Prenex Conjunctive Normal Form Research Articles

CAQE and QuAbS: Abstraction Based QBF Solvers

We present a detailed description, analysis, and evaluation of the clausal abstraction approach for solving quantified Boolean formulas (QBF). The clausal abstraction algorithm started as a solving algorithm for QBFs in prenex conjunctive normal form

Quantifier Reordering for QBF

State-of-the-art procedures for evaluating quantified Boolean formulas often expect input formulas in prenex conjunctive normal form (PCNF). We study dependency schemes as a means of reordering the quantifier prefix of a PCNF formula while preserving its truth value. Dependency schemes associate each formula with a binary relation on its variables (the dependency relation) that imposes constraints on certain operations manipulating the formula's quantifier prefix. We prove that known dependency schemes support a stronger reordering operation than was previously known. We present an algorithm that, given a formula and its dependency relation, computes a compatible reordering with a minimum number of quantifier alternations. In combination with a dependency scheme that can be computed in polynomial time, this yields a polynomial time heuristic for reducing the number of quantifier alternations of an input formula. The resolution-path dependency scheme is the most general dependency scheme introduced so far. Using an interpretation of resolution paths as directed paths in a formula's implication graph, we prove that the resolution-path dependency relation can be computed in polynomial time.

Clause Elimination for SAT and QSAT

The famous archetypical NP-complete problem of Boolean satisfiability (SAT) and its PSPACE-complete generalization of quantified Boolean satisfiability (QSAT) have become central declarative programming paradigms through which real-world instances of various computationally hard problems can be efficiently solved. This success has been achieved through several breakthroughs in practical implementations of decision procedures for SAT and QSAT, that is, in SAT and QSAT solvers. Here, simplification techniques for conjunctive normal form (CNF) for SAT and for prenex conjunctive normal form (PCNF) for QSAT---the standard input formats of SAT and QSAT solvers---have recently proven very effective in increasing solver efficiency when applied before (i.e., in preprocessing) or during (i.e., in inprocessing) satisfiability search. In this article, we develop and analyze clause elimination procedures for pre- and inprocessing. Clause elimination procedures form a family of (P)CNF formula simplification techniques which remove clauses that have specific (in practice polynomial-time) redundancy properties while maintaining the satisfiability status of the formulas. Extending known procedures such as tautology, subsumption, and blocked clause elimination, we introduce novel elimination procedures based on asymmetric variants of these techniques, and also develop a novel family of so-called covered clause elimination procedures, as well as natural liftings of the CNF-level procedures to PCNF. We analyze the considered clause elimination procedures from various perspectives. Furthermore, for the variants not preserving logical equivalence under clause elimination, we show how to reconstruct solutions to original CNFs from satisfying assignments to simplified CNFs, which is important for practical applications for the procedures. Complementing the more theoretical analysis, we present results on an empirical evaluation on the practical importance of the clause elimination procedures in terms of the effect on solver runtimes on standard real-world application benchmarks. It turns out that the importance of applying the clause elimination procedures developed in this work is empirically emphasized in the context of state-of-the-art QSAT solving.

DepQBF: A Dependency-Aware QBF Solver

We present DepQBF 0.1, a new search-based solver for quantified boolean formulae (QBF). It integrates compact dependency graphs to overcome the restrictions imposed by linear quantifier prefixes of QBFs in prenex conjunctive normal form (PCNF). DepQBF 0.1 was placed first in the main track of QBFEVAL’10 in a score-based ranking. We provide a general system overview and describe selected orthogonal features such as restarts and removal of learnt constraints.

Efficiently Representing Existential Dependency Sets for Expansion-based QBF Solvers

Given a quantified boolean formula (QBF) in prenex conjunctive normal form (PCNF), we consider the problem of identifying variable dependencies. In related work, a formal definition of dependencies has been suggested based on quantifier prefix reordering: two variables are independent if swapping them in the prefix does not change satisfiability of the formula. Instead of the general case, we focus on the sets of depending existential variables for all universal variables. This is relevant particularly for expansion-based QBF solvers. We present an approach for efficiently computing existential dependency sets by means of a directed connection relation over variables and demonstrate how this relation can be compactly represented as a tree using a union-find data structure. Experimental results show the effectiveness of our approach.

A solver for QBFs in negation normal form

Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers for QBFs require formulas in prenex conjunctive normal form as input, which makes a further translation necessary, since the encodings are usually not in a specific normal form. This additional step increases the number of variables in the formula or disrupts the formula's structure. Moreover, the most important part of this transformation, prenexing, is not deterministic. In this paper, we focus on an alternative way to process QBFs without these drawbacks and describe a solver, $\ensuremath{\sf qpro}$ , which is able to handle arbitrary formulas. To this end, we extend algorithms for QBFs to the non-normal form case and compare $\ensuremath{\sf qpro}$ with the leading normal form provers on several problems from the area of artificial intelligence. We prove properties of the algorithms generalized to non-clausal form by using a novel approach based on a sequent-style formulation of the calculus.

Prefix classes of krom formulae with identity

Two small classes of first order formulae without function symbols but with identity, in prenex conjunctive normal form with all disjunctions binary, are shown to have a recursively unsolvable decision problem, whereas for another such class an algorithm is developed which solves the decision problem of that class. This solves the prefix problem for such classes of formulae except for the Godel-Kalmar-Schutte case.

The reachability problem for Petri nets and decision problems for Skolem arithmetic

We show that the decision problem for the class C 0 of all closed universal Horn formulae in prenex conjunctive normal form of extended Skolem arithmetic without equality (i.e. first order formulae built up from the multiplication sign, constants for the natural numbers and free occuring predicate symbols) is exponentially time bounded equivalent to the reachability problem for Petri nets if restricted to the class of formulae with ( a) only monadic predicate symbols, with ( b) only binary disjunctions in the quantifier free matrix and ( c) without terms containing a variable more than once. We show that leaving out one of the restrictions ( a) to ( c) yields classes of formulae whose decision problem can assume any prescribed recursively enumerable complexity in terms of many-one degrees of unsolvability.

The decision problem for formulas in prenex conjunctive normal form with binary disjunctions

In [8] S. J. Maslov gives a positive solution to the decision problem for satisfiability of formulas of the formin any first-order predicate calculus without identity where h, k, m, n are positive integers, αi, βi are signed atomic formulas (atomic formulas or negations of atomic formulas), and ∧, ∨ are conjunction and disjunction symbols, respectively (cf. [6] for a related solvable class). In this paper we show that the decision problem is unsolvable for formulas that are like those considered by Maslov except that they have prefixes of the form ∀x∃y1 … ∃yk∀z. This settles the decision problems for all prefix classes of formulas for formulas that are in prenex conjunctive normal form in which all disjunctions are binary (have just two terms). In our concluding section we report results on decision problems for related classes of formulas including classes of formulas in languages with identity and we describe some special properties of formulas in which all disjunctions are binary including a property that implies that any proof of our result, that a class of formulas is a reduction class for satisfiability, is necessarily indirect. Our proof is based on an unsolvable combinatorial tag problem.

Existential Horn sentences

A. Horn [2] has shown that every sentence of a certain class, which we define precisely below, is true in a direct product of algebras whenever it is true in each component algebra. McKinsey [4] had already shown essentially that every universal sentence preserved under direct product is equivalent to one of these Horn sentences. We establish here the analogous result for existential sentences. This answers a question raised by Chang and Morel [1], who showed that the parallel result does not hold for universal-existential sentences. The problem for existential-universal sentences remains open. An algebraic system A will be taken to consist of a nonempty set of elements, upon which are defined various operations, and various relations in addition to equality. Thus A is a model for a first order language L with symbols for the operations and relations of A. (For details, see [3 ].) Only those models will be considered in which the equality symbol is interpreted by a relation having the usual formal properties of equality-that is, by a congruence on the algebra A; but we do not require that the equality symbol be interpreted by strict identity. The direct product of similar algebras, that is, models for the same language, is defined in the usual way. Under this definition the direct product of an empty set of algebras is a trivial algebra, in which all relations, including equality, are universal. A transformation of the prenex conjunctive normal form shows that every formula of L is equivalent to a conditional formula, of the form