normalize logical formulas

This work addresses the problem of verbose and semantically opaque partial assignments induced by CNF conversion in SAT/SMT enumeration. We systematically evaluate the suitability of Tseitin versus Plaisted–Greenbaum (PG) encodings in enumeration contexts. Theoretically and empirically, we show that Tseitin encoding inherently impedes generation of short partial assignments, whereas PG encoding—when combined with negation normal form (NNF) preprocessing—guarantees that each enumerated solution corresponds to a minimal, semantically transparent partial assignment. This synergistic approach is the first to provably eliminate encoding-induced assignment redundancy in enumeration. Evaluated on SMT-LIB benchmarks, it reduces both the number of partial solutions and total runtime by 1–3 orders of magnitude, demonstrating strong theoretical soundness and practical efficacy.

This paper addresses the isomorphism identification and canonical labeling problem for finite algebraic structures (e.g., groups, semigroups). We propose a SAT-based lexicographic normalization method that computes the lexicographically smallest representation of a structure via domain element reordering. Our approach employs a black-box SAT framework to progressively construct the minimal representative and introduces a novel constraint propagation mechanism that substantially reduces the number of SAT solver invocations. Unlike prior approaches, it supports arbitrary finite algebraic structures and achieves fully automated lexicographic canonicalization for the first time. We implement an open-source tool that processes structures of order up to 1,000 in real time. Empirical evaluation demonstrates significant speedups over naive enumeration—reducing both total solving time and SAT calls—while maintaining correctness and scalability across diverse algebraic classes, including groups, semigroups, and magmas.

This work addresses the low accuracy and unreliable verification in automatic natural-language-to-Lean-4 formalization (autoformalization). We propose a two-stage optimization framework integrating scalable type checking with self-consistency filtering. Methodologically, we combine Lean 4’s native type checker, symbolic equivalence checking, large language model (LLM) generation and re-ranking, and incorporate self-consistent sampling to enhance output reliability. Our key contributions are: (1) the first Lean-4–oriented RLM25 mathematical dataset; (2) a corrected ProofNet benchmark and a new ProofNetVerif benchmark augmented with human-verified annotations; and (3) open-sourcing of all code, a novel symbolic equivalence tool, and three benchmarks. On ProofNet, our approach achieves an +18.4% absolute accuracy gain, significantly advancing the practicality of autoformalization for theorem proving.

This work addresses the challenge of verifying equivalence between target representations—such as decision-DNNF—and their original CNF encodings in knowledge compilation. Methodologically, it introduces (1) Partitioned-Operation Graphs (POGs) as a unified intermediate representation; (2) the Certified POG (CPOG) proof framework, enabling structured, correctness-preserving compilation from CNF to POG; and (3) full formal verification in Lean 4 of the compiler, proof generator, and model counter. Contributions include: the first end-to-end, machine-checked correctness guarantee for the entire knowledge compilation pipeline; automated verification of D4-generated POGs; empirical evaluation on standard model counting benchmarks; and the first mathematically verified toolchain supporting both weighted and unweighted model counting. The framework ensures semantic equivalence at every compilation step, thereby bridging the gap between practical knowledge compilation tools and formal correctness guarantees.

This work addresses the challenge of automatically verifying strong equivalence for complex logic programs in Answer Set Programming (ASP), overcoming the limitation of the existing tool Anthem, which supports only positive programs. We introduce a novel translation σ* from equilibrium (HT) logic to classical logic and extend the τ* translation to handle negation, simple choice rules, and pool constructs. For the first time, strong equivalence of ASP programs with negation and choice is formally characterized and mechanized within a classical-logic framework. By integrating HT-semantic analysis, logical bridging, and automated theorem proving, we implement an enhanced version of Anthem. Experimental evaluation demonstrates that our tool successfully verifies strong equivalence for diverse nonmonotonic programs—including those featuring negation, choice, and pools—significantly broadening the class of verifiable programs. This advancement provides a sound and scalable foundation for industrial-strength ASP program optimization and refactoring.

This work addresses the inefficiency of query operations—such as uniform sampling, direct access, and model enumeration—on propositional formulas in conjunctive normal form (CNF) after compilation into deterministic decomposable negation normal form (d-DNNF). To overcome this limitation, the authors propose a preprocessing technique that preserves only the model count rather than full logical equivalence. Applied prior to CNF-to-d-DNNF compilation, this method optimizes the input formula while retaining essential preprocessing information to accelerate downstream queries. The study presents the first systematic evaluation of model-count-preserving preprocessors, demonstrating their ability to substantially enhance the performance of diverse query tasks on d-DNNF representations, thereby surpassing the constraints of traditional equivalence-preserving preprocessing. Extensive experiments across multiple benchmark domains confirm the approach’s efficiency and robustness.

This work addresses the high degree of manual effort and tediousness inherent in existing automated reasoning algorithms—such as those for hyper-exponential quantifier elimination—for complexity analysis. The paper proposes a higher-order abstract interpretation framework grounded in operator semantics, which automatically abstracts symbolic programs into numerical recurrence relations. By integrating termination analysis, fixed-point theory, and SMT solving techniques, the method enables fully automated derivation and verification of asymptotic upper bounds on computational complexity. This approach substantially reduces human intervention while significantly enhancing the automation, efficiency, and scalability of complexity analysis for intricate algorithms.

This study investigates the application of Craig interpolation and Beth definability to the simplification of logical expressions and database queries. By integrating model-theoretic preservation theorems with semantic-syntactic transformations, the work introduces a novel algorithmic framework that takes formal proofs as input to automatically generate interpolants or explicit definitions. Building on this foundation, it develops a new form of interpolation tailored to query rewriting in databases. The approach not only renders classical logical results effectively computable but also provides both theoretical grounding and practical algorithms for query optimization, thereby substantially expanding the applicability of interpolation and definability techniques in the database domain.