Probabilistic Aspects of Boolean Switching Functions via a New Transform

A new algorithm Js mtroduced for computing the probability expression, F = P r ( f l 1), that a Boolean functionfequals 1 as a function of the probabihUes that its inputs equal 1. It is shown that this expression ts umquely characterized by a spectrum vector S. A new matrix P which has the property that S I AP, where A is the mmterm vector of the function f, is then introduced. Next, S is related to the Reed-Muller canomc (RMC) form of the function f, and it is shown that the RMC coefficient vector a can be obtained trivially from the vector S. The reverse transformation is computationally harder. It is also shown how S and P can be used to compute the Walsh ccoefficlents o f f ~ x WORDS AND PmtAsEs: Boolean switching functions, probability expression, random testing, ReedMuller canonic form, transform techniques, Walsh transform CR CATEGORIES: 5.25, 5.5, 6.1 1. In t roduc t ion Several t ransform techniques which m a p a Boolean funct ion into various domains have been extensively studied; examples are the Reed -Mul l e r canonic (RMC) representat ion and the Walsh coefficient representat ion C. These and other transforms have been used in fault diagnosis [3, 11], in the classification o f Boolean functions [5, 6, 10], and in logic design [6-10, 12]. There has been considerable interest in the area o f r a n d o m testing o f digital circuits [1, 4, 14, 16, 17, 19, 20]. The probabi l i ty expression, F = P r ( f = 1), that a Boolean f u n c t i o n f e q u a l s 1 as a funct ion o f the probabilit ies that its inputs equal 1 is a key factor in generat ing efficient r a n d o m tests for digital circuits. The central issue in this paper deals with the generat ion o f the probabi l i ty expression F. We have identified and constructed a highly structured matr ix P with the proper ty that the spectrum S (an equivalent representat ion for F ) is given by S = A P , where A is the min te rm vector o f the func t ionf . The elegant structure o f P is helpful in identifying useful properties o f P, S, and F. W e have also discovered several relations between the spect rum vector S, the R M C coefficient vector a, and the Walsh coefficient vector C o f the Boolean funct ion f It is established in this paper that the R M C coefficient vector a can be obta ined trivially f rom the spectrum vector S. The reverse t ransformat ion is computa t ional ly harder. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the pubhcatton and its date appear, and nottce is gtven that copying is by permisston of the Associatton for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. This work was supported in part by the National Science Foundation under Grant MSC 78-26153. Authors' present addresses: S. K. Kumar, Computer Automation, Anaheim CA; M A Breuer, Department of Electrical Engineering-Systems, Umversity of Southern Cahfornla, Los Angeles, CA 90007 © 1981 ACM 0004-5411/81/0700.-0502 $00.75 Journal of the AssocaaUou for Computing Machinery, Vol 28, No 3, July 1981, pp 5022-520 Probabilistic Aspects of Boolean Switching Functions 503 In Section 2 we discuss the background material and the properties of the probability expression of Boolean functions. In Section 3 we develop the transform P used for constructing S. Properties of the matrix P used in the transformation and the properties of the Boolean function in the transformed domain are analyzed. In Sections 4 and 5 the relationship between the probability transform and the RM C and Walsh transforms are presented. 2. Probability Expression Analysis In this section we present the basic material required for the development of subsequent sections and some fundamental properties of probability expressions of Boolean functions. 2.1. MATHEMATICAL BACKGROUND. Logic signals in digital circuits are denoted by the symbols Xl, x2 . . . . . Boolean functions are denoted byfi f i , j~, . . . . Definition 1. The probability X1 = Pr(xl = 1) of a logic signal xx is a real number over the closed interval [0, 1] and denotes the probability that the logic signal x~ equals 1. Since Xl is Boolean, Pr(xl = 0) = 1 Pr(xl = 1) = 1 Xv The following lemmas establish the relationship between Boolean operations on logic signals and the corresponding operations on probabilities of logic signals. LEMMA 1 [14]. The Boolean AND of two independent signals xl, x2 in the Boolean function f = Xl X2 corresponds to the probability expression F ffi Xi Xz, where F denotes the probability that f = 1 and the implied operation is multiplication. LEMMA 2 [14]. The Boolean OR of two independent signals Xl, x2 in the Boolean function f = xl V x2 corresponds to the probability expression F = Xi + X 2 X~X2. LEMMA 3 [14]. The Boolean AND of signal xl with itself in the Boolean expression f = XlXl corresponds to the probability expression F = X1. Parker and McCluskey [14] devised two algorithms for deriving the output probability for a combinational logic circuit in terms of a set of input probabilities. Algorithm 1 requires the canonical sum-of-products form (or the minterm form) for a Boolean function, and Algorithm 2 requires a circuit description and exponent suppression. 1 Exponents arise in the calculation of probability expressions when Lemmas 1-3 are used in the presence of signals with reconvergent fanout. It has been shown informally in [14] that when the probability expression for any Boolean function is calculated by treating all signals as though they were independent and exponents in the resulting expression are suppressed, the correct probability expression results. The following example illustrates this approach for calculating the probability expression. Consider the circuit of Figure 1. Using Lemma 1, we have Pr(xlx2 = 1) = X1X2, Pr(x2x3 = 1) = X2Xs. Exponent suppression means that X~ can be replaced by X,.