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In this paper we define a model theory and give a semantic proof of cut-elimination for ICTT, an intuitionistic formulation of Church's theory of types defined by Miller et. al. and the basis for the λProlog programming language. Our approach, extending techniques of Takahashi and Andrews and tableaux machinery of Fitting, Smullyan, Nerode and Shore, is to(More)
We develop an algebraic framework, Logic Programming Doctrines, for the syntax , proof theory, operational semantics and model theory of Horn Clause logic programming based on indexed premonoidal categories. Our aim is to provide a uniform framework for logic programming and its extensions capable of incorporating constraints, abstract data types, features(More)
A new formalism, called Hiord, for defining type-free higher-order logic programming languages with predicate abstraction is introduced. A model theory, based on partial combinatory algebras, is presented , with respect to which the formalism is shown sound. A programming language built on a subset of Hiord, and its implementation are discussed. A new(More)
We propose a new framework for the syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over-categories: ÿnite product categories with canonical structure. Constraint information is directly built-in to the notion of signature via categorical syntax. Many-sorted equational are a special case of the(More)
We use formal semantic analysis based on new, model-theoretic constructions to generate intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Well-known modal semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions , and shed little(More)
We give a fully constructive semantic proof of cut elimination for intuitionistic type theory with axioms. The problem here, as with the original Takeuti conjecture, is that the impredicativity of the formal system involved makes it impossible to define a semantics along conventional lines, in the absence, a priori, of cut, or to prove completeness by(More)