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Register a new userClones and Genoids in Lambda Calculus and First Order Logic
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Added by
Zhaohua Luo
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Authors
Zhaohua Luo
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A genoid is a category of two objects such that one is the product of itself with the other. A genoid may be viewed as an abstract substitution algebra. It is a remarkable fact that such a simple concept can be applied to present a unified algebraic approach to lambda calculus and first order logic.
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The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.
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The primary goal of this paper is to present a unified way to transform the syntax of a logic system into certain initial algebraic structure so that it can be studied algebraically. The algebraic structures which one may choose for this purpose are various clones over a full subcategory of a category. We show that the syntax of equational logic, lambda calculus and first order logic can be represented as clones or right algebras of clones over the set of positive integers. The semantics is then represented by structures derived from left algebras of these clones.
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