Properties of Switch-List Representations of Boolean Functions

Abstract

In this paper, we focus on a less usual way to represent Boolean functions, namely on representations by switch-lists, which are closely related to interval representations. Given a truth table representation of a Boolean function f the switch-list representation of f is a list of Boolean vectors from the truth table which have a different function value than the preceding Boolean vector in the truth table. The main aim of this paper is to include this type of representation in the Knowledge Compilation Map by Darwiche and Marquis and to argue that switch-lists may in certain situations constitute a reasonable choice for a target language in knowledge compilation. First, we compare switch-list representations with a number of standard representations (such as CNF, DNF, and OBDD) with respect to their relative succinctness. As a by-product of this analysis, we also give a short proof of a longstanding open question proposed by Darwiche and Marquis, namely the incomparability of MODS (models) and PI (prime implicates) representations. Next, using the succinctness result between switch-lists and OBDDs, we develop a polynomial time compilation algorithm from switch-lists to OBDDs. Finally, we analyze which standard transformations and queries (those considered by Darwiche and Marquis) can be performed in polynomial time with respect to the size of the input if the input knowledge is represented by a switch-list. We show that this collection is very broad and the combination of polynomial time transformations and queries is quite unique. Some of the queries can be answered directly using the switch-list input, others require a compilation of the input to OBDD representations which are then used to answer the queries.

No Efficient Disjunction or Conjunction of Switch-Lists

“…f 1 ∧f 2 ). Chromý and Čepek [1] also considered the restricted version of bounded disjunction (resp. conjunction) in which one assumes that the involved functions f 1 , f 2 depend on the same set of variables.…”

“…, x n/2 to 1. So all models different from 0 lie in 1 We do not take into account the size of an encoding of π in this since it is the same for all switchlists in n variables and thus would only complicate the notion without giving any insights.…”

“…To systematically understand the properties of switch-lists beyond this, Chromý and Čepek [1] analyzed them in the context of the so-called knowledge compilation map. This framework, introduced in the ground-breaking work of Darwiche and Marquis [2] gives a list of standard properties which should be analyzed for languages used in the area of knowledge compilation along different axes: succinctness, queries and transformations.…”

“…Chromý and Čepek [1] analyzed switch-lists along the properties of the knowledge compilation map and got a nearly complete picture. It turns out that switch-lists, while being generally much more succinct than truth tables, have many of their good properties.…”

“…consistency, entailement and counting) can be answered in polynomial time on switch-lists and nearly all of the transformation can be performed efficiently. The only exception is that [1] leaves open if switch-lists are closed under bounded disjunction and bounded conjunction, i.e., given two Boolean functions f 1 and f 2 represented by switch-lists, can one compute a switch-list representation of f 1 ∨ f 2 , resp. f 1 ∧ f 2 , in polynomial time.…”