Neuro-symbolic approaches combining large language models (LLMs) with solvers excels in logical reasoning problems need long reasoning chains. In this paradigm, LLMs serve as translators, converting natural language reasoning problems into formal logic formulas. Then reliable symbolic solvers return correct solutions. Despite their success, we find that LLMs, as translators, struggle to handle lexical diversification, a common linguistic phenomenon, indicating that LLMs as logic translators are unreliable in real-world scenarios. Moreover, existing logical reasoning benchmarks lack lexical diversity, failing to challenge LLMs' ability to translate such text and thus obscuring this issue. In this work, we propose SCALe, a benchmark designed to address this significant gap through **logic-invariant lexical diversification**. By using LLMs to transform original benchmark datasets into lexically diversified but logically equivalent versions, we evaluate LLMs' ability to consistently map diverse expressions to uniform logical symbols on these new datasets. Experiments using SCALe further confirm that current LLMs exhibit deficiencies in this capability. Building directly on the deficiencies identified through our benchmark, we propose a new method, MenTaL, to address this limitation. This method guides LLMs to first construct a table unifying diverse expressions before performing translation. Applying MenTaL through in-context learning and supervised fine-tuning (SFT) significantly improves the performance of LLM translators on lexically diversified text. Our code is now available at https://github.com/wufeiwuwoshihua/LexicalDiver.
Process Reward Models (PRMs), which provide step-by-step feedback on the reasoning generated by Large Language Models (LLMs), are receiving increasing attention. However, two key research gaps remain: collecting accurate step-level error labels for training typically requires costly human annotation, and existing PRMs are limited to math reasoning problems. In response to these gaps, this paper aims to address the challenges of automatic dataset creation and the generalization of PRMs to diverse reasoning tasks. To achieve this goal, we propose FoVer, an approach for training PRMs on step-level error labels automatically annotated by formal verification tools, such as Z3 for formal logic and Isabelle for theorem proof, which provide automatic and accurate verification for symbolic tasks. Using this approach, we synthesize a training dataset with error labels on LLM responses for formal logic and theorem proof tasks without human annotation. Although this data synthesis is feasible only for tasks compatible with formal verification, we observe that LLM-based PRMs trained on our dataset exhibit cross-task generalization, improving verification across diverse reasoning tasks. Specifically, PRMs trained with FoVer significantly outperform baseline PRMs based on the original LLMs and achieve competitive or superior results compared to state-of-the-art PRMs trained on labels annotated by humans or stronger models, as measured by step-level verification on ProcessBench and Best-of-K performance across 12 reasoning benchmarks, including MATH, AIME, ANLI, MMLU, and BBH. The datasets, models, and code are provided at https://github.com/psunlpgroup/FoVer.
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 $\textbf{Formal Logic Deduction Diverse}$ ($\textbf{FLD}$$_{\times 2}$), comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors. Finally, we empirically show that ALT on FLD$_{\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.
Autonomous agents often face the challenge of interpreting uncertain natural language instructions for planning tasks. Representing these instructions as Linear Temporal Logic (LTL) enables planners to synthesize actionable plans. We introduce CoT-TL, a data-efficient in-context learning framework for translating natural language specifications into LTL representations. CoT-TL addresses the limitations of large language models, which typically rely on extensive fine-tuning data, by extending chain-of-thought reasoning and semantic roles to align with the requirements of formal logic creation. This approach enhances the transparency and rationale behind LTL generation, fostering user trust. CoT-TL achieves state-of-the-art accuracy across three diverse datasets in low-data scenarios, outperforming existing methods without fine-tuning or intermediate translations. To improve reliability and minimize hallucinations, we incorporate model checking to validate the syntax of the generated LTL output. We further demonstrate CoT-TL's effectiveness through ablation studies and evaluations on unseen LTL structures and formulas in a new dataset. Finally, we validate CoT-TL's practicality by integrating it into a QuadCopter for multi-step drone planning based on natural language instructions.
Game theory is a powerful framework for reasoning about strategic interactions, with applications in domains ranging from day-to-day life to international politics. However, applying formal reasoning tools in such contexts is challenging, as these scenarios are often expressed in natural language. To address this, we introduce a framework for the autoformalization of game-theoretic scenarios, which translates natural language descriptions into formal logic representations suitable for formal solvers. Our approach utilizes one-shot prompting and a solver that provides feedback on syntactic correctness to allow LLMs to refine the code. We evaluate the framework using GPT-4o and a dataset of natural language problem descriptions, achieving 98% syntactic correctness and 88% semantic correctness. These results show the potential of LLMs to bridge the gap between real-life strategic interactions and formal reasoning.
Large language models (LLMs) are increasingly explored as general-purpose reasoners, particularly in agentic contexts. However, their outputs remain prone to mathematical and logical errors. This is especially challenging in open-ended tasks, where unstructured outputs lack explicit ground truth and may contain subtle inconsistencies. To address this issue, we propose Logic-Enhanced Language Model Agents (LELMA), a framework that integrates LLMs with formal logic to enable validation and refinement of natural language reasoning. LELMA comprises three components: an LLM-Reasoner, an LLM-Translator, and a Solver, and employs autoformalization to translate reasoning into logic representations, which are then used to assess logical validity. Using game-theoretic scenarios such as the Prisoner's Dilemma as testbeds, we highlight the limitations of both less capable (Gemini 1.0 Pro) and advanced (GPT-4o) models in generating logically sound reasoning. LELMA achieves high accuracy in error detection and improves reasoning correctness via self-refinement, particularly in GPT-4o. The study also highlights challenges in autoformalization accuracy and in evaluation of inherently ambiguous open-ended reasoning tasks.
The recent proliferation of generative artificial intelligence (AI) technologies such as pre-trained large language models (LLMs) has opened up new frontiers in computational law. An exciting area of development is the use of AI to automate the deductive rule-based reasoning inherent in statutory and contract law. This paper argues that such automated deductive legal reasoning can now be viewed from the lens of software engineering, treating LLMs as interpreters of natural-language programs with natural-language inputs. We show how it is possible to apply principled software engineering techniques to enhance AI-driven legal reasoning of complex statutes and to unlock new applications in automated meta-reasoning such as mutation-guided example generation and metamorphic property-based testing.
We study a synthetic corpus based approach for language models (LMs) to acquire logical deductive reasoning ability. The previous studies generated deduction examples using specific sets of deduction rules. However, these rules were limited or otherwise arbitrary, limiting the generalizability of acquired reasoning ability. We rethink this and adopt a well-grounded set of deduction rules based on formal logic theory, which can derive any other deduction rules when combined in a multistep way. Then, using the proposed corpora, which we name FLD (Formal Logic Deduction), we first evaluate and analyze the logical reasoning ability of the latest LLMs. Even GPT-4 can solve only half of the problems, suggesting that pure logical reasoning isolated from knowledge is still challenging for the LLMs, and additional training specialized in logical reasoning is indeed essential. We next empirically verify that LMs trained on FLD corpora acquire more generalizable reasoning ability. Furthermore, we identify the aspects of reasoning ability on which deduction corpora can enhance LMs and those on which they cannot, and discuss future directions on each aspect. The released corpora serve both as learning resources and as challenging benchmarks.
We present a comprehensive study of chart visual question-answering(QA) task, to address the challenges faced in comprehending and extracting data from chart visualizations within documents. Despite efforts to tackle this problem using synthetic charts, solutions are limited by the shortage of annotated real-world data. To fill this gap, we introduce a benchmark and dataset for chart visual QA on real-world charts, offering a systematic analysis of the task and a novel taxonomy for template-based chart question creation. Our contribution includes the introduction of a new answer type, 'list', with both ranked and unranked variations. Our study is conducted on a real-world chart dataset from scientific literature, showcasing higher visual complexity compared to other works. Our focus is on template-based QA and how it can serve as a standard for evaluating the first-order logic capabilities of models. The results of our experiments, conducted on a real-world out-of-distribution dataset, provide a robust evaluation of large-scale pre-trained models and advance the field of chart visual QA and formal logic verification for neural networks in general.
Large language models (LLMs), such as GPT-3 and GPT-4, have demonstrated exceptional performance in various natural language processing tasks and have shown the ability to solve certain reasoning problems. However, their reasoning capabilities are limited and relatively shallow, despite the application of various prompting techniques. In contrast, formal logic is adept at handling complex reasoning, but translating natural language descriptions into formal logic is a challenging task that non-experts struggle with. This paper proposes a neuro-symbolic method that combines the strengths of large language models and answer set programming. Specifically, we employ an LLM to transform natural language descriptions of logic puzzles into answer set programs. We carefully design prompts for an LLM to convert natural language descriptions into answer set programs in a step by step manner. Surprisingly, with just a few in-context learning examples, LLMs can generate reasonably complex answer set programs. The majority of errors made are relatively simple and can be easily corrected by humans, thus enabling LLMs to effectively assist in the creation of answer set programs.
Translating natural language sentences to first-order logic (NL-FOL translation) is a longstanding challenge in the NLP and formal logic literature. This paper introduces LogicLLaMA, a LLaMA-7B model fine-tuned for NL-FOL translation using LoRA on a single GPU. LogicLLaMA is capable of directly translating natural language into FOL rules, which outperforms GPT-3.5. LogicLLaMA is also equipped to correct FOL rules predicted by GPT-3.5, and can achieve similar performance as GPT-4 with a fraction of the cost. This correction ability was achieved by a novel supervised fine-tuning (SFT) + reinforcement learning with human feedback (RLHF) framework, which initially trains on synthetically perturbed NL-FOL pairs to encourage chain-of-thought reasoning and then fine-tunes with RLHF on GPT-3.5 outputs using a FOL verifier as the reward model. To train LogicLLaMA, we present MALLS (large language $\textbf{M}$odel gener$\textbf{A}$ted N$\textbf{L}$-FO$\textbf{L}$ pair$\textbf{S}$), a dataset of 34K high-quality and diverse sentence-level NL-FOL pairs collected from GPT-4. The dataset was created by implementing a pipeline that prompts GPT-4 for pairs, and dynamically adjusts the prompts to ensure the collection of pairs with rich and diverse contexts at different levels of complexity, and verifies the validity of the generated FOL rules. Codes, weights, and data are available at $\href{https://github.com/gblackout/LogicLLaMA}{{\small \text{https://github.com/gblackout/LogicLLaMA}}}$.
With the rapidly increasing amount of scientific literature,it is getting continuously more difficult for researchers in different disciplines to be updated with the recent findings in their field of study.Processing scientific articles in an automated fashion has been proposed as a solution to this problem,but the accuracy of such processing remains very poor for extraction tasks beyond the basic ones.Few approaches have tried to change how we publish scientific results in the first place,by making articles machine-interpretable by expressing them with formal semantics from the start.In the work presented here,we set out to demonstrate that we can formally publish high-level scientific claims in formal logic,and publish the results in a special issue of an existing journal.We use the concept and technology of nanopublications for this endeavor,and represent not just the submissions and final papers in this RDF-based format,but also the whole process in between,including reviews,responses,and decisions.We do this by performing a field study with what we call formalization papers,which contribute a novel formalization of a previously published claim.We received 15 submissions from 18 authors,who then went through the whole publication process leading to the publication of their contributions in the special issue.Our evaluation shows the technical and practical feasibility of our approach.The participating authors mostly showed high levels of interest and confidence,and mostly experienced the process as not very difficult,despite the technical nature of the current user interfaces.We believe that these results indicate that it is possible to publish scientific results from different fields with machine-interpretable semantics from the start,which in turn opens countless possibilities to radically improve in the future the effectiveness and efficiency of the scientific endeavor as a whole.
Symbolic reasoning, rule-based symbol manipulation, is a hallmark of human intelligence. However, rule-based systems have had limited success competing with learning-based systems outside formalized domains such as automated theorem proving. We hypothesize that this is due to the manual construction of rules in past attempts. In this work, we ask how we can build a rule-based system that can reason with natural language input but without the manual construction of rules. We propose MetaQNL, a "Quasi-Natural" language that can express both formal logic and natural language sentences, and MetaInduce, a learning algorithm that induces MetaQNL rules from training data consisting of questions and answers, with or without intermediate reasoning steps. Our approach achieves state-of-the-art accuracy on multiple reasoning benchmarks; it learns compact models with much less data and produces not only answers but also checkable proofs. Further, experiments on a real-world morphological analysis benchmark show that it is possible for our method to handle noise and ambiguity. Code will be released at https://github.com/princeton-vl/MetaQNL.
Logic Artificial Intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and solved by checking their satisfiability (SAT). In contrast to machine learning approaches where the results can be approximations or local minima, Logic AI delivers formal and mathematically exact solutions to those problems. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI significantly improves the resolution of this problem, paving the path to developing more formal-based approaches in the context of quantum physics experiments.
Negation is both an operation in formal logic and in natural language by which a proposition is replaced by one stating the opposite, as by the addition of "not" or another negation cue. Treating negation in an adequate way is required for cognitive reasoning, which aims at modeling the human ability to draw meaningful conclusions despite incomplete and inconsistent knowledge. One task of cognitive reasoning is answering questions given by sentences in natural language. There are tools based on discourse representation theory to convert sentences automatically into a formal logic representation, and additional knowledge can be added using the predicate names in the formula and knowledge databases. However, the knowledge in logic databases in practice always is incomplete. Hence, forward reasoning of automated reasoning systems alone does not suffice to derive answers to questions because, instead of complete proofs, often only partial positive knowledge can be derived, while negative knowledge is used only during the reasoning process. In consequence, we aim at eliminating syntactic negation, strictly speaking, the negated event or property. In this paper, we describe an effective procedure to determine the negated event or property in order to replace it by its inverse. This lays the basis of cognitive reasoning, employing both logic and machine learning for general question answering. We evaluate our procedure by several benchmarks and demonstrate its practical usefulness in our cognitive reasoning system.