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Topological Incompleteness and Order Incompleteness of the Lambda Calculus
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2001
"... A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λterms closed under ff and ficonversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In th ..."
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A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λterms closed under ff and ficonversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce a new technique to prove the incompleteness of a wide range of lambda calculus semantics, including the strongly stable one, whose incompleteness had been conjectured by BastoneroGouy [6, 7] and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in terms of topological models whose topology satisfies a property of connectedness. In the second one we prove the incompleteness of the class of partially ordered models with finitely many connected components w.r.t. the Alexandroff topology. A further result of the paper is a proof of the completeness of the semantics of the lambda calculus given in terms of topological models whose topology is nontrivial and metrizable.
Lambda calculus: models and theories
 Proceedings of the Third AMAST Workshop on Algebraic Methods in Language Processing (AMiLP2003), number 21 in TWLT Proceedings, pages 39–54, University of Twente, 2003. Invited Lecture
"... In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories in ..."
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Cited by 2 (0 self)
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In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories induced by graph models of lambda calculus.