Showing papers on "Disjunctive normal form published in 2002"

A Logical Approach to Factoring Belief Networks.

Abstract: We have recently proposed a tractable logical form, known as deterministic, decomposable negation normal form (d-DNNF). We have shown that d-DNNF supports a number of logical operations in polynomial time, including clausal entailment, model counting, model enumeration, model minimization, and probabilistic equivalence testing. In this paper, we discuss another major application of this logical form: the implementation of multi-linear functions (of exponential size) using arithmetic circuits (that are not necessarily exponential). Specifically, we show that each multi–linear function can be encoded using a propositional theory, and that each d-DNNF of the theory corresponds to an arithmetic circuit that implements the encoded multi–linear function. We discuss the application of these results to factoring belief networks, which can be viewed as multi–linear functions as has been shown recently. We discuss the merits of the proposed approach for factoring belief networks, and present experimental results showing how it can handle efficiently belief networks that are intractable to structure–based methods for probabilistic inference.

Towards a Symmetric Treatment of Satisfaction and Conflicts in Quantified Boolean Formula Evaluation

Abstract: In this paper, we describe a new framework for evaluating Quantified Boolean Formulas (QBF). The new framework is based on the Davis-Putnam (DPLL) search algorithm. In existing DPLL based QBF algorithms, the problem database is represented in Conjunctive Normal Form (CNF) as a set of clauses, implications are generated from these clauses, and backtracking in the search tree is chronological. In this work, we augment the basic DPLL algorithm with conflict driven learning as well as satisfiability directed implication and learning. In addition to the traditional clause database, we add a cube database to the data structure. We show that cubes can be used to generate satisfiability directed implications similar to conflict directed implications generated by the clauses. We show that in a QBF setting, conflicting leaves and satisfying leaves of the search tree both provide valuable information to the solver in a symmetric way. We have implemented our algorithm in the new QBF solver Quaffle. Experimental results show that for some test cases, satisfiability directed implication and learning significantly prunes the search.

Relative Completeness of Abstraction Refinement for Software Model Checking

Abstract: Automated methods for an undecidable class of verification problems cannot be complete (terminate for every correct program). We therefore consider a new kind of quality measure for such methods, which is completeness relative to a (powerful but unrealistic) oraclebased method. More precisely, we ask whether an often implemented method known as "software model checking with abstraction refinement" is complete relative to fixpoint iteration with "oracle-guided" widening. We show that whenever backward fixpoint iteration with oracle-guided widening succeeds in proving a property ? (for some sequence of widenings determined by the oracle) then software model checking with a particular form of backward refinement will succeed in proving ?. Intuitively, this means that the use of fixpoint iteration over abstractions and a particular backwards refinement of the abstractions has the effect of exploring the entire state space of all possible sequences of widenings.

On Solving Presburger and Linear Arithmetic with SAT

Abstract: We show a reduction to propositional logic from quantifier-free Presburger arithmetic, and disjunctive linear arithmetic, based on Fourier-Motzkin elimination. While the complexity of this procedure is not better than competing techniques, it has practical advantages in solving verification problems. It also promotes the option of deciding a combination of theories by reducing them to this logic.

On Solving Temporal Logic Queries

Abstract: Temporal query checking is an extension of temporal model checking where one asks what propositional formulae can be inserted in a temporal query (a temporal formula with a placeholder) so that the resulting formula is satisfied in the model at hand.We study the problem of computing all minimal solutions to a temporal query without restricting to so-called "valid" queries (queries guaranteed to have a unique minimal solution). While this problem is intractable in general, we show that deciding uniqueness of the minimal solution (and computing it) can be done in polynomial-time.

On Normalization of Relations in Relational Databases

Abstract: A new normal form, namely, object-normal form (ONF), is introduced. It is shown that the existing definitions of the fifth normal form (5NF) are unsatisfactory. The correct definition is given for the first time. The importance of the 5NF is demonstrated. For improving data representation in a database with a 5NF schema, the notion of negating relation is introduced. It serves as a basis for the new normal form, namely, the sixth normal form. Combining requirements of the ONF and other normal forms, new normal forms are defined: the fourth object-normal form, the fifth object-normal form, and the sixth object-normal form. It is shown that the standard order of steps of the normalization procedure should be changed: first, the specific requirements of the fourth normal form should be satisfied, and only then the requirements of the second normal form, and so forth.

Heuristics for Efficient Manipulation of Composite Constraints

Abstract: Composite Symbolic Library is a symbolic manipulator for model checking systems with heterogeneous data types Our current implementation uses two basic symbolic representations: BDDs for boolean and enumerated variables, and polyhedra for (unbounded) integers These basic representations are imported to the Composite Symbolic Library using a common interface and are combined using a disjunctive composite representation In this paper, we present several heuristics for efficient manipulation of this composite representation Our heuristics make use of the following observations: 1) efficient operations on BDDs can be used to mask expensive operations on polyhedra, 2) our disjunctive representation can be exploited by computing pre and post-conditions and subset checks incrementally, and 3) size of a composite representation can be minimized by iteratively merging matching constraints and removing redundant ones We present experimental results that illustrate efficiency of our algorithms

Theory Revision with Queries: DNF Formulas

Abstract: The theory revision, or concept revision, problem is to correct a given, roughly correct concept This problem is considered here in the model of learning with equivalence and membership queries A revision algorithm is considered efficient if the number of queries it makes is polynomial in the revision distance between the initial theory and the target theory, and polylogarithmic in the number of variables and the size of the initial theory The revision distance is the minimal number of syntactic revision operations, such as the deletion or addition of literals, needed to obtain the target theory from the initial theory Efficient revision algorithms are given for three classes of disjunctive normal form expressions: monotone k-DNF, monotone m-term DNF and unate two-term DNF A negative result shows that some monotone DNF formulas are hard to revise

Test approach to the implementation of boolean functions with few zeros by disjunctive normal forms

Abstract: An algorithm is proposed that reduces the construction of a (irredundant) disjunctive normal form (DNF) of a function from its k-by-n matrix of zeros to the construction of a (irredundant) DNF of a function with a k-by-t matrix of zeros, where t ≤ 2 log 2 k + o (log 2 k) for almost all matrices. The reduction allows an effective construction of (irredundant) DNFs of functions with few zeros.

Inequalities in De Morgan systems. II

Abstract: For pt.I see ibid., p.607-9(2002). Turksen (1986) looked at the Boolean-disjunctive normal form of a term in two variables and the Boolean-conjunctive normal form of that term. He concluded that "for some cases of certain families," when these forms are interpreted in a De Morgan system by replacing the meet by a t-norm and the join by the dual t-conorm, the disjunctive normal form is contained in the conjunctive normal form. We give examples where some of these inequalities fail to hold for a large class of t-norms, and also give examples where they do hold.

Construction of disjunctive normal forms in logical recognition algorithms

Abstract: Efficient methods are proposed for constructing disjunctive normal forms (DNF) of a Boolean function based on the set of its zeros. These methods make it possible to construct the DNFs of class characteristic functions for logical recognition algorithms. The construction of an irredundant DNF of a Boolean function defined by the set of its zeros is considered.

KeyNote Policy Files and Conversion to Disjunctive Normal Form for Use in IPsec

Abstract: : We describe a utility for converting a KeyNote policy file to Disjunctive Normal Form, so that it can be further utilized in our research on Quality of Security Service for IPsec. We also provide background information on KeyNote and IPsec, on the Disjunctive Normal Form of logical expressions, as well as on the lex and yacc tools employ by our utility.

/spl alpha/-automated reasoning method based on regular generalized conjunctive normal form of LP(X)

Abstract: We focus on automated reasoning based on lattice-valued propositional logic LP(X). A new method of automated reasoning is given, and the soundness and completeness theorems of this method are also proved.

A Partially Solved Form for Heterogeneous Constraints in Disjunctive Normal Form

Abstract: In order to support constraint solving for challenging engineering applications, as e.g. accomplished by the Relational Constraint Solver (see [MST]), we need to implement join and project operators (see e.g. [AHV] or [M]) for heterogeneous constraints. The heterogeneity is due to finite domain and real-valued variables, linear and non-linear arithmetic constraints, (dis-)equations and inequalities.

An On-Line Satisfiability Algorithm for Conjunctive Normal Form Expressions with Two Literals

Abstract: This paper describes two algorithms for determining the satisfiability of Boolean conjunctive normal form expressions limited to two literals per clause (2-SAT) extending the classic effort of Aspvall, Plass, and Tarjan. The first algorithm differs from the original in that satisfiability is determined upon the presentation of each clause rather than the entire clause set. This online algorithm experimentally exhibits average run-time linear to the number of variables. This performance is achieved by performing a single depth first search of one of the incoming literals. Additional search is avoided by excluding clauses containing pure variables or variables whose truth value has been explicitly provided or can be inferred. An off-line algorithm is also described that incorporates these strategies.

Comparison of two algorithms for simplifying disjunctive normal forms

Abstract: The action of two classes of local algorithms for simplifying disjunctive normal forms (DNFs) on shorter DNFs of Boolean functions of few variables is compared. It is proved that for no less than seven variables, there exist functions whose shorter DNFs are differently simplified by algorithms of different classes, while the shorter DNF of an arbitrary function of no more than six variables is equally simplified by algorithms from these classes.