CadiBack: Extracting Backbones with CaDiCaL

The backbone of a satisfiable formula is the set of literals that are true in all its satisfying assignments. Backbone computation can improve a wide range of SAT-based applications, such as verification, fault localization and product configuration. In this tool paper, we introduce a new backbone extraction tool called CadiBack. It takes advantage of unique features available in our state-of-the-art SAT solver CaDiCaL including transparent inprocessing and single clause assumptions, which have not been evaluated in this context before. In addition, CaDiCaL is enhanced with an improved algorithm to support model rotation by utilizing watched literal data structures. In our comprehensive experiments with a large number of benchmarks, CadiBack solves 60% more instances than the state-of-the-art backbone extraction tool MiniBones. Our tool is thoroughly tested with fuzzing, internal correctness checking and cross-checking on a large benchmark set. It is publicly available as open source, well documented and easy to extend

Recognition and Exploitation of Gate Structure in SAT Solving

In der theoretischen Informatik ist das SAT-Problem der archetypische Vertreter der Klasse der NP-vollständigen Probleme, weshalb effizientes SAT-Solving im Allgemeinen als unmöglich angesehen wird. Dennoch erzielt man in der Praxis oft erstaunliche Resultate, wo einige Anwendungen Probleme mit Millionen von Variablen erzeugen, die von neueren SAT-Solvern in angemessener Zeit gelöst werden können. Der Erfolg von SAT-Solving in der Praxis ist auf aktuelle Implementierungen des Conflict Driven Clause-Learning (CDCL) Algorithmus zurückzuführen, dessen Leistungsfähigkeit weitgehend von den verwendeten Heuristiken abhängt, welche implizit die Struktur der in der industriellen Praxis erzeugten Instanzen ausnutzen. In dieser Arbeit stellen wir einen neuen generischen Algorithmus zur effizienten Erkennung der Gate-Struktur in CNF-Encodings von SAT Instanzen vor, und außerdem drei Ansätze, in denen wir diese Struktur explizit ausnutzen. Unsere Beiträge umfassen auch die Implementierung dieser Ansätze in unserem SAT-Solver Candy und die Entwicklung eines Werkzeugs für die verteilte Verwaltung von Benchmark-Instanzen und deren Attribute, der Global Benchmark Database (GBD)

Propagators and Solvers for the Algebra of Modular Systems

To appear in the proceedings of LPAR 21. Solving complex problems can involve non-trivial combinations of distinct knowledge bases and problem solvers. The Algebra of Modular Systems is a knowledge representation framework that provides a method for formally specifying such systems in purely semantic terms. Formally, an expression of the algebra defines a class of structures. Many expressive formalism used in practice solve the model expansion task, where a structure is given on the input and an expansion of this structure in the defined class of structures is searched (this practice overcomes the common undecidability problem for expressive logics). In this paper, we construct a solver for the model expansion task for a complex modular systems from an expression in the algebra and black-box propagators or solvers for the primitive modules. To this end, we define a general notion of propagators equipped with an explanation mechanism, an extension of the alge- bra to propagators, and a lazy conflict-driven learning algorithm. The result is a framework for seamlessly combining solving technology from different domains to produce a solver for a combined system.Comment: To appear in the proceedings of LPAR 2

Controlled and effective interpolation

Model checking is a well established technique to verify systems, exhaustively and automatically. The state space explosion, known as the main difficulty in model checking scalability, has been successfully approached by symbolic model checking which represents programs using logic, usually at the propositional or first order theories level. Craig interpolation is one of the most successful abstraction techniques used in symbolic methods. Interpolants can be efficiently generated from proofs of unsatisfiability, and have been used as means of over-approximation to generate inductive invariants, refinement predicates, and function summaries. However, interpolation is still not fully understood. For several theories it is only possible to generate one interpolant, giving the interpolation-based application no chance of further optimization via interpolation. For the theories that have interpolation systems that are able to generate different interpolants, it is not understood what makes one interpolant better than another, and how to generate the most suitable ones for a particular verification task. The goal of this thesis is to address the problems of how to generate multiple interpolants for theories that still lack this flexibility in their interpolation algorithms, and how to aim at good interpolants. This thesis extends the state-of-the-art by introducing novel interpolation frameworks for different theories. For propositional logic, this work provides a thorough theoretical analysis showing which properties are desirable in a labeling function for the Labeled Interpolation Systems framework (LIS). The Proof-Sensitive labeling function is presented, and we prove that it generates interpolants with the smallest number of Boolean connectives in the entire LIS framework. Two variants that aim at controlling the logical strength of propositional interpolants while maintaining a small size are given. The new interpolation algorithms are compared to previous ones from the literature in different model checking settings, showing that they consistently lead to a better overall verification performance. The Equalities and Uninterpreted Functions (EUF)-interpolation system, presented in this thesis, is a duality-based interpolation framework capable of generating multiple interpolants for a single proof of unsatisfiability, and provides control over the logical strength of the interpolants it generates using labeling functions. The labeling functions can be theoretically compared with respect to their strength, and we prove that two of them generate the interpolants with the smallest number of equalities. Our experiments follow the theory, showing that the generated interpolants indeed have different logical strength. We combine propositional and EUF interpolation in a model checking setting, and show that the strength of the interpolation algorithms for different theories has to be aligned in order to generate smaller interpolants. This work also introduces the Linear Real Arithmetic (LRA)-interpolation system, an interpolation framework for LRA. The framework is able to generate infinitely many interpolants of different logical strength using the duality of interpolants. The strength of the LRA interpolants can be controlled by a normalized strength factor, which makes it straightforward for an interpolationbased application to choose the level of strength it wants for the interpolants. Our experiments with the LRA-interpolation system and a model checker show that it is very important for the application to be able to fine tune the strength of the LRA interpolants in order to achieve optimal performance. The interpolation frameworks were implemented and form the interpolation module in OpenSMT2, an open source efficient SMT solver. OpenSMT2 has been integrated to the propositional interpolation-based model checkers FunFrog and eVolCheck, and to the first order interpolation-based model checkerHiFrog. This thesis presents real life model checking experiments using the novel interpolation frameworks and the tools aforementioned, showing the viability and strengths of the techniques

Engineering a Preprocessor for Symmetry Detection

State-of-the-art solvers for symmetry detection in combinatorial objects are becoming increasingly sophisticated software libraries. Most of the solvers were initially designed with inputs from combinatorics in mind (nauty, bliss, Traces, dejavu). They excel at dealing with a complicated core of the input. Others focus on practical instances that exhibit sparsity. They excel at dealing with comparatively easy but extremely large substructures of the input (saucy). In practice, these differences manifest in significantly diverging performances on different types of graph classes. We engineer a preprocessor for symmetry detection. The result is a tool designed to shrink sparse, large substructures of the input graph. On most of the practical instances, the preprocessor improves the overall running time significantly for many of the state-of-the-art solvers. At the same time, our benchmarks show that the additional overhead is negligible. Overall we obtain single algorithms with competitive performance across all benchmark graphs. As such, the preprocessor bridges the disparity between solvers that focus on combinatorial graphs and large practical graphs. In fact, on most of the practical instances the combined setup significantly outperforms previous state-of-the-art