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In this paper, we address the problem of applying SAT-based bounded model checking (BMC) and temporal k-induction to asynchronous concurrent systems. We investigate refinement checking in the process-algebraic setting of Communicating Sequential Processes (CSP), focusing on the CSP traces model which is sufficient for verifying safety properties. We adapt the BMC framework to the context of CSP and the existing refinement checker FDR yielding bounded refinement checking which also lays the foundation for tailoring the k-induction technique. As refinement checking reduces to checking for reverse containment of possible behaviours, we exploit the SAT-solver to decide bounded language inclusion as opposed to bounded reachability of error states, as in most existing model checkers. Due to the harder problem to decide and the presence of invisible silent actions in process algebras, the original syntactic translation of BMC to SAT cannot be applied directly and we adopt a semantic translation algorithm based on watchdog transformations. We propose a Boolean encoding of CSP processes resting on FDR’s hybrid two-level approach for calculating the operational semantics using supercombinators. We have implemented a prototype tool, SymFDR, written in C++, which uses FDR as a shared library for manipulating CSP processes and the state-of-the-art incremental SAT-solver MiniSAT 2.0. Experiments with BMC indicate that in some cases, especially in complex combinatorial problems, SymFDR significantly outperforms FDR and even copes with problems that are beyond FDR’s capabilities. SymFDR in k-induction mode works reasonably well for small test cases, but is inefficient for larger ones as the threshold becomes too large, due to concurrency.

(Labelled) transition systems are relatively common in theoretical computer science, chiefly as vehicles for operational semantics. The first part of this paper constructs a hierarchy of canonical transition systems and associated maps, aiming to give a strongly extensional theory of transition systems, where any two points with equivalent behaviors are identified. The cornerstone of the development is a notion of convergence in arbitrary transition systems, generalizing the idea of finite (n-step) approximations to a given point. In particular, our canonical transition systems are also uniform spaces. The resulting hierarchy has very rich combinatorial (and topological) structure, and a lot of the first part of the paper is devoted to its study. We also discuss fixed points in this framework. In the second part of the paper, we show how to obtain a model of set theory with Aczel's Anti-Foundation Axiom (AFA) from canonical transition systems constructed earlier. We study further the structure of the model thus obtained, and also give a few more abstract results, concerning consistency and independence in the presence of AFA.

With the flourishing development of efficient SAT-solvers, bounded model checking (BMC) has proven to be an extremely powerful symbolic model checking technique. In this paper, we address the problem of applying BMC to concurrent systems involving the interaction of multiple processes running in parallel. We adapt the BMC framework to the context of CSP and FDR yielding bounded refinement checking. Refinement checking reduces to checking for reverse containment of possible behaviours. Therefore, we exploit the SAT-solver to decide bounded language inclusion as opposed to bounded reachability of error states, as in most existing model checkers. We focus on the CSP traces model which is sufficient for verifying safety properties. We present a Boolean encoding of CSP processes resting on FDR's hybrid two-level approach for calculating the operational semantics using supercombinators. We describe our bounded refinement-checking algorithm which is based on watchdog transformations and incremental SAT-solving. We have implemented a tool, SymFDR, written in C++ which uses FDR as a shared library for manipulating CSP processes and the state-of-the-art SAT-solver MiniSAT. Experiments indicate that in some cases, especially for complex combinatorial problems, SymFDR significantly outperforms FDR.

Z3. But simple surface points are defined by means of axioms, and the axioms do not reveal what simple surface points “look like.” In this paper eight of the nine varieties of simple surface points are shown to have natural “continuous analogs,” and the one remaining variety is shown to be very different from the other types. This work yields substantial generalizations of the main theorems on simple surface points that were proved by Morgenthaler, Reed, and Rosenfeld. Q