Arnaud Mayeux and Jujian Zhang Efforts to mechanize aspects of scientific reasoning have been intertwined with the development of science from its earliest days. C1 "Whenever we have a long, difficult piece of algebra, and we have them more and more often these days, we could at least get the machine to check that the algebra was right before we went on and built further stages of derivation on top. Some people are working on such programs for algebra checking right now." C2 "Now I leave the region of known processes and enter the land of speculation. We can, I believe, reasonably expect that an algebra checking routine would not be around very long before someone would adapt the methods of heuristics that are presently being developed to the problem of doing algebra in a more creative way. The machine could supply several steps at a time, and be given only a guiding thread of a proof. The more successful the heuristics, the fewer steps we would have to supply."

Large Language Models (LLMs) have recently advanced the field of Automated Theorem Proving (ATP), attaining substantial performance gains through widely adopted test-time scaling strategies, notably reflective Chain-of-Thought (CoT) reasoning and increased sampling passes. However, they both introduce significant computational overhead for inference. Moreover, existing cost analyses typically regulate only the number of sampling passes, while neglecting the substantial disparities in sampling costs introduced by different scaling strategies. In this paper, we systematically compare the efficiency of different test-time scaling strategies for ATP models and demonstrate the inefficiency of the current state-of-the-art (SOTA) open-source approaches. We then investigate approaches to significantly reduce token usage and sample passes while maintaining the original performance. Specifically, we propose two complementary methods that can be integrated into a unified EconRL pipeline for amplified benefits: (1) a dynamic Chain-of-Thought (CoT) switching mechanism designed to mitigate unnecessary token consumption, and (2) Diverse parallel-scaled reinforcement learning (RL) with trainable prefixes to enhance pass rates under constrained sampling passes. Experiments on miniF2F and ProofNet demonstrate that our EconProver achieves comparable performance to baseline methods with only 12% of the computational cost. This work provides actionable insights for deploying lightweight ATP models without sacrificing performance.

Continuous representations of logic formulae allow us to integrate symbolic knowledge into data-driven learning algorithms. If such embeddings are semantically consistent, i.e. if similar specifications are mapped into nearby vectors, they enable continuous learning and optimization directly in the semantic space of formulae. However, to translate the optimal continuous representation into a concrete requirement, such embeddings must be invertible. We tackle this issue by training a Transformer-based decoder-only model to invert semantic embeddings of Signal Temporal Logic (STL) formulae. STL is a powerful formalism that allows us to describe properties of signals varying over time in an expressive yet concise way. By constructing a small vocabulary from STL syntax, we demonstrate that our proposed model is able to generate valid formulae after only 1 epoch and to generalize to the semantics of the logic in about 10 epochs. Additionally, the model is able to decode a given embedding into formulae that are often simpler in terms of length and nesting while remaining semantically close (or equivalent) to gold references. We show the effectiveness of our methodology across various levels of training formulae complexity to assess the impact of training data on the model's ability to effectively capture the semantic information contained in the embeddings and generalize out-of-distribution. Finally, we deploy our model for solving a requirement mining task, i.e. inferring STL specifications that solve a classification task on trajectories, performing the optimization directly in the semantic space.

Reasoning is essential for closed-domain QA systems in which procedural correctness and policy compliance are critical. While large language models (LLMs) have shown strong performance on many reasoning tasks, recent work reveals that their reasoning traces are often unfaithful - serving more as plausible justifications than as causally grounded derivations. Efforts to combine LLMs with symbolic engines (e.g., Prover9, Z3) have improved reliability but remain limited to static forms of logic, struggling with dynamic, state-based reasoning such as multi-step progressions and conditional transitions. In this paper, we propose MCFR (Model Checking for Formal Reasoning), a neuro-symbolic framework that integrates LLMs with model checking to support property verification. MCFR translates natural language into formal specifications and verifies them over transition models. To support evaluation, we introduce EduMC-QA, a benchmark dataset grounded in real academic procedures. Our results show that MCFR improves reasoning faithfulness and interpretability, offering a viable path toward verifiable QA in high-stakes closed-domain applications. In addition to evaluating MCFR, we compare its performance with state-of-the-art LLMs such as ChatGPT, DeepSeek, and Claude to contextualize its effectiveness.

Translating software between programming languages is a challenging task, for which automated techniques have been elusive and hard to scale up to larger programs. A key difficulty in cross-language translation is that one has to re-express the intended behavior of the source program into idiomatic constructs of a different target language. This task needs abstracting away from the source language-specific details, while keeping the overall functionality the same. In this work, we propose a novel and systematic approach for making such translation amenable to automation based on a framework we call program skeletons. A program skeleton retains the high-level structure of the source program by abstracting away and effectively summarizing lower-level concrete code fragments, which can be mechanically translated to the target programming language. A skeleton, by design, permits many different ways of filling in the concrete implementation for fragments, which can work in conjunction with existing data-driven code synthesizers. Most importantly, skeletons can conceptually enable sound decomposition, i.e., if each individual fragment is correctly translated, taken together with the mechanically translated skeleton, the final translated program is deemed to be correct as a whole. We present a prototype system called Skel embodying the idea of skeleton-based translation from Python to JavaScript. Our results show promising scalability compared to prior works. For 9 real-world Python programs, some with more than about 1k lines of code, 95% of their code fragments can be automatically translated, while about 5% require manual effort. All the final translations are correct with respect to whole-program test suites.

This paper presents State Algebra, a novel framework designed to represent and manipulate propositional logic using algebraic methods. The framework is structured as a hierarchy of three representations: Set, Coordinate, and Row Decomposition. These representations anchor the system in well-known semantics while facilitating the computation using a powerful algebraic engine. A key aspect of State Algebra is its flexibility in representation. We show that although the default reduction of a state vector is not canonical, a unique canonical form can be obtained by applying a fixed variable order during the reduction process. This highlights a trade-off: by foregoing guaranteed canonicity, the framework gains increased flexibility, potentially leading to more compact representations of certain classes of problems. We explore how this framework provides tools to articulate both search-based and knowledge compilation algorithms and discuss its natural extension to probabilistic logic and Weighted Model Counting.

This paper proposes a natural language translation method for machine-verifiable formal proofs that leverages the informalization (verbalization of formal language proof steps) and summarization capabilities of LLMs. For evaluation, it was applied to formal proof data created in accordance with natural language proofs taken from an undergraduate-level textbook, and the quality of the generated natural language proofs was analyzed in comparison with the original natural language proofs. Furthermore, we will demonstrate that this method can output highly readable and accurate natural language proofs by applying it to existing formal proof library of the Lean proof assistant.

Neurosymbolic (NeSy) frameworks combine neural representations and learning with symbolic representations and reasoning. Combining the reasoning capacities, explainability, and interpretability of symbolic processing with the flexibility and power of neural computing allows us to solve complex problems with more reliability while being data-efficient. However, this recently growing topic poses a challenge to developers with its learning curve, lack of user-friendly tools, libraries, and unifying frameworks. In this paper, we characterize the technical facets of existing NeSy frameworks, such as the symbolic representation language, integration with neural models, and the underlying algorithms. A majority of the NeSy research focuses on algorithms instead of providing generic frameworks for declarative problem specification to leverage problem solving. To highlight the key aspects of Neurosymbolic modeling, we showcase three generic NeSy frameworks - \textit{DeepProbLog}, \textit{Scallop}, and \textit{DomiKnowS}. We identify the challenges within each facet that lay the foundation for identifying the expressivity of each framework in solving a variety of problems. Building on this foundation, we aim to spark transformative action and encourage the community to rethink this problem in novel ways.

Trustworthy AI requires reasoning systems that are not only powerful but also transparent and reliable. Automated Theorem Proving (ATP) is central to formal reasoning, yet classical binary resolution remains limited, as each step involves only two clauses and eliminates at most two literals. To overcome this bottleneck, the concept of standard contradiction and the theory of contradiction-separation-based deduction were introduced in 2018. This paper advances that framework by focusing on the systematic construction of standard contradictions. Specially, this study investigates construction methods for two principal forms of standard contradiction: the maximum triangular standard contradiction and the triangular-type standard contradiction. Building on these structures, we propose a procedure for determining the satisfiability and unsatisfiability of clause sets via maximum standard contradiction. Furthermore, we derive formulas for computing the number of standard sub-contradictions embedded within both the maximum triangular standard contradiction and the triangular-type standard contradiction. The results presented herein furnish the methodological basis for advancing contradiction-separation-based dynamic multi-clause automated deduction, thereby extending the expressive and deductive capabilities of automated reasoning systems beyond the classical binary paradigm.

We introduce Spectral NSR, a fully spectral neuro-symbolic reasoning framework that embeds logical rules as spectral templates and performs inference directly in the graph spectral domain. By leveraging graph signal processing (GSP) and frequency-selective filters grounded in the Laplacian eigenstructure of knowledge graphs, the architecture unifies the interpretability of symbolic reasoning with the scalability and adaptability of spectral learning. Beyond the core formulation, we incorporate a comprehensive set of extensions, including dynamic graph and basis learning, rational and diffusion filters for sharper spectral selectivity, mixture-of-spectral-experts for modular specialization, proof-guided training with spectral curricula, and uncertainty quantification for calibrated confidence. Additional enhancements such as large language model coupling, co-spectral transfer alignment, adversarial robustness, efficient GPU kernels, generalized Laplacians, and causal interventions further expand the versatility of the framework. Empirical evaluation on state-of-the-art reasoning benchmarks such as ProofWriter and CLUTRR demonstrates that Spectral NSR achieves superior accuracy, faster inference, improved robustness to adversarial perturbations, and higher interpretability compared to leading baselines including transformers, message-passing neural networks, and neuro-symbolic logic programming systems. Spectral attribution and proof-band agreement analyses confirm that model decisions align closely with symbolic proof structures, while transfer experiments validate effective domain adaptation through co-spectral alignment. These results establish Spectral NSR as a scalable and principled foundation for the next generation of reasoning systems, offering transparency, robustness, and generalization beyond conventional approaches.