Weak bases of Boolean co-clones

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Highlights

  • We give weak bases of all Boolean co-clones with a finite base.

  • We prove that the relations are minimal with respect to set inclusion.

  • The weak bases provide insight into the lattice of strong partial clones.

Abstract

Universal algebra has proven to be a useful tool in the study of constraint satisfaction problems (CSP) since the complexity, up to logspace reductions, is determined by the clone of the constraint language. But two CSPs corresponding to the same clone may still differ substantially with respect to worst-case time complexity, which makes clones ill-suited when comparing running times of CSP problems. In this article we instead consider an algebra where each clone splits into an interval of strong partial clones such that a strong partial clone corresponds to the CSPs that are solvable within the same O(cn) bound. We investigate these intervals and give relational descriptions, weak bases, of the largest elements. They have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones has a simpler structure than the lattice of partial clones.

Introduction

A set of functions is called a clone if (1) it is closed under composition of functions and (2) it contains all projection functions of the form ein(x1,,xn)=xi. Dually, a set of relations is called a relational clone, or a co-clone, if it contains all relations definable through formulas built up from existential quantification, conjunction, and equality constraints, over the set in question. Clones and co-clones thus group together functions and relations which share some fundamental properties, and to better understand the structure of the full set one often considers restricted sets, bases, which are still expressive enough to preserve all properties of the full set. For any domain it is thus of interest to classify the clones and co-clones on that domain and obtain a better understanding of its lattice ordered by set inclusion. In the Boolean case this goal was achieved by Post [11] and the lattice of Boolean clones is hence known as Post's lattice. Essentially the lattice determines the expressive properties of all possible Boolean functions. Due to the Galois connection between clones and co-clones the lattice of Boolean co-clones is anti-isomorphic to Post's lattice and therefore works as a complete classification of all Boolean languages. This means that given a set of relations one can associate a clone which mirrors its structure. Note however that the ordering between the two lattices is reversed and hence the smallest co-clone in fact corresponds to the largest clone. Intuitively this holds because a small co-clone has a large associated clone. The reader is referred to Böhler et al. [3], [4] for a list of bases of Boolean clones and co-clones. The lattice of Boolean co-clones is visualized in Fig. 1. The complexity of various computational problems parameterized by constraint languages such as the constraint satisfaction problem (CSP) has been shown to be determined up to logspace reducibility by Post's lattice [2], [7]. By constraint language we here understand any finite set of Boolean relations. If one on the other hand is interested in complexity classifications based on reductions which preserve the structure of instances to a larger degree, e.g. the number of variables, Post's lattice falls short, since even logspace reductions may introduce new variables which affect the running-time.

To remedy this a more fine-grained framework which further separates constraint languages based on their expressive properties is necessary. In Jonsson et al. [8] the lattice of strong partial clones is demonstrated to have the required properties. By this we mean that constraint languages corresponding to the same strong partial clone result in CSP problems solvable within exactly the same O(cn) bound. Hence a classification of the lattice of strong partial clones similar to that of Post's lattice would provide a powerful and nuanced framework for studying complexity of CSP and related problems. We wish to emphasize that even though the lattice of partial clones is known to be uncountable [1] the same does not necessarily hold for the lattice of strong partial clones. Ideally, for each clone C, one would like to determine the interval of strong partial clones whose subset of total functions equals C. The strong partial clones in this interval are said to cover C. Even though a complete classification appears difficult a good starting point is to consider the endpoints of each interval, i.e. the largest and smallest strong partial clone corresponding to C. In Creignou et al. [5] relational descriptions known as plain bases of the smallest members of these intervals are given. In this article we give simple relational descriptions known as weak bases of the largest elements in these intervals. Our work builds on the result of Schnoor and Schnoor [13], [14] but differs in two important aspects: first, each weak base presented can in a natural sense be considered to be minimal; second, we present alternative proofs where Schnoor's and Schnoor's procedure results in relations which are exponentially larger than the bases given by Böhler et al. [4] and Creignou et al. [5], and are thus also able to cover the infinite chains in Post's lattice.

Due to the Galois connection between clones and co-clones the weak bases constitute the least expressive languages, and as such each weak base results in a CSP problem with the property that it is solvable at least as fast as any other CSP problem within the same co-clone [8]. Hence the weak bases presented in Section 3 are closely connected to upper bounds of running times for problems parameterized by constraint languages.

Section snippets

Preliminaries

In this section we introduce some basic notions from universal algebra necessary for the construction of weak bases. If f is an n-ary function and R an m-ary relation it is possible to extend f such that f(t1,,tn)=(f(t1[1],,tn[1]),,f(t1[m],,tn[m])), where ti[j] denotes the j-th element of tiR. If R is closed under f we say that f preserves R or that f is a polymorphism of R. For a set of functions F we define Inv(F) (often abbreviated as IF) to be the set of all relations preserved by all

Minimal weak bases of all Boolean co-clones

In this section we proceed by giving minimal weak bases for all Boolean co-clones with finite core-size. The results are presented in Table 1. Each entry in the table consists of a co-clone, its minimal core-size, a minimal weak base and a base of the corresponding clone. As convention we use Boolean connectives to represent relations and functions whenever this promotes readability. For example x1x2 denotes the relation {(1,1)} while x1x2 denotes the relation {(0,1),(1,0)}. We use F for the

Conclusions and future work

We have determined minimal weak bases for all Boolean co-clones with a finite base. Below are some topics worthy of future investigations.

The lattice of strong partial clones. Since the weak and plain base of a co-clone IC generates the smallest and largest elements of I(IC) it would be interesting to determine the full structure of this interval. Especially one would like to determine whether these intervals are finite, countably infinite or equal to the continuum.

Complexity of constraint

Acknowledgements

The author is grateful towards Peter Jonsson, Gustav Nordh, Karsten Schölzel and Bruno Zanuttini for helpful comments and suggestions.

References (14)

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