equivalence checking

Verifying the equivalence of implementations of the same large model across different frameworks is highly challenging due to significant discrepancies in operator decomposition, tensor layouts, and fusion strategies. This work proposes Emerge, a framework that unifies two implementations into a single e-graph representation, infers candidate equivalences guided by runtime values, and automatically synthesizes rewrite rules on demand without manual intervention. By integrating symbolic SMT-based verification with constraint-aware randomized testing, Emerge supports scenarios involving opaque operators. Experimental results demonstrate that Emerge successfully verifies equivalence for correct implementation pairs, detects 10 out of 13 known bugs, and uncovers 8 previously unknown issues confirmed by developers. The automatically generated block-level rewrite rules achieve effectiveness comparable to handcrafted ones.

This work addresses the challenge of verifying equivalence between string diagrams under different syntactic representations by proposing a normalization method based on term rewriting systems, with a focus on two key classes of string diagrams arising in quantum circuit equivalence verification. By introducing two-dimensional diagrammatic terms generated through sequential and parallel composition, and designing rewrite rules within a framework of deformation equivalence, the paper establishes—for the first time—a provably terminating and confluent normalization system for these diagram classes. The termination and confluence properties are rigorously formalized and verified using the Isabelle/HOL proof assistant, thereby providing a solid theoretical foundation and a reliable implementation pathway for automated reasoning about string diagram equivalence.

Traditional formal verification lacks mechanisms for knowledge accumulation and cross-system reuse, making it difficult to transfer specifications, contracts, and proofs. This work proposes a novel paradigm that integrates artificial intelligence with formal methods, pioneering the combination of large language models and graph-based representations to enable semantic guidance across heterogeneous notations and abstraction levels. By leveraging automated contract synthesis, semantic artifact reuse, and compositional refinement theory, the authors construct a hybrid reasoning framework that ensures formal reliability while supporting continuous synthesis and migration of verification artifacts. This approach lays the foundation for a cumulative and evolvable verification ecosystem, paving the way toward scalable, knowledge-driven next-generation verification systems.

This work addresses the challenge of automatically verifying strong equivalence for complex logic programs in Answer Set Programming (ASP), overcoming the limitation of the existing tool Anthem, which supports only positive programs. We introduce a novel translation σ* from equilibrium (HT) logic to classical logic and extend the τ* translation to handle negation, simple choice rules, and pool constructs. For the first time, strong equivalence of ASP programs with negation and choice is formally characterized and mechanized within a classical-logic framework. By integrating HT-semantic analysis, logical bridging, and automated theorem proving, we implement an enhanced version of Anthem. Experimental evaluation demonstrates that our tool successfully verifies strong equivalence for diverse nonmonotonic programs—including those featuring negation, choice, and pools—significantly broadening the class of verifiable programs. This advancement provides a sound and scalable foundation for industrial-strength ASP program optimization and refactoring.

This work proposes a hybrid concrete-symbolic interpretation method to efficiently verify semantic equivalence between original and optimized programs in MLIR, ensuring the correctness of optimization transformations. The approach supports diverse syntactic, scheduling, and memory representations and theoretically achieves linear-time complexity for equivalence checking. Building upon this method, the authors develop a formal verifier for a subset of MLIR and successfully apply it to the AMD MLIR-AIR and MLIR-AIE toolchains as well as the standard mlir-opt infrastructure. Evaluation across hundreds of benchmark variants demonstrates the verifier’s effectiveness in validating optimization pipelines, significantly enhancing the reliability of compiler optimizations within the MLIR ecosystem.

This work addresses the limited accuracy and generalization of large language models in code semantic equivalence reasoning by proposing a self-play training framework grounded in semantic equivalence. The approach uniquely integrates formal proofs from Liquid Haskell with execution-based counterexamples to construct supervision signals, employing adversarial training between a generator and an evaluator alongside a difficulty-aware curriculum learning strategy. Key contributions include a formal verification–guided supervision mechanism for semantic equivalence, the creation of the OpInstruct-HSx dataset, and substantial empirical gains: up to a 13.3 percentage point accuracy improvement on EquiBench and consistent performance gains on PySecDB, collectively demonstrating the critical role of formal semantics in enhancing model reasoning capabilities.

This work addresses the prevailing lack of systematic understanding of foundational formal theories in current AI compiler design, which hinders rigorous evaluation of the completeness and desirability of intermediate representations and compilation abstractions. For the first time, it systematically establishes precise correspondences between core mechanisms of MLIR—such as term rewriting systems, refinement calculi, and abstract interpretation—and classical formal theories. By grounding compiler abstractions in formal semantics, the paper clarifies the theoretical underpinnings of these constructs, articulates a precise notion of “design completeness,” and provides assessable criteria and guiding principles to navigate trade-offs between engineering pragmatism and theoretical ideals.

Existing program analysis techniques lack a unified mathematical framework capable of simultaneously addressing type checking, vulnerability detection, and behavioral equivalence verification. This work proposes the first unified model based on Čech cohomology, representing program semantics as a presheaf over the category of program sites. In this framework, the zeroth cohomology group \(H^0\) captures globally consistent typing, while the first cohomology group \(H^1\) characterizes errors or failures of equivalence arising from local inconsistencies. By integrating tools from algebraic geometry to model Python semantics and leveraging the Mayer–Vietoris sequence, the approach enables incremental obstruction analysis and minimal repair counting. The resulting tool, Deppy, achieves 100% vulnerability recall, 99% zero-false-positive equivalence accuracy, and 98% specification conformance accuracy on a benchmark suite of 375 programs, substantially outperforming state-of-the-art type checkers such as mypy and pyright.

Traditional algebraic rewriting is unreliable for expressions involving measurements due to domain inconsistencies arising from repeated observations and division operations. This work proposes a unified semantic framework that simultaneously tracks both the provenance and definedness of expressions, enabling sound one-way rewriting and interchangeability judgments. By introducing label-sensitive bracketing semantics, admissible domain refinement, and a relative variant of support sets, the authors develop domain-safe rewriting rules and formally prove restoration and strictness theorems. All results are fully formalized in Lean 4 without any use of `sorry`, revealing fundamental limitations: simplifications are generally irreversible, equivalence over a common domain is insufficient, and label erasure inherently causes information loss.