QSAT (QBF) INSTANCE: Boolean expression φ in CNF with variables x1, . . . , xn. QUESTION: Is there a truth value for the variable x1 such that for both truth values of x2 there is a truth value for x3 and so on up to xn, such that φ is satisfied by the overall truth assignment? Notation An instance of QSAT is written as ∃x1∀x2∃x3 · · ·Qxnφ, where Q is ∀ if n is even and ∃ if n is odd. Theorem Q...

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. We present here two results for matched CNFs: The first result is a shorter and simpler proof of the fact that Boolean minimization remains complete for the second level of polynomial hierarchy even if the input is r...

We present the Boolean dimension of a graph, we relate it with notions inner, geometric and symplectic dimensions, rank minrank graph. obtain an exact formula for tree in terms certain star decomposition. inversion index tournament.

1. Closed ideals. (1.1) Definition. Let 5 be an arbitrary set, and 2 a Boolean ring of subsets of 5. The algebra B(S, 2) consists of all uniform limits of finite linear combinations of characteristic functions of sets in 2. The norm in B(S, 2) is given by the formula ||/|| =SupseS \f(s) | • This definition is slightly more general than that given in [l ] where 2 is required to be a field of set...

Many different encodings for pseudo-Boolean constraints to the Boolean satisfiability problem have been proposed in the past. In this work we present a novel small sized and simple to implement encoding. The encoding maintains generalized arc consistency by unit propagation and results in a linear sized formula in conjunctive normal form with respect to the number of input variables. Experiment...

A complete SAT-based model checking algorithm for contextfree processes is presented. We reduce proof search in local model checking to Boolean satisfiability. Bounded proof search can therefore be performed by SAT solvers. Moreover, the completion of proof search is reduced to Boolean unsatisfiability and hence can be checked by SAT solvers. By encoding the local model checking algorithm in [1...

The satisfiability (SAT) problem is known to be NP-complete [3] and plays a central role in many domains such as computer-aided design, computing theory, and artificial intelligence. Generally, a SAT problem is defined as follows. Given a propositional formula with m clauses and n boolean variables, where each variable has value of either True or False. Negation of boolean variable xi is denote...

an immediate corollary to the main result in 2] was the theorem of Furst, Saxe, and Sipser 6] and Ajtai 1] that there is a uniform sequence of polynomial size constant depth monotone Boolean circuits that is not equivalent to whatever non-uniform sequence of polynomial size constant depth positive Boolean circuits. The thing is, we can simply translate the point-sets deenable in nite models by ...

Disjoint AND-decomposition of a boolean formula means its representation as a conjunction of two (or several) formulas having disjoint sets of variables. We show that deciding AND-decomposability is intractable in general for boolean formulas given in CNF or DNF and prove tractability of computing AND-decompositions of boolean formulas given in positive DNF, Full DNF, and ANF. The results follo...

We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new system, BPLK, is polynomially equivalent to the system G, which is based on the familiar but very different qu...