expressivity analysis

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.

Linear Temporal Logic (LTL) and Computation Tree Logic (CTL) lack direct expressiveness for hyperproperties—properties relating multiple computation traces—limiting their applicability to information-flow security and other hyperproperty verification tasks. Method: We introduce *synchronous team semantics*, the first unified semantic framework for LTL and CTL that natively supports hyperproperty reasoning. This semantics enables natural specification of trace-relational properties (e.g., noninterference) and overcomes HyperLTL’s restriction to prefix-aligned quantification, while improving both expressive flexibility and algorithmic efficiency. Results: We precisely characterize the computational complexity of key reasoning tasks: under LTL team semantics, satisfiability is PSPACE-complete and model checking EXPTIME-complete; under CTL team semantics, the complexities are reversed—EXPTIME-complete for satisfiability and PSPACE-complete for model checking. Our work establishes a novel paradigm for equipping temporal logics with rigorous, efficient hyperproperty verification capabilities.

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 paper addresses the conceptual fragmentation in sequent-style proof systems across modal, temporal, intuitionistic, conditional, and substructural logics, as well as the ill-defined distinction between “internal” and “external” calculi. We propose a unified classification framework grounded in the fundamental data structure of sequents. By systematically analyzing extant sequent calculi, we develop a hierarchical taxonomy and conduct meta-theoretic analysis of upward and downward logical translations. Crucially, we formally define—and thereby eliminate—the spurious internal/external distinction for the first time. We prove that upward translation preserves structural integrity, whereas downward translation is generally infeasible. Our results establish a principled comparability benchmark and formal assessment toolkit for cross-logical proof systems, advancing the unified modeling of structural proof theory and enabling robust automation support.

This study investigates the interplay between expressive power and computational complexity in restricted operator fragments of propositional and modal logics. By integrating Post’s lattice theory with modal logic frameworks, the work introduces the notion of “simple modal fragments,” extending Boolean clone theory to the modal setting and establishing a unified parameterized analysis methodology. The paper systematically characterizes the boundaries of these fragments—parameterized by admissible logical operators—with respect to decidability, computational complexity (including dichotomy results), and learnability (encompassing teachability and exact learnability). This approach unifies two long-standing, independent research directions, offering a cohesive theoretical framework for understanding the structure and learnability of logical fragments.

This study addresses the challenge of effectively transferring soundness and completeness from highly expressive modal logics to weaker linguistic systems. To this end, it proposes a novel strategy that leverages semantic insensitivity to carry over soundness and employs faithful translations to embed the canonical model of the target logic into the framework of normal modal logic, thereby inheriting completeness. The approach unifies various notions of operator definability and allows the standard relational semantics to be inherited without explicitly specifying an accessibility relation. Consequently, it offers a general and streamlined method for constructing semantics for weak modal languages, substantially reducing the complexity of their metatheoretic analysis.

This work presents the first systematic investigation into the capability of large language models (LLMs) to generate program specifications involving higher-order logical constructs, which are essential for expressing complex verification properties yet remain beyond the reach of existing LLMs that predominantly handle basic syntactic forms. The authors design four syntactic configurations spanning different levels of abstraction and establish a comprehensive evaluation framework to assess a range of representative LLMs on standard verification benchmarks. Experimental results demonstrate that LLMs can effectively produce valid higher-order logical expressions; moreover, integrating logical constructs with base syntax significantly enhances verification efficacy and robustness without substantially increasing verification overhead. The study also reveals distinct advantages of two refinement paradigms in specification generation.

This work addresses the lack of a unified axiomatic framework for formalizing programming languages and security protocols. We propose in Lean an intrinsically typed, many-sorted, hybrid multimodal logic that integrates ideas from algebraic specification and dynamic logic. By leveraging an intrinsic sorting mechanism, well-sorted formulas directly correspond to well-typed terms in Lean, thereby avoiding extraneous syntactic overhead. The framework supports domain-specific languages for defining signatures, axioms, and reasoning about system behavior. We formally verify the meta-theoretic correctness of the logical system and demonstrate its expressiveness and practicality through successful applications to imperative program verification, the BAN logic for security protocols, and the S5 modal system.

This work investigates the logical characterization and synthesizability of ω-regular positional properties in reactive synthesis games. Leveraging language cluster theory, game semantics, ω-automata, and model checking techniques for alternating-time temporal logic, it establishes that all such properties are expressible in linear temporal logic (LTL) and provides necessary and sufficient conditions for positionality. The main contributions include demonstrating the equivalence between ω-regular positionality and LTL expressibility, proving the nonexistence of any positional class that is both closed under Boolean operations and contains all prefix-independent properties, identifying several subclasses with favorable synthesis behavior, and isolating fragments of alternating-time temporal logic that admit efficient model checking while precisely delineating their expressive power.