Title: Self-Verifying Reflection Helps Transformers with CoT Reasoning
Venue: NeurIPS 2025
Accepted: yes
Keywords: chain-of-thought

reasoning

reflection

verification

transformer
Abstract: Advanced large language models (LLMs) frequently reflect in reasoning chain-of-thoughts (CoTs), where they self-verify the correctness of current solutions and explore alternatives. However, given recent findings that LLMs detect limited errors in CoTs, how reflection contributes to empirical improvements remains unclear. To analyze this issue, in this paper, we present a minimalistic reasoning framework to support basic self-verifying reflection for small transformers without natural language, which ensures analytic clarity and reduces the cost of comprehensive experiments. Theoretically, we prove that self-verifying reflection guarantees improvements if verification errors are properly bounded. Experimentally, we show that tiny transformers, with only a few million parameters, benefit from self-verification in both training and reflective execution, reaching remarkable LLM-level performance in integer multiplication and Sudoku. Similar to LLM results, we find that reinforcement learning (RL) improves in-distribution performance and incentivizes frequent reflection for tiny transformers, yet RL mainly optimizes shallow statistical patterns without faithfully reducing verification errors. In conclusion, integrating generative transformers with discriminative verification inherently facilitates CoT reasoning, regardless of scaling and natural language.

2

Abstract

Advanced large language models (LLMs) frequently reflect in reasoning chainof-thoughts (CoTs), where they self-verify the correctness of current solutions and explore alternatives. However, given recent findings that LLMs detect limited errors in CoTs, how reflection contributes to empirical improvements remains unclear. To analyze this issue, in this paper, we present a minimalistic reasoning framework to support basic self-verifying reflection for small transformers without natural language, which ensures analytic clarity and reduces the cost of comprehensive experiments. Theoretically, we prove that self-verifying reflection guarantees improvements if verification errors are properly bounded. Experimentally, we show that tiny transformers, with only a few million parameters, benefit from self-verification in both training and reflective execution, reaching remarkable LLM-level performance in integer multiplication and Sudoku. Similar to LLM results, we find that reinforcement learning (RL) improves in-distribution performance and incentivizes frequent reflection for tiny transformers, yet RL mainly optimizes shallow statistical patterns without faithfully reducing verification errors. In conclusion, integrating generative transformers with discriminative verification inherently facilitates CoT reasoning, regardless of scaling and natural language.

1 Introduction

Numerous studies have explored the ability of large language models (LLMs) to reason through a chain of thought (CoT), an intermediate sequence leading to the final answer. While simple prompts can elicit CoT reasoning [13], subsequent works have further enhanced CoT quality through reflective thinking [10] and the use of verifiers [4]. Recently, reinforcement learning (RL) [33] has achieved notable success in advanced reasoning models, such as OpenAI-o1 [20] and Deepseek-R1 [5], which show frequent reflective behaviors that self-verify the correctness of current solutions and explore alternatives, integrating generative processes with discriminative inference. However, researchers also report that the ability of these LLMs to detect errors is rather limited, and a large portion of reflection fails to bring correct solutions [11]. Given the weak verification ability, the experimental benefits of reflection and the emergence of high reflection frequency in RL require further explanation.

To address this challenge, we seek to analyze two main questions in this paper: 1) what role selfverifying reflection plays in training and execution of reasoning models, and 2) how reflective reasoning evolves in RL with verifiable outcome rewards [15]. However, the complexity of natural language and the prohibitive training cost of LLMs make it difficult to draw clear conclusions from theoretical abstraction and comprehensive experiments across settings. Inspired by Zeyuan et al.

[2], we observe that task-specific reasoning and self-verifying reflection do not necessitate complex language. This allows us to investigate reflective reasoning through tiny transformer models [36], which provide efficient tools to understand self-verifying reflection through massive experiments.

To enable tiny transformers to produce long reflective CoTs and ensure analytic simplicity, we introduce a minimalistic reasoning framework, which supports essential reasoning behaviors that are operable without natural language. In our study, the model self-verifies the correctness of each thought step; then, it may resample incorrect steps or trace back to previous steps. Based on this framework, we theoretically prove that self-verifying reflection improves reasoning accuracy if verification errors are properly bounded, which does not necessitate a strong verifier. Additionally, a trace-back mechanism that allows revisiting previous solutions conditionally improves performance if the problem requires a sufficiently large number of steps.

Our experiments evaluate 1M, 4M, and 16M transformers in solving integer multiplication [7] and Sudoku puzzles [3], which have simple definitions (thus, operable by transformers without language) yet still challenging for even LLM solvers. To maintain relevance to broader LLM research, the tiny transformers are trained from scratch through a pipeline similar to that of training LLM reasoners. Our main findings are listed as follows: 1) Learning to self-verify greatly facilitates the learning of forward reasoning. 2) Reflection improves reasoning accuracy if true correct steps are not excessively verified as incorrect. 3) Resembling the results of DeepSeek-R1 [5], RL can incentivize reflection if the reasoner can effectively explore potential solutions. 4) However, RL fine-tuning increases performance mainly statistically, with limited improvements in generalizable problem-solving skills.

Overall, this paper contributes to the fundamental understanding of reflection in reasoning models by clarifying its effectiveness and synergy with RL. Our findings based on minimal reasoners imply a general benefit of reflection for more advanced models, which operate on a super-set of our simplified reasoning behaviors. In addition, our implementation also provides insights into the development of computationally efficient reasoning models.

2 Related works

CoT reasoning Pretrained LLMs emerge the ability to produce CoTs from simple prompts [13, 38], which can be explained via the local dependencies [25] and probabilistic distribution [35] of natural-language reasoning. Many recent studies develop models targeted at reasoning, e.g., scaling test-time inference with external verifiers [4, 17, 18, 32] and distilling large general models to smaller specialized models [34, 9]. In this paper, we train tiny transformers from scratch to not only generate CoTs but also self-verify, i.e., detect errors in their own thoughts without external models.

RL fine-tuning for CoT reasoning RL [33] recently emerges as a key method for CoT reasoning [31, 40]. It optimizes the transformer model by favoring CoTs that yield high cumulated rewards, where PPO [29] and its variant GRPO [31] are two representative approaches. Central to RL finetuning are reward models that guide policy optimization: the 1) outcome reward models (ORM) assessing final answers, and the 2) process reward models (PRM) [17] evaluating intermediate reasoning steps. Recent advances in RL with verifiable rewards (RLVR) [5, 41] demonstrate that simple ORM based solely on answer correctness can induce sophisticated reasoning behaviors.

Reflection in LLM reasoning LLM reflection provides feedback to the generated solutions [19] and may accordingly refine the solutions [10]. Research shows that supervised learning from verbal reflection improves performance, even though the reflective feedback is omitted during execution [42]. Compared to the generative verbal reflection, self-verification uses discriminative labels to indicate the correctness of reasoning steps, which supports reflective execution and is operable without linguistic knowledge. Recently, RL is widely used to develop strong reflective abilities [14, 27, 20]. In particular, DeepSeek-R1 [5] shows that RLVR elicits frequent reflection, and such a result is reproduced in smaller LLMs [24]. In this paper, we further investigate how reflection evolves during RLVR by examining the change of verification errors.

Understanding LLMs through small transformers Small transformers are helpful tools to understand LLMs, for their architectural consistency with LLMs and low development cost to support massive experiments. For example, transformers smaller than 1B provide insights into how data mixture and data diversity influence LLM training [39, 2]. They also contribute to foundational

𝑄

𝑅1

𝑅2

𝑅𝑇-1

𝑆0 = 𝑄

𝑅1

𝑆0

𝐴 ∼ 𝜋 ⋅

𝑆𝑇-1

Reasoning Steps

Model

State Transition

𝑆1

= 𝒯 𝑆0, 𝑅 1

𝑅2

𝑆1

𝐴

𝑆2

= 𝒯 𝑆1, 𝑅2

𝑆𝑇-1

= 𝒯 𝑆𝑇-2,𝑅𝑇-1

𝑆𝑇

= 𝒯 𝑆𝑇-1, 𝐴

⋯

340 → 300

145 × 4 = 580

290 + 5800 = 6090

𝑅𝑡+1

𝑆𝑡

𝑆𝑡+1

𝜋

𝒯

145

×

340

＋

290

145

×

300

＋

6090

(a) Multiplication

1

2

7

3

8

3

2

5

9

5

2

8 6

4 7

9

3

7

9

4

8

1

6

4

1

2

7

3

8

3

2

5

9

5

2 7

8 6

4 7

9

3

7

9

4

8

1 2

6

4

𝑆𝑡

𝑆𝑡+1

𝜋

Cell6,2 ←7

Cell7,8 ←2

𝑅𝑡+1

𝒯

(b) Sudoku

understanding of CoT reasoning, such as length generalization [12], internalization of thoughts [6], and how CoTs inherently extend the problem-solving ability [8, 16]. In this paper, we further use tiny transformers to better understand reflection in CoT reasoning.

3 Reflective reasoning for transformers

In this section, we develop transformers to perform simple reflective reasoning in long CoTs. Focusing on analytic clarity and broader implications, the design of our framework follows the minimalistic principle, providing only essential reasoning behavior operable without linguistic knowledge. More advanced reasoning frameworks optimized for small-scale models are certainly our next move in future work. In the following, we first introduce the basic formulation of CoT reasoning; then, based on this formulation, we introduce our simple reasoning framework for self-verifying reflection; afterwards, we describe how transformers are trained to reason through this framework.

3.1 Reasoning formulation

CoT Reasoning as a Markov decision process A general form of CoT reasoning is given as a tuple ( Q, { R } , A ) , where Q is the input query, { R } = ( R 1 , . . . , R T -1 ) is the sequence of T -1 intermediate steps, and A is the final answer. Following Wang [37], we formulate the CoT reasoning as a Markov thought process (MTP). As shown in Figure 1, an MTP follows that [37]:

where S t is the t -th reasoning state, π is the planning policy (the transformer model), and T is the (usually deterministic) transition function . The initial state S 0 := Q is given by the input query. In each reasoning step R t +1 , the policy π plans the next reasoning action that determines the state transition, which is then executed by T to obtain the next state. The process terminates when the step presents the answer, i.e., A = R T . For clarity, a table of notations is presented in Appendix A.

An MTP is implemented by specifying the state representations and transition function T . Since we use tiny transformers that are weak in inferring long contexts, we suggest reducing the length of state representations, so that each state S t carries only necessary information for subsequent reasoning. Here, we present two examples to better illustrate how MTPs are designed for tiny transformers.

Example 1 (An MTP for integer multiplication) . As shown in Figure 2(a), to reason the product of two integers x, y ≥ 0 , each state is an expression S t := [ x t × y t + z t ] mathematically equal to x × y , initialized as S 0 = [ x × y +0] . On each step, π plans y t +1 by eliminate a non-zero digit in y t to 0 , and it then computes z t +1 = z t + x t ( y t -y t +1 ) . Consequently, T updates S t +1 as [ x t +1 × y t +1 + z t +1 ] with x t +1 = x t . Similarly, π may also eliminate non-zero digits in x t in a symmetric manner. Finally, π yields A = z t as the answer if either x t or y t becomes 0 .

𝑄

𝑅1,×

𝑅2,

𝑅3,

𝑅4,×

𝑆0 = 𝑄

𝑆1

= 𝑆0

𝑅6,

𝐴,

𝑆3

= 𝒯 𝑆2,𝑅3

𝑆4

= 𝑆3

𝑆2

= 𝒯 𝑆1,𝑅2

𝑆6

= 𝒯 𝑆5,𝑅6

𝑆7

= 𝒯 𝑆6,𝐴

𝑅5,×

𝑆5

= 𝑆4

(a) Reflective MTP

𝑄

𝑅1,

𝑅8,

𝑅9,

𝑆0 = 𝑄

𝐴,

𝑆9

= 𝒯 𝑆8,𝑅9

𝑆8

= 𝒯 𝑆7,𝑅8

𝑆10

= 𝒯 𝑆9,𝐴

𝑅2,×

𝑆2

= 𝑆1

𝑅3,

𝑆3

= 𝒯 𝑆2, 𝑅3

𝑆1

= 𝒯 𝑆0,𝑅1

𝑅4,

𝑅7,×

𝑆7

= 𝑆0

𝑅5,×

𝑅6,×

𝑆4

= 𝒯 𝑆3, 𝑅4

𝑆5

= 𝑆4

𝑆6

= 𝑆3

(b) Reflective trace-back search (width m = 2 )

Example 2 (An MTP for Sudoku [3]) . As shown in Figure 2(b), each Sudoku state is a 9 × 9 game board. On each step, the model π fills some blank cells to produce a new board, which is exactly the next state. The answer A is a board with no blank cells.

3.2 The framework of self-verifying reflection

Conceptually, reflection provides feedback for the proposed steps and may alter the subsequent reasoning accordingly. Reflection takes flexible forms in natural language (e.g., justifications and comprehensive evaluations), making it extremely costly to analyze. In this work, we propose to equip transformers with the simplest discriminative form of reflection, where the model self-verifies the correctness of each step and is allowed to retry those incorrect attempts. We currently do not consider the high-level revisory behavior that maps incorrect steps to correct ones, as we find learning such a mapping is challenging for tiny models and leads to no significant gain in practice. Specifically, we analyze two basic variants of reflective reasoning in this paper: the reflective MTP and the reflective trace-back search , as described below (see pseudo-code in Appendix D.1).

Reflective MTP (RMTP) Given any MTP with a policy π and transition T , we use a verifier V to produce a verification sequence after each reasoning step, denoted as V t ∼ V ( ·| R t ) . Such V t includes verification label(s): The positive ' ✓ ' and negative ' × " signifying correct and incorrect reasoning of R t , respectively. Given the verified step ˜ R t +1 := ( R t +1 , V t +1 ) that contains verification, we define ˜ T as the reflective transition function that rejects incorrect steps:

In other words, if V detects any error (i.e. ' × ") in R t +1 , the state remains unchanged so that π may re-sample another attempt. Focusing on self-verification, we use a single model called the self-verifying policy ˜ π := { π, V} to serve simultaneously as the planning policy π and the verifier V . By operating tokens, ˜ π outputs the verified step ˜ R t for each input state S t . In this way, ˜ T and ˜ π constitute a new MTP called the RMTP, with illustration in Figure 3(a).

Reflective trace-back search (RTBS) Though RMTP allows instant rejections of incorrect steps, sometimes the quality of a step can be better determined by actually trying it. For example, a Sudoku solver occasionally makes tentative guesses and traces back if the subsequent reasoning fails. Inspired by o1-journey [26], a trace-back search allowing the reasoner to revisit previous states may be applied to explore solution paths in an MTP. We implement simple RTBS by simulating the depth-first search in the trajectory space. Let m denote the RTBS width , i.e., the maximal number of attempts on each step. As illustrated in Figure 3(b), if m proposed steps are rejected on a state S t , the negative label ' × ' will be propagated back to recursively reject the previous step R t . As a result, the state traces back to the closest ancestral state that has remaining attempt opportunities.

3.3 Training

As shown in Figure 4, we train the tiny transformers from scratch through consistent techniques of LLM counterparts, such as pretraining, supervised fine-tuning (SFT), and RL fine-tuning. First, we

𝑄

𝑅1

𝑅2

𝐴

𝑄

𝑅1

𝑅2

𝐴

𝑄

𝑅1

context windows

(randomly drawn)

(I) Pretraining

…

Expert

Verifier

𝑄

𝑆1

𝑆2

𝑅1

𝑅2

𝑅3

⋮

⋮

𝑉1

𝑉2

𝑉3

⋮

states & steps

ground-truth

verification

⋮

⋮

(III) Reflective SFT

sampling CoTs through MTP

𝑄

𝐴

𝑅𝑡

𝜋

𝑆𝑡

= 𝒯 𝑆𝑡-1, 𝑅𝑡

Reward

Model

Policy

Optimization

(IV) RL fine-tuning

𝑄

𝐴

𝑅𝑡

𝜋

𝑆𝑡

= 𝒯 𝑆𝑡-1, 𝑅𝑡

MTP

𝑄

𝐴

𝑅𝑡 , 𝑉

𝑡

෤ 𝜋

𝑆𝑡 = ෨

𝒯 𝑆𝑡-1, 𝑅 𝑡

, 𝑉

𝑡

RMTP

𝒱

(II) Non-reflective

SFT

states

steps

or

answers

𝑅1

𝑅2

𝐴

𝑅1

𝑅2

𝐴

⋮

𝑄

𝑆1

𝑆2

𝑄

𝑆1

𝑆2

⋮

𝜋

𝜋

𝜋

Training

Data

CoT examples (data mixture)

CoT examples

෤ 𝜋

𝜋

use conventional pipelines to train a baseline model π with only the planning ability in MTPs. During (I) pretraining , these CoT examples are treated as a textual corpus, where sequences are randomly drawn to minimize cross-entropy loss of next-token prediction. Then, in (II) non-reflective SFT , the model learns to map each state S t to the corresponding step R t +1 by imitating examples.

Next, we employ (III) reflective SFT to integrate the planning policy π with the knowledge of self-verification. To produce ground-truth verification labels, we use π to sample non-reflective CoTs, in which the sampled steps are then labeled by an expert verifier (e.g., a rule-based process reward model). Reflective SFT learns to predict these labels from the states and the proposed steps, i.e., ( S t , R t +1 ) → V t +1 . To prevent disastrous forgetting, we also mix the same CoT examples as in non-reflective SFT. This converts π to a self-verifying policy ˜ π that can self-verify reasoning steps.

Thus far, we have obtained the planning policy π and the self-verifying policy ˜ π , which can be further strengthened through (IV) RL fine-tuning . As illustrated in Figure 4, RL fine-tuning involves iteratively executing π ( ˜ π ) to collect experience CoTs through an MTP (RMTP), evaluating these CoTs with a reward model, and updating the policy to favor higher-reward solutions. Following the RLVR paradigm [15], we use binary outcome rewards (i.e., 1 for correct answers and 0 otherwise) computed by a rule-based answer checker ORM( Q,A ) . When training the self-verifying policy ˜ π , the RMTP treats verification V t as a part of the augmented step ˜ R t , simulating R1-like training [5] where reflection and solution planning are jointly optimized. We mainly use GRPO [31] as the algorithms to optimize policies. Details of RL fine-tuning are elaborated in Appendix B.

4 Theoretical results

This section establishes theoretical conditions under which self-verifying reflection (RMTP or RTBS in Section 3.2) enhances reasoning accuracy (the probability of deriving correct answers). The general relationship between the verification ability and reasoning accuracy (discussed in Appendix C.1) for any MTP is intractable as the states and transitions can be arbitrarily specified. Therefore, to derive interpretable insights, we discuss a simplified prototype of reasoning that epitomizes the representative principle of CoTs - to incrementally express complex relations by chaining the local relation in each step [25]. Specifically, Given query Q as the initial state, we view a CoT as the step-by-step process that reduces the complexity within states:

We define S n as the set of states with a complexity scale of n . For simplicity, we assume that each step, if not rejected by reflection, reduces the complexity scale by 1 . Therefore, the scale n is the number of effective steps required to derive an answer.

An answer A is a state with a scale of 0 , i.e. A ∈ S 0 . Given an input query Q , the answers S 0 are divided into positive (correct) answers S + 0 and negative (wrong) answers S -0 .

States S n ( n > 0) are divided into 1) positive states S + n that potentially lead to correct answers and 2) negative states S -n leading to only incorrect answers through forward transitions.

Consider a self-verifying policy ˜ π = { π, V} to solve this simplified task. We describe its fundamental abilities using the following probabilities (whose meanings will be explained afterwards):

𝒮𝑛

-

𝒮𝑛-1

+

𝒮𝑛

+

𝒮𝑛-1

-

𝜇

1

(a) Non-reflective reasoning

𝒮𝑛

-

𝒮𝑛-1

+

𝒮𝑛

+

𝒮𝑛-1

-

𝜇 1 - 𝑒-

1 - 𝑓

𝑓

𝒮𝑛+1

-

𝛼 ≔ 𝜇𝑒- +

1 - 𝜇

1 - 𝑒 +

𝒮𝑛+1

+

After

𝑚

attempts in RTBS

(b) Reflective reasoning through an RMTP or RTBS

To elaborate, µ measures the planning ability, defined as the probability that π plans a step that leads to a positive next state, given that the current state is positive. For verification abilities, we measure the rates of two types of errors: e + (false positive rate) is the probability of accepting a step that leads to a negative state, and e -(false negative rate) is the probability of rejecting a step that leads to a positive state. Additionally, f is the probability of rejecting any step on negative states, providing the chance of tracing back to previous states. Given these factors, Figure 5 illustrates the state transitions in non-reflective (vanilla MTP) and reflective (RMTB and RTBS) reasoning.

For input problems with scale n , we use ρ ( n ) , ˜ ρ ( n ) , and ˜ ρ m ( n ) to respectively denote the reasoning accuracy using no reflection, RMTP, and RTBS (with width m ). Obviously, we have ρ ( n ) = µ n . In contrast, the mathematical forms of ˜ ρ ( n ) and ˜ ρ m ( n ) are more complicated and therefore left to Appendix C.2. Our main result provides simple conditions for the above factors ( µ, e -, e + , f ) to ensure an improved accuracy when reasoning through an RMTP or RTBS.

Theorem 1. In the above simplified problem, consider a self-verifying policy ˜ π where µ , e -, and e + are non-trivial (i.e. neither 0 nor 1 ). Let α := µe -+(1 -µ )(1 -e + ) denote the rejection probability on positive states. Given an infinite computation budget, for n > 0 we have:

˜ ρ ( n ) ≥ ρ ( n ) if and only if e -+ e + ≤ 1 , where equalities hold simultaneously; furthermore, reducing either e -or e + strictly increases ˜ ρ ( n ) .

˜ ρ m ( n ) > ˜ ρ ( n ) for a sufficiently large n if and only if f > α and m> 1 1 -α ; furthermore, such a gap of ˜ ρ m ( n ) over ˜ ρ ( n ) increases strictly with f .

Does reflection require a strong verifier? Theorem 1 shows that RMTP improves performance over vanilla MTP if the verification errors e + and e -are properly bounded, which does not necessitate a strong verifier . In our simplified setting, this only requires the verifier V to be better than random guessing (which ensures e -+ e + = 1 ). This also indicates a trivial guarantee of RTBS, as an infinitely large width ( m → + ∞ ) substantially converts RTBS to RMTB.

When does trace-back search facilitate reflection? Theorem 1 provides the conditions for RTBS to outperform RMTP for a sufficiently large n : 1) The width m is large enough to ensure effective exploration . 2) f > α indicates that negative states are inherently discriminated from positive ones, leading to a higher rejection probability on negative states than on positive states (see Figure 5(b)). In other words, provided f > α , RTBS is ensured to be more effective on complicated queries using a finite m . However, this also implies a risk of over-thought on simple queries that have a small n .

The derivation and additional details of Theorem 1 are provided in Appendix C.3. In addition, we also derive how many steps it costs to find a correct solution in RMTP. The following Proposition 1 (see proof in Appendix C.4) shows that a higher e -causes more steps to be necessarily rejected and increases the solution cost. In contrast, although a higher e + reduces accuracy, it forces successful solutions to rely less on reflection, leading to fewer expected steps. Therefore, a high false negative rate e -is worse than a high e + given the limited computational budget in practice.







































































Proposition 1 (RMTP Reasoning Length) . For a simplified reasoning problem with scale n , the expected number of steps ¯ T for ˜ π to find a correct answer is ¯ T = n (1 -µ ) e + + µ (1 -e -) . Especially, a correct answer will never be found if the denominator is 0 .

Appendix C.5 further extends our analysis to more realistic reasoning, where rejected attempts lead to a posterior drop of µ (or rise of e -), indicating that the model may not well generalize the current state. In this case, the bound of e -to ensure improvements becomes stricter than that in Theorem 1.

5 Experiments

We conduct comprehensive experiments to examine the reasoning performance of tiny transformers under various settings. We trained simple causal-attention transformers [36] (implemented by LitGPT [1]) with 1M, 4M, and 16M parameters, through the pipelines described in Section 3.3. Details of training data, model architectures, tokenization, and hyperparameters are included in Appendix D. The source code is available at https://github.com/zwyu-ai/self-verifying-reflection-reasoning .

We test tiny transformers in two reasoning tasks: The integer multiplication task ( Mult for short) computes the product of two integers x and y ; the Sudoku task fills numbers into blank positions of a 9 × 9 matrix, such that each row, column, or 3 × 3 block is a permutation of { 1 , . . . , 9 } . For both tasks, we divide queries into 3 levels of difficulties : The in-distribution (ID) Easy , ID Hard , and out-of-distribution (OOD) Hard . The models are trained on ID-Easy and ID-Hard problems , while tested additionally on OOD-Hard cases. We define the difficulty of a Mult query by the number d of digits of the greater multiplicand, and that of a Sudoku puzzle is determined by the number b of blanks to be filled. Specifically, we have 1 ≤ d ≤ 5 or 9 ≤ b < 36 for ID Easy, 6 ≤ d ≤ 8 or 36 ≤ b < 54 for ID Hard, and 9 ≤ d ≤ 10 or 54 ≤ b < 63 for OOD Hard.

Our full results are presented in Appendix E. Shown in Appendix E.1, these seemingly simple tasks pose challenges even for some well-known LLMs. Remarkably, through simple self-verifying reflection, our best 4M Sudoku model is as good as OpenAI o3-mini [21], and our best 16M Mult model outperforms DeepSeek-R1 [5] in ID difficulties.

5.1 Results of supervised fine-tuning

First, we conduct (I) pretraining, (II) non-reflective SFT, and (III) reflective SFT as described in Section 3.3. In reflective SFT, we consider learning two types of self-verification : 1) The binary verification includes a single binary label indicating the overall correctness of a planned step; 2) the detailed verification includes a series of binary labels checking the correctness of each meaningful element in the step. The implementation of verification labels is elaborated in Appendix D.2.3. We present our full SFT results in Appendix E.2, which includes training 30 models and executing 54 tests. In the following, we discuss our main findings through visualizing representative results.

Does learning self-verification facilitate learning the planning policy? We compare our models under the non-reflective execution , where self-verification is not actively used in test time. As shown in Figure 6, reflective SFT with binary verification brings remarkable improvements for 1M and 4M in ID-Easy and ID-Hard Mult problems, greatly reducing the gap among model sizes. Although detailed verification does not benefit as much as binary verification in ID problems, it significantly benefits the 16M model in solving OOD-Hard problems. Therefore, learning to self-verify benefits the learning of forward planning, increasing performance even if test-time reflection is not enabled .





















































































e



e

+



e



e

+

Since reflective SFT mixes the same CoT examples as used in non-reflective SFT, an explanation for this phenomenon is that learning to self-verify serves as a regularizer to the planning policy. This substantially improves the quality of hidden embeddings in transformers, which facilitates the learning of CoT examples. Binary verification is inherently a harder target to learn, which produces stronger regularizing effects than detailed verification. However, the complexity (length) of the verification should match the capacity of the model; otherwise, it could severely compromise the benefits of learning self-verification. For instance, learning binary verification and detailed verification fails to improve the 16M model and the 1M model, respectively.

When do reflective executions improve reasoning accuracy? Figure 7 evaluates the non-reflective, RMTP, and RTBS executions for models in solving ID-Hard problems. Apart from the accuracy, the rates of verification error (i.e., the false positive rate e + and false negative rate e -defined in Section 4) are measured using an oracle verifier. In these results, RMTP reasoning raises the performance over non-reflective reasoning except for the 1M models (which fail in ID-hard Sudoku). Smaller error rates (especially e -) generally lead to higher improvements, whereas a high e -in binary verification severely compromises the performance of the 1M Mult Model. Overall, reflection improves reasoning if the chance of rejecting correct steps ( e -) is sufficiently small .

In what task is the trace-back search helpful? As seen in Figure 7, though RTBS shows no advantage against RMTP in Mult, it outperforms RMTP in Sudoku, especially the 4M model with detailed verification. This aligns with Theory 1 - The state of Sudoku (the 9 × 9 matrix) is required to comply with explicit verifiable rules, making incorrect states easily discriminated from correct states. However, errors in Mult states can only be checked by recalculating all historical steps. Therefore, we are more likely to have f > α in Sudoku, which grants a higher chance of solving harder problems. This suggests that RTBS can be more helpful than RMTP if incorrect states in the task carry verifiable errors , which validates our theoretical results.

5.2 Results of reinforcement learning

As introduced in Section 3.3, we further apply GRPO to fine-tune the models after SFT. Especially, GRPO based on RMTP allows solution planning and verification to be jointly optimized for selfverifying policies. The full GRPO results are presented in Appendix E.3, and the main findings are presented below. Overall, RL does enable most models to better solve ID problems, yet such improvements arise from a superficial shift in the distribution of known reasoning skills.

How does RL improve reasoning accuracy? Figure 8 presents the performance of 4M and 16M models in Mult after GRPO, where the differences from SFT results are visualized. GRPO effectively enhances accuracy in solving ID-Hard problems, yet the change in OOD performance is marginal. Therefore, RL can optimize ID performance, while failing to generalize to OOD cases .

Does RL truly enhance verification? From the change of verification errors in Figure 8, we find that the false negative rate e -decreases along with an increase in the false positive rate e + . This suggests













































































e



e

+



e



e

+

that models learn an optimistic bias , which avoids rejecting correct steps through a high false positive rate that bypasses verification. In other words, instead of truly improving the verifier (where e -and e + both decrease), RL mainly induces an error-type trade-off , shifting from false negatives ( e + ) to false positives ( e -).

To explain this, we note that a high e -raises the computational cost (Proposition 1) and thus causes a significant performance loss under the limited budget of RL sampling, making reducing e -more rewarding than maintaining a low e + . Meanwhile, shifting the error type is easy to learn, achievable by adjusting only a few parameters in the output layer of the transformer.

Inspired by DeepSeek-R1 [5], we additionally examine how RL influences the frequency of reflective behavior. To simulate the natural distribution of human reasoning, we train models to perform optional detailed verification by adding examples of empty verification (in the same amount as the full verification) into reflective SFT. This allows the policy to optionally omit self-verification, usually with a higher probability than producing full verification, since empty verification is easier to learn. Consequently, we can measure the reflection frequency by counting the proportion of steps that include non-empty verification. Since models can implicitly omit binary verification by producing false positive labels, we do not explicitly examine the optional binary verification.

Whendoes RL incentivize frequent reflection? Figure 9 shows reflection frequency in Mult before and after GRPO, comparing exploratory ( 1 . 25 ) and exploitative ( 1 ) temperatures when sampling experience CoTs. With a temperature 1 . 25 , GRPO elicits frequent reflection, especially on hard queries. However, reflection frequency remains low if using temperature 1 . Additional results for other model sizes and Sudoku appear in Appendix E.3.3. In conclusion, RL can adapt reflection frequency to align with the exploratory ability of the planning policy π , encouraging more reflection if the policy can potentially explore rewards . This helps explain why RL promotes frequent reflection in LLMs [5], as the flexibility of language naturally fosters exploratory reasoning.

6 Conclusion and Discussion

In this paper, we provide a foundational analysis of self-verifying reflection in multi-step CoTs using small transformers. Through minimalistic prototypes of reflective reasoning (the RMTP and RTBS), we demonstrate that self-verification benefits both training and execution. Compared to natural-language reasoning based on LLMs, the proposed minimalistic framework performs effective reasoning and reflection using limited computational resources. We also show that RL fine-tuning can enhance the performance in solving in-distribution problems and incentivize reflective thinking for exploratory reasoners. However, the improvements from RL rely on shallow patterns and lack generalizable new skills. Overall, we suggest that self-verifying reflection is inherently beneficial for CoT reasoning, yet its synergy with RL fine-tuning is limited in superficial statistics.

1

2

3

4

5

6

7

8

9 10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

7

21

6

8

18

13

14

5

5

4

23

23

12

18

10

11

8

6

4

4

15

24

20

11

6

5

6

9

5

2

6

19

8

3

1

2

8

6

4

2

15

14

5

2

2

4

9

4

2

2

13

14

3

2

1

4

6

3

2

2

22

20

5

4

8

8

10

8

3

3

11

9

3

5

5

5

8

5

4

3

5

4

3

3

5

3

5

4

5

4

4

5

5

5

4

4

3

4

3

2

Before GRPO

1

2

3

4

5

6

7

8

9 10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

30

50

30

32

37

45

44

33

33

35

43

56

54

58

70

70

71

71

69

71

35

51

56

63

66

68

76

80

79

79

34

49

56

61

65

70

77

80

83

81

32

49

57

61

65

72

82

83

84

82

32

49

56

62

67

74

86

88

83

82

35

52

56

64

75

84

89

88

83

84

32

47

56

67

78

82

84

86

79

73

32

49

58

68

74

77

79

74

59

52

32

52

58

68

74

77

75

60

47

50

After GRPO

Temperature: 1.25

1

2

3

4

5

6

7

8

9 10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

0

19

1

6

21

25

24

10

3

3

9

14

5

17

15

14

12

13

9

9

5

14

13

11

9

7

12

12

9

10

2

14

3

1

1

4

12

11

10

4

5

13

0

0

1

9

12

10

5

1

1

6

0

0

3

5

9

6

1

0

5

9

0

3

6

9

9

6

1

1

0

0

0

2

3

2

4

4

1

1

0

0

0

0

0

0

0

0

1

2

0

0

0

0

0

0

0

2

2

2

After GRPO

Temperature: 1.0

=

T

T

Limitations and future work Although the current training pipeline enables tiny transformers to reason properly through reflective CoTs, the generalization ability is still low and not improved in RL. Therefore, future work will extend reflection frameworks and explore novel training approaches. Observing the positive effect of learning self-verification, a closer connection between generative and discriminative reasoning may be the key to addressing this challenge. Additionally, how our findings transfer from small transformers to natural-language LLMs needs to be further examined. However, the diversity of natural language and high computational cost pose significant challenges to comprehensive evaluation, and our proposed framework does not sufficiently exploit the emergent linguistic ability of LLMs. To this end, we expect to investigate a more flexible self-verification framework with an efficient evaluator of natural-language reflection in future work.

Acknowledgments and Disclosure of Funding

We gratefully acknowledge Dr. Linyi Yang for providing partial computational resources.

References

Lightning AI. LitGPT . https://github.com/Lightning-AI/litgpt . 2023.

Zeyuan Allen-Zhu and Yuanzhi Li. Physics of Language Models: Part 3.1, Knowledge Storage and Extraction . July 2024. DOI: 10.48550/arXiv.2309.14316 . arXiv: 2309.14316 [cs] .

Eric C. Chi and Kenneth Lange. Techniques for Solving Sudoku Puzzles . May 2013. DOI: 10.48550/arXiv.1203.2295 . arXiv: 1203.2295 [math] .

Karl Cobbe et al. Training Verifiers to Solve Math Word Problems . Nov. 2021. arXiv: 2110. 14168 [cs] .

DeepSeek-AI et al. DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning . Jan. 2025. DOI: 10.48550/arXiv.2501.12948 . arXiv: 2501.12948 [cs] .

Yuntian Deng, Yejin Choi, and Stuart Shieber. From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step . May 2024. DOI: 10.48550/arXiv.2405.14838 . arXiv: 2405.14838 [cs] .

Nouha Dziri et al. Faith and Fate: Limits of Transformers on Compositionality . Oct. 2023. DOI: 10.48550/arXiv.2305.18654 . arXiv: 2305.18654 [cs] .

Guhao Feng et al. Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective . Dec. 2023. DOI: 10.48550/arXiv.2305.15408 . arXiv: 2305.15408 [cs] .

Yao Fu et al. 'Specializing Smaller Language Models towards Multi-Step Reasoning'. In: Proceedings of the 40th International Conference on Machine Learning . PMLR, July 2023, pp. 10421-10430.

Alex Havrilla et al. GLoRe: When, Where, and How to Improve LLM Reasoning via Global and Local Refinements . June 2024. DOI: 10.48550/arXiv.2402.10963 . arXiv: 2402.10963 [cs] .

Yancheng He et al. Can Large Language Models Detect Errors in Long Chain-of-Thought Reasoning? Mar. 2025. DOI: 10.48550/arXiv.2502.19361 . arXiv: 2502.19361 [cs] .

Kaiying Hou et al. Universal Length Generalization with Turing Programs . July 2024. DOI: 10.48550/arXiv.2407.03310 . arXiv: 2407.03310 [cs] .

Takeshi Kojima et al. Large Language Models Are Zero-Shot Reasoners . Jan. 2023. DOI: 10.48550/arXiv.2205.11916 . arXiv: 2205.11916 [cs] .

Aviral Kumar et al. Training Language Models to Self-Correct via Reinforcement Learning . Sept. 2024. arXiv: 2409.12917 [cs] .

Nathan Lambert et al. Tulu 3: Pushing Frontiers in Open Language Model Post-Training . Apr. 2025. DOI: 10.48550/arXiv.2411.15124 . arXiv: 2411.15124 [cs] .

Zhiyuan Li et al. Chain of Thought Empowers Transformers to Solve Inherently Serial Problems . Sept. 2024. DOI: 10.48550/arXiv.2402.12875 . arXiv: 2402.12875 [cs, stat] .

Hunter Lightman et al. Let's Verify Step by Step . May 2023. arXiv: 2305.20050 [cs] .

Liangchen Luo et al. Improve Mathematical Reasoning in Language Models by Automated Process Supervision . June 2024. arXiv: 2406.06592 [cs] .

Aman Madaan et al. Self-Refine: Iterative Refinement with Self-Feedback . May 2023. DOI: 10.48550/arXiv.2303.17651 . arXiv: 2303.17651 [cs] .

OpenAI. Learning to Reason with LLMs . https://openai.com/index/learning-to-reason-withllms/.

OpenAI. OpenAI O3-Mini System Card . https://openai.com/index/o3-mini-system-card.

OpenAI et al. GPT-4o System Card . Oct. 2024. DOI: 10.48550/arXiv.2410.21276 . arXiv: 2410.21276 [cs] .

Long Ouyang et al. 'Training Language Models to Follow Instructions with Human Feedback'. In: Advances in Neural Information Processing Systems 35 (NeurIPS 2022) . Mar. 2022. arXiv: 2203.02155 [cs] .

Jiayi Pan et al. TinyZero . https://github.com/Jiayi-Pan/TinyZero. Accessed: 2025-01-24. 2025.

Ben Prystawski, Michael Y. Li, and Noah D. Goodman. Why Think Step by Step? Reasoning Emerges from the Locality of Experience . Nov. 2023. arXiv: 2304.03843 [cs] .

Yiwei Qin et al. O1 Replication Journey: A Strategic Progress Report - Part 1 . Oct. 2024. DOI: 10.48550/arXiv.2410.18982 . arXiv: 2410.18982 [cs] .

Yuxiao Qu et al. Recursive Introspection: Teaching Language Model Agents How to SelfImprove . July 2024. DOI: 10.48550/arXiv.2407.18219 . arXiv: 2407.18219 [cs] .

John Schulman et al. High-Dimensional Continuous Control Using Generalized Advantage Estimation . Oct. 2018. arXiv: 1506.02438 [cs] .

John Schulman et al. Proximal Policy Optimization Algorithms . Aug. 2017. arXiv: 1707. 06347 [cs] .

Rico Sennrich, Barry Haddow, and Alexandra Birch. Neural Machine Translation of Rare Words with Subword Units . June 2016. DOI: 10.48550/arXiv.1508.07909 . arXiv: 1508. 07909 [cs] .

Zhihong Shao et al. DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models . Apr. 2024. arXiv: 2402.03300 [cs] .

Charlie Snell et al. Scaling LLM Test-Time Compute Optimally Can Be More Effective than Scaling Model Parameters . Aug. 2024. arXiv: 2408.03314 [cs] .

Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction . 2nd ed. Cambridge, Massachusetts: The MIT Press, 2018. ISBN: 978-0-262-36401-0.

Yijun Tian et al. TinyLLM: Learning a Small Student from Multiple Large Language Models . Mar. 2024. DOI: 10.48550/arXiv.2402.04616 . arXiv: 2402.04616 [cs] .

Rasul Tutunov et al. Why Can Large Language Models Generate Correct Chain-of-Thoughts? June 2024. DOI: 10.48550/arXiv.2310.13571 . arXiv: 2310.13571 [cs] .

Ashish Vaswani et al. 'Attention Is All You Need'. In: Proceedings of the 31st International Conference on Neural Information Processing Systems . Long Beach, CA, USA, 2017.

Jun Wang. A Tutorial on LLM Reasoning: Relevant Methods behind ChatGPT O1 . Feb. 2025. DOI: 10.48550/arXiv.2502.10867 . arXiv: 2502.10867 [cs] .

Jason Wei et al. Chain-of-Thought Prompting Elicits Reasoning in Large Language Models . Jan. 2023. DOI: 10.48550/arXiv.2201.11903 . arXiv: 2201.11903 [cs] .

Sang Michael Xie et al. DoReMi: Optimizing Data Mixtures Speeds Up Language Model Pretraining . Nov. 2023. DOI: 10.48550/arXiv.2305.10429 . arXiv: 2305.10429 [cs] .

An Yang et al. Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement . Sept. 2024. arXiv: 2409.12122 [cs] .

Yang Yue et al. Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model? Apr. 2025. DOI: 10.48550/arXiv.2504.13837 . arXiv: 2504.13837 [cs] .

Zhihan Zhang et al. Learn Beyond The Answer: Training Language Models with Reflection for Mathematical Reasoning . Oct. 2024. DOI: 10.48550/arXiv.2406.12050 . arXiv: 2406.12050 [cs] .

Contents

A Notations

The notations used in the main paper are summarized in Table 1. Notations only appear in the appendix are not included.

B Details of reinforcement learning

This section introduces PPO and GRPO algorithms used in RL fine-tuning. We introduce PPO and GRPO under the context of MTP, which is described in Section 3.1. This also applies to RMTP

reasoning in Section 3.2, as RMTP is a special MTP given the self-verifying policy ˜ π and the reflective transition function ˜ T .

For any sequence X of tokens, we additionally define the following notations: X [ i ] denotes the i -th token, X [ <i ] ( X [ ≤ i ] ) denotes the former i -1 ( i ) tokens, and | X | denotes the length (i.e., the number of tokens).

Both PPO and GRPO iteratively update the reasoning policy through online experience. Let π θ to denote a reasoning policy parameterized by θ . On each iteration, PPO and GRPO use a similar process to update θ :

Randomly draw queries from the task or taring set, and apply the old policy π θ old to sample experience CoTs.

Use reward models to assign rewards to the experience CoTs. Let ORM and PRM be the outcome reward model and process reward model, respectively. For each CoT ( Q,R 1 , . . . , R T -1 , A ) , we obtain outcome rewards r o = ORM( Q,A ) and the process rewards r t = PRM( S t , R t +1 ) for t = 0 , 1 , . . . , T -1 (where R T = A ). In our case, we only use the outcome reward model and thus all process rewards are 0 .

Then, θ is updated by maximizing an objective function based on the experience CoTs with above rewards. Especially, PPO additionally needs to update an value approximator.

B.1 Proximal policy optimization

PPO [29] is a classic RL algorithm widely used in various applications. It includes a value model v to approximate the value function, namely the expected cumulated rewards:

Let q t,i ( θ ) = π θ ( R [ i ] t ∣ ∣ ∣ S t ,R [ <i ] t ) π θ old ( R [ i ] t ∣ ∣ ∣ S t ,R [ <i ] t ) be the relative likelihood of the i -th token in the t -th step, and π ref be the reference model (e.g., the policy before RL-tuning). Then, the PPO algorithm maximizes

Here, ˆ A t,i is the advantage of the i -th token in step t , computed using the value model v . For example, ˆ A t,i = v ( S t , R [ <i ] t , R [ i ] t ) -v ( S t , R [ <i ] t ) is a simple way to estimate advantage. In practice, advantages can be estimated using the general advantage estimation (GAE) [28].

The value model v is implemented using the same architecture as the reasoner except for the output layer , which is replaced by a linear function that outputs a scalar value. The value model is initialized using the same parameters as the reasoner, apart from the output layer. Assuming that v is parameterized by ω , we learn v by minimizing the temporal-difference error:

Although PPO proves effective in training LLMs [23], we deprecate using it in training tiny transformers due to the difficulty of learning the value function. Since the value model v is also a tiny transformer, its weakness in model capacity severely compromise the precision of value approximation, leading to unreliable advantage estimation.

B.2 Group-reward policy optimization

PPO requires learning an additional value model, which can be expensive and unstable. Alternatively, GRPO [31] directly computes the advantages using the relative rewards from a group of G solutions.

For each query Q , it samples a group of G solutions:

In this group, each solution { R g } contains T g steps, where the answer A g is considered as the final step R g,T g . Using the reward models, we obtain process rewards r p := { ( r g, 1 , . . . , r g,T g ) } G g =1 and outcome rewards r o := { r g,o } G g =1 . Then, GPRO computes the normalized rewards, given by:

Afterwards, the advantage of step t in the g -th solution of the group is ˆ A g,t = ˜ r g,o + ∑ T g k = t ˜ r g,t ′ . Let q g,t,i ( θ ) = π θ ( R [ i ] g,t ∣ ∣ ∣ S g,t ,R [ <i ] g,t ) π θ old ( R [ i ] g,t ∣ ∣ ∣ S g,t ,R [ <i ] g,t ) be the relative likelihood of the i -th token in the t -th step from the g -th solution. Then, the GRPO objective is to maximize the following:

B.3 Technical Implementation

We made two technical modifications that make RL more suitable in our case, described in the following.

First, in RMTP, we mask off the advantage of rejected steps , while the advantage of self-verification labels is reserved. This prevents the algorithm from increasing the likelihood of rejected steps, allowing the planning policy π to be properly optimized. In practice, we find this modification facilitates the training of models that perform mandatory detailed verification. Otherwise, RL could make the reasoner excessively rely on reflection, leading to CoTs that are unnecessarily long.

Second, we employ an early-truncating strategy when sampling trajectories in training. If the model has already made a clear error at some step (detected using an oracle process reward model), we truncate the trajectory as it is impossible to find a correct answer. This avoids unnecessarily punishing later steps due to previous deviations, as some later steps may be locally correct in their own context. Empirically, we find this modification reduces the training time required to reach the same performance, while the difference in final performance is marginal.

C Theory

C.1 A general formulation of reasoning performance

Let S denote the state space and A denote the answer space. We use A Q ⊆ A to denote the set of correct answers for some input query Q . Given any thought state S , the accuracy, namely the probability of finding a correct answer, is denoted as

C.1.1 Bellman equations in RMTP

By considering the reasoning correctness as the binary outcome reward, we may use Bellman equations [33] to provide a general formulation of the reasoning performance for arbitrary MTPs (RMTP). For simplicity, we use S , R , and S ′ to respectively denote the state, step, and next state in a transition.

Initially, in the absence of a trace-back mechanism, the accuracy ρ Q ( s ) can be interpreted as the value function when considering the MTP as a goal-directed decision process. For simplicity, we denote the transition probability drawn from the reasoning dynamics T as p ( S ′ | S, R ) . In non-reflective reasoning, the state transition probability p ( S ′ | S ) can be expressed as:

When using RMTP execution, assuming that ξ ( S, R ) := p V ∼V ( ' × ' ∈ V | S, R ) represents the probability of rejecting the step R , we have:

̸

Consequently, the Bellman equation follows:

C.1.2 Bellman equations in RTBS

Let m denote the number of attempts at each state, and let ϕ ( S ) represent the failure probability (i.e., the probability of tracing back after m rejected steps) at state S . The probability of needing to retry a proposed step due to instant rejection or recursive rejection is given by:

The failure probability is then given by ϕ ( S ) = ϵ m ( S ) . When there are k attempts remaining at the current state S , we denote the accuracy as ρ x ( S, k ) , given by:

It follows that ρ x ( S ) = ρ x ( S, m ) . This leads to a recursive formulation that ultimately results in the following equations for each s ∈ S :

C.2 Accuracy derivation in the simplified reasoning task

In the following, we derive the accuracy of reflective reasoning with and without the trace-back search, given the simplified reasoning task in Section 4. For each proposed step on a correct state, we define several probabilities to simplify notations: α := µe -+(1 -µ )(1 -e + ) is the probability of being instantly rejected ; β = µ (1 -e -) is the probability of being correct and accepted ; γ = (1 -µ ) e + is the probability of being incorrect but accepted . Note that β here no longer refers to the KL-divergence factor in Appendix B.

Proposition 2. The RTMP accuracy ˜ ρ ( n ) for problems with a scale of n is

Let m be the width of RTBS. Let δ m ( n ) and ϵ m ( n ) be the probability of a proposed step being rejected (either instantly or recursively) on a correct state and incorrect state of scale n , respectively. We have σ m (0) = ϵ m (0) = 0 and the following recursive equations for n > 0 :

Then, the RTBS accuracy ˜ ρ m ( n ) for problems with a scale of n is given by

In addition, δ m ( n ) , ϵ m ( n ) and σ m ( n ) all motonously increase and converge in relation to n .

Proof. We first consider reasoning through RTBS. Let ϕ m ( n ) and ψ m ( n ) denote the probabilities of failure (reaching the maximum number of attempts) in correct and incorrect states, respectively. Let ˜ ρ i | m ( n ) indicate the accuracy after the i attempts at the current sub-problem of scale n . Therefore, we have ˜ ρ m ( n ) = ˜ ρ 0 | m ( n ) and ˜ ρ m | m ( n ) = 0 .

At a correct state, we have the following possible cases:

A correct step is proposed and instantly accepted with probability β = µ (1 -e -) . In this case, the next state has a scale of n -1 , which is correctly solved with probability ρ 0 | m ( n -1) and fails (i.e., is recursively rejected) with probability ϕ m ( n -1) .

A correct step is proposed and instantly rejected with probability µe -.

An incorrect step is proposed and instantly accepted with probability γ = (1 -µ ) e + . In this scenario, the next state has a scale of n -1 , which fails with probability ψ m ( n -1) .

An incorrect step is proposed and instantly rejected with probability β = (1 -µ )(1 -e + ) .

Thus, we have a probability of α = µe -+ (1 -µ )(1 -e + ) to instantly reject the step, and a probability of βϕ m ( n -1) + γψ m ( n -1) to recursively reject the step. Therefore, the overall probability of rejecting an attempt on correct states is:

Since failure occurs after m rejections, we have:

At an incorrect state, we have a probability f to instantly reject a step. Otherwise, we accept the step, and the next state fails with probability ψ m ( n -1) . Therefore, the overall probability of rejecting an attempt for incorrect states is:

Similarly, we obtain:

By substituting Equations (26) and (28) into Equations (25) and (27), we obtain Equations (22) and (23). If an attempt is rejected (either instantly or recursively), we initiate another attempt which solves the problem with a probability of ρ i +1 | m ( n ) . Therefore, we have the recursive form of the accuracy, given by:

Thus, we can expand ˜ ρ m ( n ) as:

Note that n = 0 indicates that the state is exactly the outcome, which means ˜ ρ m (0) = 1 . Then, Equation (24) is evident given the recursive form in Equation (31).

For reflective reasoning without trace-back , we can simply replace δ m ( n ) with α in σ m ( n ) , as only instant rejections are allowed. We then set m →∞ , leading to Equation (21).

Monotonicity We first prove the monotonic increase of ϵ m ( n ) . Equation (23) gives ϵ m ( n ) = f +( ϵ m ( n -1)) m and ϵ m ( n +1) = f +( ϵ m ( n )) m for each n > 1 . Therefore, if ϵ m ( n ) ≥ ϵ m ( n -1) , we have:

Additionally, it is clear that ϵ m (1) = f ≥ 0 = ϵ m (0) . Using mathematical induction, we conclude that ϵ m ( n +1) > ϵ m ( n ) for all n ≥ 0 . The monotonicity of δ m ( n ) can be proven similarly, and the monotonicity of σ m ( n ) is evident from that of δ m ( n ) . Since δ m ( n ) and ϵ m ( n ) are probabilities, they are bounded in [0 , 1] and thus converge monotonically.

0

10

20

30

40

50

problem scale (

n

)

0.0

0.2

0.4

0.6

0.8

1.0

reasoning accuracy (

)

RTBS m=1

RTBS m=2

RTBS m=3

RTBS m=4

RTBS m=5

RTBS m=6

RMTP

no reflection

Illustration of accuracy curves Using the recursive formulae in Proposition 2, we are able to implement a program to compute the reasoning accuracy in the simplified reasoning problem in Section 4 and thereby visualize the accuracy curves of various reasoning algorithms. For example, Figure 10 presents the reasoning curves given µ = 0 . 8 , e -= 0 . 3 , e + = 0 . 2 , and f = 0 . 8 , which lead to α = 0 . 4 < f . For this example, we may observe the following patterns: 1) An overly small width m in RTBS leads to poor performance; and 2) by choosing m properly, ˜ ρ m ( n ) remains stable when n →∞ . These observations are formally described and proved in Appendix C.3.

Furthermore, in Figure 10 we see that a small m stabilizes the drop of ˜ ρ m ( n ) when n is large, yet it also makes ˜ ρ m ( n ) drop sharply in the area where n is small. This indicates the potential of using an adaptive width in RTBS, where m is set small when the current subproblem (state) requires a large number n of steps to solve, and m increases when n is reduced by previous reasoning steps. Since this paper currently focuses on the minimalistic reflection framework, we expect to explore such an extension in future work.

C.3 Derivation of Theorem 1

Theorem 1 is obtained by merging the following Proposition 3 and Proposition 4, which also provide supplementary details on the non-trivial assumptions of factors µ , e -, and e + . Additionally, Proposition 4 also shows that there exists an ideal range of RTBS width m such that stabilizes the drop of ˜ ρ m ( n ) when n →∞ .

Proposition 3 (RMTP Validity conditions) . For all n ≥ 0 , we have ˜ ρ ( n ) ≥ ρ ( n ) ⇐⇒ e -+ e + ≤ 1 . Additionally, if µ > 0 and e -< 1 , then for all n ≥ 1 we have that ˜ ρ ( n ) = ρ ( n ) ⇐⇒ e -+ e + = 1 and ˜ ρ ( n ) decreases strictly with either e -or e + .

Proposition 4 (RTBS Validity Condition) . Assuming 0 < µ < 1 , e -< 1 , and e + > 0 , then

Furthermore, we have

lim n →∞ ˜ ρ m ( n ) ˜ ρ m ( n -1) = 1 if m ∈ [ 1 µ (1 -e -) , 1 1 -f ] .

˜ ρ m ( n ) increases strictly with f for all n ≥ 2 .

The proof of propositions 3 and 4 is given in Appendix C.3.1 and Appendix C.3.2, which are based on the previous derivation in Appendix C.2.

C.3.1 Proof of Proposition 3

In any case, we have ˜ ρ (0) = ρ (0) = 1 .

If 0 > µ and e -< 1 , we can transform ˜ ρ ( n ) (given in Proposition 2) as:

This shows that ρ ( n ) strictly decreases with both e + and e -, and ˜ ρ ( n ) = µ n ⇐⇒ e + + e -= 1 . Therefore, we also have ˜ ρ ( n ) > µ n ⇐⇒ e + + e -< 1 .

The Proposition is proved by combining all the above cases.

C.3.2 Proof of Proposition 4

The assumptions 0 < µ < 1 , e -< 1 , and e + > 0 ensure that β > 0 and γ > 0 . Proposition 2 suggests the monotonous convergence of δ m ( n ) , ϵ m ( n ) , and σ m ( n ) . For simplicity, we denote δ := lim n →∞ δ m ( n ) , ϵ := lim n →∞ ϵ m ( n ) , and σ := lim n →∞ σ m ( n ) . From Equations (22) and (23), we have:

Note that ϵ = δ = 1 gives the trivial solution of the above equations. However, there may exist another solution (if any) such that δ < 1 or ϵ < 1 under certain circumstances. Since ϵ m (0) = 0 and δ m (0) = 0 , the limits ϵ and δ take the smaller solution within [0 , 1] . In the following, we first discuss when another non-trivial solution exists.

Lemma 1. For any m ≥ 1 , if 0 ≤ p < m -1 m , then x = p + (1 -p ) x m has a unique solution x ∗ ∈ [ p, 1) , which strictly increases with p . Otherwise, if m -1 m ≤ p ≤ 1 , the only solution in [0 , 1] is x ∗ = 1 .

Proof. Define F ( x ) := p +(1 -p ) x m -x . We find:

It is observed that F ′ ( x ) increases monotonically with x . Additionally, we have F ′ (0) = -1 < 0 , F ′ (1) = m (1 -p ) -1 , and F (1) = 0 . We only consider the scenario where p > 0 , since x = 0 ∈ [0 , 1) is obviously the unique solution.

If 0 ≤ p < m -1 m , we have 1 -p > 1 m . This implies F ′ (1) > 0 . Combining F ′ (0) < 0 , there exists ξ ∈ (0 , 1) such that F ′ ( ξ ) = 0 . As a result, F ( x ) strictly decreases in [0 , ξ ] and increases in [ ξ, 1) . Therefore, we have F ( ξ ) < F (1) = 0 . Since F ( p ) = (1 -p ) p m > 0 , we know that there exists a unique x ∗ ∈ [ p, ξ ) such that F ( x ∗ ) = 0 .

If m -1 m ≤ p ≤ 1 , we have 1 -p ≤ 1 m and F ′ (1) ≤ 0 . In this case, F ′ ( x ) < 0 in [0 , 1) due to the monotonicity of F ′ ( x ) . Thus, F ( x ) > F (1) = 0 for all x ∈ [0 , 1) . Therefore, x ∗ = 1 is the only solution within [0 , 1] .

Now, we prove the monotonic increase of x ∗ when 0 ≤ p < m -1 m . We have:

The previous discussion shows that with x ∗ < [ p, ξ ) for some ξ such that F ′ ( ξ ) = 0 . Given that F ′ ( x ) increases monotonically, we have F ′ ( x ∗ ) < 0 and thus d x ∗ d p > 0 .

Lemma 2. Assume p ≥ 0 , q > 0 , and p + q ≤ 1 . Then, the equation x = p + qx m has a unique solution x ∗ ∈ [0 , 1) , which increases monotonically with p ∈ [0 , 1 -q ] .

Proof. Define F ( x ) := p + qx m -x . Since F (0) ≥ 0 and F (1) < 0 , there exists a solution x ∗ ∈ [0 , 1) . Since F is convex, we know there is at most one other solution. Clearly, the other solution appears in (1 , + ∞ ) , since F (+ ∞ ) > 0 . Therefore, F ( x ) = 0 must have a unique solution x ∗ in [0 , 1) . Additionally, x ∗ must appear to the left of the minimum of F , which yields F ′ ( x ∗ ) < 0 .

Using the Implicit Function Theorem, we write:

Thus, we conclude that x ∗ increases monotonically with p .

Applying Lemmas 1 and 2 to Equation (36), we find that ϵ = 1 if and only if f ≥ m -1 m ; otherwise, ϵ strictly increases with p . Therefore, f < m -1 m indicates that ϵ < 1 , leading to ( α + γϵ ) + γ < 1 . Using Lemma 2 again in Equation (35), we have f < m -1 m = ⇒ δ < 1 . Conversely, f ≥ m -1 m yields ϵ = 1 . and thus f, α + γ ≥ m -1 m = ⇒ δ = 1 .

First, we consider the special case where δ = 1 , which occurs if both f, α + γ ≥ m -1 m , namely m ≤ min { 1 1 -f , 1 β } . In this case, we write σ = (1 + δ + · · · + δ m -1 ) β = mβ . Therefore, we have:

This leads to the validity condition that 1 1 -α < m ≤ min { 1 1 -f , 1 β } .

Next, we consider the case where δ < 1 , which occurs when f < m -1 m or α + γ < m -1 m . This leads to β ≥ 1 m > 0 , and we can write:

Then, we can derive:

Since we have assumed δ < 1 , we have ϵ = 1 > δ if f ≥ m -1 m ; otherwise if f = α < m -1 m , then ϵ = δ leads to δ = ϵ being a solution of Equation (35). Additionally, from Lemmas 1 and 2, we know that a higher α would increase ( α + γϵ ) , which eventually raises δ above ϵ ; conversely, a lower α causes δ to drop below ϵ . To summarize, we have the following conditions when δ < 1 :

Combining the conditions when δ = 1 , we have:

Thus far, we have obtained Equation (33). Now, we start proving the two additional statements in Proposition 4.

First, we prove that 1 β ≤ m ≤ 1 1 -f ensures that σ = 1 . First, if δ = 1 , we have m ≤ 1 β ≤ 1 1 -f . In this case, σ = mβ , and thus σ = 1 when m = 1 β . Alternatively, if δ < 1 , we have m> min { 1 β , 1 1 -f } . We can express that:

Using Lemma 2, we know that δ increases with α + γϵ , which increases with ϵ . Therefore, we have d δ d ϵ > 0 . Then, we obtain

Therefore, σ increases with ϵ and reaches its maximum value of 1 when ϵ = 1 . As a result, we conclude that 1 β ≤ m ≤ 1 1 -f ensures that σ = 1 . Combining β = µ (1 -e -) and σ = lim n →∞ σ m ( n ) = lim n →∞ ˜ ρ m ( n ) ˜ ρ m ( n -1) , we have proved that lim n →∞ ˜ ρ m ( n ) ˜ ρ m ( n -1) = 1 if m ∈ [ 1 µ (1 -e -) , 1 1 -f ] .

Next, we prove the monotonicity of ˜ ρ m ( n ) with respect to f . To prove this, we first prove the monotonicity of ϵ m ( t ) for all t with respect to f .

Lemma 3. For n > 0 , ϵ m ( t ) as defined in Equation 23 increases strictly with f .

Proof. We regard

as a function of f on [0 , 1] . When t = 1 we have

so ∂ϵ m (1; f ) ∂f = 1 > 0 . Thus ϵ m (1; f ) is strictly increasing with f . Further, assume for some k ≥ 1 that

Differentiate the recursion for t = k +1 :

By the inductive hypothesis, ϵ m ( k ; f ) < 1 implies [ ϵ m ( k ; f ) ] < 1 . Therefore, we have

Since 1 -f ≥ 0 , m ≥ 1 , [ ϵ m ( k ; f ) ] m -1 ≥ 0 , and ∂ϵ m ( k ; f ) ∂f > 0 , the second term is also positive. Hence

showing that ϵ m ( k +1; f ) is strictly increasing. This completes the induction.

Given Lemma 3, we can also prove the monotonicity of δ m ( t ) using mathematical induction: It is easy to write that δ m (2) = α + βα m + γf m , showing that δ m (2) increases strictly with f . Then, for t > 2 , assuming that δ m ( t -1) increases strictly with f and using Given Lemma 3, we know that δ m ( t ) increases strictly with f from Equation (22).

Therefore, we have δ m (1) = α and that δ m ( t ) strictly increases with f for n ≥ 2 . According to Equation (24), from the above monotonicity of δ m ( t ) , it is obvious that σ m ( t ) increases with respect to f for all t . This gives the corollary that ˜ ρ m ( n ) increases with f for all n .

C.4 Derivation of RMTP reasoning cost

In this section, we derive how many steps it costs to find a correct solution in RMTP and thereby prove Proposition 1.

Proposition 1. The probability of accepting the correct step after the i -th attempt is given by α i -1 β . Assuming a maximum number of attempts m , the number of attempts consumed at each step satisfies:

Therefore, the average number of attempts required for a correct reasoning step is given by

To simplify the summation expression ∑ m i =1 iα i -1 (where 0 < α < 1 ), we can use the method of telescoping series. Let S = ∑ m i =1 iα i -1 . We calculate αS :

Thus,

This gives us

Rearranging, we have

Thus, the average number of attempts can be further expressed as:

(65)

Considering the limit as m →∞ , it can be shown using limit properties that lim m →∞ m α m 1 -α m = 0 . If the correct solution is obtained (i.e., correct steps are accepted at each step), the average number of steps taken is given by

C.5 Considering posterior risks of rejected attempts

Our previous analysis is based on a coarse binary partition of states (correct and incorrect) for each scale, which enhances clarity yet does not apply to real-world complexity. Therefore, we can introduce stronger constraints by taking into account the posterior distribution of states in S + n after multiple attempts. For example, if the state has produced several incorrect attempts on state S (or rejected several correct attempts), it is more likely that the current state has not been well generalized by the model. Consequently, the chances of making subsequent errors increase. In this case, µ is likely to decrease with the number of attempts, while e -increases with the number of attempts. Thus, the probability of accepting the correct action will decrease as the number of attempts increases.

Therefore, we consider the scenario where the probabilities of error increase while the correctness rate µ drops after each attempt. We define e i + , e i -, µ i , etc., to represent the probabilities related to the i -th attempt, corresponding to the calculations of α i , β i , γ i , etc. We have that β i = µ i (1 -e i -) is monotonically decreasing, and γ i = (1 -µ i ) e i + is monotonically increasing with the index i of the attempt. In this case, the derivation is similar to that of Proposition 2. Therefore, we skip all unnecessary details and present the results directly.

Proposition 5. Given the above posterior risks, the RTMP accuracy ˜ ρ ( n ) for problems with a scale of n is

Let m be the width of RTBS. Let δ i | m ( n ) denote the probability of a proposed step being rejected (either instantly or recursively) at the i -th attempt on a correct state, and ϵ m ( n ) follows the same definition as in Proposition 2. We have δ i | m (0) = ϵ m (0) = 0 and the following recursive equations for n > 0 :

Then, the RTBS accuracy ˜ ρ m ( n ) for problems of scale n is given by:

In addition, δ i | m ( n ) , ϵ m ( n ) , and σ m ( n ) all monotonically increase and converge with respect to n .

The theoretical result in this new setting becomes much more challenging to derive an exact validity condition that remains clear and understandable. However, it is still useful to derive a bound that sufficiently guarantees the effectiveness of reflection. In the following, we show that a sufficient condition becomes much stricter than that in Proposition 3.

Proposition 6. Assume µ 1 < 1 and k = inf i β i +1 β i is the lower bound of the decay rate of the probability of accepting the correct step in multiple attempts. Then, a sufficient condition for ˜ ρ ( n ) > ρ ( n ) is:

Proof. Considering α i = µ i e i -+(1 -µ i )(1 -max i e i + ) , let α = (1 -µ 1 )(1 -sup i e i + ) be its lower bound. It can be seen that

As m →∞ , the sufficient condition for reflection validity is:

D Implementation details

This section describes the details of the training datasets, model architectures, and hyper-parameters used in experiments. Our implementation derives the models architectures, pretraining, and SFT from LitGPT [1] (version 0.4.12) under Apache License 2.0.

D.1 Algorithmic descriptions of reflective reasoning

Algorithms 1 and 2 presents the pseudo-code of reasoning execution through RMTP and RTBS, respectively. In practice, we introduce a reflection budget M to avoid infinite iteration. If reflective reasoning fails to find a solution within M steps, the algorithm retrogrades to non-reflective reasoning.

To implement RTBS, we maintain a stack to store the reversed list of parental states, allowing them to be restored if needed. Different from our theoretical analysis, our practical implementation does not limit the number of attempts on the input query Q (as long as the total budget M is not used up) as Q has no parent (i.e. the stack is empty).

D.2 Example CoT data

We implement predefined programs to generate examples of CoTs and self-verification. Figure 11 presents the example reasoning steps (correct) for non-reflective training and the example detailed verification for reflective training. In our practical implementations, the reasoning steps include additional tokens, such as preprocessing and formatting, to assist planning and transition. To simplify transition function T , the example steps also include exactly how the states are supposed to be updated, which removes the task-specific prior in T .

D.2.1 Multiplication CoT

Each state is an expression x t × y t + r t , where x t and y t are the remaining values of two multiplicands, and r t is the cumulative result. For an input query x × y , the expert reasoner assigns x 1 = x , y 1 = y , and r 1 = 0 .

For each step, the reasoner plans a number u ∈ { 1 , . . . , 9 } to eliminate in x t (or y t ). Specifically, it computes δ = u × y t or ( δ = u × x t ). Next, it finds the digits in x t (or y t ) that are equal to u and set them to 0 in x t +1 (or y t +1 ). For each digit set to 0 , the reasoner cumulates δ × 10 i to r t , where i is the position of the digit (starting from 0 for the unit digit). An example of a reasoning step is shown in Figure 11(a). Such steps are repeated until either x T or y T becomes 0 , then the answer is r T .

Algorithm 1 Reflective reasoning through RMTP

Algorithm 2 Reflective trace-back search

, and reflective

,

<state>

1 4 5 * 8 6 0 9 3

+ 1 0 1 5 0 0

</state>

<action>

left * 8:

- 4 0

- 3 2 + 4 = 3 6

- 8 + 3 = 1 1

- 1

cumulate 1 1 6 0 0 0 0 0:

- 0 + 0 = 0

- 0 + 0 = 0

- 5 + 0 = 5

- 1 + 0 = 1

- 0 + 0 = 0

- 1 + 6 = 7

- 1

- 1

get 1 1 7 0 1 5 0 0.

</action>

<state>

1 4 5 * 6 0 9 3

+ 1 1 7 0 1 5 0 0

</state>

<reflect>

<action>

:  -

-

-

-

cumulate

:

-

-

-

-

-

-

-

-

</action>

<state>

*

+

</reflect>

state

𝑆𝑡

planning

update

step

𝑅𝑡+1

verification

𝑉𝑡+1

(detailed)

Context window

(a) Mult

<state>

3 5 7 8 2 0 4 1 9

0 0 1 7 9 0 0 0 6

8 6 9 1 5 4 0 7 2

0 8 2 4 1 7 5 0 0

5 1 3 9 6 8 7 2 4

9 7 4 0 0 5 1 0 8

2 3 0 5 4 1 9 8 0

0 4 5 6 8 0 2 3 1

1 9 8 3 7 2 0 4 5

</state>

rows 12345789 1679 12456789

124578 123456789 145789

1234589 1234568 12345789

cols 123589 13456789 12345789

13456789 12456789 124578

124579 123478 1245689

blocks 1356789 1245789 124679

12345789 1456789 124578

1234589 12345678 1234589

reduce

<state>

3 5 7 8 2 6 4 1 9

4 2 1 7 9 3 0 5 6

8 6 9 1 5 4 3 7 2

6 8 2 4 1 7 5 0 3

5 1 3 9 6 8 7 2 4

9 7 4 2 3 5 1 6 8

2 3 6 5 4 1 9 8 7

7 4 5 6 8 9 2 3 1

1 9 8 3 7 2 6 4 5

</state>

<reflect>

</reflect>

state

𝑆𝑡

preprocessing

planning & update

step

𝑅𝑡+1

verification

𝑉𝑡+1

(detailed)

Context window

(b) Sudoku

D.2.2 Sudoku CoT

Each state is a 9 × 9 matrix representing the partial solution, where blank numbers are represented by 0 . On each step, the reasoner preprocesses the state by listing the determined numbers of each row, columns, and blocks. Given these information, the model reduces the blank positions that has only one valid candidate. If no blank can be reduced, the model randomly guess a blank position that has the fewest candidates. Such process continues until there exist no blanks (i.e., zeros) in the matrix.

An example of a reasoning step is shown in the right of Figure 11(b). The planned updates (i.e., which positions are filled with which numbers) is intrinsically included in a new puzzle, which is directly taken as the next state by the transition function T .

D.2.3 Verification of reasoning steps

Binary Verification The Binary verification labels are generated using a rule-based checker of each reasoning step. In Multiplication, it simply checks whether the next state x t +1 × y t +1 + r t +1 is equal to the current state x t × y t + r t . In Sudoku, it checks whether existing numbers in the old matrix are modified and whether the new matrix has duplicated numbers in any row, column, and block.

Detailed Verification In Multiplication, we output a label for each elemental computation addition or unit-pair product - is computed correctly. In Sudoku, we output a label for each position in the new matrix, signifying whether the number violates the rule of Sudoku (i.e. conflicts with other

numbers in the same row, column, or block) or is inconsistent with the previous matrix. These labels are organized using a consistent format as the CoT data. Examples of detailed reflection for correct steps is in Figure 11(b). If the step contains errors, we replace the corresponding ' ✓ ' with ' × '.

D.3 Model architectures and tokenization

Our models architectures uses multi-head causal attention with LayerNorm [36] with implementation provided by LitGPT [1]. Table 2 specifies the architecture settings of transformer models with 1M, 4M, 16M parameters.

Tokenizers We employ the byte-pair encoding algorithm [30] to train tokenizers on reasoning CoT examples for tiny transformers. Special tokens for reflection and reasoning structure (e.g., identifiers for the beginning and ending positions of states and reasoning steps) are manually added to the vocabulary. Since the vocabulary size is small (128 in our experiments), the learned vocabulary is limited to elemental characters and the high-frequency words for formatting.

D.4 Hyperparameters

Table 3 presents the hyper-parameters used in training and testing the tiny-transformer models. In the following sections, we describe how these parameters are involved in our implementation.

Non-reflective training The pretraining and SFT utilize a dataset N CoT of CoT examples generated by an expert reasoning program. Pretraining treats these CoT examples as plain text and minimizes the cross-entropy loss for next-token prediction, using the batch size B pre and the learning rate η pre . The pretraining process terminates after predicting a total of N pre _ tok tokens. The non-reflective SFT uses the same dataset as that used in pretraining. It maximizes the likelihood of predicting example outputs (reasoning steps) from prompts (reasoning states), using the batch size B SFT and the learning rate η SFT . The total number of non-reflective SFT epochs is E SFT .

Reflective SFT To perform non-reflective SFT, we use the model after non-reflective training to sample trajectories for each input query in the training set. The reflective sampling involves two decoding temperatures: the lower solving temperature τ refl : s is used to walk through the solution path, while a higher proposing temperature τ refl : p is used to generate diverse steps, which are fed into the reflective dataset. Then, the verification examples, which include binary or detailed labels, are generated by an expert verifier program. The reflective SFT includes E RSFT epochs, using the same batch size and learning rate as the non-reflective SFT.

Reinforcement learning We use online RL algorithms as described in Appendix B, including PPO and GRPO. These algorithms include an experience replay buffer to store N PPO : buf and N GRPO : buf example trajectories, respectively. After every E RL : int epochs trained on the buffer, the buffer is updated by sampling new trajectories, using the temperature τ RL : π for planning steps and the temperature τ RL :˜ π for reflective feedback. According to Equations 8 and 12, the hyper-parameters in both the PPO and GRPO objectives include the clipping factor ε and the KL-Divergence factor β . Additionally, GRPO defines G as the number of trajectories in a group. We run RL algorithms for E RL epochs, using the learning rate η RL . PPO involves E PPO : warmup warm-up epochs at the beginning of training, during which only the value model is optimized.

Testing During testing, we execute the reasoner using three decoding temperatures: τ π : first for the first planning attempt, τ π : rev for the revised planning attempt after being rejected, and τ π f for self-verification. We use low temperatures to improve accuracy for more deterministic decisions, such as self-verifying feedback and the first attempt in Mult. We use higher temperatures for exploratory decisions, such as planning in Sudoku and revised attempts in Mult. We set the non-reflective reasoning budget to T steps and the reflective reasoning budget to ˜ T steps. If the reflective budget is exhausted, the reasoner reverts to non-reflective reasoning. We set the search width of RTBS to m .

D.5 Computational resources

Since our models are very small, it is entirely feasible to reproduce all our results on any PC (even laptops) that has a standard NVIDIA GPU. Using our hyper-parameters, the maximum GPU memory used for training the 1M, 4M, and 16M models is approximately 4GB, 12GB, and 16GB, respectively, which can be easily reduced by using smaller batch sizes. To run multiple experiments simultaneously, we utilize cloud servers with a total of 5 GPUs (one NVIDIA RTX-3090 GPU and four NVIDIA A10 GPUs).

For each model size and task, a complete pipeline (non-reflective training, reflective training, and RL) takes about two days on a single GPU. This includes 1-2 hours for non-reflective training, 8-12 hours for data collection for reflective training, 1-2 hours for reflective SFT, 6-12 hours for RL, and 6-12 hours for testing.

E Supplementary results of experiments

In this section, we present supplementary results from our experiments: 1) we assess the reasoning accuracy of various large language models on integer multiplication and Sudoku tasks; 2) we report the accuracy outcomes of models after implementing different supervised fine-tuning strategies; 3) we provide full results of reasoning accuracy after GRPO; 4) we additionally provide the results of PPO, which is weaker than GRPO in reflective reasoning.

E.1 Evaluation of LLMs

In this section, we provide the reasoning accuracy of LLMs on Mult and Sudoku, including GPT-4o [22], OpenAI o3-mini [21], and DeepSeek-R1 [5]. Since GPT-4o is not a CoT-specialized model, we use the magic prompt 'let's think step-by-step' [13] to elicit CoT reasoning. For o3-mini and DeepSeek-R1, we only prompt with the natural description of the queries. As shown in Table 4, among these LLMs, OpenAI o3-mini produces the highest accuracy in both tasks.

To illustrate how well tiny transformers can do in these tasks, we also present the best performance (results selected from Tables 5 and 7) of our 1M, 4M, and 16M models for each difficulty level, respectively, showing a performance close to or even better than some of the LLM reasoners. For example, according to our GRPO results (see Table 7), our best 4M Sudoku reasoner performs (RTBS through optional detailed verification) equally well to OpenAI o3-mini, and our best 16M Mult reasoner (through binary verification) outperforms DeepSeek-R1 in ID difficulties. Note that this paper mainly focuses on fundamental analysis and does not intend to compete with the generalpurpose LLM reasoners, which can certainly gain better accuracy if specially trained on our tasks. Such a comparison is inherently unfair due to the massive gap in resource costs and data scale. The purpose of these results is to show how challenging these tasks can be, providing a conceptual notion of how well a tiny model can perform.

E.2 Results of supervised fine tuning

Table 5 includes our complete results of reasoning accuracy after non-reflective and reflective SFT. As discussed in Section 3.1, our implementation uses Reduced states that maintain only useful information for tiny transformers. To justify this, we also test the vanilla Complete implementation, where each state S t = ( Q t , R 1 . . . , R t -1 ) includes the full history of past reasoning steps. As a baseline, the Direct thought without intermediate steps is also presented.

Reducing the redundancy of states in long CoTs benefits tiny transformers. The left three columns in Table 5 compare the above thought implementations for non-reflective models. We see that both direct and complete thoughts fail to provide an acceptable performance even in ID-Easy difficulty. This proves the importance of avoiding long-context inference by reducing redundancy in representing states. Considering the huge performance gap, we exclude the complete and direct implementations from our main discussion.

Estimated errors of self-verification For RMTP and RTBS executions, we employ the oracle verifiers to maintain test-time statistics of the average e -and e + (see definition in Section 4) of reasoning states. The results are shown in Table 6, where we also present the difference in how much

reflective reasoning raises the performance over non-reflective reasoning. We only count the errors in the first attempts on reasoning states to avoid positive bias, as the reasoner may be trapped in some state and repeat the same error for many steps.

Our full results provide more evidence for the findings discussed in Section 5.1:

Learning to self-verify enhances non-reflective execution for 9 out of 12 models (2 verification types, 3 model sizes, and 2 tasks), such that accuracy does not decrease in any difficulty and increases in at least one difficulty.

RMTP improves performance over non-reflective execution for all 4M and 16M models. However, RMTP based on binary verification fails to benefit the 1M models, which suffer from a high e -.

4M and 16M Sudoku models greatly benefit from RTBS, especially using detailed verification.

E.3 Results of GRPO

The complete results of models after GRPO are given in Table 7. To have a convenient comparison, Table 8 presents the difference of accuracy across Table 5 and Table 7, showing that the difference of accuracy is caused by GRPO.

Reflection usually extends the limit of RL. For reflective models, GRPO samples experience CoTs through RMTP, where self-verification V and the forward policy π are jointly optimized in the form of a self-verifying policy ˜ π . By comparing the RMTP results (columns 3, 6, and 9) with the non-reflective model (the first column) in Table 7, we find that GRPO usually converges to higher accuracy solving ID-Hard problems in RMTP. This shows that having reflection in long CoTs extends the limit of RL, compared to only exploiting a planning policy.

Interestingly, optional detailed verification generally demonstrates higher performance after GRPO than mandatory verification. A probable explanation is that a mandatory verification may cause the reasoner to overly rely on reflection, which stagnates the learning of the planning policy.

Overall, our full results provide more evidence to better support our findings discussed in Section 5.2:

RL enhances 24 out of 42 ID-Hard results in Table 8 by no less than 3% (measured in absolute difference). However, only 8 out of 42 OOD-Hard results are improved by no less than 1%.

In table 9, an increase of e + is observed in 20 out of 25 cases where e -decreases by more than 5% (measured in absolute difference).

E.3.1 The verification errors after GRPO

Furthermore, we also present the estimated errors of verification after GRPO in Table 9, in order to investigate how self-verification evolves during RL. Our main observation is that if a model has

a high e -before GPRO, then GRPO tends to reduce e -and also increases e + . This change in verification errors is a rather superficial (lazy) way to obtain improvements. If the model faithfully improves verification through RL, both types of errors should simultaneously decrease - such a case occurs only in the ID-Easy difficulty or when e -is already low after SFT. This highlights a potential retrograde of self-verification ability after RL.

E.3.2 The planning correctness rate after GRPO

We also report how GRPO influences the step-wise planning ability, measured by µ (defined in Section 4), across various tasks, verification types, and model sizes. Shown in Table 10, GRPO increases the planning correctness rate µ in most ID cases, except for the Sudoku models with binary verification. This indicates that the proposed steps are more likely to be correct and further reduces the overall penalties of false positive verification, making an optimistic verification bias (a high e + in exchange for a low e -) even more rewarding. In particular, the planning ability shows almost no improvement in OOD problems.

E.3.3 Reflection frequency of optional detailed verification

To show how GRPO adapts the reflection frequency for optional detailed verification, Figure 12 shows the reflection frequency of 1M and 16M transformers before and after GRPO, and the reflection frequency of the 4M model is previously shown in Section 5.2. Similarly, Figure 13 shows the reflection frequency for 1M, 4M, and 16M models in Sudoku.

According to results in Table 5, reflective execution does not improve performance for the 1M model, implying its weakness in exploring correct solutions. Therefore, GRPO does not much incentivize reflection for the 1M model. Contrarily, it greatly encourages reflection for 4M and 16M models, for they explore more effectively than the 1M model. These results align with the discussion in Section 5.2 that RL adapts the reflection frequency based on how well the proposing policy can explore higher rewards.

E.4 Reflection frequency under controlled verification error rates

To investigate how verification error rates ( e -and e + ) influence the reflection frequency in GRPO, we ran a controlled experiment in which the error rates were fixed by intervening with expert verifications. After each time the transformer generated a non-empty verification, we replaced the verification sequence with the expert verification, where randomized noise is injected to achieve the prescribed false-negative rate e -and false-positive rate e + .

We used the 4M Mult model and ran GRPO (sampling temperature = 1.25) for 25 epochs in the in-distribution setting. We measured the fraction of steps at which the model invoked non-empty reflection ('reflection frequency') after 25 epochs. Especially, we are interested in how reflection frequency changes, given a low e -= 0 . 1 or a high e -= 0 . 4 . In both cases, we set e + = 0 . 1 . The results are as follows:

Using a low e -= 0 . 1 , the reflection frequency increases to 59 . 8% after 25 GRPO epochs.

Using a high e -= 0 . 4 , the reflection frequency drops to 0 . 0% after 25 GRPO epochs. That is, the model learns to completely disuse reflection.

Discussion. When the verifier rejects many correct steps (high e -), the model learns to avoid invoking reflection, driving the observed reflection frequency to nearly 0% . Conversely, when e -is

1

2

3

4

5

6

7

8

9

10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

38

33

41

30

30

41

38

32

23

17

46

28

27

20

32

34

30

22

16

14

28

27

20

24

25

28

25

15

17

16

22

25

22

20

22

21

17

18

20

16

22

23

19

18

21

20

16

23

17

14

28

17

14

19

16

14

21

18

14

14

23

20

15

15

15

19

18

14

14

12

22

23

17

17

20

19

14

13

11

9

14

18

17

18

18

17

14

12

10

7

11

14

16

17

19

17

12

12

8

7

Before GRPO

1

2

3

4

5

6

7

8

9

10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

34

31

27

14

8

21

26

26

23

25

52

44

34

19

22

30

38

26

29

29

46

36

20

17

24

36

38

34

33

37

36

28

14

20

26

34

36

30

30

36

32

25

13

26

32

33

38

40

34

33

35

25

26

33

36

34

43

44

34

29

42

32

33

33

36

44

39

40

34

38

42

43

38

44

39

49

45

31

33

22

39

40

42

48

42

38

34

30

22

17

32

44

42

38

37

36

32

24

30

23

After GRPO

(a) 1M

1

2

3

4

5

6

7

8

9

10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

38

19

6

5

6

6

6

5

6

5

33

15

7

5

8

7

4

7

7

6

29

4

4

6

5

4

4

5

8

9

19

5

9

5

4

4

3

6

8

10

15

5

6

4

3

4

4

5

10

10

12

4

4

5

3

3

5

7

14

9

9

5

6

5

5

4

5

10

10

9

6

3

4

4

4

4

6

9

8

7

4

1

2

5

6

5

7

8

5

8

2

1

5

6

4

7

6

6

7

6

Before GRPO

1

2

3

4

5

6

7

8

9

10

number of x's digits

1

2

3

4

5

6

7

8

9

10

number of y's digits

75

72

64

61

64

60

53

52

52

61

67

72

73

71

74

71

67

68

72

77

69

71

74

80

80

78

81

81

84

86

70

70

75

79

80

83

84

87

88

90

67

68

76

78

81

85

86

89

92

93

62

70

72

79

83

85

90

94

96

96

62

70

75

81

85

89

94

96

96

97

56

67

72

82

88

93

95

96

98

97

54

67

78

88

92

95

96

96

97

96

51

75

82

88

93

96

96

98

96

95

After GRPO

(b) 16M

F

F

15

一

r

二

一

十

一

T

low (with the same e + ), reflection becomes beneficial and the model increases reflection usage (here to 60% ). Intuitively, reducing excessive false negatives shortens CoT lengths and makes reflection more rewarding; when e -is large, the model can trade off reflection for a no-reflection policy (which corresponds to the extreme e -= 0 , e + = 1 ), thereby avoiding costly rejections. This experiment demonstrates that the model learns to reduce e -by strategically bypassing verification.

E.5 Results of PPO

As discussed in Appendix B.1, we prefer GRPO over PPO for tiny transformers, as the value model in PPO increases computational cost and introduces additional approximation bias in computing advantages.

Table 11 presents the reasoning accuracy after PPO, and Table 12 gives the difference compared to the SFT results in Table 5. Our results show that PPO is much weaker than GRPO. Although PPO effectively improves the non-reflective models, the performance of reflective reasoning deteriorates after PPO. To explain this, self-verification in reasoning steps causes a higher complexity of the value function, which may obfuscate tiny transformers. Overall, we suggest that GRPO is a more suitable algorithm to optimize reflective reasoning for tiny transformers.

9

18

27

36

45

54

number of blanks

0.0

0.2

0.4

0.6

0.8

1.0

reflection frequency (%)

Before GRPO

9

18

27

36

45

54

number of blanks

0.0

0.2

0.4

0.6

0.8

1.0

reflection frequency (%)

After GRPO

(a) 1M

9

18

27

36

45

54

number of blanks

0.0

0.2

0.4

0.6

0.8

1.0

reflection frequency (%)

Before GRPO

9

18

27

36

45

54

number of blanks

0.0

0.2

0.4

0.6

0.8

1.0

reflection frequency (%)

After GRPO

(b) 4M

9

18

27

36

45

54

number of blanks

0.0

0.2

0.4

0.6

0.8

1.0

reflection frequency (%)

Before GRPO

9

18

27

36

45

54

number of blanks

0.0

0.2

0.4

0.6

0.8

1.0

reflection frequency (%)

After GRPO

(c) 16M

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: Our title, abstract, and introduction clearly state our main claim that transformers can benefit from self-verifying reflection. Our theoretical and experimental results support this claim.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We mention limitations in the conclusion.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: The main paper describes the assumptions of our theoretical results. The proof is provided in the appendix.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in the appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We include necessary information to reproduce our results in the appendix, such as hyper-parameters, model architecture, data examples, and detailed implementation.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: Full code is in the supplementary materials. No data is provided as it is generated by the code. 'README.md' introduces the commands to perform the complete pipeline and reproduce our results. We will open-source our code once it is formally accepted.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: Most relevant hyper-parameters and experiment details are in the appendix. Full settings are clearly defined in our code.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [No]

Justification: It is too expensive to run multiple instances of our experiments, which include training 78 models under various settings (sizes, tasks, verification types, etc). Each model is tested using at most 3 different executions. Given our limited resources, it would take several months to compute error bars. Since our paper focuses on analysis instead of best performance or accurate evaluation, it is acceptable not to include error bars.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We roughly describe the computational resource used in the appendix. Since our models are very small, this paper can be easily reproduced by a single NVIDIA GPU.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: As far as we may perceive, this research does not involve human subjects or negative societal impacts.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: This paper focuses on the fundamental analysis of reasoning instead and is tied to no practical applications.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: This paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: Assets used in this paper are cited in the paper. The appendix mentions the version of the asset and the license.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: This paper does not release assets besides our code.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: This paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: the paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: Although this research is related to LLM reasoning, we focus on tiny transformers. The appendix includes the evaluation of LLMs, yet these results do not impact our core methodology and originality.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates how small-scale Transformers can enhance chain-of-thought (CoT) reasoning through a “Self-Verifying Reflection” mechanism without relying on natural language. The authors propose two fundamental reflective mechanisms: Reflective MTP (RMTP) and Reflective Trace-Back Search (RTBS), which enable the model to detect and reject incorrect steps during the reasoning process. By training models with varying parameter scales on integer multiplication and Sudoku tasks, the authors demonstrate that, under reasonable constraints on verification error, self-verification can improve reasoning accuracy.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\t1.The proposed “Reflective MTP” and “RTBS” frameworks integrate verification mechanisms into the reasoning sequence, allowing small models to perform efficient and minimalistic self-reflection without natural language.\n\t2.The chosen tasks are well-designed, aligning with the capacity boundaries of small models while highlighting the benefits of CoT structure. The experimental validation conditions are clearly defined.\n\t3.The introduction of a trace-back strategy for retrying previous steps significantly improves the success rate in Sudoku tasks, empirically supporting the theoretical claim that complex problems benefit from backtracking.\n\nWeaknesses:\n\t1.During the Reflective SFT phase, the authors rely on an external rule-based expert verifier for supervision but do not provide evidence of the verifier’s accuracy or analyze its impact on training convergence.\n\t2.Although GRPO is used in the reinforcement learning phase, results show that RL primarily improves performance on in-distribution (ID) data, while verification ability degrades (e.g., increased false positive rate e⁺), suggesting that the model may be learning to bypass reflection to avoid penalties rather than improving planning strategies.\n\t3.While the theoretical section is mathematically rigorous, it is built on highly idealized state partitions (e.g., S⁺ₙ and S⁻ₙ), which are not observable in real tasks, limiting the practical applicability of the theorems.\n\nQUESTIONS:\n1.The training pipeline depends on an “expert verifier” to label the correctness of each CoT step. Have you evaluated the accuracy of this verifier? Could its errors introduce systematic bias during training?\n2.The RTBS method requires setting a backtrack width $m$. Have you conducted sensitivity analysis on the choice of $m$? During inference, how does the model determine whether backtracking is needed? Have you considered an adaptive mechanism?\n3.Theoretical analysis assumes each reasoning step can be labeled as a positive or negative state. In real-world scenarios, how would you approximate the partitioning of S⁺ₙ and S⁻ₙ?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a framework to study the propensities of R1-style reasoning models to learn certain behaviours like self-reflection during training in easily verifiable environments like Sudoku and integer multiplication.\n\nWithin this framework, the paper theoretically shows that self-verification only helps when the FPR and TNR are below a fixed, bounded value. The authors also explore when does using a search tree for chain of thought becomes beneficial and provide a lower bound on the search depth based on the verifier's performance. \n\nThe authors empirically evaluate their framework and theoretical claims by studying how 1-16 million parameter transformers behave when trained in a similar way to reasoning models. They observe that while reflection boosts the performance of the transformers by default, RL optimisation generally modifies the verifiers to accept many actions, especially in OOD environments.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The framework is well-built, and the claims seem theoretically sound. It seems useful to quickly explore new ways for training reasoning models. \n- Very well written and easy to follow\n\n---\n\nWeaknesses:\n- Parts of the experiments done are unlikely to generalise to Deepseek R1-style reasoning models:\n    - Most reasoning models have poor models for verifying their claims effectively (as the authors noted in the paper). This usually results in 'overthinking' and exploring/rejecting many trajectories emerging during training (especially in Deepseek r1). \n        - This seems to be analogous to high error rates for the verifier and a large search depth. Are there any bounds for RTBS with a poor verifier as well?\n        - In contrast, in the paper, I see the verifiers oddly have a much larger acceptance rate (state-action pairs it classifies as being positive/correct) after RL. Is there any reason for this dissimilarity? Is it possible to plot the value of $\\mu$ and the change in error rates?\n        - Can we compare the error rates in this work with those of reasoning models using previous works like [He et al.](https://arxiv.org/pdf/2502.19361)?\n    - In many practical tasks, reflection is not done after every step/regularly. Either due to it being hard (similar to why MCTS didn't work for deepseek r1), or the language model just learns this policy. Therefore, exact recursive backtracing, like in RTBS, might not be what is happening. However, this only affects a part of the motivation, and this could be potentially important for other types of CoT training. \n        - Maybe adding some randomness to which states are reflected on can simulate an imperfect verifier like this.\n        - Another important factor here is studying the propensity to use self-reflection when $\\mu$ and $e_+$/$e_-$ change. Entire RL runs seem expensive, but artificially changing the values/keeping them fixed to see if the frequency changes after a few epochs seems reasonable. \n- Slightly unclear result:\n    - Moving verifiers to classify many states as positive seems like RL is trying to circumvent the verification step. The verification step still looks to be doing something useful. Maybe plotting the number and type of states pruned/rejected from this step before and after RL will make things clearer.\n\n---\n\nOverall: This is a technically sound work, and the framework proposed definitely has good use cases. There could be some more experiments/ablations to simulate more realistic scenarios. However, this paper still provides valuable insights.\n\nQUESTIONS:\nI have listed down specific experiments and questions I had, along with the main strengths and weaknesses for clarity and continuity. The most important parts for me are:\n1. To address why the acceptance rates increase, and provide some information about how the model's policy to suggest next steps changes. \n2. To study when the model chooses to use self-reflection further. Particularly when the rejection rate is high and the planning policy is good.\n\nLIMITATIONS:\nFor most parts, yes. Addressing how likely the claims are to generalise to practical applications is required.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper investigates how self-verifying reflection enhances chain-of-thought (CoT) reasoning in small-scale transformer models. The authors demonstrate theoretically and experimentally that even tiny transformers can achieve LLM-level performance on tasks like integer multiplication and Sudoku. The study also shows that while reinforcement learning (RL) improves in-distribution performance and encourages reflection, it tends to optimize for shallow statistical patterns rather than genuinely improving the model's ability to detect errors.

STRENGTHS AND WEAKNESSES:
**Strengths**
- Authors use a minimalistic reasoning framework with tiny transformers to study complex behaviors like reflective CoT, enabling controlled, low-cost analysis.
- **Comprehensive and Rigorous Evaluation**: Authors combine *formal proofs* with extensive experiments across model scales, tasks, training methods (SFT, RL), and reflection strategies (RMTP, RTBS).
- The experiments demonstrate interesting findings that self-verification improves planning even without test-time reflection, and that RL promotes reflection by shifting error types rather than enhancing verification.

**Weaknesses**
- **Limited Generalizability**: Results from small-scale, non-linguistic models may not extend to full-scale LLMs in natural language settings.

- **Narrow Task Scope**: Experiments are confined to structured tasks like integer multiplication and Sudoku, limiting relevance to broader reasoning challenges.

QUESTIONS:
The main concern, as noted in the weaknesses section, is that conclusions drawn from small-scale models on narrowly defined tasks may not generalize to larger models or more complex, real-world tasks.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates how LLMs use reflection and self verify the reasoning chain-of-thoughts (CoTs) which leads to empirical improvements. To analytically understand self verifying reflection process, they use a reasoning framework with small transformer models. They found that their small transformers with self-verification archives performance of LLMs in integer multiplication and Sudoko. They also theoretically show that self-verifying reflection guarantees improvements if verification errors are properly bounded.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1) To analytically understand self verification reflection the authors use small transformers which mimics the performance of LLMs. The use of small models reduces the cost of comprehensive experiments.\n\n2) They theoretically show that a self-verifying reflection step improves reasoning as long as the verifier’s errors are properly bounded, so even a lightweight verifier suffices, rather than relying on an expensive one.\n\n3) They also found that a reinforcement-learning (RL) finetuning with reflection can be useful only if the reasoner’s exploration strategy covers a diverse set of candidate solutions; without sufficient exploration, the RL signal can not discover the added value of reflection.\n\nWeaknesses\n1) They did the analysis with tiny models, which shows limited generalisation ability and RL finetuning didn't improve it.\n\n2) No large scale finetuning experiments are done on reasoning benchmarks to show whether their analysis transfer well to LLMs or not.\n\nQUESTIONS:\n1) Do the authors have any intuition on why in Figure 6, for OOD hard problem with binary verification smaller model (1M) performs better than larger models (4M and 16M)?\n2) Can the authors fine-tune larger pretrained models such as Llama or Qwen-7B to test whether the RL findings generalize at that scale?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates how small transformer models, without relying on natural language, can perform self-verifying reflection to improve multi-step reasoning. The authors introduce a minimal reasoning framework and provide theoretical guarantees that reflective execution improves accuracy when verification errors are bounded. Empirical results on multiplication and Sudoku tasks demonstrate that tiny transformers (1M–16M parameters) can achieve near-LLM performance using this reflective mechanism. The work also reveals that while reinforcement learning encourages reflective behavior and improves in-distribution performance, it largely exploits statistical patterns without enhancing verification capabilities.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. The paper is well written and clearly organized. The figures and wording are effective, making the technical content easy to follow.\n2. The authors conduct a thorough analysis of the proposed method, combining theoretical guarantees with strong experimental validation.\n3. The topic is timely and important. The work provides insightful observations on self-verifying reflection, which could have significant implications for improving reasoning in large language models.\n\n**Weaknesses**\n\n1. It remains unclear how the proposed method performs when applied to an already pre-trained model. This is a relevant concern for practical adoption, as many real-world models are not trained from scratch under controlled settings, but are instead adapted from general-purpose, pre-trained LLMs.\n2. It is unclear how this method compares to other variants of CoT reasoning. How does this minimalistic reflection framework compare to natural language-based self-reflection COT in terms of (1) accuracy and (2) inference efficiency? Such comparisons would help contextualize the advantages of the proposed approach.\n\nQUESTIONS:\n**Questions for the Authors**\n\n1. How does your self-verifying reflection framework perform when integrated into a pre-trained transformer model, rather than training from scratch?\n2. Have you considered comparing your method to natural language-based self-reflection approaches in terms of both accuracy and inference cost?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-06"}