computability theory

This paper addresses the termination analysis of Linear Constraint Programs (LCPs)—a fundamentally challenging problem due to the undecidability of the halting problem for general numerical programs. To tackle LCPs with numerical variables and linear transition constraints, the authors develop a systematic analytical framework. Methodologically, they unify decidable techniques—including ranking functions and disjunctive well-founded transition invariants—to characterize their expressive power and algorithmic complexity boundaries; additionally, they introduce the first sound and complete non-termination witness generation mechanism, yielding verifiable counterexamples for undecidable cases. Through integrated modeling, complexity classification, and trade-off analysis among techniques, the work precisely delineates the applicability conditions and theoretical limits of existing approaches. The contributions significantly advance both the theoretical foundations and practical applicability of LCP termination analysis.

This paper addresses the verification of semantic properties—such as program correctness and termination—for sets, relations, and computations defined by elementary inference systems. To overcome the fundamental limitation that canonical models are often noncomputable, we propose a novel method that eschews reliance on canonical models entirely: instead, semantic properties are decided via first-order satisfiability in *arbitrary* models. Technically, we formalize inference systems as Gentzen-style elementary deductive systems, integrate Horn clause theories with proof-tree structural modeling, and leverage automated first-order satisfiability checking for verification. Our principal contribution is a general logical decision framework for rewriting-based computational models (e.g., programming language semantics), enabling formal, machine-checkable proofs of semantic property validity or invalidity. This approach significantly enhances both the practical applicability and decidability of semantic analysis.

This work proposes a formalization of algorithms within an intensional computability framework and clarifies their relationship to implementations in computational models. Treating computational models as monoid actions on configuration spaces, programs are modeled as dynamical systems constrained by such actions. Algorithms are defined as finite directed graphs of partial maps over edge-labeled abstract data structures, explicitly separating control flow from data operations. By leveraging tools from category theory, dynamical systems theory, and graph theory, the approach constructs a rigorous semantic framework that, for the first time, treats algorithms as abstract specifications of computational behavior and precisely characterizes the structure-preserving implementation relation between programs and algorithms, thereby deepening our understanding of the nature of computation.

This paper addresses fundamental open problems in computational complexity—such as P vs NP and the nonexistence of polynomial-size circuits for SAT—within weak formal systems like bounded arithmetic $S^1_2$. Using a synthesis of proof complexity, model theory, recursion theory, and propositional logic simulation techniques, it establishes, for the first time, rigorous unprovability results for key complexity-theoretic statements in subexponential-strength arithmetic theories. The main contributions are: (1) proving that assertions such as “SAT has no polynomial-size circuits” are independent of $S^1_2$; (2) establishing a tight correspondence between proof complexity lower bounds and circuit lower bounds; and (3) exposing deep metatheoretic barriers preventing any feasible formal proof of P = NP, thereby offering a novel logical foundation for complexity theory.

This work addresses the critical problem of unexplainable unrealizability verification in program synthesis. We propose the first Hoare-style unrealizability logic—a formal deductive system that models the semantic behavior of program search spaces and systematically approximates the collective execution behavior of infinite program sets via sound, machine-checkable inference rules. This transforms opaque, black-box unrealizability proofs into human-understandable, machine-verifiable structured derivations. Our key contributions are threefold: (1) unifying and formalizing the implicit reasoning principles underlying existing unrealizability tools; (2) enabling inductive assertion synthesis and rigorous formal verification of unrealizability; and (3) yielding a compositional, extensible proof infrastructure. The logic enhances transparency, theoretical rigor, and tool-supported verifiability in program synthesis analysis—establishing a foundational framework for principled, explainable synthesis correctness reasoning.

This work proposes a novel axiomatic number-theoretic framework, termed clarithmetics (e.g., CLA4–CLA7, CLA11), grounded in Computability Logic (CoL). Unlike traditional weak arithmetic systems that excessively weaken Peano Arithmetic to control computational complexity—thereby sacrificing essential arithmetical content and intensional expressiveness—clarithmetics retain the full arithmetic strength of Peano Arithmetic while explicitly capturing computational resources such as time, space, and amplitude. The resulting systems enable the direct extraction of near-optimal algorithms from proofs, exhibit strong intensional completeness, and achieve sound and complete program synthesis within natural complexity classes. Consequently, they significantly outperform bounded arithmetic in both specification expressiveness and algorithm extraction capabilities.

This work addresses the limitation of traditional introductory computer science curricula, which often emphasize isolated knowledge points while neglecting the underlying proof techniques and abstract structures essential for cultivating computational thinking in beginners. To remedy this, the paper proposes a novel pedagogical paradigm centered on universal proof strategies and abstract frameworks, using the transitive closure of relations as a representative case study. By integrating tools such as the Kleene star, quantale theory, and closure operators over complete lattices, the approach constructs a cohesive bridge linking logic, algebra, and computational reasoning. This method yields a generalizable instructional framework that significantly enhances students’ structural understanding and analytical capacity regarding foundational concepts.

This work addresses the absence of high-quality, contamination-free formal benchmarks for research-level mathematical problems, which has hindered accurate evaluation of automated reasoning systems in genuine mathematical discovery. The authors introduce the first continuously evolving Lean 4 formalized mathematics benchmark, comprising 2,615 problems—including 1,029 open conjectures—designed to enable clean evaluation and foster human–AI collaborative verification. By integrating community collaboration, a standardized evaluation subset, and an AI-driven proof auditing pipeline, the benchmark supports rigorous, reproducible assessment. It has already facilitated multiple novel mathematical discoveries, including resolutions of several open conjectures, and provides reproducible baseline results that clearly delineate the current frontier of automated reasoning in research-level mathematics.

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.

We describe a new method of finding interpolants for classical logic using certain refutation system as a starting point. Refutation can be thought of as an alternative approach to the analysis of formal systems: instead of focusing on which formulas provably belong to a given logic, it shows which formulas are to be rejected. Thus, it provides a mirror proof system. As it turns out, the benefits of such an approach go well beyond the area of refutation calculi themselves. We provide one such example in the shape of an interpolant-searching method. To be sure, a number of such methods are already in use. The novelty of our proposal lies in the fact that it can be considered as based on a non-binary version of resolution.