Speeding up the constraint-based method in difference logic

"The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-319-40970-2_18"Over the years the constraint-based method has been successfully applied to a wide range of problems in program analysis, from invariant generation to termination and non-termination proving. Quite often the semantics of the program under study as well as the properties to be generated belong to difference logic, i.e., the fragment of linear arithmetic where atoms are inequalities of the form u v = k. However, so far constraint-based techniques have not exploited this fact: in general, Farkas’ Lemma is used to produce the constraints over template unknowns, which leads to non-linear SMT problems. Based on classical results of graph theory, in this paper we propose new encodings for generating these constraints when program semantics and templates belong to difference logic. Thanks to this approach, instead of a heavyweight non-linear arithmetic solver, a much cheaper SMT solver for difference logic or linear integer arithmetic can be employed for solving the resulting constraints. We present encouraging experimental results that show the high impact of the proposed techniques on the performance of the VeryMax verification systemPeer ReviewedPostprint (author's final draft

Tri-State Boolean Satisfiability with Commit: An Efficient Partial Solution Using Hyperlogic

We present two implementation enhancements for the Boolean satisfiability problem and one visualization technique. The first is an expansion to a tri-nary logic system with a commit phase. The three states are (1) true, (2) false, and (3) don\u27t care. We abstracted the operations of AND and OR to this hyperlogic system in a novel way. The commit phase works on one variable at a time and transitions values from temporary to permanent whenever possible. We viewed tri-state logic as a hyperspace above the binary (Boolean) logic. The second improvement is algorithmic. We modified the semantics of the classic 3 Conjunctive Normal Form Problem in order to develop a polynomial time algorithm for a simplified normal form - avoiding the need to examine all combinatoric limitations. In particular, we abandoned 3 CNF and used an unstructured left to right associativity. We do not claim that this new semantic is comprehensive. We do claim that it is simpler. Lastly, we introduced a node analogy to help us understand the algorithm itself

Modeling and Solving a Resource Allocation Problem with Soft Constraint Techniques

We study a resource allocation problem, which is a central piece of a real-world crew scheduling problem. We first formulate the problem as a hybrid soft constraint satisfaction and optimization problem and show that its worst-case complexity is NP-complete. We then propose and study a set of decision and optimization modeling schemes for the problem. We consider the expressiveness of these modeling schemes for the problem. We consider the expressiveness of these modeling methods. Specifically, we experimentally investigate how these modeling schemes interplay with the best existing systematic search and local search methods. Our experimental results show that soft constraint techniques can be effective on large resource allocation problem instances, and an optimization approach is more efficient than a model checking approach based on decision models

A lower bound on CNF encodings of the at-most-one constraint

Constraint "at most one" is a basic cardinality constraint which requires that at most one of its $n$ boolean inputs is set to $1$. This constraint is widely used when translating a problem into a conjunctive normal form (CNF) and we investigate its CNF encodings suitable for this purpose. An encoding differs from a CNF representation of a function in that it can use auxiliary variables. We are especially interested in propagation complete encodings which have the property that unit propagation is strong enough to enforce consistency on input variables. We show a lower bound on the number of clauses in any propagation complete encoding of the "at most one" constraint. The lower bound almost matches the size of the best known encodings. We also study an important case of 2-CNF encodings where we show a slightly better lower bound. The lower bound holds also for a related "exactly one" constraint.Comment: 38 pages, version 3 is significantly reorganized in order to improve readabilit

X and more Parallelism. Integrating LTL-Next into SAT-based Planning with Trajectory Constraints while Allowing for even more Parallelism

Linear temporal logic (LTL) provides expressive means to specify temporally extended goals as well as preferences. Recent research has focussed on compilation techniques, i.e., methods to alter the domain ensuring that every solution adheres to the temporally extended goals. This requires either new actions or an construction that is exponential in the size of the formula. A translation into boolean satisfiability (SAT) on the other hand requires neither. So far only one such encoding exists, which is based on the parallel $\exists$-step encoding for classical planning. We show a connection between it and recently developed compilation techniques for LTL, which may be exploited in the future. The major drawback of the encoding is that it is limited to LTL without the X operator. We show how to integrate X and describe two new encodings, which allow for more parallelism than the original encoding. An empirical evaluation shows that the new encodings outperform the current state-of-the-art encoding

Pseudo-Boolean Constraint Encodings for Conjunctive Normal Form and their Applications

In contrast to a single clause a pseudo-Boolean (PB) constraint is much more expressive and hence it is easier to define problems with the help of PB constraints. But while PB constraints provide us with a high-level problem description, it has been shown that solving PB constraints can be done faster with the help of a SAT solver. To apply such a solver to a PB constraint we have to encode it with clauses into conjunctive normal form (CNF). While we can find a basic encoding into CNF which is equivalent to a given PB constraint, the solving time of a SAT solver significantly depends on different properties of an encoding, e.g. the number of clauses or if generalized arc consistency (GAC) is maintained during the search for a solution. There are various PB encodings that try to optimize or balance these properties. This thesis is about such encodings. For a better understanding of the research field an overview about the state-of-the art encodings is given. The focus of the overview is a simple but complete description of each encoding, such that any reader could use, implement and extent them in his own work. In addition two novel encodings are presented: The Sequential Weight Counter (SWC) encoding and the Binary Merger Encoding. While the SWC encoding provides a very simple structure – it is listed in four lines – empirical evaluation showed its practical usefulness in various applications. The Binary Merger encoding reduces the number of clauses a PB encoding needs while having the important GAC property. To the best of our knowledge currently no other encoding has a lower upper bound for the number of clauses produced by a PB encoding with this property. This is an important improvement of the state-of-the art, since both GAC and a low number of clauses are vital for an improved solving time of the SAT solver. The thesis also contributes to the development of new applications for PB constraint encodings. The programming library PBLib provides researchers with an open source implementation of almost all PB encodings – including the encodings for the special cases at-most-one and cardinality constraints. The PBLib is also the foundation of the presented weighted MaxSAT solver optimax, the PBO solver pbsolver and the WBO, PBO and weighted MaxSAT solver npSolver

Constraint satisfaction problems in clausal form

This is the report-version of a mini-series of two articles on the foundations of satisfiability of conjunctive normal forms with non-boolean variables, to appear in Fundamenta Informaticae, 2011. These two parts are here bundled in one report, each part yielding a chapter. Generalised conjunctive normal forms are considered, allowing literals of the form "variable not-equal value". The first part sets the foundations for the theory of autarkies, with emphasise on matching autarkies. Main results concern various polynomial time results in dependency on the deficiency. The second part considers translations to boolean clause-sets and irredundancy as well as minimal unsatisfiability. Main results concern classification of minimally unsatisfiable clause-sets and the relations to the hermitian rank of graphs. Both parts contain also discussions of many open problems.Comment: 91 pages, to appear in Fundamenta Informaticae, 2011, as Constraint satisfaction problems in clausal form I: Autarkies and deficiency, Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structur