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...

This work describes the proof and uses of a theorem allowing definition of recursive functions over the type of λ-calculus terms, where terms with bound variables are identified up to α-equivalence. The theorem embodies what is effectively a principle of primitive recursion, and the analogues of this theorem for other types with binders are clear. The theorem’s side-conditions require that the ...

‖ The methods that were used to prove the consistency of formalized theories from the finitist standpoint can be surveyed according to the following classification. ‖1. Method of valuation. It has obtained its essential development by Hilbert’s procedure of trial valuation. Using this procedure Ackermann and v. Neumann demonstrated the consistency of number theory—admittedly, under the restrict...

A theory satis es the k variable property if every rst order formula is equivalent to a formula with at most k bound variables possibly reused Gabbay has shown that a model of temporal logic satis es the k variable property for some k if and only if there exists a nite basis for the temporal connectives over that model We give a model theoretic method for establishing the k variable property in...

In this paper we describe how a combination of the classical “universal” E-unification and “rigid” Eunification, called “mixed” E-unification, can be used to efficiently handle equality in universal formula semantic tableaux, that are an extension of free variable tableaux.