Closure Certificates

Vishnu Murali; Ashutosh Trivedi; Majid Zamani

Home/Archive/arXiv

arXiv · 2305.17519

Closure Certificates

Vishnu Murali·Ashutosh Trivedi·Majid Zamani

View original source ↗Read source document ↗Export metadata ↓

Abstract

A barrier certificate, defined over the states of a dynamical system, is a real-valued function whose zero level set characterizes an inductively verifiable state invariant separating reachable states from unsafe ones. When combined with powerful decision procedures such as sum-of-squares programming (SOS) or satisfiability-modulo-theory solvers (SMT) barrier certificates enable an automated deductive verification approach to safety. The barrier certificate approach has been extended to refute omega-regular specifications by separating consecutive transitions of omega-automata in the hope of denying all accepting runs. Unsurprisingly, such tactics are bound to be conservative as refutation of recurrence properties requires reasoning about the well-foundedness of the transitive closure of the transition relation. This paper introduces the notion of closure certificates as a natural extension of barrier certificates from state invariants to transition invariants. We provide SOS and SMT based characterization for automating the search of closure certificates and demonstrate their effectiveness via a paradigmatic case study.

Explore related subjects

cs.LO cs.SY eess.SY

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Vishnu Murali, Ashutosh Trivedi, Majid Zamani. 2024-03-05. Closure Certificates. https://arxiv.org/abs/2305.17519

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Differential Equations as Fixpoints and Games

Games and fixpoints are unified by proving that first-order game logic GL and the first-order modal mu-calculus L_mu are proved to be equiexpressive and equivalent, thereby fully aligning their expressive and deductive power. That is, there is a semantics-preserving translation from GL to L_mu, and vice versa. And both translations are provability-preserving, while equivalence with there-and-back-again roundtrip translations are provable in both calculi. This is to be contrasted with the propositional case, where game logic is strictly less expressive than the modal mu-calculus (without adding sabotage games). The extensions with differential equations, differential game logic (dGL) and differential modal mu-calculus, are also proved equiexpressive and equivalent. Moreover, as the continuous dynamics are definable by fixpoints or via games, ODEs can be axiomatized completely and, as a consequence, infinitesimally robust properties of ODEs can be decided via proof search. Rational gameplay provably collapses the games into single-player games to yield a strong arithmetical completeness theorem for dGL with rational-time ODEs.

cs.LO↗

arXiv2026

Self-extensional logics of formal inconsistency: Decidability and limits for paraconsistency

RmbC is a self-extensional paraconsistent logic in the family of Logics of Formal Inconsistency (LFIs). This system is obtained from mbC (the basic LFI) by adding the replacement property via two global inference rules. RmbC is characterized by a non-explosive negation $\neg$ and a consistency operator $\circ$, which recovers the principle of explosion in a controlled way. Together with its principal axiomatic extensions, RmbC admits a standard Lindenbaum-Tarski algebraization, with Boolean algebras with LFI operators (BALFIs) as its algebraic semantics. In this paper, we study how far this self-extensional paraconsistent behavior can be extended axiomatically, starting from RmbC. We classify pairs of very natural consistency axioms according to whether they preserve paraconsistency or force classical collapse; identify six minimal explosive combinations that collapse to a single algebraic core; and isolate a separate structural obstruction for the combination of excluded middle for $\neg$ with an involutive negation. We also investigate, for the first time, the decidability of this family of self-extensional LFIs. As a first result, we prove the finite model property for RmbC with respect to BALFI semantics via an algebraic filtration, which yields decidability, and transfer this result to several paraconsistent axiomatic extensions of RmbC. Finally, we establish a coNEXPTIME upper bound for the validity problem of RmbC and a coNP-hardness lower bound, and prove coNP-completeness for the principal extensions containing one of the six minimal explosive pairs.

cs.LO↗

arXiv2026

The Stochastic Target Discounted-Sum Problem

The target discounted-sum problem (TDS) asks, given a finite integer alphabet $Σ$, a rational discount factor $λ$, and a rational target $t$, whether some infinite sequence over $Σ$ has discounted sum exactly $t$. This problem remains open and underlies several open questions in automata theory, games, and Markov decision processes. We introduce and solve its stochastic counterpart, the stochastic target discounted-sum problem, which replaces existence by computation of the probability. We show that the probability that a random sequence generated by a finite Markov chain has discounted sum $t$ is rational and computable in pseudo-polynomial time. We further show how to decide, in polynomial time, whether the discounted-sum distribution of a Markov chain is atomless, and how to approximate to an arbitrary precision the probability that the discounted sum exceeds a rational threshold. Our techniques for the stochastic TDS problem allow us to make progress on TDS objectives in stochastic games, which are known to be as hard as the TDS problem. Restricting the maximizing player to finite-memory strategies, while allowing the minimizing player to use arbitrary strategies, we reduce the value problem and the synthesis problem to corresponding problems for safety objectives in stochastic games. This yields computable optimal values and deterministic optimal strategies with pseudo-polynomially bounded memory for stochastic games, and results in pseudo-polynomial-time algorithms for special cases of Markov decision processes and deterministic two-player games.

cs.LO↗