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  1. A barrier certificate, defined over the states of a dynamical system, is a real-valued function whose zero level set characterizes an in- ductively 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 de- ductive verification approach to safety. The barrier certificate ap- proach has been extended to refute LTL and l -regular specifications by separating consecutive transitions of corresponding l -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 aug- ment these definitions with SOS and SMT based characterization for automating the search of closure certificates and demonstrate their effectiveness over some case studies. 
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    Free, publicly-accessible full text available May 14, 2025
  2. Notions of transition invariants and closure certificates have seen recent use in the formal verification of controlled dy- namical systems against ω-regular properties. The existing approaches face limitations in two directions. First, they re- quire a closed-form mathematical expression representing the model of the system. Such an expression may be difficult to find, too complex to be of any use, or unavailable due to security or privacy constraints. Second, finding such invari- ants typically rely on optimization techniques such as sum-of- squares (SOS) or satisfiability modulo theory (SMT) solvers. This restricts the classes of systems that need to be formally verified. To address these drawbacks, we introduce a notion of neural closure certificates. We present a data-driven algo- rithm that trains a neural network to represent a closure cer- tificate. Our approach is formally correct under some mild as- sumptions, i.e., one is able to formally show that the unknown system satisfies the ω-regular property of interest if a neural closure certificate can be computed. Finally, we demonstrate the efficacy of our approach with relevant case studies. 
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    Free, publicly-accessible full text available February 20, 2025
  3. Bouajjani, A ; Holík, L. ; Wu, Z. (Ed.)
    This paper presents an optimization based framework to automate system repair against omega-regular properties. In the proposed formalization of optimal repair, the systems are represented as Kripke structures, the properties as omega-regular languages, and the repair space as repair machines—weighted omega-regular transducers equipped with Büchi conditions—that rewrite strings and associate a cost sequence to these rewritings. To translate the resulting cost-sequences to easily interpretable payoffs, we consider several aggregator functions to map cost sequences to numbers—including limit superior, supremum, discounted-sum, and average-sum—to define quantitative cost semantics. The problem of optimal repair, then, is to determine whether traces from a given system can be rewritten to satisfy an omega-regular property when the allowed cost is bounded by a given threshold. We also consider the dual challenge of impair verification that assumes that the rewritings are resolved adversarially under some given cost restriction, and asks to decide if all traces of the system satisfy the specification irrespective of the rewritings. With a negative result to the impair verification problem, we study the problem of designing a minimal mask of the Kripke structure such that the resulting traces satisfy the specifications despite the threshold-bounded impairment. We dub this problem as the mask synthesis problem. This paper presents automata-theoretic solutions to repair synthesis, impair verification, and mask synthesis problem for limit superior, supremum, discounted-sum, and average-sum cost semantics. 
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