characterize definable functions

This study addresses the fine-grained characterization of the arithmetic fragment $S_1$ within the linear-time hierarchy of bounded arithmetic. Methodologically, it introduces novel syntactic classes $Sigma^{ ext{UT}}_i$ and $Sigma^{ ext{IT}}_i$ to stratify sharply bounded quantifier blocks, thereby constructing systems $check{S}^i_1$, $TLS^i_1$, and $TSC^i_1$, and systematically analyzing their inclusion relations and relative strengths. The main contributions are threefold: (i) For the first time in bounded arithmetic, it precisely captures the function classes $mathbf{FLOGSPACE}$ and $mathbf{FSC}$—i.e., multi-valued functions computable in deterministic logarithmic space and in SC (polynomial time and polylogarithmic space), respectively—via the equivalences $TLS^1_1 equiv mathbf{FLOGSPACE}$ and $TSC^1_1 equiv mathbf{FSC}$; (ii) It establishes independence results linking function definability with closure under complementation in complexity classes, using length induction, the axiom of dependent choice, and witness oracle models; (iii) It refines the logical boundaries of the MRDP theorem under sublinear resource constraints.

This paper investigates the definability and separability problems for formulas with counting quantifiers in first-order logic (FO) and its modal counterparts: determining whether a counting formula is equivalent to one in a counting-free fragment, or whether two disjoint formulas can be separated by a counting-free formula. Focusing on the two-variable fragment FO² and the graded modal logic with inverses, nominals, and universal modalities, we establish that separability is undecidable in both logics—the first such result for these settings. We systematically analyze how individual modal operators affect computational complexity and show that definability reduces in polynomial time to the validity problem of the base logic. Furthermore, we pinpoint the exact complexity of separability across key fragments: undecidability, coNExpTime-completeness, and 2ExpTime-completeness.

This paper investigates the decidability of the widely applicable query containment problem in first-order logic, focusing on cases admitting structurally simple countermodels—characterized by bounded treewidth, cliquewidth, and a newly introduced width measure, partitionwidth. We introduce the notion of “width-bounded universal model sets” and develop a unified framework grounded in partitionwidth, systematically integrating model-theoretic methods, graph width theory, and existential rule techniques. Partitionwidth is employed as the central width parameter for the first time, subsuming and extending classical decidable classes such as Datalog± and guarded rules. We establish decidability for several classes of homomorphism-closed queries under finite-partitionwidth rule sets. Furthermore, we expose inherent limitations of finite-unification sets and propose principled repairs to restore decidability.

This paper addresses the precise expressibility characterization of Monadic Second-Order (MSO) and Guarded Second-Order (GSO) logic over finite structures in terms of Datalog programs. To establish expressibility criteria, we introduce an existential pebble game, combined with homomorphism-closure analysis, Constraint Satisfaction Problem (CSP) modeling, and countably categorical structure theory. This yields the first necessary and sufficient conditions for MSO/GSO definability by Datalog. In particular, we prove that every complement-closed, homomorphism-closed GSO class must be a finite union of countably categorical CSPs. We further propose the notion of *canonical Datalog programs*, enabling the construction of width-bounded (l,k)-optimal inference programs for homomorphism-closed MSO/GSO classes—programs that are sound and maximally complete among all sound Datalog programs. Our results establish a tight correspondence between logical expressibility and Datalog’s computational power.

解决模态语言等价性刻画问题，采用同态计数不可区分性方法，结合适当半环结构，针对特定标记转移系统类，涵盖多种模态逻辑及其扩展。

This work investigates the impact of unary and binary structure operations—such as disjoint union and Cartesian product—that are definable in first-order logic on the recognizability of classes of finite structures. By leveraging a backward translation theorem and a splitting theorem, it reduces first-order properties of output structures to finitely many first-order properties of input structures, preserving quantifier depth in the quantifier-free case and extending to logical fragments enriched with modulo-counting existential quantifiers. Combining first-order transductions with tree automata techniques over structures of bounded treewidth or cliquewidth, the paper establishes the recognizability of such finite structure classes under these operations and provides effective automata-based decision procedures, thereby forging a novel connection between structural recognizability and automata theory.

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 definability of nine function properties in the modal-temporal language \(L_{T\times W}\), which combines modal operator \(\Box\) with Priorian temporal operators \(G\) and \(H\), across five classes of ordered structures. By integrating modal logic and Priorian tense logic, the work introduces ordered frame semantics, minimal function frames (the \(O^2\) family), indexed languages, and a uniform domain condition to systematically analyze expressive power under both standard and strict semantics. The key findings reveal that function multiplicity is the primary constraint on definability; once multiplicity is controlled, strict semantics can define properties such as injectivity over reflexive orders, whereas the absence of connectedness in non-linear orders poses an inherent obstacle. The research further shows that in the original multi-flow setting, the language is weakly expressive and the two semantics coincide, yet under restricted frames most properties become definable, with three multiplicity-control mechanisms yielding consistent definability patterns.

This work addresses the long-standing lack of a rigorous foundational characterization of Hoare Logic and corrects erroneous claims in prior literature regarding its relationship with second-order logic. By establishing a precise equivalence between the partial correctness assertions of iterative programs and provability in a restricted form of standard second-order logic—where second-order quantification is limited to first-order predicates—the study provides the first formally sound logical foundation for Hoare Logic. The proof is carried out within a second-order logical system equipped with suitably constrained comprehension axioms, thereby demonstrating the exact correspondence between derivability in Hoare Logic and provability in this restricted second-order framework. In doing so, the paper rectifies two historically persistent misstatements and delivers the first reliable and exact theoretical underpinning for Hoare Logic.