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In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with interval-based arithmetic constraint solving. Our approach deviates substantially from lazy theorem proving approaches in that it directly controls arithmetic constraint propagation from the SAT solver rather than delegating arithmetic decisions to a subordinate solver. Through this tight integration, all the algorithmic enhancements that were instrumental to the enormous performance gains recently achieved in propositional SAT solving carry over smoothly to the rich domain of non-linear arithmetic constraints. As a consequence, our approach is able to handle large constraint systems with extremely complex Boolean structure, involving Boolean combinations of multiple thousand arithmetic constraints over some thousands of variables.

In this paper we present HySAT, a bounded model checker for linear hybrid systems, incorporating a tight integration of a DPLL-based pseudo-Boolean SAT solver and a linear programming routine as core engine. In contrast to related tools like MathSAT, ICS, or CVC, our tool exploits the various optimizations that arise naturally in the bounded model checking context, e.g. isomorphic replication of learned conflict clauses or tailored decision strategies, and extends them to the hybrid domain. We demonstrate that those optimizations are crucial to the performance of the tool.

Aiming at automatic verification and analysis techniques for hybrid discrete-continuous systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODE-LP, as a state-of-the-art interval solver for ODEs, and with bracketing systems, which exploit monotonicity properties allowing to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply the combined iSAT-ODE solver to the analysis of a variety of non-linear hybrid systems by solving predicative encodings of reachability properties and of an inductive stability argument, and evaluate the impact of the different enclosure methods, decision heuristics and their combination. Our experiments include classic benchmarks A preliminary version of this paper appeared in [6]. from the literature, as well as a newly-designed conveyor belt system that combines hybrid behavior of parallel components, a slip-stick friction model with non-linear dynamics and flow invariants and several dimensions of parameterization. In the paper, we also present and evaluate an extension of VNODE-LP tailored to its use as a deduction mechanism within iSAT-ODE, to allow fast re-evaluations of enclosures over arbitrary subranges of the analyzed time span.

Abstract. The analysis of hybrid systems exhibiting probabilistic behaviour is notoriously difficult. To enable mechanised analysis of such systems, we extend the reasoning power of arithmetic satisfiability-modulotheory solving (SMT) by a comprehensive treatment of randomized (a.k.a. stochastic) quantification over discrete variables within the mixed Booleanarithmetic constraint system. This provides the technological basis for a fully symbolic analysis of probabilistic hybrid automata. Generalizing SMT-based bounded model-checking of hybrid automata [2, 11], stochastic SMT permits the direct and fully symbolic analysis of probabilistic bounded reachability problems of probabilistic hybrid automata without resorting to approximation by intermediate finite-state abstractions.

Abstract. In order to facilitate automated reasoning about large Boolean combinations of non-linear arithmetic constraints involving ordinary differential equations (ODEs), we provide a seamless integration of safe numeric overapproximation of initial-value problems into a SATmodulo-theory (SMT) approach to interval-based arithmetic constraint solving. Interval-based safe numeric approximation of ODEs is used as an interval contractor being able to narrow candidate sets in phase space in both temporal directions: post-images of ODEs (i.e., sets of states reachable from a set of initial values) are narrowed based on partial information about the initial values and, vice versa, pre-images are narrowed based on partial knowledge about post-sets. In contrast to the related CLP(F) approach of Hickey and Wittenberg [10], we do (a) support coordinate transformations mitigating the wrapping effect encountered upon iterating interval-based overapproximations of reachable state sets and (b) embed the approach into an SMT framework, thus accelerating the solving process through the algorithmic enhacements of recent SAT solving technology.

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