Positional Games and QBF: The Corrective Encoding

Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBF) such that a game instance admits a winning strategy for first player if and only if the corresponding formula is true. Our approach improves over previous QBF encodings of games in multiple ways. First, it is generic and lets us encode other positional games, such as Hex. Second, structural properties of positional games together with a careful treatment of illegal moves let us generate more compact instances that can be solved faster by state-of-the-art QBF solvers. We establish the latter fact through extensive experiments. Finally, the compactness of our new encoding makes it feasible to translate realistic game problems. We identify a few such problems of historical significance and put them forward to the QBF community as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT2020

Linear Encodings of Bounded LTL Model Checking

We consider the problem of bounded model checking (BMC) for linear temporal logic (LTL). We present several efficient encodings that have size linear in the bound. Furthermore, we show how the encodings can be extended to LTL with past operators (PLTL). The generalised encoding is still of linear size, but cannot detect minimal length counterexamples. By using the virtual unrolling technique minimal length counterexamples can be captured, however, the size of the encoding is quadratic in the specification. We also extend virtual unrolling to Buchi automata, enabling them to accept minimal length counterexamples. Our BMC encodings can be made incremental in order to benefit from incremental SAT technology. With fairly small modifications the incremental encoding can be further enhanced with a termination check, allowing us to prove properties with BMC. Experiments clearly show that our new encodings improve performance of BMC considerably, particularly in the case of the incremental encoding, and that they are very competitive for finding bugs. An analysis of the liveness-to-safety transformation reveals many similarities to the BMC encodings in this paper. Using the liveness-to-safety translation with BDD-based invariant checking results in an efficient method to find shortest counterexamples that complements the BMC-based approach.Comment: Final version for Logical Methods in Computer Science CAV 2005 special issu