Sufficient conditions are provided for quantifier elimination to hold in a first-order theory. The conditions have two main purposes: (1) to ensure that satisfaction of existential formulas is reflected by an embedding; and (2) to guarantee the existence of a “minimal" model of the theory extending a model of the universal formulas entailed by the theory. The first goal is obtained by requiring...

A binary language-theoretic operation is proposed, which is dual to the concatenation of languages in the same sense as the universal quantifier in logic is dual to the existential quantifier; the dual of Kleene star is defined accordingly. These operations arise whenever concatenation or star appear in the scope of negation. The basic properties of the new operations are determined in the pape...

This paper is a sequel to the papers [4, 6] in which an alternative skolemization method called ekolemization was introduced that, when applied to the strong existential quantifiers in a formula, is sound and complete for constructive theories. Based on that method an analogue of Herbrand’s theorem was proved to hold as well. In this paper we extend the method to universal quantifiers and show ...

We introduce the recursive, rule-based RDF query language RDFLog. RDFLog extends previous RDF query languages by arbitrary quantifier alternation: blank nodes may occur in the scope of all, some, or none of the universal variables of a rule. In addition RDFLog is aware of important RDF features such as the distinction between blank nodes, literals and URIs or the RDFS vocabulary. The semantics ...

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, not including Q, for every ε > 0, there exists a set of primes S of natural density exceeding 1− ε, such that Z can be defined as a s...

We shall discuss several situations in which it is possible to extract from a proof, be it a proof in a first-order theory or a propositional proof, some feasible computational information about the theorem being proved. This includes extracting feasible algorithms, deterministic or interactive, for witnessing an existential quantifier, a uniform family of short propositional proofs of instance...

SPARQL has become the gold-standard for RDF query languages. Nevertheless, we believe there is further room for improving RDF query languages. In this chapter, we investigate the addition of rules and quantifier alternation to SPARQL. That extension, called SPARQLog, extends previous RDF query languages by arbitrary quantifier alternation: blank nodes may occur in the scope of all, some, or non...

We will not prove the theorem here, but we should step back and take a minute to think about the meaning of this theorem. A naive NL algorithm for Conn is to take the same NTM for Conn but flip its answers. Thus, we will accept, if there exists a wrong guess connecting s to t. However, for any (s, t), there exists a wrong guess, and we will end up accepting everything! Essentially, if (G, s, t)...

The generalized quantifier theory does not apply to Japanese quantifiers since: i) the number of NP argument is unspecified; and, ii) quantities are often expressed by predicative adjectives. The word order changes the interpretation, e.g., non-split quantifiers correspond to definite NPs with uniqueness and maximality conditions while split NPs are wide scope indefinites. This paper shows that...