The power of primitive positive definitions with polynomially many variables

Two well-studied closure operators for relations are based on existentially quantified conjunctive formulas, primitive positive (p.p.) definitions, and primitive positive formulas without existential quantification, quantifier-free primitive positive definitions (q.f.p.p.) definitions. Sets of relations closed under p.p. definitions are known as co-clones and sets of relations closed under q.f.p.p. definitions as weak partial co-clones. The latter do however have limited expressivity, and the corresponding lattice of strong partial clones is of uncountably infinite cardinality even for the Boolean domain. Hence, it is reasonable to consider the expressiveness of p.p. definitions where only a small number of existentially quantified variables are allowed. In this article, we consider p.p. definitions allowing only polynomially many existentially quantified variables, and say that a co-clone closed under such definitions is polynomially closed, and otherwise superpolynomially closed. We investigate properties of polynomially closed co-clones and prove that if the corresponding clone contains a k-ary near-unanimity operation for k amp;gt;= 3, then the co-clone is polynomially closed, and if the clone does not contain a k-edge operation for any k amp;gt;= 2, then the co-clone is superpolynomially closed. For the Boolean domain we strengthen these results and prove a complete dichotomy theorem separating polynomially closed co-clones from superpolynomially closed co-clones. Using these results, we then proceed to investigate properties of strong partial clones corresponding to superpolynomially closed co-clones. We prove that if Gamma is a finite set of relations over an arbitrary finite domain such that the clone corresponding to Gamma is essentially unary, then the strong partial clone corresponding to Gamma is of infinite order and cannot be generated by a finite set of partial functions

Weak Bases of Boolean Co-Clones

Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For classifications where primitive positive definitions are unsuitable, such as size-preserving reductions, weaker closure operations may be necessary. In this article we consider strong partial clones which can be seen as a more fine-grained framework than Post's lattice where each clone splits into an interval of strong partial clones. We investigate these intervals and give simple relational descriptions, weak bases, of the largest elements. The weak bases 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 is considerably simpler than the full lattice of partial clones

Clones from Creatures

A clone on a set X is a set of finitary operations on X which contains all the projections and is closed under composition. The set of all clones forms a complete lattice Cl(X) with greatest element O, the set of all finitary operations. For finite sets X the lattice is "dually atomic": every clone other than O is below a coatom of Cl(X). It was open whether Cl(X) is also dually atomic for infinite X. Assuming the continuum hypothesis, we show that there is a clone C on a countable set such that the interval of clones above C is linearly ordered, uncountable, and has no coatoms.Comment: LaTeX2e, 20 pages. Revised version: some concepts simplified, proof details adde

Equivalence of operations with respect to discriminator clones

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.Comment: 17 page

A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page