In this paper we explore a proof-theoretic approach to the combination of generic proof-search and theory-specific procedures, in presence of quantifiers. Forming the basis of tableaux methods, the sequent calculus is refined with meta-variables (a.k.a existential variables or free variables) to delay the choice of witnesses, and parameterised by theory-specific features based on a system of co...

We present an extension of first-order predicate logic with a novel predicate ‘at t’ meaning intuitively “this term is a variable symbol”. We give simple sequent proof-rules for it, we demonstrate cut-elimination for the resulting logic, and we give a semantics for which the logic is sound and complete. Because we can now make assertions about what would normally be considered an intensional pr...

Cats are nice. (: : :) because cats never overlap. Abstract In this paper we discuss two approaches to adverbial quantiication, the so-called`bound variable approach' and thèsituation-based approach', and we address the suggestion that has been made in the literature that the latter reduces to the former as soon as the underlying structure of situations has been characterized at the required le...

We present a procedure for translating standard predicate logic into English. The procedure generates both referring expressions and non-referring expressions, including both referential and bound variable anaphora. Non-referring expressions correspond to short-term discourse referents [1], which present a special set of challenges for a natural language generation system: (i) they have limited...

1. Let us agree on the following notation. Lower-case Latin letters from a to n (inclusively) with indices and without them will be used as variables for nonnegative integers, the remaining lower-case Latin letters will be used as variables for integers. Analogously, lower case Greek letters from to will be used as metavariables for nonnegative integers, and the rest of the Greek letters will b...

We argue that the semantics of a null complement spot in such sentences is given by a free variable, which may either occur free (so that the null complement is interpreted like a deictic) or bound (so that the reading of the whole utterance is quantificational). We also propose that the syntax is what one sees on the surface: there aren't empty categories which are complements to the verbs, an...

The standard second-order matching algorithm by Huet may be expansive in matching a exible-rigid pair. On one hand, many fresh free variables may need to be introduced; on the other hand, attempts are made to match the heading free variable on the exible side with every \top layer" on the rigid side and every argument of the heading free variable with every subterm covered by the \top layer". W...

This paper addresses the problem of how to represent free variables and subexpressions involving-bindings. The aim is to apply what is known as higher-order abstract syntax to higher-order term rewriting systems. Directly applying-reduction for the purpose of subterm-replacement is incompatible with the requirements of term-rewriting. A new meta-level representation of subterms is developed tha...

Let (∗l∈IA, ∗l∈I(φl, ψl)), be the conditionally free product of unital free *algebras Al, where φl, ψl are states on Al, l ∈ I. We construct a sequence of noncommutative probability spaces (Ã(m), Φ̃(m)), m ∈ N, where Ã(m) = ⊗ l∈I Ã ⊗m l and Φ̃ (m) = ⊗ l∈I φ̃l ⊗ ψ̃ ⊗(m−1) l , m ∈ N, Ãl = A ∗ C[t], and the states φ̃l, ψ̃l are Boolean extensions of φl, ψl, l ∈ I, respectively. We define unital *-homomor...

We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The exec...