Tobias Philipp

Soundness of inprocessing in clause sharing SAT solvers

We present a formalism that models the computation of clause sharing portfolio solvers with inprocessing. The soundness of these solvers is not a straightforward property since shared clauses can make a formula unsatisfiable. Therefore, we develop characterizations of simplification techniques and suggest various settings how clause sharing and inprocessing can be combined. Our formalization models most of the recent implemented portfolio systems and we indicate possibilities to improve these. A particular improvement is a novel way to combine clause addition techniques --- like blocked clause addition --- with clause deletion techniques --- like blocked clause elimination or variable elimination.

A More Compact Translation of Pseudo-Boolean Constraints into CNF Such That Generalized Arc Consistency Is Maintained

In this paper we answer the open question for the existence of a more compact encoding from Pseudo-Boolean constraints into CNF that maintains generalized arc consistency by unit propagation, formalized by Bailleux et al. in [21]. In contrast to other encodings our approach is defined in an abstract way and we present a concrete instantiation, resulting in a space complexity of \(\mathcal{O}(n^2 \text{\,log}^2(n)\text{\,log}(w_{\mathsf{max}}))\) clauses in contrast to \(\mathcal{O}(n^3 \text{\,log}(n)\text{\,log}(w_{\mathsf{max}}))\) clauses generated by the previously best known encoding that maintains generalized arc consistency.

PBLib – A Library for Encoding Pseudo-Boolean Constraints into CNF

PBLib is an easy-to-use and efficient library, written in C++, for translating pseudo-Boolean (PB) constraints into CNF. We have implemented fifteen different encodings of PB constraints. Our aim is to use efficient encodings, in terms of formula size and whether unit propagation maintains generalized arc consistency. Moreover, PBLib normalizes PB constraints and automatically uses a suitable encoder for the translation. We also support incremental strengthening for optimization problems, where the tighter bound is realized with few additional clauses, as well as conditions for PB constraints.

DRAT Proofs for XOR Reasoning

Unsatisfiability proofs in the DRAT format became the de facto standard to increase the reliability of contemporary SAT solvers. We consider the problem of generating proofs for the XOR reasoning component in SAT solvers and propose two methods: direct translation transforms every XOR constraint addition inference into a DRAT proof, whereas T-translation avoids the exponential blow-up in direct translations by using fresh variables. T-translation produces DRAT proofs from Gaussian elimination records that are polynomial in the size of the input CNF formula. Experiments show that a combination of both approaches with a simple prediction method outperforms the BDD-based method.

An Expressive Model for Instance Decomposition Based Parallel SAT Solvers

SAT solvers are highly efficient programs that decide the satisfiability problem for propositional formulas in conjunctive normal form. Contemporary SAT solvers combine many advanced techniques such as clause sharing and inprocessing. Clause sharing is a form of cooperation in parallel SAT solvers based on clause learning, whereas inprocessing simplifies formulas in a satisfiability-preserving way. In this paper, we present the instance decomposition formalism ID that models parallel SAT solvers with label-based clause sharing and inprocessing. We formally prove soundness of ID and show that the concept of labels can be used to ensure satisfiability-preserving operations. Moreover, we develop a new proof format for SAT solvers based on this approach, which is derived from ID.

Formula Simplifications as DRAT Derivations

Many real world problems are solved with satisfiability testing (SAT). However, SAT solvers have documented bugs and therefore the answer that a formula is unsatisfiable can be incorrect. Certifying algorithms are an attractive approach to increase the reliability of SAT solvers. For unsatisfiable formulas an unsatisfiability proof has to be created. This paper presents certificate constructions for various formula simplification techniques, which are crucial to the success of modern SAT solvers.