On Davis–Putnam reductions for minimally unsatisfiable clause-sets

"Minimally unsatisfiable clause-sets" are fundamental building blocks of satisfiability (SAT) theory. In order to establish a structural theory about them,elimination of certain types of degenerations via "Davis-Putnam (DP) reductions" are essential. These DP-reductions have been used at many placessince more than 50 years, and we now show that we have certain forms of confluence, that is, that the applications of DP-reductions are independent oftheir implementation, to a certain degree

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails

Irreducible Subcube Partitions

A \emph{subcube partition} is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it "mentions" all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of $\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.Comment: 39 page

Resolution-based methods for linear temporal reasoning

The aim of this thesis is to explore the potential of resolution-based methods for linear temporal reasoning. On the abstract level, this means to develop new algorithms for automated reasoning about properties of systems which evolve in time. More concretely, we will: 1) show how to adapt the superposition framework to proving theorems in propositional Linear Temporal Logic (LTL), 2) use a connection between superposition and the CDCL calculus of modern SAT solvers to come up with an efficient LTL prover, 3) specialize the previous to reachability properties and discover a close connection to Property Directed Reachability (PDR), an algorithm recently developed for model checking of hardware circuits, 4) further improve PDR by providing a new technique for enhancing clause propagation phase of the algorithm, and 5) adapt PDR to automated planning by replacing the SAT solver inside with a planning-specific procedure. We implemented the proposed ideas and provide experimental results which demonstrate their practical potential on representative benchmark sets. Our system LS4 is shown to be the strongest LTL prover currently publicly available. The mentioned enhancement of PDR substantially improves the performance of our implementation of the algorithm for hardware model checking in the multi-property setting. It is expected that other implementations would benefit from it in an analogous way. Finally, our planner PDRplan has been compared with the state-of-the-art planners on the benchmarks from the International Planning Competition with very promising results.Das Ziel dieser Doktorarbeit ist es, das Potential resolutionsbasierter Methoden zur linearer, temporaler Beweisführung zu untersuchen. Von einem abstrakten Gesichtspunkt aus gesehen bedeutet dies, neue Algorithmen über die Eigenschaften von sich zeitlich entwicklenden Systemen im Bereich des automatischen Theorembeweisens zu entwickeln. Konkreter gesagt werden wir 1) aufzeigen, wie sich das Rahmenprogramm der Superposition so anpassen lässt, damit es Theoreme in propositionaler Linear Temporal Logic (LTL) beweist, 2) eine Verbindung zwischen der Superposition und dem CDCL-Kalkül moderner SAT-Solver nutzen, um mit einem effizienten LTL-Prover aufzuwarten, 3) das Vorangegangene auf Erreichbarkeitseigenschaften spezialisieren, und eine starke Verbindung zu der Property Directed Reachability (PDR), einem jüngst eintwickeltem Model-Checking-Algorithmus für Hardware-Schaltkreise, aufzudecken, 4) PDR durch die Einführung neuer Technik verbessern, die die Clause-Propagation-Phase des Algorithmus beschleunigt, und 5) PDR für das automatisierte Planen anpassen, indem wir den inneren SAT-Solver durch eine planungsspezifische Prozedur ersetzen. Wir haben die vorgeschlagenen Ideen implementiert, und es werden experimentelle Ergebnisse angegeben, die das praktische Potential dieser Ideen auf repräsentativen Benchmarks aufzeigt. Es hat sich herausgestellt, dass unser System LS4 der staerkste öffentlich zugängliche LTL-Prover ist. Die erwähnte Erweiterung von PDR verbessern die Leistungsfähigkeit unserer Implementierung des Hardware-Model-Checking-Algorithmus substantiell im Bereich der Multi-Property-Einstellungen. Wir erwarten, dass andere Implementierungen in ähnlicher Weise profitieren würden. Schließlich haben wir viel versprechende Ergebnisse durch den Vergleich unser Planer PDRplan mit anderen state-of-the-art Planer auf den Benchmarks der International Planning Competition erzielt