A specification formalism with parameterisation of an arbitrary order is presented. It is given a denotational-style semantics, accompanied by an inference system for proving that an object satisfies a specification. The inference system incorporates, but is not limited to, a clearly identified type-checking component. Special effort is made to carefully distinguish between parameterised specifications, which denote functions yielding classes of objects, and specifications of parameterised objects, which denote classes of functions yielding objects. To deal with both of these in a uniform framework, it was convenient to view specifications, which specify objects, as objects themselves, and to introduce a notion of a specification of specifications. The formalism includes the basic specification-building operations of the ASL specification language. This choice, however, is orthogonal to the new ideas presented. The formalism is also institution-independent, although this iss...

Abstract. We define infinitary combinatory reduction systems (iCRSs). This provides the first extension of infinitary rewriting to higher-order rewriting. We lift two well-known results from infinitary term rewriting systems and infinitary λ-calculus to iCRSs: 1. every reduction sequence in a fully-extended left-linear iCRS is compressible to a reduction sequence of length at most ω, and 2. every complete development of the same set of redexes in an orthogonal iCRS ends in the same term. 1

a mis hermanas, Patricia y Paula, y a mi sobrino y ahijado, Nicol'as. Acknowledgements First of all, I would like to express my gratitude to my supervisor, Eike Ritter, for his wisdom, insight, uncountably many discussions, and invaluable friendship. I am indebted to my tutor, Valeria de Paiva, who also believed in me from the very beginning, encouraged me to work in this area, showed me the beauty of logic, and, above all, honoured me with her friendship. This thesis would not exist if it were not for their constant support. Thanks to my old friends, Cecilia C. Crespo, Santiago M. Peric'as, and, especially, Mat'ias Giovannini, for being always a wonderful critic of my work. Many thanks to Mathias Kegelmann for showing me the thrill of theorem proving; and to my former supervisor, Achim Jung, for introducing me to semantics.

Introduction Consider a structure A = hA; s; k; \Deltai, where A is some set containing the distinguished elements s; k, equipped with a binary operation \Delta on A, called application, which may be partial. Notation 1.1. 1 Instead of a \Delta b we write ab; and in writing applicative expressions, the usual convention of association to the left is employed. So for elements a; b; c 2 A, the expression aba(ac) is short for ((a \Delta b) \Delta a) \Delta (a \Delta c). 2 ab # will mean that ab is defined; ab " means that ab is not defined. Obviously, an applicative expression

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