apply coinductive reasoning

Verifying temporal hyperactivity properties—such as “on every execution trace, there exists a subtrace satisfying a safety condition”—expressed as alternated universal–existential quantification (∀*∃*ψ) remains highly challenging. To address this, we propose an approximation-based verification method grounded in coinductive relations. This work introduces the first systematic application of coinduction to hyperactivity property verification and presents HyCo: a fully Coq-formalized, sound reasoning framework for hyperproperties. HyCo integrates coinductive logic, temporal hyperlogic, safety relation modeling, and imperative program semantics—including support for nondeterminism and I/O. Experimental evaluation demonstrates that HyCo effectively verifies canonical reactive-system hyperactivity properties, such as noninterference and observational termination. The framework significantly enhances both the intuitiveness and mechanizability of hyperproperty reasoning, advancing the formal verification of complex temporal hyperproperties.

This work addresses the absence of a causal stream semantics in symmetric monoidal categories equipped with a monad. We introduce *monoidal streams*, a novel categorical structure providing a unified semantic framework for process theories based on such categories. Methodologically, we (i) give the first coinductive definition of monoidal streams; (ii) generalize classical Cartesian stream semantics to arbitrary symmetric monoidal categories with a monad; and (iii) construct a category of monoidal streams endowed with a feedback operator, augmented by a coinductive string diagram calculus for formal reasoning about feedback systems. Our principal contributions are: (i) establishing a rigorous categorical semantics for stochastic dataflow languages; (ii) unifying purely functional and probabilistic processes within a single compositional model; and (iii) enabling graphical, verifiable reasoning about feedback systems—including signal flow graphs—via the introduced string diagrammatic syntax and its coinductive semantics.

This work addresses the formal benchmarking problem for concurrent calculi by introducing, for the first time, a formalization of strong barbed similarity in the π-calculus extended with replication. Leveraging the Beluga proof assistant, the approach combines higher-order abstract syntax (HOAS) with copattern-based coinductive reasoning to uniformly characterize barb observations and internal transitions in a coinductive framework, thereby defining behavioral equivalence. This methodology yields a concise and compositional proof structure, successfully establishing the context lemma and compatibility properties required for strong barbed precongruence. The result provides a novel technical pathway for the formal verification of concurrent systems.

This work addresses the longstanding challenge of unifying inductive and coinductive reasoning by proposing a novel cyclic induction method grounded in the principle of cyclicity, which is seamlessly integrated with existing cyclic coinduction into a unified framework. For the first time, the principle of cyclicity is systematically applied to inductive proof, enabling flexible combinations of inductive and coinductive strategies within formal verification. The algorithm has been implemented in the CIRC prover, and experimental results demonstrate its expressiveness, flexibility, and competitive performance across a range of scenarios, significantly enhancing the capabilities of automated reasoning.

This paper addresses the decidability of proof search in intuitionistic logic. Method: It extends the coinductive proof search (CoIPS) methodology from minimal implicational logic to polarized intuitionistic logic LJP. Specifically, it establishes, for the first time, a coinductive characterization of focused proofs in LJP alongside an equivalent inductive syntactic representation; introduces a decidability framework based on recursive predicates; and constructs a faithful negative translation from full intuitionistic propositional logic LJT into LJP. Contributions/Results: It achieves full decidability of both existence and finiteness of focused witnesses in LJP. Consequently, it yields novel, unified decidability proofs for inhabitation and finiteness in LJT—enhancing algorithmic reliability through integrated treatment. Moreover, it provides a transferable polarized modeling pathway applicable to variants such as LJQ.

Safety verification of complex systems is often hindered by the difficulty of constructing inductive invariants, intricate Boolean structures, and extensive quantifier alternations. This work proposes an incremental safety proof method that integrates forward reasoning, backward reasoning under time reversal, and a prophecy variable mechanism to decompose global invariants into simpler subgoals. Without expanding the set of candidate invariant formulas, the approach strictly enhances proof power while substantially reducing the logical complexity of required invariants. Experiments on Paxos, its variants, and the Raft protocol demonstrate that the method effectively eliminates complex Boolean structures, reduces quantifier usage and alternation depth, and significantly shrinks the invariant search space.

Standard simulation techniques struggle to verify liveness properties, and existing notions of fair simulation are often too complex for interactive verification. This work proposes a family of “approximate fair simulation” relations tailored to transition systems equipped with Büchi fairness conditions. By simplifying the nested inductive–coinductive structures inherent in traditional approaches, our method introduces a stronger and more user-friendly reasoning mechanism. We formalize this framework within a fixed-point logic and develop a corresponding deductive system, which we mechanize and prove correct in the Rocq proof assistant. Case studies demonstrate the effectiveness and practicality of our approach for interactive verification of fairness properties.

This work addresses the challenge of automatically inferring loop invariants in programs with multiple interacting loops. It proposes a novel neurosymbolic framework that uniquely integrates obligation-guided reasoning with weakest precondition refinement, explicitly modeling inter-loop dependencies through loop-level abstractions and propagating proof obligations accordingly. By synergistically combining large language models with formal verification techniques, the approach introduces a deductive feedback mechanism to iteratively refine candidate invariants. Evaluated on a new benchmark comprising classical algorithms, the method successfully solves 72 out of 82 multi-loop problems, substantially outperforming existing approaches, while also maintaining state-of-the-art performance on single-loop tasks.

This work addresses the bidirectional translation between non-well-founded and cyclic proofs in Linear Temporal Logic (LTL) by proposing a unified framework based on linear nested sequent calculi. The central challenge lies in identifying and unfolding cycles, for which the authors introduce a saturated recursive normal form to effectively recognize cyclic structures and devise a rule-forwarding mechanism to syntactically reconstruct cyclic proofs into non-well-founded ones. This approach establishes a precise correspondence between the two proof systems, resolving a long-standing problem of mutual translatability in LTL. Moreover, it enriches the proof-theoretic toolkit for multi-conclusion sequent calculi and lays the groundwork for more expressive reasoning systems in temporal logic.