Enhancing Reasoning Capabilities of LLMs via Principled Synthetic Logic Corpus
Authors: Terufumi Morishita
Gaku Morio
Atsuki Yamaguchi
Yasuhiro Sogawa
Venue: NeurIPS 2024 poster (2024)
Keywords: large language model
artificial intelligence
reasoning
logical reasoning
math
coding
synthetic corpus
Primary area: natural_language_processing
TL;DR: We enhanced LLM's reasoning capabilities by principled synthetic corpus.

Abstract: Large language models (LLMs) are capable of solving a wide range of tasks, yet they have struggled with reasoning.
To address this, we propose $\textbf{Additional Logic Training (ALT)}$, which aims to enhance LLMs' reasoning capabilities by program-generated logical reasoning samples.
We first establish principles for designing high-quality samples by integrating symbolic logic theory and previous empirical insights.
Then, based on these principles, we construct a synthetic corpus named $\textbf{Formal} \ \textbf{Logic} \ \textbf{\textit{D}eduction} \ \textbf{\textit{D}iverse}$ (FLD$ _{\times2}$), comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors.
Finally, we empirically show that ALT on FLD$ _{\times2}$ substantially enhances the reasoning capabilities of state-of-the-art LLMs, including LLaMA-3.1-70B.
Improvements include gains of up to 30 points on logical reasoning benchmarks, up to 10 points on math and coding benchmarks, and 5 points on the benchmark suite BBH.

Enhancing Reasoning Capabilities of LLMs via Principled Synthetic Logic Corpus
Terufumi Morishita
Gaku Morio 1+
Atsuki Yamaguchi
Yasuhiro Sogawa
'Advanced Al Innovation Center, Hitachi
The University Sheffield
Abstract
Large language models (LLMs) are capable of solving wide range of tasks, yet they have struggled with reasoning. To address this, propose Additional Logic Training (ALT), which aims to enhance LLMs' reasoning capabilities by program- generated logical reasoning samples. We first establish principles for designing high-quality samples by integrating symbolic logic theory and previous empirical insights. Then, based on these principles, construct synthetic corpus named Formal Logic Deduction Diverse (FLDx2). comprising numerous samples multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors. Finally, we empirically show that ALT on FLD x2 substantially enhances the reasoning capabilities of state-of-the-an LLMs, including LLaMA-3.1 1-7BB. Improvements include gains of up to 30 points on logical reasoning benchmarks, up 10 points on math and coding benchmarks, and 5 points on the benchmark suite BBH.
Introduction
Knowledge and reasoning have long been considered essential elements for achieving artificial intelligence (McCarthy, 1959; Weizenbaum, 1966; Winograd, 1971; Colmerauer and Roussel, 1973; Shortliffe, 1976; Elkan and Greiner, 1993). Knowledge refers to facts about the world, e.g., "objects with mass generate a gravitational field" and "the Earth has mass." Reasoning involves combining multiple facts according to specific rules to obtain new knowledge. For example, the new knowledge that "the Earth generates a gravitational field" can be derived from the aforementioned two facts.
Recent observations suggest that LLMs can solve problems using memorized knowledge of similar samples seen during pre-training, but they cannot solve novel, unknown problems that require reasoning (Hodel and West, 2023; Dasgupta et al., 2023; Zhang et al., 2024). For instance, LLMs can solve famous arithmetic problems as but not when the numbers et al., 2022; Mirzadeh al., 2024), and they can solve coding tests from past years before the "knowledge cutoff" but not from the present year (Mitchell, 2023). This bias towards knowledge has been observed even in state-of-the-art LLMs such as GPT-4 (Liu al., 2023b; Wu et al., 2023; Dziri et al., 2023).
names are changed (Razeghi
LLMs' poor reasoning capabilities can stem from the lack of high- quality reasoning samples in the pre-training corpus, which primarily consists of numan--ritten texts (Betz al., 2021: Morishita et al., 2023). Indeed, reasoning samples in human-written texts often exhibit low quality, evidenced by fallacies and biases commonly found online debates (Hansson, 2004; Guiaşu and Tindale, 2018; Cheng et al., 2017). This is unsurprising given that humans usually think reflexively rather than through rigid reasoning (Kahneman, 2011; Sunstein and Hastie, 2015; Paglieri, 2017). Thus,
Equal Contribution
Work done at Hitachi
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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75
2.1-0-
50
80
70
50-
40-
70
60
69
66
00
66
55
10-
50
Logi Math
Code
NL
Others
Ca
CaT
Pro
Benchmark (5-shot)
BBH (3-shot)
BBH (0-shot)
MMLU (S-shot)
Figure 1: The performance gains to LLaMA-3. 3-70 by Additional Logic Training (ALT) on the proposed synthetic corpus, FLDxx2 x2 (Formal Logic Deduction Diverse). Each benchmark set, such as "Logic" and "Math", comprises various benchmarks in that domain. Tables 2, 4 shows the details.
straightforward strategy to improve LLMs' reasoning capabilities is prepare many high-quality reasoning samples and train LLMs on them.
We propose one such approach, Additional Logic Training (ALT), which utilizes high-quality samples of logical reasoning, the most fundamental form of reasoning. To prepare such samples, we utilize synthetic generation (Clark et al., 2021; Betz et al., 2021; Tafjord et al., 10 Morishita et al., 2023), where computer programs generate deductive reasoning samples in which given hypothesis is proven disproven by combining given facts following rigid reasoning rules. We overview ALT in Figure 2.
In synthetic generation, computer programs generate samples according to pre-designed patterns, so this design largely determines the quality of these samples by nature. Thus, we start by discussing what is the ideal design for synthetic logic samples, incorporating symbolic logic theory and empirical findings (Section 2). The essence of logical reasoning lies in its ability to handle unknown facts, unlike knowledge, which deals solely with established facts, such as commonsense facts; therefore, samples must cover reasoning with unknown facts. Samples must include both illogical and logical reasoning to enable LLMs to distinguish between them. The samples must cover various patterns regarding comprehensive set of reasoning aspects, such as reasoning rules and linguistic expressions of logical statements. We summarize these discussions into design principles, which guide the design of synthetic logic samples. Finally, based on these principles, we construct a synthetic corpus named Formal Logic Deduction Diverse (FLD x2) ) comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors (Section 3).
We then empirically verify that ALT can enhance LLMs' reasoning capabilities (Sections 4, 5). Using 31 benchmarks covering diverse tasks, we observed that ALT on FLD x2 substantially boosts state-of-the-an LLMs' reasoning capabilities. Even LLaMA-3.J 1-70B. the largest LLM pre-trained on over 15 trillion tokens, shows substantial improvements with ALT (Figure 1). Among synthetic logic corpora with different sample designs, FLDxx1 yielded the largest performance gains, validating our proposed design principles. Moreover, prevention method during ALT is critically important, as it likely prevents the LLM's knowledge of established facts from being displaced by the unknown facts included in synthetic logic corpora.
discovered that employing
mooweddge-orreetinn
Finally, we analyze which fask-solving capabilities ALT can enhance and why (Section 6). We ob- served a substantial improvement of up to 30 points on logical reasoning tasks (Table 4a). Surprisingly. we also observed improvements in abductive reasoning tasks, which go beyond the synthetic logic corpora's 's original deductive reasoning tasks. Case analyses indicate that these improvements result from LLMMs having acquired the fundamentals of the logic reflected in the design principles. We also observed improvements of up to 10 points on math and coding tasks, indicating the generalizability of the obtained reasoning capabilities (Tables 4b, 4c). We also observed improvements of up to 6 points on natural language inference (NLJ) tasks (Table 4d). Case analyses suggest that LLMs successfully integrated the commonsense knowledge they had originally acquired during pre-training with the logical reasoning capabilities newly acquired from ALT.
Improvements across various other tasks (Table 4e) demonstrate the broad benefits of the obtained reasoning capabilities beyond standard reasoning tasks, though the modest improvements of up to 2 points indicate the need for future research on more effective application of these capabilities.

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Sample Generator
Logical Reasoning Sample
oetregatien lac4:
Multi-step Deductive Ressoning
exis star also (l =13
Input
Design Principles
and cutt
ceitts
Induude unirssees facls
[P 3C)
614
Indlude vainss
reauornngg ng tuko.
2
0
Indluce nathoss
cstss,
652 star ousts
seprestions
6
30
Hypothesis:
Jaz star eents 1..
Logical steps:
Comphetenes Theeeen
14 sittep dediution by the sanmec can eqpees sny other
xc) -
Assume that Foo star exitt
Gate slar sossts (7-mS)
Gaz star exsts B
Fact? > Har stor edsts + if Foo star edsts
Output
Nemove cos.mptonn]
Her star wants
Fact! 9 Jux star exints.
Figure 2: Our proposed Additional Logic Training (ALT) aims to enhance LLMs' reasoning capa- bilities through training on many synthetically generated logical reasoning samples. Our sample generator (left) first generates sample of multi-step deductive reasoning and then converts it into a deduction sample written in English (right). LLMs must generate logical steps to derive a given hypothesis from provided facts. The sample generator adheres to theoretically and empirically grounded design principles discussed in Section 2. Refer to Figure D.3 for a real sample.
Our contributions are summarized follows:
We propose Additional Logic Training (ALT) and empirically verify that it can enhance the reasoning capability of state-of-the-art LLMs across various sizes, from 7B to 70B.
We establish systematic design principles for synthetic logic samples; then, we construct a synthetic corpus named Formal Logic Deduction Diverse (FLDx2), comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors. We empirically verify that Formal Logic Deduction Diverse indeed leads to the largest improvements among corpora with different sample designs.
We demonstrate that LLMs enhanced by ALT can solve not only the original logical reasoning tasks present synthetic logic corpora but also other tasks, such as math and coding tasks, and notably NLI tasks, which require integrating knowledge and reasoning. This finding underscores the potential for advancing truly versatile Al possessing both knowledge and reasoning capabilities. We release the corpus, code, and the trained model under permissive license !
2 How Should Synthetic Logic Samples Be Designed?
In synthetic generation, computer programs generate samples according pre-deni patterns, so this design largely determines the quality of the samples. While Previous studies have examined several designs (Clark et al., 2021; Betz al., 2021; Tafford et al., 2021; Morishita al., 2023), these designs were not systematically discussed, so they may not be the most effective ones.
Thus, we start by discussing how optimally design synthetic logic samples. To this end, we consider symbolic logic theory as suggested by Morishita et (2023) and integrate empirical findings from previous studies. First, observe that the essence of logical reasoning, based solely the logical relationships between facts, lies in its ability to handle unknown facts, unlike knowledge, which by definition deals solely with established facts (Section 2).). Therefore, argue that samples should cover reasoning with unknown facts to represent this essential aspect of logical reasoning. We also observe that logical reasoning involves various other aspects, such as illogical reasoning, reasoning rules, and linguistic expressions that represent logical statements (sections 2.2 to 2.4). The samples should cover various patterns regarding these aspects to enable L.MMs to solve various reasoning problems. We summarize these discussions into the following design principles, which guide the design of synthetic logic samples.
lattps: http://ttthhbbmmmhttte// nlp//LD
3

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2.1 Teaching Reasoning with Unknown Facts
We first explore the essence of logical reasoning that differentiates itself from knowledge. Consider the following logical step:
The Earth orbits the Sun.
If the Earth orbits the sun, the Earth has four seasons.
(1)
The Earth has four seasons.
This step is valid because the conclusion is logically derived from the two premises. Next, consider another logical step:
The Earth orbits the Sun.
If the Earth orbits the sun, the Earth does not have four seasons. The Earth does not have four seasons.
(2)
The second premise and consequently, the conclusion, is factually wrong, Nevertheless, if the premise was hypothetically correct, the conclusion could be logically derived. Therefore, step (2) is also logically valid. Finally:
1. A Foo star exists.
2. If Foo star exists, Bar star also exists.
(3)
A Bar star exists.
"Foo star" and "Bar star" are unknowns; nonetheless, we can still determine that step (3) is logically valid. Steps (1) (3) above can be abstracted into deduction rule, i.e., modus ponens, using symbols:
F F + G modus ponens
(4)
As we have seen, the logical validity of a deduction rule depends solely whether the conclusion is logically derived from the premises, not on the factual correctness of the contents of F and g. Therefore, the contents of F and G can be arbitrary.
Now, we consider what kind of samples would be needed to teach the deduction rule (4) to LLMs. We assume a task to generate the conclusion given the premises as prompt inputs. If the learner were human, they would be able infer the underlying deduction rule (4) by observing samples such as (1) to (2). As result, they would become able to solve the unknown problem (3).
However, from a purely inductive perspective, samples (1) to (2) cannot simply be generalized the deduction rule (4). This is because the samples (1) to (2) themselves do not contain the information that the contents of F and are arbitrary. fact, one could generalize samples (1) (2) other rules; for example, the conclusion 9 can be derived if F and JF + & are given as premises and F and G include 'Earth' as their contents. Innumerable such deduction rules can be inductively inferred from the given samples. In other words, induction has arbitrariness (Hume, 1748; Goodman, 1954; Quine, 1969).
Humans prefer simpler rules (Bertrand: Wittgenstein, 1922), so they boldly induce up to the deduction rule (4). However, is unclear how purely inductive learners such LLMs, which extract only what can be inferred from samples without prior preferences, induce up to (4). For example, if only specific contents such as "Alice is kind" and "Bob is smart" are assigned F and G in training samples, an LLM could develop into a machine that generates the conclusion G only when the input contains the specific contents. In order for LLMs accurately induce that F and are indeed arbitrary:
Design Principle 1 (Reasoning with Unknown Facts). Prepare many samples assigning arbitrary contents to F and G. They will make LLMs accurately induce F and G are indeed arbitrary, ultimately enabling them to reason with unknown facts.
2.2 Teaching Illogical Reasoning
Suppose we have LLMs trained on large number of samples as follows:
FAG
(F H
(5)
H.
where A denotes logical conjunction, and arbitrary contents are assigned to F,G, H. Suppose that we give this LLM a problem such as:
F
(FA9)
(6)
??
Since the premises are insufficient for logically deducting the conclusion, outputting nothing the correct answer.

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Unfortunately, an LLM could output H, which was indeed often observed in our preliminary experi- ments. This is because while the LLMs can induce from sample (5) that it can generate the conclusion H when the two premises of (5) are given, the LLMS cannot induce from the sample that it is not allowed to generate the conclusion H when the premises of (6) are given, as such information not included in the sample (5) itself. Therefore,
Design Principle (Illogical Reasoning). Include negative samples such as (6). These samples will make LLMs induce that conclusions cannot be derived from insufficient premises.
2.3
Teaching Diverse Reasoning Rules
Deduction rules other than (4) exist:
(FAg)
(FAG)
g)A(G +H) syllogism
Aelimination
F
G
F-7 (FV9) -(FNg) De Morgan's laws
F G
contraposition
(7)
9 FF
-FA-G -FV-G
where V denotes logical disjunction and - negation. Since there are infinitely many possible logical formulas that appear premises and conclusions, there infinitely many deduction rules. Providing LLMs with these infinite deduction rules is obviously intractable.
Instead of directly providing these infinite deduction rules, we can take another approach. Consider multi-step deductive reasoning (Figure left), where multiple deduction rules derive conclusion. Notice that the syllogism in (7) can be expressed by multi-step deductive reasoning using more "atomic" deduction rules. Indeed, there exists a set of atomic deduction rules called the axioms that satisfies the following:
Theorem 2.1 (Completeness of First-Order Predicate Logic Gödel (1930).. Any valid deduction rule can be expressed by multistep deductive reasoning constructed from the axioms.
In contrast to the axioms, the 'compound' deduction rules, such as syllogism, contraposition, and De Morgan's 's laws, called theorems. According to the completeness Theorem 2.1, if we can handle the axioms, we can effectively handle other deduction rules as well. Indeed, Morishita et al. (2023) empirically verified that language model trained the axioms generalizes handle other deduction rules more effectively than those trained on non-axiom deduction rules. Therefore,
Design Principle 3 (Diverse Reasoning Rules). Samples should express multi-step deduction con- structed from from 2023)
the axioms. They will effectively teach LLMs diverse deduction rules (Morishita et al.,
In multi-step deductive reasoning, the number of logical steps from premises conclusion can vary largely depending on the problem. Therefore:
Design Principle 3' (Diverse Reasoning Rules). Samples should include diverse numbers of logical steps 8.
Ideally, this would be sufficient, but empirical evidence has shown that LLMs struggle with construct- ing multi-step deductive reasoning with large steps s (Gontier et al., 2020; Morishita et al., 2023). Consequently, LLMs would not excel at handling theorems that require a large number of steps when expressed by the axioms. Therefore, as an additional countermeasure:
Design Principle 3 (Diverse Reasoning Rules). Samples should also include representative theorems, such as syllogism, contraposition, and Morgan's laws.
2.4 Teaching Diverse Linguistic Expressions that Represent Logical Statements
There are various linguistic structures for expressing the logical relationship F - G, such as "If F then g", "F leads to G", and "F results in g If we only include specific expressions the corpora, LLMs may only learn to react to these specific expressions, which has been observed in previous experiments (Zhang et al., 2022; Yuan al., 2023). To prevent this,
Design Principle (Diverse Linguistic Expressions). Samples should include diverse linguistic expressions that represent logical statemenis.
In this chapter, we have established the principles to guide the design of synthetic logic samples. Next, we construct synthetic logic corpus based these principles.
5

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Table I: Synthetic logic corpora compared in this study, with their features categorized according to our proposed design principles (DP). Note that the last row of the ablation corpora lists variations of FLD xi, each of which differs from the original regarding one of the design principles.
DP1
DP2
DP3
DP4
vocahulary size
distractors
deduction rules
logical steps
expressions per formula
RuleTaker 2011)
100
random formula
1-5
0(1)
(RT)
(hami-selecte))
(implication
PARARULE-PP (lanetal, 2032)
100
random
1-5
o(1)
(PRP)
(hmi-selected)
formula
(implication)
15k
random
13 (axioms)
FL.D (Muridina 2019)
1-8
10~100
(WordNet, subset)
formula
100k
adversarial formula
50
10-100
FLDx2
(WordNet, Full)
(axioms and theorems)
(more extensive than FI.D)
FLDxa ablation corpora
100
used
2 (implication)
w/o DP1
-> w/o DP2
w/o Dul.s
- wlo Stt.steps
- w/o DP4
3 Creating a Synthetic Corpus based on Design Principles
To prepare diverse samples reflecting the design principles 1 to 4 (DP1-4), we built novel sample generator by extending the previous one by Morishita al. (2023) and then generated the synthetic logic corpus named FLDxa (Formal Logic Deduction Diverse). Figure 2 shows schematic of our generator and deduction sample. Table 1 compares FLDxx2 with existing corpora. Figure D.3 provides an actual deduction sample included in FI.Dx1-
More specifically, our generator generates deduction samples through the following steps. First, the generator randomly generates a sample of multi-step deductive reasoning written in logical formulas, as shown on the left side of Figure 2, where conclusion is derived from premises using multiple deduction rules (See Appendix D.3 for more details of this generation procedure). At this time, the generator also generates 'distractor' logical formulas, which express negative premises of DP2. Next, the generator converts each logical formula into English expressions. To achieve this, the generator first randomly selects a template from pre-defined options, such as "If F. then G." "F leads to g." or "F results in G," for the logical formula "F + G." It then assigns English content randomly constructed from a vocabulary, such as "(that) a Foo star exists" and "(that) a Bar star exists," " to each symbol, such as F and G. Finally, it converts the multi-step deduction into deduction sample (right side of Figure 2) by using the premises as 'facts', the conclusion as 'hypothesis', and the intermediate logical steps as 'logical steps'. The deduction sample requires LLMs to generate logical steps that derive a given hypothesis based on the given facts.
Table 1 outlines the comparison of FLDxa x2 with other existing corpora (Clark et al., 2021; Bao et al., 2022: Morishita et al., 2023) in terms of DP1-4, which detailed as follows:
DPI: We assign F and 9 content randomly constructed from a vocabulary. While the existing corpora used small-sized vocabulary of up to 15k, we large vocabulary of around 100k words built from WordNet (Miller. 1995). This will teach LLMs that F and 9 are truly arbitrary, ultimately enabling them to reason with unknown facts.
DP2: The existing corpora used randomly generated logical formulas as distractors. In contrast, we implement adversarial distractors. For example, for a premise F A G. we use F with missing information (see Equations (5), (6)), and for a premise F > H. we use information as distractors. These distractors teach LLMs precisely when a conclusion can and cannot be derived. As with previous corpora, we include a variable number of distractors in each sample, randomly chosen from range of ( to 20.
G - H with missing
D:3-3": While the existing corpora used a small number of deduction rules of up to 13 (refer to Figure B. of Morishita et al. (2023)), we include diverse deduction rules, encompassing the axioms and representative theorems, such as modus ponens, syllogisms, and contraposition, totaling about 50 rules. We include samples with up to s = 8 logical steps, following (Morishita et al., 2023).
DP4: We manually craft several more English templates per logical formulas than those used in FLD. Since the templates have a nested structure, they yield combinatorially more diverse English expressions. While counting the exact number of the resulting expressions is intractable, we observed at least dozens of expressions per logical formula, including minor variations. See Appendix D.4 for details.

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4 Experimental Setup
We briefly explain the experimental settings. Refer to Appendix for the details.
Synthetic Logic Corpora: We examine the proposed FLDxx xI and previous corpora (Table 1).
LLMs: We used the state-of-the-art art LLM, LLaMA-3.1 (8B and 70B) (AI@ Meta, 2024).
Training Settings: We trained the LLMs by a method similar to supervised fine-tuning; as illustrated in Figure 2. we used the facts and hypothesis as inputs and logical steps and additional answer label (see Appendix D.I) as outputs. We excluded loss computation for the inputs to prevent LLMs from learning to generate unknown facts. We trained the LLMs for 1 epoch on 100k samples (~ O.IB tokens) from the training split of each corpus, with a batch size of 256, resulting in 390 steps, with a linear warmup for 200 steps. We used the learning rate of 2c-05 for the 8B model and 3e-06 for the 70B model. We used Huggingface (Wolf et al., 2020) for implementation.
Prevention of Knowledge Forgetting by Recall Adam Optimizer: Synthetic logic corpora include many samples with unknown facts, so training on them should cause LLMs to forget their knowledge of existing facts. To prevent this, we employed the Recall Adam optimizer (Chen et al., 2020), which regularizes parameter updates to avoid deviating too far from the pre-training parameters. Recall Adam stands out for LLM training for several reasons (see Appendix E.0.1 for details). We used our re-implemented version" 2. The hyperparameters were: 31 = 0.9, 32 = 0.999, e 10 fisher coefficient 4000 for the 8B model and 2000 for the 70B model.
Benchmarks: We evaluated the trained LLMs on 31 benchmarks shown in Table E.7 using 5-shot in-context learning, except for BBH and AbuductionRules, which used 3-shot in-context learning. These benchmarks cover a wide range of tasks and are prominent in LLM evaluation. Note that we excluded the synthetic logic corpora used for training, as training on them often leads to overfitting to their superficial and statistical cues (Zhang et al., 2022; Yuan et al., 2023), failing to measure truly generalizable reasoning capabilities. We used in-evallutioon-hanees (Gao et al., 2023) and eagcode-evaluation--amne (Ben Allal et al., 2022) for the implementation.
5 Can Additional Logic Training Enhance LLMs' Capabilities?
Table 2 show the performance LLMs before and after ALT. Most LLMs trained with ALT outperformed their counterparts without ALT. Notably, ALT yielded substantial gains of up to 10 points even for 1.aMA-3311705B the largest L.M pre-trained on over 15 trillion tokens. These results verify that ALT can enhance the capabilities of state-of-the-art LLMs.
Among the LLMs trained with ALT, the one trained on FLD x2 (i.e., @ALT-FLD, achieved the highest generalization performance across the benchmarks. Table 3 shows the performance of the LLMs trained on ablated FLDx> corpora, each of which lacks one the design principles. As seen, ablating any design principle almost always led to performance degradation. These results demonstrate that the proposed design principles are critical to obtaining the maximum possible gain from ALT, and each principle is indispensable.
Table F.8 shows that the LLMs trained with ALT without preventing knowledge forgetting by Recall Adam optimizer underperformed compared to their counterparts trained with knowledge forgetting prevention and even the LLM without ALT. This behavior presumably occurred because the unknown facts included in synthetic logic corpora displaced the LLM's knowledge of existing facts. Therefore, knowledge-forgetting prevention critically important for the success of ALT.
6
What Capabilities Can Additional Logic Training Enhance and Why?
We analyze the results on each benchmark or each case and discuss whether and why the LLM's capabilities to solve the tasks can or cannot be enhanced by ALT.
6.1 Logical Reasoning Tasks
Table 4a shows that ALT substantially boosted LLaMA-3.1M 3..1-1..1.33 performance by up to 30 points on various benchmarks dealing with logical reasoning tasks. Surprisingly. we also observed im- provements on abductive reasoning tasks, which go beyond the original deductive reasoning tasks
pattps: //gtt//btttttt////t wttpiithub..mmmiiie com/hitachi wwwechiinppp/eeeee nlp/rec-edar adam
7

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Table 2:
5-shot performance of LLMS before and after ALT. @ALT-2 denotes the LLM trained with ALT on the synthetic logic corpus x from Table 1. The color shows the rank in each column (darker is better). Each benchmark set, such "Logic" and "Math", comprises various benchmarks in that domain (see Table E.7). "Avg." represents the micro-average of all the benchmarks.
(a) LLaMA-3.1-8B.
Avg.
Logic
Math
Code
NLI
Others
BBH (3-shot)
BBH (0-shot)
MMLU
CoT
CoT
Pro
LLaMA-3, 3.1-88 1B
47.9
_2.8_a
39.64es
35.4
65.4
60.7
44,9
61.9
8.24sa
36.5
65.3
35.8
@ALT-PPP
48.1
43.7
39.2
35.7
65.6gm
60.8
44.9
61.8
8.2
36.4gm
65.3gr
35.3
SALTRT
50.1
46.8
42.4.s:
36.5
68.6.a1
61.3
46.9.o
63.5
13.7
38.4.12
65.3
35.7
@ ALTFFLD
51.6
43.4
20.
26.7
24.9
11.9
39.6
6r.4
26.2
BALT-FLDxa
52.0
52.2.au
43.2
38.0
70.7
61.5
46.5.o
65.3
11.3
38.7
65.5
36.4
(b) LLaMA-3.1-70B.
Avg.
Logic
Math
Code
Others
BBH (3-shot)
BBH (0-shot)
MMLU
CUT
CuT
Pro
LLaMA-3.1
37.4.04
60.0.cs
73.7
67.7
82.1.ea
6.5
50.1
78.7
50.7 ...
SALT-PRP
60.4
57.7
59.8.as
49.2
73.5
67.6
60.4...
82.2.u
6.0
50.1
78.7
50.9...
@ALT-RT
61.4
75.4
68.4
64.1gm
82.5gr
11.5
59.2
79.0
@ALT-FLD
64.2
65.7.m:
63.62c2
52.0
25.
68.5
65.0.s:
83.6.m
12.
59.9
79.3.m1
54.4..>
GALT-FLDDEE
64.4
66.1
63.3
52.4
16.
68.5
65.4.02
83.6.
11.4.00
60.8 oz
79.5.m.
$4.4 .2
Table LLaMA-3.1-8B trained on the ablation corpora.
Avg.
Logic
Math
Code
NLI
Others
BBH (3-shot)
BBH (0-shot)
MMLU
CoT
CoT
Pro
BALT-FLD
52.0
52.2
43.2
38.0
70.7 mm
61.5
46.5
65.3
11.3 au
38.7
65.5
36.4
w/o DPI
51.4
52.2.
43.1
39.2
70.0
59.4.m
46.7.m:
64.7am:
11.5
38.9
65.4
36.l
w/o DP2
50.6
49.9
43.1
38.1
71.1
59.3
16.
46.
64.6
10.4_a:
37.4
65.4gm
35.7
w/o Du3.rules
50.7
50.4g
42.8
38.3
69.5
59.4
46.42s
64.0az
11.8
38.3
65.6
36.22e7
w/o DP3.steps
21.5
43.1
38.7
69.6gm
59.5
46.8.a3
65.0.sa
12.3
38.8
65.6
36.32eo
w/o DP4
51.3
52.2
42.8.aa
38.4
70.3
59.5
16.
64.8.s1
12.8
39.3
65.5
36.3.02
in synthetic logic corpora. Abductive reasoning involves guessing the missing premises that caused the observed conclusion rather than deriving a conclusion from the premises. For example, from the observed conclusion, "the window glass at home was broken and the room ransacked," guess the premise "a burglar broke in." The improvements would be due to the fact that, while the surface form of abductive reasoning problems differs from that of deductive reasoning, they share the fundamentals of logic reflected in the design principles.
Next, conduct case analyses see whether the LLM enhanced by ALT acquired the abilities intended by the proposed design principles (DP1-4). Table shows problems where LLaMA-3.1 70B's errors have been corrected by ALT. The first problem is very simple, it is surprising that LLaMA-3.1 failed to solve it, indicating the inherent difficulty of learning logical reasoning solely from pre-training. In contrast, #ALT-FLDxx: which solved the problem correctly. The premises of the problem are randomly constructed to express unknown facts. Therefore, the result suggests that @ALT-FLD, x2 acquired genuine logical reasoning ability, which can handle unknown facts (DP1).
additionally trained FLD x2+
In the second problem, AALT-FLDD correctly answered "neutral", indicating that it successfully learned that conclusions cannot be derived from insufficient facts (DP2).
The third problem comes from the FOLIO benchmark. To solve this problem, LLMs must use syllogism at the first step as follows: "All eels fish, and fish plants. Therefore, all ells are not plants." @ALT-FLD, x2 answered this problem correctly, suggesting that it successfully learned diverse deduction rules (DP3).
FOLIO problems are created based on Wikipedia topics, describing them in more natural and realistic linguistic expressions than in other benchmarks. As seen in the fourth problem, @ALT-FLD. understands such expressions, suggesting the effect of diverse expressions from DP4 and/or that LLMSs integrate their original linguistic ability with the newly acquired logical reasoning ability.
8

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Table Benchmark- 5-shot performance of LLaMA-3.1-700 before and after ALT on FLDx2- Refer to Table F.9 for LLaMA-3.1 1-BB results. Table E.7 details each benchmark.
(a) Logic.
bAbiD FOLIO LogieNLI RobustLR AR-LSAT LogiQA ReClor AbductionR
ART
LLaMA-3. 3.1--00 83.8-12 58.9
34,9
49 19,6a09
21.5
64.3
33.7
84.0z07 95.7
85,4 85.5
@ALT-FLL
x 83.5 66.7=as
50.9
81.6
25.0
69.4
36.3
(b) Math.
GSM8k
MATH MathQA
CoT
CoT (0-shot)
LLaMA-3.1-27 3.0BBB
80,9
75.2+1a
65 4-.4
23.7
55.0
@ALT-FIL FLF,
83.3
80.4
73.0
24.4
55.4
(c) Code
Eval
MBPP
MBPP+
MultiPL-E (cpp)
MultiPL-E (go)
LLaMA-3,1-708 ALT-FLLDxx
32.3
43.4
48.7
29.8
76.6
42.6
49.5
52.5
38.7
78.6
(d) Natural language inference (NLI).
HELP
MNLI
RTE
SNLI
LLaMA-3.1-70B
45.8
82.2
84.0 87.2 m03
82.6 82.3
ALT-FLDD2s
51.3 102
83.7 1uz
(c) Others.
CommonscnsoQA HellaSwag SQuAD WinoGrande ARCo
ARCc GPQA OpenBookQA
SciQ
LIMMA-3.1-P0 70B
81.2g
69.2
38.5
85.6gie
89. 1on. 65.3g 40.7
41.4g
98.5
GALT-FLD
69.6
40.1
86.1.s4
89.4 66.7 40.6
42.8
v8.5
6.2
Math and Coding Tasks
Tables 4b, 4c shows that ALT substantially boosted the LLaMA-3.1-70F performance by up to and 10 points math and coding tasks, respectively. The math improvements are reasonable, as understanding predicate logic is prerequisite for solving mathematical problems. For coding, some recent studies have verified the opposite direction, namely, that training coding data improves logical reasoning abilities (Jiang et al., 2024b; MA et al., 2024; Uchiyama et al., 2024).
6.3 NLI Tasks
Table 4d shows that ALT substantially boosted the LLaMA-3.1170BB performance by to 6 points on various natural language inference (NLI) benchmarks. NLI is similar deductive reasoning in assessing whether premise supports or contradicts hypothesis. However, the main difference is that this judgment requires rich of commonsense knowledge beyond the given premise.
Consider the fifth problem in Table 5: by supplementing the given fact "An Indian woman dancing with her partner" with the commonsense knowledge "If someone is dancing, then he/she moving.". we can derive the hypothesis "A woman is moving." The sixth problem more challenging as we have trace multiple logical steps while supplementing with sufficient commonsense knowledge as follows: "a church choir sings at church," "baseball is often played at baseball field," "a person cannot be two or more places at the same time." "therefore, a church choir cannot sing for baseball."
Since synthetic logic corpora only contain unknown facts, LLMs cannot acquire new knowledge from them. Therefore, the commonsense knowledge used to solve the above problems must have been acquired by the LLMs from pre-training. This suggests that LLMs can integrate their original knowledge with the logical reasoning capabilities newly acquired from ALT to solve problems.

---

Table 5: Problems where LLaMA-3. 1...MMA-3.11770 3.1--0B initially answered incorrectly and then correctly after training with ALT on FLDx2. Red highlights the premises related to the hypothesis.
benchmark premises
hypotesis
(LLaMA-3.1 3.Dooll)
answer
required ability
Mice afruid of wolves. Cuts afraid sheep.
L
Winoma is woll' Shoep are afraid of cats.
lessica a
are afraid of cats.
Jessica afraid of sheep.
noutral/ entailment
DPI
Rhett is not modest. Vivian is confosed.
Rhett is lay. If someone is modest or not confosed, then he is not eager.
Rhett confased.
estailmens/ neutral
DP2
All cels fish. No fish plants.
POLJO
All animals displayed sea oel is displayed in the collectionn
Everything displayed in collection cither a plant animal.
the collection are multicelllllr.
sea multicellular or is bacteria.
neurra// exaiimmnt
DP3
The col is an cel or animal not plant.
Common utilities include water, electricity, gas, heating, Many apartment rents cover the cost water and clectricity.
trash, and recycling.
Nosh both need to pay the heating bill
Susin lives in an apartment where the remi covers all uiilities
neutral/ enaiilment
DP4
The roni of the mpnttment where Asa lives does not cover any unility expenses.
Nosh lives
aparmmett where the ront does not cover heating.
neutral/ entailment
An Indian woman dancing with her partner.
woman is moving.
reasoning with commonsense knowledge
This church choir sings to the musses as they sing joyous songs from the book a church.
A choir is singing al buseball game.
catailmens/ contrudiction
Table 6: Problems that -LaMA-3.1-708 trained with ALT on FLDx2 x2 still cannot solve.
benchmark question
answer
ARC (challenge)
end result in the process of photosynthesis is the production of sugar and oxygen. Which step signals the beginning of photosynthesis?
Chlorophyll leaf captures light energy.
A spin-half' purtiele is in linear superposition 0.8) f) 0.6] 4 of its spin-up and spin-down states. If| f) and () are the eigenstates oz. then what the expectation value up one decimal place, of the operator 10z, 5a,?
GPQA
-0.7
ARC (challenge)
Beavers build their homes in ponds and streams. Which characteristic least critical to building homes in an aquatic environment?
(A) waterproof fur (B) webbed hind feet
(C) arge, sharp teeth (D) flat, wide tail
6.4 Other Tasks
Improvements across various other tasks (Table 4e) demonstrate the broad benefits of the obtained reasoning capabilities beyond standard reasoning tasks; though the improvements were modest at up to percentage points, which may be due to the following reasons. First, these benchmarks include problems that purely test knowledge, such the first one in Table 6. Since ALT does not aim to provide new knowledge, the ability to solve such problems does not improve by nature. Next, some problems may require knowledge that is too advanced for LLMs, so potential improvements by the enhanced reasoning capabilities may be bottlenecked. For example, the second problem does involve reasoning but requires sufficient quantum mechanics knowledge knowledge-related issues should be solved by improving the quantity and quality of pre-training.
prerequisite. However, these
Finally, LLMs may not be able to fully utilize the potential of enhanced reasoning capabilities for problems that require complex procedures. To solve the third problem, LLMs first must attempt reasoning related to each choice as follows: "To build homes in an aquatic environment, one needs to maintain body heat and insulation despite being frequently submerged in cold water. Therefore, the waterproof fur of (A) is essential", and "To build... ... one must gather and process natural materials like wood. Large, sharp teeth of (C) are critical as they allow beavers to cut down trees and shape branches." Next, while reasoning traces (A) to (D) all seem reasonable, LLMs must choose the single best answer, considering the subtle nuance of the question context, as follows: "Since the question emphasizes the aquatic environment, the least related reasoning trace should be (C)." This complex procedure contrasts with logical reasoning and NLI problems, where LLMs can directly obtain an answer from a single reasoning trace. Previous studies also observed that such procedure on multiple-choice QA problems are challenging for LLMs (Robinson and Wingate, 2023; Zheng et al., 2024; Wang et al., 2024a). Since ALT alone does not teach LLMs such task-specific procedures, additional training on these procedures should be necessary to solve these problems.
7 Conclusion
Towards versatile artificial intelligence with reasoning capabilities, we proposed Additional Logic Training on synthetic logic samples. We established systematic design principles well- rell-grounded on symbolic logic theory and previous empirical findings. We constructed corpus named Formal Logic Deduction Diverse (FLDx2) based on the design principles. We empirically showed that ALT on FLD, x2 substantially enhances the capabilities of -tate-of-the-art LLMs.
10

---

Acknowledgement
Computational resources of AI Bridging Cloud Infrastructure (ABCI) provided by the National Institute of Advanced Industrial Science and Technology (AIST) were used. We thank Dr. Masaaki Shimizu at Hitachi for the convenience of additional computational resources. We thank Dr. Naoaki Okazaki, professor at the Tokyo Institute of Technology, for the keen comments.
References
Al@Meta. 2024. Llama 3 model card.
Aida Amini, Saadia Gabriel, Shanchuan Lin, Rik Koncel-Kedziorski, Yejin Choi, and Hannaneh Hajishirzi. 2019. MathQA: Towards interpretable math word problem solving with operation- based formalisms. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume I (Long and Short Papers), pages 2357-2367, Minneapolis, Minnesota. Association for Computational Linguisties.
Risako Ando, Takanobu Morishita, Hirohiko Abe, Koji Mineshima, and Mitsuhiro Okada. 2023. Evaluating large language models with NeuBAROCO: Syllogistic reasoning ability and human-like biases. In Proceedings of the 4th Natural Logic Meets Machine Learning Workshop, pages 1-11, Nancy, France. Association for Computational Linguistics.
Yoichi Aoki, Keito Kudo, Tatsuki Kuribayashi, Shusaku Sone, Masaya Taniguchi, Keisuke Sakaguchi, and Kentaro Inui. 2024. First heuristic then rational: Dynamic use of heuristics in language model reasoning.
Amanda Askell. 2020. Gpt-3: Towards renaissance models. Daily Nous Blog: Philosophers On GPT-3.
Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, et al. 2021. Program synthesis with large language models. Xrxiv preprint arXiv:2108.011
Qiming Bao, Alex Yuxuan Peng. Tim Hartill, Neset Tan, Zhenyun Deng. Michael Witbrock, and natural language: An empirical study
Jiamou Liu. 2022. Multi-step deductive reasoning out-of- distribution generalisation. In Proceedings of the 16th International Workshop on Neural- Symbolic Learning and Reasoning as part of the 2nd International Joint Conference on Learning & Reasoning (IJCLR 2022), pages 202-217, Cumberland Lodge, Windsor Great Park, United Kingdom.
Loubna Ben Allal, Anton Lozhkov, Guilherme Penedo, Thomas Wolf, and Leandro von Werra. 2024. Cosmopedia.
Loubna Ben Allal, Niklas Muennighoff, Logesh Kumar Umapathi, Ben Lipkin, and Leandro von Werra. 2022. A framework for the evaluation of code generation models. https:/ http:/iggode-proooo
/bigcode evaluation harness.
Luisa Bentivogli, Ido Dagan, Hoa Trang Dang, Danilo Giampiccolo, and Bernardo Magnini. 2009. The fifth pascal recognizing textual entailment challenge. In Text Analysis Conference.
Leonardo Bertolazzi, Albert Gatt, and Raffaella Bernardi. 2024. systematic analysis of large language models as soft reasoners: The case of syllogistic inferences.
Russell Bertrand. A history of western philosophy.
Gregor Betz, Christian Voigt, and Kyle Richardson. 2021. Critical thinking for language models. In Proceedings of the 14th International Conference on Computational Semantics (/WCS), pages 63-75, Groningen, The Netherlands (online). Association for Computational Linguistics.
Chandra Bhagavatula, Ronan Le Bras, Chaitanya Malaviya, Keisuke Sakaguchi, Ari Holtzman, Han- nah Rashkin, Doug Downey, Scott Wen-tau Yih, and Yejin Choi. 2019. Abductive commonsense reasoning, arXiv preprint 1rXiv:1908.05777 05739.

---

Neeladri Bhuiya, Viktor Schlegel, and Stefan Winkler. 2024. Seemingly plausible distractors in multi-hop reasoning: Are large language models attentive readers?
Kaj Bostrom, Xinyu Zhao, Swarat Chaudhuri, and Greg Durrett. 2021. Flexible generation of natural language deductions. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 6266-6278 Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
Samuel R Bowman, Gabor Angeli, Christopher Potts, and Christopher D Manning. A large annotated corpus for learning natural language inference.
Federico Cassano, John Gouwar, Daniel Nguyen, Sydney Nguyen, Luna Phipps-Costin, Donald Pinckney, Ming-Ho Yee, Yangtian Zi, Carolyn Jane Anderson, Molly Q Feldman, Arjun Guha, Michael Greenberg, and Abhinav Jangda. 2023. Multipl-e: A scalable and polyglot approach to benchmarking neural code generation. IEEE Transactions on Software Engineering, 49(7):3675- 3691.
Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel Herbert- Voss. William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin. Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojeiech Zaremba. 2021. Evaluating large language models trained on code.
Sanyuan Chen, Yutai Hou, Yiming Cui, Wanxiang Che, Ting Liu, and Xiangzhan Yu. 2020. Recall and learn: Fine-tuning deep pretrained language models with less forgetting, In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1870-7881, Online. Association for Computational Linguistics.
Xinyun Chen, Ryan Andrew Chi, Xuezhi Wang, and Denny Zhou. 2024. Premise order matters in reasoning with large language models. In Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 6596-6620. PMLR.
Cheng, M. Bernstein, C. Danesca-Nieulescu-Mini and J. Leskovec. 2017. Anyone can become troll: Causes of trolling behavior in online discussions. CSCW: Proceedings of the Conference on Computer-Supporree Cooperative Work. Conference on Computer-Supporter Cooperative Work, 2017.
Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and Oyvind Tafjord. 2018. Think you have solved question answering? try are, the ai2 reasoning challenge. arXiv preprint arXiv:1803.05457.
Peter Clark, Oyvind Tafjord, and Kyle Richardson. 2021. Transformers as soft reasoners over language. In Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence, pages 3882-3890.
Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. 2021. Training verifiers to solve math word problems. arXiv preprint arXiv:21 14.166.
A. Colmerauer and P Roussel. 1973. The birth of prolog. The ALP Newsletter.
Ido Dagan, Oren Glickman, and Bernardo Magnini. 2005. The PASCAL. recognising textual entail- ment challenge. pages 177-190.
12

---

Bhavana Dalvi, Peter Jansen, Oyvind Tafjord, Zhengnan Xie, Hannah Smith, Leighanna Pi- patanangkura, and Peter Clark. 2021. Explaining answers with entailment trees. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 7358 -770, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
Ishita Dasgupta, Andrew K. Lampinen, Stephanie C. Y. Chan, Hannah R. Sheahan, Antonia Creswell, Dharshan Kumaran, James I., McClelland, and Felix Hill. 2023. Language models show human-like content effects on reasoning tasks.
John Dougrez-Lewis, Mahmud Elahi Akhter, Yulan He, and Maria Liakata. 2024. Assessing the reasoning abilities of chatgpt in the context of claim verification.
Nouha Dziri, Ximing Lu, Melanie Sclar, Xiang Lorraine Li, Liwei Jiang, Bill Yuchen Lin, Peter West, Chandra Bhagavatula, Ronan Le Bras, Jena D. Hwang, Soumya Sanyal, Sean Welleck, Xiang Ren, Allyson Ettinger, Zaid Harchaoui, and Yejin Choi. 2023. Faith and fate: Limits of transformers on compositionality.
Tiwalayo Eisape, Michael Tessler, Ishita Dasgupta, Fei Sha, Sjoerd Steenkiste, and Tal Linzen. 2024. A systematic comparison of syllogistic reasoning in humans and language models. In Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), pages 8425-8444, Mexico City, Mexico. Association for Computational Linguistics.
Charles Elkan and Russell Greiner. 1993, Building large knowledge-based systems: Representation and inference in the сyс project: Db lenat and rv guha.
Leo Gao, Jonathan Tow, Baber Abbasi, Stella Biderman, Sid Black, Anthony DiPofi, Charles Foster, Laurence Golding, Jeffrey Hsu, Alain Le Noac'h, h, Haonan Li, Kyle McDonell, Niklas Muennighoff, Chris Ociepa, Jason Phang. Laria Reynolds, Hailey Schoelkopf, Aviya Skowron, Lintang Sutawika, Eric Tang, Anish Thite, Ben Wang, Kevin Wang, and Andy Zou. 2023. A framework for few-shot language model evaluation.
Danilo Giampiccolo, Bernardo Magnini, Ido Dagan, and William B Dolan. 2007. The third PASCAL recognizing textual entailment challenge. In ACL-PACALLL Workshop on Textual Entailment and Paraphrasing, pages 1-9.
Kurt Gödel. 1930. Uber die vollständigkeit des logikkalkiils. Ph.D. thesis, Ph. D. dissertation, University of Vienna.
Nicolas Gontier, Koustuv Sinha, Siva Reddy, and Chris Pal. 2020. Measuring systematic generaliza- tion in neural proof generation with transformers. Advances in Neural Information Processing Systems, 33:22231-22242.
Nelson Goodman. 1954. Fact, fiction, and forecast. london: University of london.
Radu Comel Guiaşu and Christopher W Tindale. 2018. Logical fallacies and invasion biology. Biology philosophy, 33(5-6):34.
Ivan Habernal, Henning Wachsmuth, Iryna Gurevych, and Benno Stein. 2018. The argument reasoning comprehension task: Identification and reconstruction of implicit warrants. Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Techmologies, Volume I (Long Papers). pages 1930-1940, New Orleans, Louisiana. Association for Computational Linguistics.
Pengrui Han, Peiyang Song, Haofei Yu, and Jiaxuan You. 2024. In-context learning may not clicit trustworthy reasoning: A-not-b errors in pretrained language models.
Simeng Han, Hailey Schoelkopf, Yilun Zhao, Zhenting Qi, Martin Riddell, Luke Benson, Lucy Sun, Ekaterina Zubova, Yujic Qiao, Matthew Burtell, et al. 2022. Folio: Natural language reasoning with first-order logic. arXiv e-prints, pages arXiv-2209.
Sven Ove Hansson. 2004. Fallacies of risk. Journal of Risk Research, 7(3)3533-60.
13

---

Dan Hendrycks, Collin Burns, Steven Basart. Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt. 2021a. Measuring massive multitask language understanding. Proceedings of the International Conference Learning Representations (ICIR).
Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. 2021b. Measuring mathematical problem solving with the math dataset. NeurlPS.
Namgyu Ho, Laura Schmid, and Se- Ye-Yo Yun. 2023. Large language models are reasoning teachers.
Damian Hodel and Jevin West. 2023, Response: Emergent analogical reasoning in large language models.
Ruixin Hong, Hongming Zhang, Xinyu Pang, Dong Yu, and Changshui Zhang. 2024. A closer look at the self-verification abilities of large language models in logical reasoning. In Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguisties: Human Language Technologies (Volume 1: Long Papers), pages 900-99 920, Mexico City, Mexico. Association for Computational Linguistics.
Peng Hu, Changjiang Gao, Ruiqi Gao, Jiajun Chen, and Shujian Huang. 2024. Large language models are limited in out-of-context knowledge reasoning.
Jie Huang, Xinyun Chen, Swaroop Mishra, Huaixiu Steven Zheng, Adams Wei Yu, Xinying Song, and Denny Zhou. 2024. Large language models cannot self-correct reasoning yet. In The Twelfth International Conference Learning Representations.
David Hume. 1748. An enquiry concerning human understanding (section iv). Recuperado de http://www. clorenzano, com, ar.
Bowen Jiang, Yangxinyu Xie. Zhuoqun Hao, Xiaomeng Wang, Tanwi Mallick, Weijie Su, Camillo J. Taylor, and Dan Roth. 2024a. A peek into token bias: Large language models are not yet genuine reasoners.
Jin Jiang, Yuchen Yan, Yang Liu, Yonggang Jin, Shuai Peng, Mengdi Zhang, Xunliang Cai, Yixin Cao. Liangeai Gao, and Zhi Tang. 2024b. Logicpro: Improving complex logical reasoning via program-guided learning.
Daniel Kahneman. 2011. Thinking, fast and slow. Macmillan.
Tamera Lanham, Anna Chen, Ansh Radhakrishnan, Benoit Steiner, Carson Denison, Danny Her- nandez, Dustin Li, Esin Durmus, Evan Hubinger, Jackson Kernion, Kamilé Lukošiūté, Karina Nguyen, Newton Cheng, Nicholas Joseph, Nicholas Schiefer, Oliver Rausch, Robin Larson, Sam McCandlish, Sandipan Kundu, Saurav Kadavath, Shannon Yang. Thomas Henighan, Timothy Maxwell, Timothy Telleen-Lawton, Tristan Hume, Zac Hatfield-Dodds Jared Kaplan, Jan Brauner, Samuel R. Bowman, and Ethan Perez. 2023. Measuring faithfulness in chain-of of-thought reasoning.
Liunian Harold Li, Jack Hessel, Youngjae Yu, Xiang Ren, Lai-Wei Chang, and Yejin Choi. 2023. Symbolic chain-of thoough distillation: Small models can also "think" step-by-step. In Proceedings of the s1 Annual Meeting of the Association for Computational Linguistics (Volume I: Long Papers), pages 2665-2679, Toronto, Canada. Association for Computational Linguistics.
Shiyang Li, Jianshu Chen, Yelong Shen, Zhiyu Chen, Xinlu Zhang. Zekun Li, Hong Wang, Jing Qian, Baolin Peng, Yi Mao, Wenhu Chen, and Xifeng Yan. 2022. Explanations from large language models make small reasoners better.
Hanmeng Liu, Jian Liu, Leyang Cui, Zhiyang Teng, Nan Duan, Ming Zhou, and Yue Zhang. 2023a. Logiqa 2.0-an improved dataset for logical reasoning in natural language understanding. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 31:2947-2962.
Hanmeng Liu, Ruoxi Ning, Zhiyang Teng, Jian Liu, Qiji Zhou, and Yue Zhang. 2023b. Evaluating the logical reasoning ability of chatgpt and gpt-4.
Hanmeng Liu, Zhiyang Teng, Leyang Cui, Chaoli Zhang, Qiji Zhou, and Yue Zhang. 2023c. LogiCoT: Logical ffain-of-thoughh instruction tuning. In Findings of the Association for Computational Lin- guistics: EMNLP 2023, pages 2908-2924 Singapore. Association for Computational Linguistics.
14

---

Jian Liu, Leyang Cui, Hanmeng Liu, Dandan Huang, Yile Wang, and Yue Zhang. 2020. Logiqa: A challenge dataset for machine reading comprehension with logical reasoning. In Proceedings of the Twenty-Ninth Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, pages 8622-3628. International Joint Conferences on Artificial Intelligence Organization. Main track.
Jiawei Liu, Chunqiu Steven Xia, Yuyao Wang, and Lingming Zhang. 2023d. Is your code generated by chatGPT really correct? rigorous evaluation of large language models for code generation. In Thirty-seventh Conference on Neural Information Processing Systems.
Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. RoBERTa: A robustly optimized BERT pretraining approach. arXiv preprint arXiv: 1692.
Ziyi Liu, Isabelle Lee, Yongkang Du, Soumya Sanyal, and Jieyu Zhao. 2024. Self-contradictryy reasoning evaluation and detection.
Zimu Lu, Aojun Zhou, Houxing Ren, Ke Wang, Weikang Shi, Junting Pan, Mingjie Zhan, and Hongsheng Li. 2024. Mathgenie: Generating synthetic data with question back-translation for enhancing mathematical reasoning of Ilms.
YINGWEI MA, Yue Liu, Yue Yu, Yuanliang Zhang, Yu Jiang, Changjian Wang, and Shanshan Li. 2024. At which training stage does code data help LLMs reasoning? In The Twelfth International Conference on Learning Representations.
Lucie Charlotte Magister, Jonathan Mallinson, Jakub Adamek, Eric Malmi, and Aliaksei Severyn. 2023. Teaching small language models to reason. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pages 1773-1781, Toronto, Canada. Association for Computational Linguistics.
John W. McCarthy, 1959. Programs with common sense. In Proc. Tedding Conf. on the Mechanization of Thought Processes, pages 75-91.
Todor Mihaylov, Peter Clark, Tushar Khot, and Ashish Sabharwal. 2018. Can suit of armor conduct electricity? a new dataset for open book question answering. In EMNLP.
George A Miller. 1995. 38(11):39-41
Wordnet: a lexical database for english. Commumications of the ACM,
Iman Mirzadeh, Keivan Alizadeh, Hooman Shahrokhi, Oncel Tuzel, Samy Bengio, and Mchrdad Farajtabar. 2024. Gsm-symbolic: Understanding the limitations of mathematical reasoning in large language models.
Melanie Mitchell, 2023. Can large language models reason? https://aggidde.substtkk htt.://iidd..uuusttcc.o/o///ppee Lareea--laag--aagugg---oooo wowpotage-oooll---eeeeee
blog, pages
Arindam Mitra, Luciano Del Corro, Shweti Mahajan, Andres Codas, Clarisse Simoes, Sahaj Agarwal, Xuxi Chen, Anastasia Razdaibiedina, Erik Jones, Kriti Aggarwal, Hamid Palangi, Guoqing Zheng, Corby Rosset, Hamed Khanpour, and Ahmed Awadallah. 2023. Orca 2: Teaching small language models how reason.
Philipp Mondorf and Barbara Plank. 2024. Liar, liar, logical mire: A benchmark for suppositional reasoning in large language models.
Terufumi Morishita, Gaku Morio, Atsuki Yamaguchi, and Yasuhiro Sogawa. 2023. Learning deductive reasoning from synthetic corpus based on formal logic. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 25254-25274. PMLR.
Terufumi Morishita, Atsuki Yamaguchi, Gaku Morio, Hikaru Tomonari, Osamu Imaichi, and Yasuhiro Sogawa. 2024. JFLD: A Japanese benchmark for deductive reasoning based on formal logic. In Proceedings of the 2024 Joint International Conference Computational Linguistics, Language Resources and Evaluation (I.RECCCOLING 2024), pages 9526-9535, Torino, Italia. EL R.R and ICCL.
15

---

Aliakbar Nafar, K. Brent Venable, and Parisa Kordjamshidi. 2024. Teaching probabilistic logical reasoning transformers. In Findings of the Association for Computational Linguisties: EACL 2024, pages 1615-1632, St. Julian's, 8, Malta. Association for Computational Linguistics.
Kentaro Ozeki, Risako Ando, Takanobu Morishita, Hirohiko Abe, Koji Mineshima, and Mitsuhiro Okada. 2024, Exploring reasoning biases in large language models through syllogism: Insights from the NeuBAROCO dataset. In Findings of the Association for Computaaionnal Linguistics ACL 2024, pages 16063-16077, Bangkok, Thailand and virtual meeting. Association for Computational Linguisties.
Fabio Paglieri. 2017. A plea for ecological argument technologies. Philosophy & Technology. 30(2):209-25
Mihir Parmar, Nisarg Patel, Neeraj Varshney, Mutsumi Nakamura, Man Luo, Santosh Mashetty, Arindam Mitra, and Chitta Baral. 2024. LogicBench: Towards systematic evaluation of logical reasoning ability of large language models. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume Long Papers), pages 13679-13707, Bangkok, Thailand, Association for Computational Linguistics.
Nisarg Patel. Mohith Kulkarni, Mihir Parmar, Aashna Budhiraja, Mutsumi Nakamura, Neeraj Varshney, and Chitta Baral. 2024. Multi-logieval: Towards evaluating multi-step logical reasoning ability of large language models.
Xinyu Pi, Qian Liu, Bei Chen, Morteza Ziyadi, Zeqi Lin, Qiang Fu, Yan Gao, fian-Guang Lou, and Weizhu Chen. 2022. Reasoning like program executors. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, pages 761-779, Abu Dhabi, United Arab Emirates. Association for Computational Linguisties.
Willard Van Orman Quine. 1969. Epistemology naturalized. ontological relativity and other essays. New York: Columbia UP.
Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019. Language models are unsupervised multitask learners.
Jack W Rae, Sebastian Borgeaud, Trevor Cai, Katie Millican, Jordan Hoffmann, Francis Song, John Aslanides, Sarah Henderson, Roman Ring, Susannah Young, al. 2021. Scaling language models: Methods, analysis & insights from training gopher. arXiv preprint arXiv: 12. 1446..
Pranav Rajpurkar, Robin Jia, and Percy Liang. 2018. Know what you don't 't know: Unanswerable questions for SQuAD. In Proceedings of the 56th Annual Meeting of the Association for Computa- fional Linguistics (Volume 2: Short Papers), pages 784-789, Melbourne, Australia. Association for Computational Linguistics.
Yasaman Razeghi, Robert L Logan IV, Matt Gardner, and Sameer Singh. 2022. Impact of pre- training term frequencies few-shot numerical reasoning. In Findings of the Association for Computational Linguistics: EMNLP 2022, pages 840-854.
David Rein, Betty Li Hou, Asa Cooper Stickland. Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R Bowman. 2023. Gpqa: A graduate-level google-proof q&a benchmark. arXiv preprint arXiv:2311.12022.
Joshua Robinson and David Wingate. 2023. Leveraging large language models for multiple choice question answering. In The Eleventh International Conference on Learning Representations.
Mohammed Saeed, Naser Ahmadi, Preslav Nakov, and Paolo Papotti. 2021. RuleBERT: Teaching soft rules to pre-trained language models. In Proceedings of the 2021 I Conference on Empirical Methods in Natural Language Processing, pages 1460-1476, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
Swarnadeep Saha, Sayan Ghosh, Shashank Srivastava, and Mohit Bansal. 2020. PRover: Proof generation for interpretable reasoning over rules. In Proceedings of the 2020 Conference on Empirical Methods Natural anguage Processing (EMNLP), pages 122-136, Online. Association for Computational Linguistics.
16

---

Keisuke Sakaguchi, Ronan Le Bras, Chandra Bhagavatula, and Yejin Choi. 2021. Winogrande: An adversarial winograd schema challenge at scale. Communications of the ACM. 64(9):99-106.
Soumya Sanyal, Zeyi Liao, and Xiang Ren. 2022a. Robustlr: Evaluating robustness to logical perturbation in deductive reasoning. arXiv preprint 1rXiv:2205./2598
Soumya Sanyal, Harman Singh, and Xiang Ren. 2022b. Fairr: Faithful and robust deductive reasoning over natural language. In Proceedings of the 60th Anmual Meeting of the Association for Computational Linguistics (Volume I: Long Papers), pages 1075-1093
eh Shortliffe. 1976. Computer based medical consultations: Mycin. Elsevier.
Kumar Shridhar, Alessandro Stolfo, and Mrinmaya Sachan. 2023. Distilling reasoning capabilities into smaller language models. In Findings of the Association for Computational Linguisties: ACL 2023, pages 7059-7073, Toronto, Canada, Association for Computational Linguistics.
Damien Sileo. 2024. Scaling synthetic logical reasoning datasets with context-sensitive declarative grammars.
Zayne Rea Sprague, Xi Ye, Kaj Bostrom, Swarat Chaudhuri, and Greg Durrett. 2024. MuSR: Testing the limits of chain-of-thought with multistep soft reasoning. In The Twelfth International Conference on Learning Representations.
Cass Sunstein and Reid Hastie. 2015. Wiser: getting beyond groupthink to make groups smarter. Harvard Business Review Press, Boston.
Mirac Suzgun, Nathan Scales, Nathanael Schärli, Sebastian Gehrmann, Yi Tay, Hyung Won Chung, Aakanksha Chowdhery, Quoe V Le, Ed Chi, Denny Zhou, and Jason Wei. 2022. Challenging big-bench tasks and whether chain-of-thought can solve them. arXiv preprint arXiv:2210.09261.
Oyvind Tafjord, Bhavana Dalvi, and Peter Clark. 2021. ProofWriter: Generating implications, proofs, and abductive statements over natural language. In Findings of the Association for Computa- fional Linguistics: ACL-IJCNLP 2021, pages 3621-3634, Online. Association for Computational Linguisties.
Alon Talmor, Jonathan Herzig, Nicholas Lourie, and Jonathan Berant. 2018. Common- senseqa: question answering challenge targeting commonsense knowledge. arXiv preprint 1rXiv:1811.00933
Jidong Tian, Yitian Li, Wenqing Chen, Liqiang Xiao, Hao He, and Yaohui Jin. 2021. Diagnosing the first-order logical reasoning ability through logienli. Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 3738-3747.
Trieu H Trinh, Yuhuai Wu, Quoe V Le, He He, and Thang Luong. 2024. Solving olympiad geometry without human demonstrations. Nature, 625(7995):476-4482
Miles Turpin, Julian Michael, Ethan Perez, and Samuel R. Bowman. 2023. Language models don't 't always say what they think: Unfaithful explanations in chain-of of-in-of-thuhhh prompting.
Fumiya Uchiyama, Takeshi Kojima, Andrew Gambardella, Qi Cao, Yusuke Iwasawa, and Yutaka Matsuo. 2024. Which programming language and what features at pre-training stage affect downstream logical inference performance?
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, t. ukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In Advances in Neural Information Processing Systems, volume 30.
Yuxuan Wan, Wenxuan Wang. Yiliu Yang. Youliang Yuan, Jen tse Huang. Pinjia He, Wenxiang Jiao, and Michael R. Lyu. 2024. Logicasker: Evaluating and improving the logical reasoning ability of large language models.
Haochun Wang, Sendong Zhao, Zewen Qiang, Bing Qin, and Ting Liu. 2024a. Beyond the answers: Reviewing the rationality of multiple choice question answering for the evaluation of large language models.
17

---

System Peifeng Wang, Zhengyang Wang, Zheng Li, Yifan Gao, Bing Yin, and Xiang Ren. 2023. SCOTT: Self-consistent chain-of-thoughe oh-thuught distillation. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguisties (Volume I: Long Pupers), pages Canada. Association for Computational Linguistics.
Toronto,
Siyuan Wang, Zhongyu Wei, Yejin Choi, and Xiang Ren. 2024b. Can LLMs reason with rules? logic scaffolding for stress-testing and improving LLMs. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume I: Long Papers), pages 7523-7543, Bangkok, Thailand. Association for Computational Linguistics.
Yubo Wang, Xueguang Ma, Ge Zhang. Yuansheng Ni, Abhranil Chandra, Shiguang Guo, Weiming Ren, Aaran Arulraj, Xuan He, Ziyan Jiang. Tianle Li, Max Ku, Kai Wang, Alex Zhuang, Rongqi Fan, Xiang Yue, and Wenhu Chen. 2024c. Mmlu-pro: more robust and challenging multi-task language understanding benchmark (published at neurips 2024 track datasets and benchmarks).
Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, brian ichter, Fei Xia, Ed H. Chi, Quoc V Le, and Denny Zhou. 2022. Chain of thought prompting elicits reasoning in large language models. In Advances in Neural Information Processing Systems.
Joseph Weizenbaum. 1966. Eliza-a a computer program for the study of natural language communi- cation between man and machine. Communications of the ACM, 31):36-45.
Johannes Welbl, Nelson F. Liu, and Matt Gardner. 2017. Crowdsourcing multiple choice science questions. In Proceedings of the 3rd Workshop Noisy User-generated Text, pages 94-106, Copenhagen, Denmark. Association for Computational Linguistics.
Jason Weston, Antoine Bordes, Sumit Chopra, Alexander M Rush, Bart Van Merriënboer, Armand Joulin, and Tomas Mikolov, 2015. Towards ai-complete question answering: A set of prerequisite toy tasks. arXiv preprint 100iv:1502.05669
Adina Williams, Nikita Nangia, and Samuel R Bowman. 2018. A broad-coverage challenge corpus for sentence understanding through inference. In Proceedings of NAACL-IILT, pages 1112-1122.
T Winograd. 1971.
Procedures as representation for data in computer program for understanding natural language, mit ai technical report 235.
Ludwig Wittgenstein. 1922. Tractatus Logico Philosophicus: Logical- Philloophhil Treatise. Really Simple Media.
Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020. Transformers: State-of-the-art natural language processing. In Empirical Methods in Natural Language Processing: Demonstrations, pages 45.
Zhaofeng Wu, Linlu Qiu, Alexis Ross, Ekin Akyürek, Boyuan Chen, Bailin Wang, Najoung Kim, Jacob Andreas, and Yoon Kim. 2023. Reasoning or reciting? exploring the capabilities and limitations of language models through counterfactual tasks.
Hitomi Yanaka, Koji Mineshima, Daisuke Bekki, Kentaro Inui, Satoshi Sekine, Lasha Abzianidze, and Johan Bos. 2019. Help: A dataset for identifying shortcomings of neural models in monotonicity reasoning. arXiv preprint 194.v:190..12166.
Nathan Young, Qiming Bao, Joshua Bensemann, and Michael Witbrock. 2022. Abductionrules: Training transformers to explain unexpected inputs. In Findings of the Association for Computa- tional Linguistics: ACL 2022, pages 218-221 227.
Weihao Yu, Zihang Jiang, Yanfei Dong, and Jiashi Feng. 2020. Reclor: A reading comprehension dataset requiring logical reasoning. In International Conference on Learning Representations (ICLR).
18

---

Zhangdie Yuan, Songbo Hu, Ivan Vulić, Anna Korhonen, and Zaiqiao Meng. 2023. Can pretrained language models (yet) reason deductively? In Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics, pages 1439-1454.
Rowan Zellers, Ari Holtzman, Yonatan Bisk, Ali Farhadi, and Yejin Choi. 2019. Hellaswag: Can machine really finish your sentence? In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 4791-4800
Honghua Zhang, Liunian Harold Li, Tao Meng, Kai-Wei Chang, and Guy Van den Broeck. 2022. On the paradox of learning to reason from data.
Hugh Zhang, Jeff Da, Dean Lee, Vaughn Robinson, Catherine Wu, Will Song, Tiffany Zhao, Pranav Raja, Dylan Slack, Qin Lyu, Sean Hendryx, Russell Kaplan, Michele Lunati, and Summer Yue. 2024. A careful examination of large language model performance on grade school arithmetic.
Jun Zhao, Jingqi Tong, Yurong Mou, Ming Zhang, Qi Zhang, and Xuanjing Huang. 2024a. Exploring the compositional deficiency of large language models in mathematical reasoning.
Wenting Zhao, Justin Chiu, Jena Hwang, Faeze Brahman, Jack Hessel, Sanjiban Choudhury, Yejin Choi, Xiang Li, and Alane Suhr. 2024b. UNcommonsense reasoning: Abductive reasoning about uncommon situations. In Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume I: Long Papers), pages 8487-8505, Mexico City, Mexico. Association for Computational Linguistics.
Chujie Zheng, Hao Zhou, Fandong Meng, Jie Zhou, and Minlie Huang. 2024. Large language models are not robust multiple choice selectors. In The Twelfth International Conference on Learning Representations.
Wanjun Zhong, Siyuan Wang, Duyu Tang, Zenan Xu, Daya Guo, Jiahai Wang, Jian Yin, Ming Zhou, and Nan Duan. 2021. Ar-lsat: Investigating analytical reasoning of text. arXiv preprint arXiv:2/04.06598
Jin Peng Zhou, Charles E Staats, Wenda Li, Christian Szegedy, Kilian Q Weinberger, and Yuhuai Wu. 2024a. Don't 't trust: Verify grounding LLM quantitative reasoning with autoformalization. In The Twelfth International Conference Learning Representations.
Yue Zhou, Yada Zhu, Diego Antognini, Yoon Kim, and Yang Zhang. 2024b. Paraphrase and solve: Exploring and exploiting the impact of surface form on mathematical reasoning in large language models. In Proceedings of the 2024 Conference of the North American Chupter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), pages 2793-2804 Mexico City, Mexico. Association for Computational Linguistics.
Kaijie Zhu, Jiaao Chen, Jindong Wang, Neil Zhenqiiang Gong, Diyi Yang, and Xing Xie. 2024. Dyval: Dynamic evaluation of large language models for reasoning tasks. In The Twelfth International Conference on Learning Representatios.
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A Related Work
A.1 Investigation of Reasoning Capabilities of LLMs
Many studies examine LLMs' reasoning capabilities (Askell, 2020; Rac et al., 2021; Razeghi et al., 2022; Liu et al., 2023b; Turpin et al., 2023; Lanham et al., 2023; Wu et al., 2023; Hodel and West, 2023; Dziri et al., 2023; Dasgupta ct al., 2023). Patel et al. (2024) observed LLMs' performance significantly declines as reasoning steps increase in multi-step logical reasoning tasks. Dougrez- Lewis et al. (2024) revealed ChatGPT struggles with abductive reasoning when verifying claims by decomposing their evidence into atomic reasoning steps. Wang al. (2024b) found that GeT-series models showed significant gaps compared to humans in dealing with inference rules. Parmar et al. (2024) introduced LogicBench and showed that existing LLMS struggle with instances involving complex reasoning and negations. Wan et al. (2024) introduced LogicAsker, which assesses whether LLMs can employ set of atomic reasoning skills grounded in propositional and predicate logic and found significant gaps in LLMs' learning of logical rules. Bhuiya et al. (2024) proposed a challenging multi-hop reasoning benchmark with seemingly plausible but incorrect multi-hop reasoning chains and found that statc-of-the art LLMs' capabilities to perform multi-hop reasoning is affected by such chains. Mondorf and Plank (2024) introduced TruthQuest, which assesses LLMs' capabilities to conduct suppositional reasoning, i.e., reasoning where each statement can be false, and found that LLMs exhibit significant difficulties solving these tasks. Sprague al. (2024) introduced a complex multi-step reasoning benchmark, MuSR, and characterized the gaps that remain for techniques like chain-of-thought to perform robust reasoning.
Biases and Errors Ando et al. (2023); Ozeki et al. (2024); Bertolazzi et al. (2024); Eisape et al. (2024) found that LLMs exhibit human-like reasoning biases in syllogistic arguments. Jiang et al. (2024a) found that LLMs exibit "token-biases in solving logical reasoning problems. Aoki et al. (2024) revealed that LMs rely heavily on heuristics, such as lexical overlap. in the earlier stages of reasoning. Zhao et al. (2024a) constructed a MATUTRAP with carefully designed logical traps into the problem descriptions of MATH and GSM8K and found that while LLMs possess the knowledge required to solve these traps. they do not spontaneously use such knowledge them to handle the problems. Han et al. (2024) found that LLMs exhibit A-Not-B errors similar to human infants, failing to suppress the previously established response pattern during ICL. Liu et al. (2024) found that LLMs often contradict themselves in reasoning tasks involving contextual information understanding or commonsense. Zhou et al. (2024b) found that subtle alterations in the surface form can significantly impact the answer distribution, suggesting that LLMs solve reasoning problems using surface cues. Chen et al. (2024) found that the reasoning performance of LLMs is affected by the order of the premises. Hong et al. (2024); Huang et al. (2024) found that LLMs struggle to identify fallacious reasoning steps accurately, suggesting challenges in self-verification methods.
Reasoning in Unknown Situation Zhao al. (2024b) found that LLMs struggle with reasoning in uncommon situations. Zhu et al. (2024) introduced a framework to dynamically generate reasoning samples, and LLMs perform worse those samples. Hu et al. (2024) found that while LLMs can conduct reasoning when relevant knowledge is given in context, they are not proficient at reasoning with knowledge embedded in the training data.
A.2 Synthetic Logic Corpus for Training LLMs
RuleTaker Clark et al. (2021) proposed a deduction corpus composed of synthetically generated mul- tistep deductive proofs written in natural languages. Each deductive proof (dis-)proves a hypothesis by applying deduction rules multiple times to a given set of facts. They showed that Transformer Vaswani et al. (2017) LMs can solve these problems in the sense that they can predict the final answer (i.e., "proved", "disproved", or "unknown") of each deductive proof given the fact set. Later studies Saha et al. (2020); Dalvi et al. (2021); Tafjord et al. (2021); Sanyal et al. (2022b) showed that generative LMs can generate even the intermediate proofs as well as the final answer. Later studies (Saha et al., 2020; Dalvi et al., 2021; Tafjord et al., 2021; Sanyal et al., 2022b) showed that T5 can generate even the intermediate logical steps as well as the final answer.
PARARULE-PAE (Bao et al., 2022) is the enhanced version of PARARULE (Clark et al., 2021), variation of RuleTaker, that includes more samples and more logical steps. RoBERTa (Liu et al., 2019) trained on PARARULE- Plus outperformed the models trained on RuleTaker.
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Artificial Argument Corpus (Betz et al., 2021) includes single-step deductive reasoning samples constructed from send-selected deduction rules useful for critical thinking. They showed that the GPT-2 (Radford et al., 2019) trained on this corpus can generalize to solve NLI tasks. However, at the same time, they found that the LMM does not generalize well to solve more challenging reasoning tasks such as ARC (Habernal et al., 2018) and LogiQA (Liu et al., 2020).
FLD by Morishita et al. (2023, 2024) is the first synthetic logic corpus based on formal logic theory. It includes multistep deductive reasoning samples constructed from the axioms of first-order predicate logic, which can express any deduction rule due the completeness theorem. Due to this nature, T5 trained on FLD generalizes most effectively to other synthetic logic corpora, compared to models trained on other corpora.
Gontier et al. (2020) investigated the deductive reasoning capabilities of LMs on a corpus composed of a specific type of multistep inference, kinship relationships on synthetic kinship graphs. They found that LMS can solve this task when there are relatively few proof steps, but it is difficult for them to generalize solve proof steps longer than those shown in training data. Bostrom al. (2021) studied how to create realistic natural language expressions that represent deduction rules. To this end, they scraped sentences from Wikipedia using a template-basee They showed that training on this corpus helps solve real-world deductive reasoning problems such as EntailmentBank (Dalvi et al., 2021). Pi al. (2022) used synthetic data from program executors. most notably SQL programs. They verified that this data can enhance numerical reasoning, logical reasoning, and multi-hop reasoning abilities. Trinh et al. (2024) generated 100 million geometry problems and verified that the capability of artificial intelligence can be enhanced to to pass the bronze medal threshold of the International Mathematics Olympiad. Saeed et al. (2021); Nafar et (2024) created soft reasoning rules involving with probabilistic logic, instead of hard-logic examined by the aformentioned studies. Sileo (2024) introduced a simpler and more general declarative framework for synthetic generation, and verified its effectiveness. Zhou et al. (2024a) synthetically generated large dataset of mathematics, and gained over 12 points on GSM8K.
method and paraphrased them.
While these studies partly examined the effect of synthetic logic corpora, whether this approach is promising remains an open question. It has been unexplored whether the capabilities obtained from synthetic logic corpora generalizes to solve various tasks beyond the original tasks in these corpora. Additionally, the effect of these corpora has only been examined for small L.S trained on small pre-training corpora such as T5 and RoBERTa; it has been highly questionable whether they can still benefit state-of-thh-an LLMs trained on huge pre-training corpus. Furthermore, even if their benefits were verified, it remains unclear which design of synthetic logic samples yields the largest benefits due to the lack of systematic discussions on sample designs and empirical verification of these designs. We aimed to answer these questions in this paper and demonsirate the potential of synthetic logic corpora.
A.3 Distilling Reasoning Traces from Very Large LLMs
Recent approaches (Ho et al., 2023; Magister et al., 2023; Li et al., 2022, 2023; Shridhar et al., 2023; Wang et al., 2023; Mitra et al., 2023; Liu et al., 2023c; Ben Allal et al., 2024; Lu et al., 2024) utilize very large LLMs, such as GPT-4, to prepare synthetic reasoning datasets to train smaller LLMs. A typical procedure is as follows: (i) prepare existing reasoning problems, (ii) prompt large LLMs to generate reasoning traces to solve these problems using techniques such as chain-of-thougeee prompting (Wei et al., 2022), and (iii) train smaller LLMs on these reasoning traces.
The distillation approach and the synthetic logic corpora approach examined in this paper have specific advantages and disadvantages, as follows.
The advantage of the distillation approach is its immediate practical effect, it directly teaches LLMs solutions to various existing problems. The disadvantages could be that (i) it is non-trivial for specific solutions to specific problems to generalize to other problems, (ii) the number of training samples is limited to existing problems in nature, (iii) the correctness and faithfulness of the reasoning traces not guaranteed; indeed, some studies (Turpin et al., 2023; Lanham et al., 2023) suggest that large LLMs do not always faithfully follow the "reasoning traces" they themselves generate, and (iv) it cannot enhance the very large L.LMs themselves by nature.
The advantages of synthetic logic corpus approaches are that (i) since they teach the fundamentals of reasoning, such as deductive reasoning, they have the potential to generalize to various problems,
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(ii) they can generate an unlimited number of new samples, and (iii) the correctness of the reasoning traces guaranteed by nature. The disadvantage this approach is that, as only teaches the basics of reasoning, additional training may be needed to solve more complex real-world problems, as suggested in Section 6.4.
We hypothesize that integrating both approaches could be promising. That is, we first train LLMs using ALT to make them understand the fundamentals of reasoning through high-quality samples and then train them using more realistic reasoning traces to solve complex real-world problems.
B
Limitations
We only used deductive reasoning samples for ALT. Future work should examine other reasoning samples, e.g., abductive and inductive reasoning.
We only examined the first-order predicate logic system. Future work should examine other logic systems, such as modal and linear logic.
C Ethics and Social Impacts
The ultimate goal of the direction of this study is to develop an AI capable of reasoning logically step by step. If Al make a decision one logical step at time, it would be highly explainable and transparent to users. Furthermore, the user would be able to trace the Al's errors. We believe that our study is step towards such AI that will positively impact society.
D Details of Formal Logic Deduction Diverse
Figure D.3 shows a real sample from FLD x2. Below, We briefly explain our sample generator. Please refer to Morishita et al. (2023) for the details.
D.1 Answer Labels
In addition to the logical steps, the samples of FLD x2 and previous corpora include answer labels (Figure D.3): "proved" indicating that the hypothesis can be proved by the logical steps, "disproved" indicating that the hypothesis can be disproved, and "unknown" indicating that the given facts are insufficient for either proving or disproving the hypothesis, For samples with "unknown" labels, the logical steps are "None.". FLDx2 have uniform distribution over the labels.
D.2 Splits
FLD x2 includes 100k/5k/5k samples for train/valid/ten splits.
D.3
Generation of Multistep Deduction
Our sample generator first randomly generates examples of multistep deduction by forward- and backward random deduction, using the deduction rules specified by a user.
The forward random deduction is done as follows. The generator first chooses deduction rule randomly and forms the initial tree where the root node is the conclusion the chosen deduction rules and the child nodes are the premises of the chosen deduction rule. The generator next randomly chooses another deduction rule that can be "jointed" to the root note of the tree. A deduction rule can be jointed to the root node of tree if one of the premises of that deduction rule can be identified with the root node. Then, the generator updates the tree by jointing this chosen deduction rule. The generator continues this step multiple times until the tree achieves the required depth.
The backward random deduction is done as follows. For each step, the generator randomly chooses leaf node of the tree. Then, the generator randomly chooses a deduction rule that can be jointed to leaf node. Here, a deduction rule can be jointed to the leaf node if the deduction rule's conclusion can be identified with the leaf node. Then, the generator updates the tree by jointing this chosen deduction rule. The generator continues this step multiple times until the complexity of branches achieves the required level.
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Facts
fact1: This speck is a kind of Hittite.
fact2: If that aevsperson is not googol then this coralwood is not Hittite but the wan is steadying.
fact3:
norgue fact4: That newsperson is a Blattidhe.
Hittite if that norgue neutralizes Mrbli.
fact5: If soneone is not oesophagitis then the one is dysplastic and nounted.
fect6: The fact
Hittite but the fact7: If this speck is anticancer the fact that that norgue does not tax Eclan and the one is not kind of a refractiv
norgue
refractiveness false if this speck is an
fact8: the Eelan taxes speck.
fact9:
speck
Eelan
anticancer.
|fact10: This speck taxes Eelan.
factil: That sorgue is not refractiveness if the fact that this speck is anticancer and/or does not tax Eelan is incor
fact12: sundress not oesophag(tis
histological if she/he facti3: the fact that this holograph is not Hors and does not crean Thessalonikt does not hold the fact that this
fact14: If that newsperson is kind of Blattidae then that that newsporson does not spawn Malacostraca and does resor
fact15: fact that soneone is
Mors
croon Thessalonik
wrong
dysplastic.
fact16: that sonebody does not spawn Malacos traca and resorts ts does not hold then it is not a kind of googol.
fact1?: If this speck anticancer that that norgue is not Hittite and the one is not refractiveness does not hold.
Hypothesis
That norgue is not Hittite and not refractivaness.
Logical steps
fact10 fact17
fact9
intl:
This speck is anticancer.
hypothesks
[' DISPROVED
Figure D.3: A real deduction sample included in Formal Logic Deduction Diverse. Facts and hypothesis are given to L.Mss, then the LLMs are required to generate logical steps to (dis-)prove the hypothesis based on the facts, and an answer label (see Appendix D.2).
D.4 Linguistic Expressions
We prepared linguistic templates for each logical formula, exemplified as follows:
((A A B) C)) IF ((A A B)
then (C.predicate, phrase).
((A A 11,.oon_pharree (cause synnss_synmmms) (C.noun_phan camas_phrsee... (...)
((A A B) prediedicttte_ppun _hhass)
A (occur synonyms) also B sycour_syaooyms).
A and also B (occur_synonyms).
Both and B (eecur_synomyms)
(...)
(C.predicate, ppaaase (occur_synonyme
(...)
sycour_synonym) occur
happen
take place
(...)
((A A nnumu pharse)
A and B
also
Both A and B
That A and (ocour_synonyms)
(...)
(cause_ synonyms) cause
result in
lead to
bring about (...)
(...)
(D.I)
As can be seen, the templates can be nested deeply, yielding combinatorially diverse linguistic expressions.
Expanding these templates beforehand is intractable due to the combinatorial explosion, so expand these templates on the fly to randomly sample a single expression at a time. Estimating the exact number of expressions is intractable for the same reason.
We manually crafted several additional English templates per logical formula (i.e., the left-hand sides of (D.1)) compared to those used in FLD, which yield combinatorially more diverse English
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expressions. We observed that at least dozens of expressions, including minor variations, are yielded for each formula.
E Details of Experimental Setup
E.0.1 Prevention of Knowledge Forgetting by Recall Adam Optimizer
We employed the Recall Adam (RecAdam) optimizer (Chen et al., 2020), which regularizes parameter updates to prevent them from being too far from the pre-training parameters. Recall Adam stands out for LLM training it does not require access to the pre- training corpus, which is often inaccessible or too huge to handle, nor does require changes to the model architecture, and has a proven track record of usage in language models such as BERT.
E.1 Benchmarks
Table E.7 details the benchmarks used the experiments.
E,2
Experimental Runs
We show the average and standard deviations over five seeds.
E.3
Computational Resources
The entire experiment, including preliminary ones, took about week 128 NVIDIA H100 GPUs of our own.
Results without using Recall Adam
Table F.8 shows the results of LLMs trained without using Recall Adam.
24

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Table E.7: 31 benchmarks used in the experiments. These benchmarks cover wide range of tasks and are prominent for LLL evaluation. We also show the form reasoning and the type of knowledge required to solve the problems in each benchmark.
Reasoning form
Required knowledge
Set
Benchmarks
bAhi deduction al., 2015),
FOLIO al., 2022)
LogicNLI (Tian et al., 2021)
(not required)
RobustLR (Sanyal et al, 2022a)
deduction
Logic
AR-LSAT (Zhong et al. 2021)
LogiQA2 (Lin et al., 20236)
commonsense
ReClor (Yn
AbductionRules (Young et al., X122)
abduction
ART (Bhagavatula 2019)
commonsense
HELP (Yanaka et al., 2019)
validate conclusion based on given premises
MultiNLI (Williams et 2018)
NL.
RTE (Dugan
Champiccolo 2007; Bentivogli al., 2009)
commonsense
SNLI
GSM8k (Chbbe
Math
MATH (Hendrycks al., 2021b)
Math
Math
MathQA (Amani et 2019)
HumanEval (Chen et al., 2021)
MBPP (Austin et al., 2021)
Coding
Coding
Coding
MBPP+ at, 2023d)
MultiPL-E (cpp/go) (Cassano
CommonsenseQA (Talmner 2018)
HellaSWAG
2019)
SQuAD2 (Raipurkar al. 2018)
commonsense
WinoGrande (Sukaguchi et al., 2021)
complicated pracedures
Others
ARC (easy/challenge) (Clark et 201,
GPQA (Rein et 2023)
science
OpenBookQA (Mihaylov
SciQ (Wolll
MMLU (Heodrycks et.l. 2021a)
aggregated
MMLU-PPr (Wang et 2024g)
various
various
BBH (Suzgun et al., 2022)
25

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Table F.8: 5-shot performance of LLMs before and after ALT. @ALT-r denotes the LLM trained with ALT on the synthetic logic corpus x from Table 1. Color shows the rank in each column (darker is better). "Logic", "Math", "Code", , and "Others" each comprises various benchmarks (see Table E.7). "Avg." represents the micro-avee average of all the benchmarks. "w/o RecAdam" denotes that LLM was trained without knowledge forgetting prevention by Recall Adam optimizer.
(a) LLaMA-3.1-888
Avg.
Logic
Code
Others
BBH (3-shot)
(0-shot)
MMLU
CoT
CoT
Pro
LLaMA-3.1-BB
47.9
42.8.c.
39,6
35.4
65.4.0
60.7.oo
44.9
61.9
8.2
36.5.o
65.31s
35.8
@ALT-PRP we Rocidees
39.5..a
29.
57.8
61.0.oz
40.5.xa
47.0,a:
3.9.0
6.3
64.9.aa
,4.0
6LTT-PRP
48.1
43.7
39.2
35.7
65.6.00
60.8
44.9
61.8
8.2
36.4.02
65 65.3
35.3
BALT-RT
46.82m
42.4
68.6.m
61.3
46.92m
63.5
13.7
38.4.0
65.3
35.7
ALT-FLD
51.9
51.6.
13.4
38.1
70.1
61.3
46.7
64.9
11.9.a
39.6.
65.4
36.2.1:
BALT-FLD.
52.0
52.2am
43.2
38.0
20.7
61.5.m
46.5az
65.3
11.3
38.7
65 65.5
(b) LLaMA-3.1-70B.
Avg.
Logic
Others
(3-shot)
(0-shot)
CaT
CaT
Pro
1LaMA-3.1-700
60.0
57 Aum
60.0
46.2
73.7
67.7.m>
60.4.sa
82.1
6,5su
50.1
78.7
50,7...
@ALTT-PR PRP san RecAber
54.3
59.2gmm
72.7
65. gm
60.4
81.5
6.1
48.3
78.5
50.7
@ALT-PRP
60.4
57.7
59.8
49.2
73.5.mm
67.6.o>
60.4...
82.2.
6.0,az
50.
78.7
50.9...
4mm
62.1 g
50.8
75.4.p
68.4gm
14.
82.5g
11.5g
59.2
79.0
52.4
GALT-FLD
64.2
65.7
63.6
52.0
75.3
68.5
65.0.a
83.6...
12.1
59.9
79.3..:
54.4..:
63.3
am
6_.5
65.4g
23.6
11.4
60.8
79.5
24.4
Table F9: Benchmark-wise 5-shot performance of LLaMA-3.1-8B before and after ALT on FLDx2-
(a) Logic.
bAbiD FOLIO LogieNII RobustLR AR-LSAT LogiQA ReClor AbductionR
ART
LLaMA-3.1 48.7 50.0
28.5
43.2
20.7
28.7 42.8 gm 29.4
39.6
52.4 85.5
13.4 76.1 1o4
ALT-FLD, xa 55.8 Los 54.5 gas
42.0_m
62.6 100
21.1
(b) Math.
GSM8k
MATH MathQA
CoT
CoT (0-shot)
1LMAA-3.1-8B 50.2
51.5
39.5
14.1 14.1_ms
42.8
#ALT-FLD22 53.6gns 56.4
48.4
14.3
43.3
(c) Coding.
HumanEyal
MBPP
MBPP+
MultiPL.EE (cpp)
MultiPI-FE (go)
LLaMA-3.1-8B
22.6
31.6
38.1
21.7
63.0
DALT-FLD.
25.9
34.0
39.9
23.0
67.1
(d) Natural language inference (NLI).
HELP
MNLI
RTE
SNLI
LLaMA-3.1 3-B-8B-
46.4+05
68.1
74.6 83.1
72.6
GALT-FLD
47.9 +02
75.3
76.5
(c) Others.
CommonsenseeA HellaSwag SQuAD WinoGrande ARCe
ARCs
GPQA OpenBookQA
SciQ
LLaMA-3.1-8B
73.9
61.2
30.8
77.4n1
84.2
$4.7 31.1
35.3
97.7
OALT-FLD,
74.8
61.5
33.5
78.1
85.0
55.62ms 31.1
36,3
97.6
26

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Reviews:
[{"review_id": "bETXLRY7yP", "reviewer": "Reviewer_aRM4", "summary": "The paper discusses a novel approach to improve the logical reasoning capabilities of large language models (LLMs). The authors propose a method called Additional Logic Pre-Training (ALPT), which involves training LLMs on a synthetic corpus named PureLogicDiverse. This corpus is designed to include high-quality, program-generated reasoning samples that adhere to strict logical principles.", "questions": "Authors discussed about the performance on other tasks like CommonsenseQA and Hellaswag, the results show that there is no substantial improvement on these tasks. I am curious to see the performance on other tasks like math or coding.", "limitations": "The approach may not fully address tasks requiring complex procedural understanding or multiple-choice questions that involve nuanced reasoning. Further research is needed to integrate ALPT with other methods to enhance performance on these types of tasks.\nhe effectiveness of ALPT heavily depends on the quality and comprehensiveness of the design principles used to create the synthetic corpus. Any biases or limitations in these principles could affect the overall performance improvements and generalizability of the models.", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The approach significantly enhance LLMs in logical reasoning abilities. Empirical results show that models pre-trained with ALPT on the PureLogicDiverse corpus perform much better, especially on benchmarks like BBH and NLI tasks. This demonstrates that ALPT can make LLMs more versatile and capable of handling various reasoning tasks.\n\nThe systematic design of the PureLogicDiverse corpus ensures that the reasoning samples are high quality, covering a wide range of logical rules and linguistic expressions. This comprehensive method not only boosts logical reasoning but also helps models integrate existing knowledge with new reasoning skills. As a result, they become more effective in tasks requiring both logic and knowledge.", "weaknesses": "There is a risk of models overfitting to the synthetic logic patterns, which could limit their generalizability to real-world applications. The success of ALPT heavily depends on the quality of the design principles used to create the synthetic corpus, and any deficiencies in these principles could reduce its effectiveness. Therefore, while ALPT shows great promise, it comes with challenges that need careful management to maximize its potential benefits.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your constructive feedback.\nBelow, we will address your questions to the best of our ability.\n\n> I am curious to see the performance on other tasks like math or coding.\n\nWe conducted additional experiments on math and coding tasks.\nTable F.9 (in the newly attached PDF on OpenReview) shows that **ALPT significantly enhances the LLM's performance on various math and coding benchmarks.**\nThese results are very interesting to us; since our samples use first-order predicate logic, which differs much from math and coding, we did not expect improvements in math and coding tasks.\n\nWe will include these findings in the paper's final version to further strengthen our argument.\nThank you very much for the suggestion!\n\n\n> (...) Therefore, while ALPT shows great promise, it comes with challenges that need careful management to maximize its potential benefits.\n\nWhile our experiments (Table 3) showed no notable performance degradation across various tasks, we agree that ALPT's success generally depends on sample quality.\nExplicitly enumerating and discussing design principles, as demonstrated in our study, should help maintain high-quality samples.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FyR3XUNmph", "reviewer": "Reviewer_zCCP", "summary": "This work proposes Additional Logic Pre-Training (ALPT) to enhance logical reasoning abilities using synthetic rule-based data. The paper first discusses the design principles for creating a logical corpus and subsequently builds PureLogicDiverse (PLD). By training on PLD with RecAdam, models demonstrate improved performance in logical reasoning across various tasks and datasets, effectively integrating enhanced logical reasoning capabilities with their inherent knowledge. The results confirm the effectiveness of the proposed dataset and the ALPT strategy.", "questions": "1. In dataset creation (Line 188), when sampling from the predefined logical formulas, is there a prior distribution from which to sample, or are the formulas evenly distributed? Do you have statistics on the operators and formulas of the created dataset? \n2. Since the nouns and predicates are randomly composed to form logical expressions, they lack practical meanings. Intuitively, the coverage of rules and formulas plays a more important role than vocabulary coverage. Considering this, is covering a wide range of 15k vocabulary really necessary? As shown in Figure 2 (w/o DP1), restricting the vocabulary size appears to have minimal impact on the final performance and could also help decrease the dataset size, potentially improving data efficiency. Do you have any insights on this?", "limitations": "The authors have reasonably discussed the limitations and potential societal impacts of their research.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well-written and presents a fluent narrative from the design principles to the creation of the dataset.\n2. Compared to previous logical datasets, PureLogicDiverse contains more comprehensive and complex scenarios, serving as a useful resource for the community.\n3. After additional logical pre-training on PLD, different models achieve better performance on both logical reasoning and NLI tasks, with no loss on other tasks. This training strategy together with the PLD dataset can be of interest for the development of future models.\n4. The experiments are solid and sound. The comparisons and ablations are comprehensive, providing clear analyses of the effects of different design principles.", "weaknesses": "The overall experiments are well-established. However, when comparing the ablation results with those of previous work (Table 2a and Figure 2), RuleTakers appears to perform well enough given the simplicity of its design. For example, comparing ‘ALPT-RT’(Table 2a) with ‘w/o DP2’(Figure 2), despite ‘w/o DP2’ containing much bigger vocabulary size and more extensive rules and expressions, these two settings gain similar performances.  \n\nTo clarify these comparisons, it would be helpful if the authors could provide statistics on the different dataset settings, such as the number of steps contained in the sampled training splits of each corpus and the average number of rules/steps per sample.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your constructive feedback.\nBelow, we will address your questions to the best of our ability.\n\n\n> (...) restricting the vocabulary size appears to have minimal impact on the final performance and could also help decrease the dataset size, potentially improving data efficiency. Do you have any insights on this?\n\nWe agree on the importance of dataset efficiency.\nWe conducted additional experiments to investigate whether restricting vocabulary (i.e., w/o DP1) has minimal impact, and whether we can reduce the dataset size.\nTable F.9 (in the PDF newly attached to OpenReview) shows that:\n1. w/o DP1 significantly degraded the performance of LLMs other than Llama3-8B, which was originally shown in the paper. This indicates that vocabulary size is important.\n1. Reducing the dataset size also degraded performance significantly. This suggests a trade-off between dataset size and final performance, making it challenging to reduce dataset size without performance degradation.\n\nWhile we maintain the full-vocabulary version as the official corpus, we will allow users to choose their preferred trade-off by using our generator's \"--limit-vocab-size\" option (https://anonymous.4open.science/r/PLD-generator/scripts/create_corpus.py).\n\nWe appreciate the suggestion for new experiments and will include these additional results in the paper's final version.\n\n\n\n> To clarify these comparisons, it would be helpful if the authors could provide statistics on the different dataset settings,\n\nWe calculated some statistics for PLD using 100,000 examples:\n\n| total labels                                       | vocab size | total distractors | unique rules | total steps |\n|----------------------------------------------------|------------|-------------------|--------------|-------------|\n| proved(33,391)/ disproved(33,610)/ unknown(32,999) | 100,000    | 1,008,725         | 50           | 187,416     |\n\nWe will enumerate all other PLD attributes, such as unique rules and logical operators, and include them in the paper's final version.\nWe will attempt to count similar statistics for the baseline corpora.\nIdeally, we will add statistics for all corpora, including PLD and the baselines, to provide a comprehensive comparison.\n\n\n\n> when sampling from the predefined logical formulas, is there a prior distribution from which to sample, or are the formulas evenly distributed?\n\nTo avoid overly complex formulas, we use logical formulas with up to three predicates in a sentence, such as \"ForAll(x): F(x)\", \"ForSome(x): (F(x) AND ^G(x)) -> H(x)\", and \"^(F(a) OR G(b)) -> H(c)\", where \"^\" indicates negation.\nThis setting aligns with the previous studies.\nWe sample these formulas evenly.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OD8NJspo1H", "reviewer": "Reviewer_BW1m", "summary": "This paper aims to improve LLM's logical reasoning ability by constructing synthetic data used in continual training. This work is largely built upon FLD, and proposes four other design principles for the synthetic dataset. Namely, reasoning with unknown facts, illogical reasoning, diverse reasoning rules, and linguistic expressions.  By training llama-7b and 70b models on the synthetic dataset, the authors proved the effectiveness of their methods by achieving improvement on various benchmarks such as BBH.", "questions": "1. I think the training process using the constructed synthetic data is more like a continual training or SFT setting. Calling it additional \"pre-training\" could be a little misleading.", "limitations": "There is a limitation section in the appendix.", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The experiments are pretty solid. The authors tested ALPT on different scales of LLMs on various NLP tasks such as logical reasoning, reading comprehension, etc.\n2. According to Table 2, the performance gain on 70b model is even larger than 7b model, which shows the potential of ALPT.\n3. The paper is organized well.\n4. The authors provide an anonymous link for all the code, model, and data to ensure reproducibility.", "weaknesses": "1.  I am mostly concerned about the contribution of this paper. It seems to me that the four design principles are incremental improvements based on [1] by increasing the vocab size and enriching the diversity of logical rules and linguistic expressions. The results in Table 2 seem incremental compared with FLD baselines either. Also, the writing of some parts of the paper is very similar to [1], with even the same examples.\n2. The current writing makes it hard for non-experts to understand. Specifically, the description of how the generator works is listed in the appendix. If one hasn't read [1] or the appendix, it would be very hard to understand the overall workflow.\n3. It would strengthen the paper if the authors could also provide the results for FLD in Table 3 as a baseline for comparison.\n\n\n[1] Morishita, Terufumi, et al. \"Learning deductive reasoning from synthetic corpus based on formal logic.\" International Conference on Machine Learning. ICML, 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your constructive feedback.\nBelow, we will address your questions to the best of our ability.\n\n> the training process is more like a continual training\n\nWe greatly appreciate your observation.\nIndeed, *pre*-training doesn't make sense much.\nWe will rename it to **C**ontinual **L**ogic **T**raining.\n\n> the description of how the generator works is listed in the appendix.\n\nWe will add a new subsection in Section 3 to explain the workflow.\nWe will include content from the appendix and a new diagram illustrating the workflow.\n\n> It would strengthen the paper if the authors could also provide the results for FLD in Table 3\n\nWe will add the task-specific results (i.e., Table 3) of baseline corpora (RT, PRP and FLD) to the main text if space permits; otherwise, we will include them in the appendix.\n\n> I am mostly concerned about the contribution of this paper. It seems to me that the four design principles are incremental improvements based on [1] by increasing the vocab size and enriching the diversity of logical rules and linguistic expressions. The results in Table 2 seem incremental compared with FLD baselines either.\n\n* We first emphasize the higher-level, more fundamental contribution of this study beyond updates from the previous studies: it is the first comprehensive study to verify the promise of synthetic logic corpora to enhance LLMs' reasoning capabilities (detailed below in `D.1`).\n* Regarding the design principles (DPs), while the final suggestions in Table 1 may seem incremental, the comprehensive and systematic analyses of these principles (Sec2 and the experiments) are novel and crucial in the field of synthetic logic corpus, establishing a foundation for future research (detailed below in `D.2`).\n* We consider the Llama-3-70B results in Table 2 (b) significant improvements over the FLD corpus.\n\n# D.1: Our study is the first to verify the promise of synthetic logic corpus through comprehensive experiments.\nRuleTaker (2020,[14]) initially proposed a synthetic logic corpus approach.\nFollow-up studies proposed various corpora, including ProofWriter [63], PARARULE-Plus [3], AACorpus [7], and FLD [51].\nThese studies aim to enhance reasoning capabilities of language models with high-quality samples, ultimately achieving versatile AI.\n\nDespite conceptual promise, the effectiveness of the synthetic logic corpus remains uncertain due to lack of comprehensive empirical verification.\nPrevious studies examined small language models (SLMs) trained on limited pre-training corpora, such as T5.\nThus, a fundamental question remains: **RQ1: \"Can a synthetic logic corpus enhance reasoning capabilities of state-of-the-art large language models (LLMs) trained on massive pre-training corpora?\"**\nThis question is non-trivial because massive pre-training corpora might have already taught LLMs reasoning, potentially negating the effect of synthetic logic corpora.\nMoreover, previous studies examined a limited range of evaluation tasks, raising **RQ2: \"Can capabilities obtained from synthetic logic corpora generalize to different tasks?\"**\nAnswering these questions is crucial to advancing the synthetic logic corpus approach further.\n\nOur study addressed these questions through comprehensive experiments, as follows:\n* RQ1: Synthetic logic corpora significantly enhance state-of-the-art LLMs' capabilities, including Llama-3-70B, one of the largest LLMs trained on over 15 trillion tokens (Table 2).\n* RQ2: Capabilities obtained from synthetic logic corpora generalize to various task types (Table 3, and Table F.9 in the newly attached PDF on OpenReview, which includes math and coding tasks).\n\nWe emphasize that we demonstrated the positive effect of the synthetic logic corpus approach *as a whole*, involving *not only ALPT-PLD but also ALPT-RT and ALPT-FLD*.\n\nIn summary, our study's fundamental contribution provides comprehensive verification of the synthetic logic corpus approach.\nThis establishes a promising direction for developing versatile AI with reasoning capabilities.\n(We sincerely apologize for including these important discussions only in Appendix A1; we will include them in the main text.)\n\n# D.2: DPs provide the first comprehensive and systematic analyses of synthetic corpus design, accelerating future research in this field.\nIn synthetic generation, computer programs create samples based on pre-designed patterns, which significantly influence sample quality.\nPrevious studies developed several corpora with different designs [14,7,71], but lack comprehensive and systematic analyses on these designs.\nImportant questions remain, such as **\"What aspects are crucial for the design?\"** and **\"*Why* are these aspects important?\"**\n\nWe addressed these questions by enumerating crucial design aspects as principles and discussing why each principle is important, considering symbolic logic theory and integrating empirical insights from previous studies.\nFor example:\n* A large vocabulary is important *because* it teaches reasoning with unknown facts (DP1).\n* Hard distractors are important *because* they teach when *not* to derive conclusions (DP2).\n* For reasoning rules, while axioms are theoretically sufficient [23,51], theorems are also important *due to* LLMs' limited capability to handle long logical proofs, which was observed in several studies [24] (DP3).\n* We need to include diverse linguistic expressions for logical formulas *because* otherwise, LLMs could overfit to specific linguistic expressions in the corpus, which was observed in several studies [90,92] (DP4).\n\nWe also verified the importance of each principle through ablation experiments (Figure 2) and analyzed how these principles influence LLMs' output generations through case studies (Table 4).\n\nOur comprehensive and systematic analyses are novel in the field of synthetic logic corpus.\nThese analyses offer insights into future research in this field. For example, researchers can apply, critique, or update the proposed principles.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their valuable feedback!
We will update the paper to address the reviewers' suggestions as follows:

* **Additional experiments on math and coding suggested by reviewer aRM4 show that ALPT significantly enhances LLMs' capabilities in various tasks in these domains** (Table F.9 in the newly attached PDF on OpenReview). Combined with the original paper's results, ALPT demonstrates improvement in various LLM capabilities, from reasoning in natural language and NLI to math and coding. We will include these interesting results to strengthen the paper.
* We will include additional experiments on smaller dataset sizes (Table F.10 in the PDF), as suggested by reviewer zCCP, to provide insights on the trade-off between dataset efficiency and performance.
* To make the paper more self-contained, we will include:
    1. Dataset statistics of PLD and baseline corpora (suggested by reviewer zCCP)
    2. The dataset generator workflow (suggested by reviewer BW1m)
* We will emphasize our study's important position in the context of synthetic logic corpora, currently only shown in Appendix A.1, as replied to reviewer BW1m.

Meta-review: This is an interesting work to enhance reasoning ability of LLMs using synthetic rule-based data. By training on the additional data with as continual pretraining, LLMs show better performance in logical reasoning without forgetting the pretrained knowledge. Two of the three reviewers are convinced with the contribution of the paper, while the other reviewer who was against it, didn't participate in the rebuttal.
Decision: Accept (poster)