The complexity of definability by open first-order formulas

In this article, we formally define and investigate the computational complexity of the definability problem for open first-order formulas (i.e. quantifier free first-order formulas) with equality. Given a logic L, the L-definability problem for finite structures takes as an input a finite structure A and a target relation T over the domain of A and determines whether there is a formula of L whose interpretation in A coincides with T. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem and prove that if the size and the arity of the target relation T are taken as parameters, then open definability is coW[1]-complete for every vocabulary τ with at least one, at least binary, relation.Fil: Areces, Carlos Eduardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Campercholi, Miguel Alejandro Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Penazzi, Daniel Eduardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Ventura, Pablo Gabriel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentin

Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Fixed-parameter tractability, definability, and model checking

In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.Comment: To appear in SIAM Journal on Computin

First Order Theories of Some Lattices of Open Sets

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic

Multidimensional extension of the Morse--Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of \ZZ^d definable by a first order formula in the Presburger arithmetic . With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $\ZZ^d$ definable in in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often

An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction

We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the known complexity classification of quantified constraint satisfaction on equality templates