termination analysis

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 work addresses the challenge of verification failures in loop invariant synthesis caused by local reasoning errors in large language models (LLMs). To this end, the authors propose LORIS, a novel framework that integrates formal verification of natural-language reasoning steps with feedback-driven iterative refinement. LORIS automatically translates LLM-generated natural language invariants into first-order logic and employs formal verification to detect logical inconsistencies, which are then used to generate targeted feedback for guiding the model to correct its reasoning trajectory. Experimental results demonstrate that LORIS achieves a 93.1% success rate on a benchmark of 460 C programs and exhibits strong robustness on 50 challenging programs involving nonlinear properties, substantially enhancing the reliability of LLM-based reasoning in program verification.

Existing approaches struggle to automatically disprove (positive) almost-sure termination of probabilistic term rewriting systems. This work proposes a novel method that extends qualitative cycle detection to quantitative analysis by embedding a random walk model into computation paths, thereby integrating cycle existence with expected step count estimation to automatically refute non-terminating behaviors. The approach is fully implemented in the tool AProVE and successfully applied to various classes of probabilistic term rewriting systems, significantly enhancing both the automation capability and applicability of termination analysis.

This paper addresses the lack of a unified metatheoretic characterization for program logics handling multi-branching effects—such as nondeterminism and probabilism. We propose a novel program logic framework centered on algebraic choice structures. Methodologically, we are the first to embed algebraic effects modeling directly into the core of Hoare logic, integrating modal semantics with a relatively complete proof system that supports general loops and uniform reasoning across effect types (e.g., nondeterministic and probabilistic). Our main contributions are: (1) the first relatively complete proof system for Hoare logic strictly extending it to cover multiple branching effects; (2) a unified metatheoretic account of multi-result programs; and (3) formal support for cross-model reuse of proof fragments—enabling verification transfer between distinct semantic models (e.g., relational, probabilistic, or game-based interpretations).

Termination analysis of programs has long relied on disparate, specialized logics, making it difficult to uniformly characterize termination, non-termination, and partial correctness—especially for nondeterministic and probabilistic programs. To address this, we propose the first unified program logic framework that simultaneously models and reasons about total correctness, partial correctness, and non-termination within a single formalism. Our logic extends Hoare logic by integrating incorrectness logic and semantics for nondeterminism and probability, and establishes a rigorous metatheory—including soundness and relative completeness. We validate its expressiveness and practicality through multiple case studies, demonstrating support for compositional verification across programming paradigms. The framework significantly enhances the generality, unification, and engineering applicability of termination analysis.

This work addresses the theoretical complexity and lack of a unified, reproducible formal foundation for the chase algorithm in existential rule reasoning by developing a formal framework within the Lean theorem prover. The framework clarifies subtle discrepancies in existing chase definitions, unifies various termination criteria—including the MFA class—and, for the first time in a formal system, supports existential rules with constants while characterizing the chase result as the core model in the absence of “homomorphic matches.” Leveraging Lean’s dependent type theory and libraries such as Mathlib and CSlab, the project formally verifies key properties of the chase-generated universal model, completes an initial proof of the core model theorem, and delivers a reusable, machine-checkable formal library for future research on existential rules.

This work formally verifies the correctness, completeness, and termination of DPLL-based SAT solvers. The authors model the DPLL procedure as a rule-driven state transition system and formalize the syntax and semantics of propositional logic in the Rocq proof assistant. They define both a basic DPLL system and an extended variant incorporating the pure literal rule, introducing a strategy abstraction to uniformly characterize solver behavior. Notably, this is the first complete formal verification in Rocq of an abstract DPLL system that includes the pure literal rule. Termination is established using well-founded relations and structural induction. The proposed framework enables the automatic derivation of terminating and sound solvers from strategies satisfying specified conditions, thereby unifying theoretical verification with concrete implementation.

This study addresses the challenge of program termination analysis—a classically undecidable problem that remains difficult in practical software verification—by presenting the first systematic evaluation of large language models (LLMs) on predicting termination of C programs. Leveraging the SV-Comp 2025 Termination Competition dataset and employing test-time scaling techniques, we assess state-of-the-art models including GPT-5, Claude Sonnet-4.5, and Code World Model (CWM). Our results show that GPT-5 and Claude Sonnet-4.5 achieve performance comparable to the competition’s top-performing tool, with CWM closely following the runner-up. However, LLMs struggle to generate valid termination witnesses, and their prediction accuracy degrades significantly as program length increases. This work thus reveals both the promising potential and current limitations of LLMs in formal verification tasks.

This work addresses the absence of a mechanized formal verification framework for primal-dual algorithm analysis. It presents the first systematic formalization in Isabelle/HOL that supports rigorous correctness and performance verification of such algorithms, unifying a diverse range of instances—from the classical Hungarian algorithm to modern Adwords algorithms—within a single coherent framework. By achieving machine-checked proofs for multiple primal-dual algorithms, this study not only establishes their formal correctness but also develops a reusable library of verified components. The resulting infrastructure offers a novel paradigm for trustworthy verification of combinatorial optimization algorithms, enhancing both reliability and reusability in formal methods applied to algorithmic analysis.

This work addresses the challenge of linear temporal logic (LTL) model checking for infinite-state systems by introducing the MoAT framework, which innovatively reduces LTL verification to a fair termination problem. By leveraging existing termination analysis tools—KoAT and LoAT—the approach enables efficient solving of the reduced problem. MoAT uniquely integrates automata-theoretic techniques with fair termination reductions, representing the first systematic application of termination analyzers to LTL model checking. Experimental evaluation demonstrates that MoAT achieves performance on par with state-of-the-art tools for LTL verification over infinite-state systems.