Quantifier-free First-order Formulas Research Articles

Efficient simplicial replacement of semialgebraic sets

Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .

The complexity of definability by open first-order formulas

Abstract 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 $\boldsymbol{\mathcal{L}}$, the $\boldsymbol{\mathcal{L}}$-definability problem for finite structures takes as an input a finite structure $\boldsymbol{A}$ and a target relation $T$ over the domain of $\boldsymbol{A}$ and determines whether there is a formula of $\boldsymbol{\mathcal{L}}$ whose interpretation in $\boldsymbol{A}$ 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 $\textrm{coW}[1]$-complete for every vocabulary $\tau $ with at least one, at least binary, relation.

Deciding Quantifier-free Definability in Finite Algebraic Structures

This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present a decision algorithm based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.

Schaefer's Theorem for Graphs

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems : it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Φ of statements (“constraints”) about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifier-free first-order formulas; the question is whether Φ is satisfiable in a graph. We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method for classifying the computational complexity of those CSPs is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.

The validity degree vectors of formulae in two-valued predicate logic

AbstractBy means of infinite product of uniformly distributed probability spaces of cardinal n, the concept of n-validity degrees and validity degree vectors of formulae in two-valued predicate logic are introduced. It is proved that the validity degree vectors of formulae can preserve the logical relation between formulae. Moreover, a consistency theorem is obtained which says that the n-validity degree τn(A) of the quantifierfree first-order formula A without any repeated predicate symbols or terms is independent of the natural number n, and is a constant equal to the validity degree τ (A0) of the corresponding proposition A0 in classical propositional logic.

Proof-Guided Test Selection from First-Order Specifications with Equality

This paper deals with test case selection from axiomatic specifications whose axioms are quantifier-free first-order formulas with equality. We first prove the existence of an ideal exhaustive test set to start the selection from. We then propose an extension of the test selection method called axiom unfolding, originally defined for algebraic specifications, to quantifier-free first-order specifications with equality. This method basically consists of a case analysis of the property under test (the test purpose) according to the specification axioms. It is based on a proof search for the different instances of the test purpose. Since the calculus is sound and complete, this allows us to provide a full coverage of this property. The generalisation we propose allows to deal with any kind of predicate (not only equality) and with any form of axiom and test purpose (not only equations or Horn clauses). Moreover, it improves our previous works with efficiently dealing with the equality predicate, thanks to the paramodulation rule.

Games and total Datalog [formula omitted] queries

We show that the expressive power of Datalog ¬ programs under the well-founded semantics does not decrease when restricted to total programs thereby affirmatively answering an open question posed by Abiteboul et al. (Foundations of Databases, Addison-Wesley, Reading, MA, 1995). In particular, we show that for every such program there exists an equivalent total program whose only recursive rule is of the form win( X ̄ ) ← move( X ̄ , Y ̄ ), ¬win( Y ̄ ), where move is definable by a quantifier-free first-order formula. Also, for the noninflationary semantics we derive a new normal form whose only recursive rule simulates a version of the game of life.

Simplification of Quantifier-free Formulae over Ordered Fields

Given a quantifier-free first-order formula over the theory of ordered fields, our aim is to find an equivalent first-order formula that issimpler. The notion of a formula being simpler will be specified. An overview is given over various methods combining elements of field theory, order theory, and logic. These methods do not require a Boolean normal form computation. They have been developed and implemented inreducefor simplifying intermediate and final results of automatic quantifier elimination by elimination sets. Their applicability is certainly not restricted to the area of quantifier elimination.

Henkin quantifiers and complete problems

We analyze computational aspects of partially ordered quantification in first-order logic. Show that almost any non-linear quantifier, applied to quantifier-free first-order formula suffices to express an NP -complete predicate. The remaining non-linear quantifiers experiment exactly co- NL predicates.

On the Borel classification of the isomorphism class of a countable model.

Introduction For p, a countable similarity type, let Xp be the space of structures of similarity type p whose universe is ω (see [13], Section 3). For any element & of Xp, let \d/ be the set of all elements of Xp which are isomorphic to CO. Scott [10] showed that \β/ is a Borel subset of Xp. In fact, he showed that for any such (2/ there is a sentence θ of Lωχω such that \C0 is exactly the set of elements of Xp which are models of θ (see [ 1 ], Ch. VII, for a good write-up of Scott sentences). In [13] Vaught considerably strengthened Scott's result. There is a natural hierarchy of formulas of Lωχω. Let ΠQ = ΣQ be the quantifier-free first-order formulas. For any a > 1 the Π° formulas are those of the form: