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OrderIncompleteness and Finite Lambda Models (Extended Abstract)
 Eleventh Annual IEEE Symposium on Logic in Computer Science
, 1996
"... Peter Selinger Department of Mathematics University of Pennsylvania 209 S. 33rd Street Philadelphia, PA 191046395 selinger@math.upenn.edu Abstract Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about or ..."
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Peter Selinger Department of Mathematics University of Pennsylvania 209 S. 33rd Street Philadelphia, PA 191046395 selinger@math.upenn.edu Abstract Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi or fijconversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the typefree lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal'cev operators, of those lambda theories which cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the typefree lambda calculus. We introduce partial syntactical lambda models, which are derived from Plotkin's syntactical models of redu...
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ii COPYRIGHT ..."
version in that it has been copyedited and reformatted. OrderIncompleteness and Finite Lambda Reduction Models
"... Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory ofβ orβηconversion, the partial order is ..."
Abstract
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Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory ofβ orβηconversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the untyped lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal’cev operators, of those lambda theories which cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the untyped lambda calculus, or more precisely, finite models of reduction. We demonstrate how such models can be used as practical tools for giving finitary proofs of term inequalities. 1