Matti Järvisalo

Inprocessing rules

Decision procedures for Boolean satisfiability (SAT), especially modern conflict-driven clause learning (CDCL) solvers, act routinely as core solving engines in various real-world applications. Preprocessing, i.e., applying formula rewriting/simplification rules to the input formula before the actual search for satisfiability, has become an essential part of the SAT solving tool chain. Further, some of the strongest SAT solvers today add more reasoning to search by interleaving formula simplification and CDCL search. Such inprocessing SAT solvers witness the fact that implementing additional deduction rules in CDCL solvers leverages the efficiency of state-of-the-art SAT solving further. In this paper we establish formal underpinnings of inprocessing SAT solving via an abstract inprocessing framework that covers a wide range of modern SAT solving techniques.

Blocked clause elimination

Boolean satisfiability (SAT) and its extensions are becoming a core technology for the analysis of systems. The SAT-based approach divides into three steps: encoding, preprocessing, and search. It is often argued that by encoding arbitrary Boolean formulas in conjunctive normal form (CNF), structural properties of the original problem are not reflected in the CNF. This should result in the fact that CNF-level preprocessing and SAT solver techniques have an inherent disadvantagecompared to related techniques applicable on the level of more structural SAT instance representations such as Boolean circuits. In this work we study the effect of a CNF-level simplification technique called blocked clause elimination (BCE). We show that BCE is surprisingly effective both in theory and in practice on CNFs resulting from a standard CNF encoding for circuits: without explicit knowledge of the underlying circuit structure, it achieves the same level of simplification as a combination of circuit-level simplifications and previously suggested polarity-based CNF encodings. Experimentally, we show that by applying BCE in preprocessing, further formula reduction and faster solving can be achieved, giving promise for applying BCE to speed up solvers.

The International SAT Solver Competitions

The International SAT Solver Competition is today an established series of competitive events aiming at objectively evaluating the progress in state-of-the-art procedures for solving Boolean satisfiability (SAT) instances. Over the years, the competitions have significantly contributed to the fast progress in SAT solver technology that has made SAT a practical success story of computer science. This short article provides an overview of the SAT solver competitions.

Complexity-sensitive decision procedures for abstract argumentation

Abstract argumentation frameworks (AFs) provide the basis for various reasoning problems in the areas of Knowledge Representation and Artificial Intelligence. Efficient evaluation of AFs has thus been identified as an important research challenge. So far, implemented systems for evaluating AFs have either followed a straight-forward reduction-based approach or been limited to certain tractable classes of AFs. In this work, we present a generic approach for reasoning over AFs, based on the novel concept of complexity-sensitivity. Establishing the theoretical foundations of this approach, we derive several new complexity results for preferred, semi-stable and stage semantics which complement the current complexity landscape for abstract argumentation, providing further understanding on the sources of intractability of AF reasoning problems. The introduced generic framework exploits decision procedures for problems of lower complexity whenever possible. This allows, in particular, instantiations of the generic framework via harnessing in an iterative way current sophisticated Boolean satisfiability (SAT) solver technology for solving the considered AF reasoning problems. First experimental results show that the SAT-based instantiation of our novel approach outperforms existing systems.

Clause elimination procedures for CNF formulas

We develop and analyze clause elimination procedures, a specific family of simplification techniques for conjunctive normal form (CNF) formulas. Extending known procedures such as tautology, subsumption, and blocked clause elimination, we introduce novel elimination procedures based on hidden and asymmetric variants of these techniques. We analyze the resulting nine (including five new) clause elimination procedures from various perspectives: size reduction, BCP-preservance, confluence, and logical equivalence. For the variants not preserving logical equivalence, we show how to reconstruct solutions to original CNFs from satisfying assignments to simplified CNFs. We also identify a clause elimination procedure that does a transitive reduction of the binary implication graph underlying any CNF formula purely on the CNF level.

Unrestricted vs restricted cut in a tableau method for Boolean circuits

AbstractThis paper studies the relative efficiency of variations of a tableau method for Boolean circuit satisfiability checking. The considered method is a nonclausal generalisation of the Davis–Putnam–Logemann–Loveland (DPLL) procedure to Boolean circuits. The variations are obtained by restricting the use of the cut (splitting) rule in several natural ways. It is shown that the more restricted variations cannot polynomially simulate the less restricted ones. For each pair of methods T, T′, an infinite family

Simulating Circuit-Level Simplifications on CNF

\{\mathcal{C}_{n}\}

A Module-Based Framework for Multi-language Constraint Modeling

of circuits is devised for which T has polynomial size proofs while in T′ the minimal proofs are of exponential size w.r.t. n, implying exponential separation of T and T′ w.r.t. n. The results also apply to DPLL for formulas in conjunctive normal form obtained from Boolean circuits by using Tseitin’s translation. Thus DPLL with the considered cut restrictions, such as allowing splitting only on the variables corresponding to the input gates, cannot polynomially simulate DPLL with unrestricted splitting.

Clause elimination for SAT and QSAT

Boolean satisfiability (SAT) and its extensions have become a core technology in many application domains, such as planning and formal verification, and continue finding various new application domains today. The SAT-based approach divides into three steps: encoding, preprocessing, and search. It is often argued that by encoding arbitrary Boolean formulas in conjunctive normal form (CNF), structural properties of the original problem are not reflected in the CNF. This should result in the fact that CNF-level preprocessing and SAT solver techniques have an inherent disadvantage compared to related techniques applicable on the level of more structural SAT instance representations such as Boolean circuits. Motivated by this, various simplification techniques and intricate CNF encodings for circuit-level SAT instance representations have been proposed. On the other hand, based on the highly efficient CNF-level clause learning SAT solvers, there is also strong support for the claim that CNF is sufficient as an input format for SAT solvers. In this work we study the effect of CNF-level simplification techniques, focusing on SatElite-style variable elimination (VE) and what we call blocked clause elimination (BCE). We show that BCE is surprisingly effective both in theory and in practice on CNF formulas resulting from a standard CNF encoding for circuits: without explicit knowledge of the underlying circuit structure, it achieves the same level of simplification as a combination of circuit-level simplifications and previously suggested polarity-based CNF encodings. We also show that VE can achieve many of the same effects as BCE, but not all. On the other hand, it turns out that VE and BCE are indeed partially orthogonal techniques. We also study the practical effects of combining BCE and VE for reducing the size of formulas and on the running times of state-of-the-art SAT solvers. Furthermore, we address the problem of how to construct original witnesses to satisfiable CNF formulas when applying a combination of BCE and VE.

Theory and Applications of Satisfiability Testing – SAT 2013

We develop a module-based framework for constraint modeling where it is possible to combine different constraint modeling languages and exploit their strengths in a flexible way. In the framework a constraint model consists of modules with clear input/output interfaces. When combining modules, apart from the interface, a module is a black box whose internals are invisible to the outside world. Inside a module a chosen constraint language (approaches such as CP, ASP, SAT, and MIP) can be used. This leads to a clear modular semantics where the overall semantics of the whole constraint model is obtained from the semantics of individual modules. The framework supports multi-language modeling without the need to develop a complicated joint semantics and enables the use of alternative semantical underpinnings such as default negation and classical negation in the same model. Furthermore, computational aspects of the framework are considered and, in particular, possibilities of benefiting from the known module structure in solving constraint models are studied.