Generic Expression Hardness Results for Primitive Positive Formula Comparison

We study the expression complexity of two basic problems involving the comparison of primitive positive formulas: equivalence and containment. In particular, we study the complexity of these problems relative to finite relational structures. We present two generic hardness results for the studied problems, and discuss evidence that they are optimal and yield, for each of the problems, a complexity trichotomy

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Searching for an algebra on CSP solutions

The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism ${\mathbf R}\to {\bf \Gamma}$ between two relational structures, where ${\mathbf R}$ is defined over a domain $V$ and ${\bf \Gamma}$ is defined over a domain $D$. In a fixed template CSP, denoted $CSP({\bf \Gamma})$, the right side structure ${\bf \Gamma}$ is fixed and the left side structure ${\mathbf R}$ is unconstrained. We consider the following problem: given a prespecified finite set of algebras ${\mathcal B}$ whose domain is $D$, is it possible to present the solutions set of a given instance of $CSP({\bf \Gamma})$ (which is an input to the problem) as a subalgebra of ${\mathbb A}_1\times ... \times {\mathbb A}_{|V|}$ where ${\mathbb A}_i\in {\mathcal B}$? We study this problem and show that it can be reformulated as an instance of a certain fixed-template CSP, over another template ${\bf \Gamma}^{\mathcal B}$. First, we demonstrate examples of ${\mathcal B}$ for which $CSP({\bf \Gamma}^{\mathcal B})$ is tractable for any, possibly NP-hard, ${\bf \Gamma}$. Under natural assumptions on ${\mathcal B}$, we prove that $CSP({\bf \Gamma}^{\mathcal B})$ can be reduced to a certain fragment of $CSP({\bf \Gamma})$. We also study the conditions under which $CSP({\bf \Gamma})$ can be reduced to $CSP({\bf \Gamma}^{\mathcal B})$. Since the complexity of $CSP({\bf \Gamma}^{\mathcal B})$ is defined by $Pol({\bf \Gamma}^{\mathcal B})$, we study the relationship between $Pol({\bf \Gamma})$ and $Pol({\bf \Gamma}^{\mathcal B})$. It turns out that if $\mathcal{B}$ is preserved by $p\in Pol({\bf \Gamma})$, then $p$ can be extended to a polymorphism of ${\bf \Gamma}^{\mathcal B}$. In the end to demonstrate usefulness of our definitions we study one case when ${\bf \Gamma}$ is not of bounded width, but ${\bf \Gamma}^{\mathcal B}$ is of bounded width (i.e. has a richer structure of polymorphisms).Comment: 34 page

On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded relational width

Eilenberg theorems for many-sorted formations

A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts $S$ and a fixed $S$-sorted signature $\Sigma$, the concepts of formation of congruences with respect to $\Sigma$ and of formation of $\Sigma$-algebras, we prove that the algebraic lattices of all $\Sigma$-congruence formations and of all $\Sigma$-algebra formations are isomorphic, which is an Eilenberg's type theorem. Moreover, under a suitable condition on the free $\Sigma$-algebras and after defining the concepts of formation of congruences of finite index with respect to $\Sigma$, of formation of finite $\Sigma$-algebras, and of formation of regular languages with respect to $\Sigma$, we prove that the algebraic lattices of all $\Sigma$-finite index congruence formations, of all $\Sigma$-finite algebra formations, and of all $\Sigma$-regular language formations are isomorphic, which is also an Eilenberg's type theorem.Comment: 46 page

Disjoint-union partial algebras

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjoint-union partial algebra of sets but prove that no finite axiomatisation exists. We do the same for other signatures including one or both of disjoint union and subset complement, another partial binary operation we define. Domain-disjoint union is a partial binary operation on partial functions, returning the union if the arguments have disjoint domains and undefined otherwise. For each signature including one or both of domain-disjoint union and subset complement and optionally including composition, we consider the class of partial algebras isomorphic to a collection of partial functions closed under the operations. Again the classes prove to be axiomatisable, but not finitely axiomatisable, in first-order logic. We define the notion of pairwise combinability. For each of the previously considered signatures, we examine the class isomorphic to a partial algebra of sets/partial functions under an isomorphism mapping arbitrary suprema of pairwise combinable sets to the corresponding disjoint unions. We prove that for each case the class is not closed under elementary equivalence. However, when intersection is added to any of the signatures considered, the isomorphism class of the partial algebras of sets is finitely axiomatisable and in each case we give such an axiomatisation.Comment: 30 page