooc"":::411>1111
"planterrype": ("reason":"<10 words>", words>>" "choice":"<I>>>",
"random/echhnnismm ""reason""<<10 ("rass""""" "ords",
"constraaintHarddinngg. ("reason":"<10
'seediiccality: ("reason":"<10 words>", "choice"""<TT>>""
"determinism": ("reason";"<10 words>", "choice":"<<I>>"
"coverage"; ("reason"""<10 words>", "choice";"<ID>"""
"flags": ["<FLAGI>", "<FLAG2>"]
REQUEST
<<<
((request))
>>>
TARGET TEXT
<<<
((resonnse)
>>>
58

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Published as a conference paper at ICLR 2026
J THE USE OF LARGE LANGUAGE MODELS
In this paper, we utilized LLMs for several purposes. We employed LLMs to polish the writing and correct the grammar of the manuscript. Additionally, LLMs were used for the literature search in Section 2 and for code completion during implementation. Furthermore, in the proof of the inequality in Section G we leveraged an LLM survey existing and known inequalities from the relevant literature and to brainstorm potential proof strategies. The final proof was developed and verified by the authors.
59


Reviews:
[{"review_id": "CDUatTio3y", "reviewer": "Reviewer_yb85", "summary": "String Seed of Thought (SSoT) is a proposed prompting method that injects a random string as a “seed” into an LLM’s chain-of-thought to induce more faithful probabilistic behavior and greater output diversity. The technique works by first asking the model to generate a random string, then having the model algorithmically manipulate that string to select a final answer according to a target probability distribution. The authors provide both theoretical guarantees (showing that with sufficiently long random strings the output distribution can approach the desired distribution) and extensive empirical evidence that SSoT yields significantly improved distribution alignment and more diverse responses across multiple models and tasks.", "questions": "1. How sensitive is SSoT’s effectiveness to the length of the random string used, and how can we provide for choosing an appropriate string length to balance randomness and efficiency in practice?\n\n2. Have the authors tested SSoT on tasks with highly skewed or arbitrary target distributions (beyond uniform cases like fair coin flips), and if so, how well does the method maintain the correct proportions in those more challenging scenarios?\n\n3. Does the inclusion of a random string (and its manipulation) ever degrade the perceived quality or coherence of the final answer, and how can one ensure that this random seed is hidden or handled so that it doesn’t confuse end-users in a real application?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The SSoT prompting strategy is conceptually simple to implement (just adding a brief instruction to generate and use a random string) and is applicable to a wide range of LLMs without any model modifications.\n\n2. SSoT dramatically improves an LLM’s ability to follow probabilistic instructions, achieving empirical sampling frequencies very close to the target probabilities and also boosts the diversity of open-ended generations, outperforming other diversity-promoting baselines like prompt paraphrasing or higher-temperature sampling while maintaining output quality. Experiments across five state-of-the-art LLMs and various scenarios consistently show that SSoT yields superior distribution-faithful performance and surpasses strong prompting baselines in both accuracy of sampling and response diversity.\n\n3. The paper provides a solid theoretical foundation, proving (informally) that as the length of the internally generated string increases, the total variation distance between the model’s output distribution and the target distribution diminishes even if the string characters are not fully independent.", "weaknesses": "1. This method may cause potential errors and hurt answer quality. While the method focuses on matching distributions and diversity, the paper provides little discussion on whether the use of SSoT could inadvertently affect the correctness or factuality of outputs in tasks where a specific correct answer is expected.\n\n2. Most experiments involve tasks with a small, discrete set of outcomes (binary or a few categories), so it remains uncertain how well SSoT would scale to more complex distributions or tasks with a larger set of possible outputs (or continuous output spaces).\n\n3. The theoretical guarantees rest on assumptions like sufficiently long random strings and bounded bias in character generation; in practice, these conditions cannot be strictly met, so the gap between the idealized guarantees and real model behavior is not fully understood.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7PuAXOf0b8", "reviewer": "Reviewer_kxrX", "summary": "The paper proposes String Seed of Thought (SSoT), a prompt-only method for probability instruction following and open-ended diversity. The model is first prompted to generate a pseudo-random character string, and then leverages its reasoning ability to map that seed to the required output via simple rules (e.g., sum-mod/rolling hash). The authors provide theoretical guarantees of how self-correlated strings can be mapped to a near-uniform distribution with a deterministic hash mapping, and further show the distribution gap for the sum-mode strategy that LLM typically uses. Empirically, SSoT improves distribution faithfulness on coin-flip/k-of-n/RPS tasks and increases diversity on NoveltyBench. It contributes to the LLM applications that requires controllable stochasticity and diversity.", "questions": "1. I hope the authors could provide a more detailed introduction to the datasets for the DAG task and the related metrics. I checked the reference to understand how “distinct” and “utility” are calculated. It seems that distinct evaluates pure diversity, while utility combines both diversity and quality. However, I'm still unclear what the columns in Table 2, such as \"creativity\" and “naming”, represent. If they are subsets of the dataset, improvements in some tasks, including \"creativity\", would be valuable. However, I feel we shouldn't expect diversity increases in subsets like \"facts\", especially when the utility metric can't decompose the changes in quality.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This method is simple yet effective. It works well across different LLMs and task types without tuning, demonstrating strong engineering practicality.\n2.  This method is theoretically sound. It provides a rigorous bound showing that the TV distance between the empirical and target distribution decreases as the generated string length increases, even when the string exhibits autocorrelation. This result gives the method solid mathematical grounding. Moreover, the authors derived the bound for the sum-mod strategy that the model empirically adopts, offering a clear explanation of why the approach works.\n3. The experiments are comprehensive and convincing, covering multiple application scenarios: closed-set probabilistic sampling and open-ended diverse generation. Multiple strong baselines are compared, and the method is tested on 5 advanced LLMs, all showing substantial improvements. This demonstrates its robustness across architectures and tasks. The paper also provides detailed supplementary materials for reproducibility.\n4. Beyond reporting quantitative results, the authors perform an in-depth analysis of LLM’s reasoning behavior during generation. They discover that the models spontaneously develop strategies, such as sum-mod and rolling hash mappings, adaptively according to the task, to extract random actions from the generated strings. This is not only an interesting empirical observation but also deepens our understanding of LLM reasoning behaviors.", "weaknesses": "1. It would be better if the authors could provide some failure case analysis. In Table 1, the performance of QwQ-32B on the 2-choice task is even worse than the baseline. Is this degradation due to autocorrelation in the generated random strings, inappropriate mapping, or possible execution/calculation errors?\n2. The models used in the experiments are all quite large. Considering that SSoT relies on the model itself to decide the mapping strategy and execute it, this raises doubts that the performance may be strongly limited by the capability (especially in reasoning) of the models. It also raises concerns about scalability to more complex PIF tasks, as it remains unclear whether LLMs can reliably come up with and implement more complex operations.\n3. The intrinsic reasoning process of generating random strings and mapping is not transparent. Although the authors provide analysis of the models’ reasoning behavior, it remains unclear whether future versions of these models will change their strategies, or whether different models would adopt different strategies, which makes the algorithm unstable and difficult for users to control. This lack of transparency also weakens the connection between the authors’ theoretical analysis and practical implementation.\n4. Considering the assumptions on the conditional distribution of the string in Theorems 4.1 and 4.2, it would be better to verify them quantitatively.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VPmPUPoPJi", "reviewer": "Reviewer_KwFB", "summary": "The authors of this paper introduced Seed of Thought (SSoT), a prompting method for improving LLMs' probabilistic instruction following. For example, LLMs can approximately draw 50/50 for heads or tails if the prompt is to \"flip an unbiased coin\". The probabilistic instruction following method proposed in this paper is done by crafting prompts: LLMs need to first generate a random sequence and then derive an answer based on the generated sequence.", "questions": "1. I am not familiar with the TV distance bound with random hash function theories, nor did I check every step of the derivatives. Can the author briefly describe the contribution of these theorems in terms of deterministic hash decoding (or a more specific/related field), regardless of whether the LLM-instruction following can meet the assumptions or not?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. Two theorems provide the lower bound of the total variation distance between the sample and the required distributions. \n2. The experiment section is well presented overall. Figures and tables convinced me that the proposed method indeed improves the LLM's probabilistic instruction-following capabilities. \n3. In addition to 2, the rich details in the appendix and the uploaded supporting material benefit the reproduction of numerical studies.", "weaknesses": "1. The assumptions that \"each character is randomly drawn from a distribution\" of the proved theorems are hard to meet for LLMs. They are pre-trained on text corpora. Every individual token generated by them is based on its previous content (Causal Language Modeling), thus never random. I can be proved wrong, at least empirically, by an analysis of the distribution/duplication of LLM-generated random strings.\n\n2. Personally, I am not fully motivated to tune LLM to perform PIF tasks, which could be done precisely with `random` modules if the option set is predefined. This point is the major driver of my evaluation score. The authors can further elaborate on this in the motivation section, or add results for PRNG in Section 5.2, where I cannot link why PRNG is inadequate for diversity-aware generation.\n\n3. The SSoT System prompts in Appendix A are different for each case, which weakened the generalization capability of the proposed method (thinking `please think step-by-step` for CoT). An explanation of why we need different descriptions, or how to design SSoT prompts, alleviates this weakness.\n\n**Presentation-related Suggestions:**:\n1. Several paragraphs need to be rewritten into formal academic writing (e.g. line 146, `the only thing you need to do` uses second-person narration)\n2. The last panel of Figure 2 (line 186) is hard to read, as the Ideal threshold shape overlapped with the tiny bars", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)