Recognition of Positive 2-Interval Boolean Functions

Boolean function f on n variables is represented by set I of disjoint intervals of non-negative n-bit integers, if the following condition holds: vector x of length n is a truepoint of f , if and only if integer x, binary representation of which corresponds to vector x, belongs to some interval from I. In this paper we present a polynomial-time algorithm recognizing, whether a Boolean function given by a positive prime DNF can be represented by two intervals. Introduction Interval Boolean functions were introduced in [Schieber et al., 2005], where the minimal DNF representations of these functions were studied. The motivation of this research lies in the branch of hardware verification ([Chandra and Iyengar, 1992], [Lewin et al., 1995]) and software testing ([DeMillo and Offutt, 1991]), where more compact DNF representations of interval functions can be used to generate test data faster. In this paper we generalize the results from [Čepek et al., 2006], where we presented (together with other results) a polynomial algorithm recognizing, whether a given positive DNF represents a function, which can be represented by a single interval. Here we formulate a polynomial algorithm recognizing positive 2-interval functions. Since 1-interval functions constitute a (proper) subclass of 2-interval functions and the algorithm presented here has the same asymptotic time complexity as the algorithm from [Čepek et al., 2006] recognizing positive 1-interval functions, our new algorithm extends the previous result. Representation of a Boolean function using a small number of intervals constitutes a compact and efficient representation of the function. Therefore results (such as the one of ours) concerning recognizing whether a given function can be represented by a small number of intervals contribute to the problem of Boolean function minimization, i.e., problem when we are searching for a minimal representation of a Boolean function. This paper is structured as follows. In the rest of this section we introduce all the necessary notation and definitions. In the next section we develop the algorithm recognizing positive 2-interval functions. We give several concluding remarks in the last section. Let us start with some basic definitions. ABoolean function, or a function in short, on n propositional variables is a mapping f : {0, 1} 7→ {0, 1}. A Boolean vector of length n is an n-tuple of Boolean values 0 and 1 (usually denoted by false and true). Boolean vectors (or vectors for short) will be denoted by x,y . . .. If f(x) = 1 (0, resp.), then x is called a true (false, resp.) vector of f (sometimes called truepoint resp. falsepoint). The set of all true vectors (false vectors) is denoted by T (f) (F (f)). For function f on n variables and v ∈ {0, 1} we denote by f [xi := v] the function on (n − 1) variables, which is formed from f by fixing the value of i-th variable to v. Propositional variables x1, . . . , xn and their negations x1, . . . , xn are called literals (positive and negative literals, respectively). An elementary conjunction of literals