algebraic characterization

This paper addresses the lack of systematic support for theory morphisms and logical relations in the $λΠ$-calculus modulo rewriting framework. Methodologically, it introduces a unified extension mechanism that formally integrates both concepts for the first time within this framework and designs a pattern-based invariant verification procedure, reducing the proof of translation invariants to finite, decidable propositional checks. The main contributions are: (1) a structurally clear, machine-verifiable formalization of inductive translations—e.g., type erasure; (2) the first fully verified type-erasure instance in $λΠ$-calculus modulo rewriting; and (3) a reusable methodology for rigorously verifying the correctness of translations between formal systems.

Traditional accounts of “function” narrowly construe it as either a mapping or a structure-preserving property, failing to capture its generative role in syntactic construction and semantic interpretation. Method: The paper proposes a generative reconceptualization—defining function as a dual-structured generative principle capable of both syntactic item formation and semantic interpretation. To formalize this, it introduces Construction-Defining Functions (CDFs), establishing the first axiomatic framework that jointly ensures generativity and compositionality. The framework is modeled categorically using initial algebras and endofunctors, integrating type theory, model theory, and formal semantics. Contribution/Results: This work achieves the first cross-paradigmatic unification of structural generation mechanisms across logic, linguistic semantics, and computability theory. It provides a foundational redefinition of “function,” advancing deep integration between logical and linguistic theories while offering a rigorous, mathematically grounded basis for functional abstraction in formal systems.

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 study investigates the classes of regular languages maintainable in Patnaik and Immerman’s dynamic descriptive complexity framework DynFO when restricted to unary auxiliary relations, focusing on fragments of first-order logic—specifically, quantifier-free, positive existential, and those allowing a single alternation of quantifiers. By integrating algebraic theory of formal languages with fine-grained logical analysis, we provide the first precise algebraic characterizations of the language classes maintainable in the quantifier-free and positive existential fragments. Moreover, we strengthen Hesse’s result by showing that first-order formulas with a single quantifier alternation, together with unary auxiliary relations, suffice to maintain all regular languages. This work systematically elucidates the intricate correspondence among logical expressiveness, constraints on auxiliary relations, and the algebraic structure of regular languages.

Automated verification in separation logic (SL) has long relied on ad hoc heuristics, lacking a systematic metatheory and suffering from poor scalability. Method: This paper establishes the first general SL metatheory grounded in category theory and algebraic structures—specifically functors, homomorphisms, and modules over rings—systematically integrating abstract algebra into SL automation. The framework supports compositional model instantiation and modular predicate synthesis for any data structure admitting an algebraic characterization. All results are formally verified in Isabelle/HOL, and an automatic algebraic instantiation algorithm is developed. Contribution/Results: Experiments demonstrate fully automated algebraic modeling of complex imperative program semantics—including lists, trees, and graphs—and yield inference engines whose performance matches state-of-the-art hand-crafted systems. This approach decisively overcomes the scalability limitations inherent in heuristic-based methods.

This study addresses the problem of full definability in profunctor-based semantic models over groupoids—namely, ensuring that every semantic element is denoted by a proof net of multiplicative linear logic (MLL). To this end, the work introduces stability into profunctor semantics for the first time as a key criterion for definability, combining logical relations with categorical semantics to fully characterize definable profunctors. The main contribution establishes that every stable and total family of logical profunctors can be precisely defined by MLL proof nets augmented with the MIX rule, thereby forging a rigorous correspondence between stability and proof nets. This result confirms the model’s capacity for highly refined expressiveness in capturing program semantics.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

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.

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.

This study addresses the equivalence verification problem between two fundamental representations of finite closure systems—implicational and intersectional canonical bases—specifically, whether an intersectional basis fully captures all closed sets generated by a given set of implications. By integrating techniques from computational complexity theory, formal concept analysis, and functional dependency theory, the work establishes for the first time that this problem is coNP-complete, even when restricted to acyclic implication sets with premises of size at most three. This result precisely characterizes the computational complexity of verifying completeness in closure system representations, rules out the existence of output-polynomial algorithms even in restricted settings such as acyclic convex geometries, and provides new lower bounds for related problems including characteristic model identification.