On variables with few occurrences in conjunctive normal forms

We consider the question of the existence of variables with few occurrences in boolean conjunctive normal forms (clause-sets). Let mvd(F) for a clause-set F denote the minimal variable-degree, the minimum of the number of occurrences of variables. Our main result is an upper bound mvd(F) <= nM(surp(F)) <= surp(F) + 1 + log_2(surp(F)) for lean clause-sets F in dependency on the surplus surp(F). - Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets. - For the surplus we have surp(F) <= delta(F) = c(F) - n(F), using the deficiency delta(F) of clause-sets, the difference between the number of clauses and the number of variables. - nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of natural numbers all numbers of the form 2^n - 1. We conjecture that this bound is nearly precise for minimally unsatisfiable clause-sets. As an application of the upper bound we obtain that (arbitrary!) clause-sets F with mvd(F) > nM(surp(F)) must have a non-trivial autarky (so clauses can be removed satisfiability-equivalently by an assignment satisfying some clauses and not touching the other clauses). It is open whether such an autarky can be found in polynomial time. As a future application we discuss the classification of minimally unsatisfiable clause-sets depending on the deficiency.Comment: 14 pages. Revision contains more explanations, and more information regarding the sharpness of the boun

Incremental QBF Solving

We consider the problem of incrementally solving a sequence of quantified Boolean formulae (QBF). Incremental solving aims at using information learned from one formula in the process of solving the next formulae in the sequence. Based on a general overview of the problem and related challenges, we present an approach to incremental QBF solving which is application-independent and hence applicable to QBF encodings of arbitrary problems. We implemented this approach in our incremental search-based QBF solver DepQBF and report on implementation details. Experimental results illustrate the potential benefits of incremental solving in QBF-based workflows.Comment: revision (camera-ready, to appear in the proceedings of CP 2014, LNCS, Springer

Sibling Rivalry among Paralogs Promotes Evolution of the Human Brain

Geneticists have long sought to identify the genetic changes that made us human, but pinpointing the functionally relevant changes has been challenging. Two papers in this issue suggest that partial duplication of SRGAP2, producing an incomplete protein that antagonizes the original, contributed to human brain evolution

Binary clause reasoning in QBF

Abstract. Binary clause reasoning has found some successful applications in SAT, and it is natural to investigate its use in various extensions of SAT. In this paper we investigate the use of binary clause reasoning in the context of solving Quantified Boolean Formulas (QBF). We develop a DPLL based QBF solver that employs extended binary clause reasoning (hyper-binary resolution) to infer new binary clauses both before and during search. These binary clauses are used to discover additional forced literals, as well as to perform equality reduction. Both of these transformations simplify the theory by removing one of its variables. When applied during DPLL search this stronger inference can offer significant decreases in the size of the search tree, but it can also be costly to apply. We are able to show empirically that despite the extra costs, binary clause reasoning can improve our ability to solve QBF.

Quantified constraint satisfaction, maximal constraint languages, and symmetric polymorphisms

The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the quantified constraint satisfaction problem (QCSP), a more general framework in which variables can be quantified both universally and existentially. We study the complexity of restricted cases of the QCSP where the types of constraints that may appear are restricted by a constraint language. We give a complete complexity classification of maximal constraint languages, the largest possible languages that can be tractable. We also give a complete complexity classification of constraint languages arising from symmetric polymorphisms