Implementation of propositional temporal logics using BDDs

This tutorial intends to convey the step-by-step process by which computer programs for model checking and satisfiability testing for temporal logics may be derived from the theory. The idea is to demonstrate that it is very well possible to implement such a program in an efficient way without sacrificing a correct-by-construction approach. The tutorial will be fully self-contained, only a general knowledge of programming and propositional logic is assumed

Minimal Proof Search for Modal Logic K Model Checking

Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K. While the model checking problems for LTL and to a lesser extent ATL have been very active research areas for the past decades, the model checking problem for the more basic Multi-agent Modal Logic K (MMLK) has important applications as a formal framework for perfect information multi-player games on its own. We present Minimal Proof Search (MPS), an effort number based algorithm solving the model checking problem for MMLK. We prove two important properties for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal cost for a general definition of cost, and MPS is an optimal algorithm for finding (dis)proofs of minimal cost. Optimality means that any comparable algorithm either needs to explore a bigger or equal state space than MPS, or is not guaranteed to find a (dis)proof of minimal cost on every input. As such, our work relates to A* and AO* in heuristic search, to Proof Number Search and DFPN+ in two-player games, and to counterexample minimization in software model checking.Comment: Extended version of the JELIA 2012 paper with the same titl

Tableaux and witnesses for the my--calculus

Symbolic temporal logic model checking is an automatic verification method. One of its main features is that a counterexample can be constructed when a temporal formula does not hold for the model. Most model checkers so far have restricted the type of formulae that can be checked and for which counterexamples can be constructed to fair CTL formulae. This paper shows how counterexamples and witnesses for the whole µ-Calculus can be constructed. The witness construction presented in this paper is polynomial in the model and the formula

Local model checking in Park\u27s my--calculus

Temporal logic model checking is an automatic verification method for finite-state systems. In global model checking, the truth of a formula (and its subformulae) is determined for all the states in the model. Local model checking procedures are designed for proving that a specific state of the model satisfies a given formula. This may avoid the exhaustive traversal of a model. Also, the proof tree constructed during local model checking can serve as a witness (counterexample) which demonstrates the error in the design and can thus help locating errors. In \cite{StiWal91} it was shown how local model checking can be performed in the modal $\mu$-calculus. In this paper, we introduce a tableau system and thus a local model checking method for the more expressive $\mu$-calculus of Park \cite{Par76} and prove its soundness and completeness

On the Proof Theory of Regular Fixed Points

International audienceWe consider encoding finite automata as least fixed points in a proof theoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to proof-theory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular

A Compositional Proof System for the Modal mu-Calculus

We present a proof system for determining satisfaction between processes in a fairly general process algebra and assertions of the modal mu-calculus. The proof system is compositional in the structure of processes. It extends earlier work on compositional reasoning within the modal mu-calculus and combines it with techniques from work on local model checking. The proof system is sound for all processes and complete for a class of finite-state processes

The Power of Proofs: New Algorithms for Timed Automata Model Checking (with Appendix)

This paper presents the first model-checking algorithm for an expressive modal mu-calculus over timed automata, $L^{\mathit{rel}, \mathit{af}}_{\nu,\mu}$, and reports performance results for an implementation. This mu-calculus contains extended time-modality operators and can express all of TCTL. Our algorithmic approach uses an "on-the-fly" strategy based on proof search as a means of ensuring high performance for both positive and negative answers to model-checking questions. In particular, a set of proof rules for solving model-checking problems are given and proved sound and complete; we encode our algorithm in these proof rules and model-check a property by constructing a proof (or showing none exists) using these rules. One noteworthy aspect of our technique is that we show that verification performance can be improved with \emph{derived rules}, whose correctness can be inferred from the more primitive rules on which they are based. In this paper, we give the basic proof rules underlying our method, describe derived proof rules to improve performance, and compare our implementation of this model checker to the UPPAAL tool.Comment: This is the preprint of the FORMATS 2014 paper, but this is the full version, containing the Appendix. The final publication is published from Springer, and is available at http://link.springer.com/chapter/10.1007%2F978-3-319-10512-3_9 on the Springer webpag