certificate construction

To address the trustworthiness challenge in verifying multi-objective reachability, invariance, and long-run average rewards simultaneously in Markov decision processes (MDPs), this paper introduces the first logical query framework supporting existential and universal quantification over multiple objectives. Methodologically, we design a certified linear programming–based algorithm that jointly generates independently verifiable mathematical certificates, diagnostic schedulers, and minimal counterexample systems. Our approach transcends conventional single-objective certification by unifying formal correctness guarantees with human-interpretable explanations. The implemented prototype tool demonstrates, across multiple benchmarks, certificate compactness (averaging <5% of the state space), witness inspectability (100% manually verifiable), and computational feasibility on models with up to 10⁶ states.

This work addresses the challenge that counterexamples generated by formal verification often consist of numerous low-level Boolean variables, rendering them difficult for developers to interpret at the application-domain level. To bridge this gap, the paper proposes a novel hierarchical explanation method that integrates predicate relevance metrics with dependency graph analysis—a first-time fusion of these two techniques—to automatically extract human-readable, domain-oriented explanations from logical formulas. By leveraging formal modeling and a dedicated explanation-generation algorithm, the approach produces concise and semantically clear descriptions of failure causes across multiple case studies. Empirical results demonstrate that the method significantly outperforms existing techniques, offering effective support for fault localization in practical verification tasks.

This work establishes lower bounds on certificate size for local certification of distributed graph properties. Focusing on classical properties—including connectivity, matching, and coloring—it introduces the first local hardness reduction framework. By constructing local graph gadgets and applying information-theoretic analysis within the local model, the framework systematically transfers certificate-size lower bounds from one class of properties to another, overcoming the limitations of property-specific proofs. This approach achieves the first transferable lower-bound results in local certification. It uniformly establishes polynomial-scale (Ω(n)) lower bounds for multiple fundamental properties, exposing their inherent local complexity. The framework provides a general tool for local certification theory and advances the understanding of the fundamental limits of distributed verification efficiency.

This work studies local certification of $P_5$-free graphs—graphs containing no induced path on five vertices—with the goal of designing a subquadratic distributed verification scheme. To overcome the $Omega(n^2)$ certificate-size bottleneck of conventional approaches, we propose a novel certification framework based on structural graph decomposition, modular encoding, and localized label propagation. Our method achieves, for the first time, an $O(n^{3/2})$ upper bound on the certificate size. Each vertex verifies $P_5$-freeness using only its local neighborhood and an $O(n^{3/2})$-bit global certificate, with verification completing in constant rounds. We provide a rigorous theoretical analysis proving that this $O(n^{3/2})$ bound is asymptotically tight, thereby establishing an optimal complexity threshold for local certification of sparse graph properties. This result significantly advances the state of the art in distributed graph property testing.

This paper addresses the undecidability of knowledge-related problems—namely, deduction and static equivalence—in symbolic analysis of security protocols. To tackle this challenge, we introduce *graph-embedding term rewriting systems* and, for the first time, define their *contracting-convergent* subclass. While general graph-embedding systems are proven undecidable, contracting-convergent systems enforce structural constraints that ensure decidability of knowledge problems. We establish a decidability theory for this subclass and formally verify that several real-world protocol rule sets—including those modeling encryption, hashing, and digital signatures—belong to it. Furthermore, we develop a sound composition theorem integrating permutation equational theories, substantially broadening the class of analyzable protocols. Our core contributions are threefold: (i) the first formal framework of graph-embedding rewriting for protocol analysis; (ii) the novel notion of contracting convergence as a decidability-enforcing paradigm; and (iii) the unification of theoretical decidability with practical applicability for knowledge reasoning in security protocols.

This work addresses the efficient computation of sum-of-squares multipliers (i.e., certificates) for non-negative univariate polynomials within Archimedean saturated quadratic modules, thereby verifying their membership. To this end, the authors propose a novel symbolic algorithm that leverages the natural generators introduced by Kuhlmann and Marshall, incorporates the Basic Lemma to decompose non-negative factors, and employs a systematic case analysis to achieve, for the first time, a constructive transformation from natural to primitive generators. This approach establishes a complete framework for certificate construction in univariate Archimedean saturated quadratic modules. Implementation in Maple demonstrates the algorithm’s effectiveness and superiority, successfully handling several instances where RealCertify fails.

This work addresses the quotient admissibility problem over finite graph window rows, aiming to make maximally informative and guard-compatible decisions on evidence atoms under a given evidence partition. By introducing admissible evidence mappings, semantic labels, witness-support hypergraphs, and atom-level admissibility predicates, the authors devise a refinement-free atomic decision mechanism that outputs one of four outcomes: certificate, residual, low-confidence, or blockage. Key innovations include a union characterization of identifiable atom classes, a witness-hypergraph-based guard mechanism for certificate admissibility, and a formal characterization of blockage caused by projected label conflicts. The proposed algorithm achieves expected time complexity $O(B + I + n)$ and deterministic time complexity $O(B + I + n \log n)$ in the key-linear comparison model—where $n$ is the number of rows, $B$ the total evidence encoding length, and $I$ the hyperedge incidence size—matching theoretical optimality, while also establishing an indistinguishability lower bound for evaluators relying solely on residual magnitude.

This work addresses the verification of polynomial nonnegativity and optimization over compact semialgebraic sets. It proposes a disjunctive sum-of-squares approach that constructs a convergent hierarchy of lower bounds using multiple low-degree algebraic identities, each yielding semidefinite constraints of fixed size and solvable in parallel, without requiring additional optimization steps. The core contributions include two novel disjunctive Positivstellensatz theorems—one facilitating low-degree certificates of nonnegativity and the other enabling the construction of the hierarchy—and an extension of the framework to matrix copositivity testing. Experimental results demonstrate that the method is both effective and scalable across polynomial optimization, copositivity verification, and combinatorial optimization problems.

This work addresses the local certification of $k$-vertex-connectivity and $k$-edge-connectivity in graphs, where short certificates are assigned to vertices so that each node can verify global connectivity using only local neighborhood information. The authors present the first general-purpose local certification scheme applicable to arbitrary $k$, overcoming prior limitations restricted to small values of $k$. By integrating combinatorial tools such as branch decompositions, Eulerian subgraphs, and independent spanning trees, they devise efficient protocols grounded in structural graph properties. For $k \geq 3$, they establish a tight $O_k(\log n)$-bit upper bound for edge-connectivity certification in general graphs, matching the known lower bound; constant-size certificates are achieved for sparse graph classes. Additionally, they prove that certifying 2-vertex-connectivity in general graphs requires certificates of size $\Omega(\log \log^* n)$.

This work demonstrates a fundamental limitation of unified formal verification methods within the standard Turing model when applied to nontrivial semantic invariants. By formalizing “acceptable verification schemes” as generator–verifier pairs through a model-theoretic lens, the study integrates Rice’s theorem with formal verification frameworks to prove that such schemes implicitly induce undecidable decision procedures. Crucially, this impossibility stems from the computational behavior inherent to the verification mechanism itself, rather than from unprovable complexity-theoretic assumptions. Leveraging computability theory, model theory, and Coq-based formalization, the authors construct an extended structural model capturing semantic–syntactic interactions and rigorously establish that properties related to P vs NP and cryptographic assumptions such as one-way functions cannot be certified by any such unified method. A complete Coq implementation accompanies the theoretical results.