Set-theoretical and other elementary models of the &lgr;-calculus
Theoretical Computer Science - A collection of contributions in honour of Corrado Bo¨hm on the occasion of his 70th birthday
Journal of Computer and System Sciences
Full abstraction in the lazy lambda calculus
Information and Computation
Information and Computation
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
Lambda abstraction algebras: coordinatizing models of Lambda calculus
Fundamenta Informaticae
Theoretical Computer Science - Modern algebra and its applications
On the algebraic models of Lambda calculus
Theoretical Computer Science - Modern algebra and its applications
Stable Models of Typed lambda-Calculi
Proceedings of the Fifth Colloquium on Automata, Languages and Programming
Order-Incompleteness and Finite Lambda Models, Extended Abstract
LICS '96 Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science
A Continuum of Theories of Lambda Calculus without Semantics
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
All Compact Hausdorff Lambda Models Are Degenerate
Fundamenta Informaticae
Graph models of $\lambda$-calculus at work, and variations
Mathematical Structures in Computer Science
Theoretical Computer Science - Algebraic methods in language processing
From λ-Calculus to Universal Algebra and Back
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
On the completeness of order-theoretic models of the λ-calculus
Information and Computation
Effective λ-models versus recursively enumerable λ-theories
Mathematical Structures in Computer Science
Order structures on böhm-like models
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
Infinitary rewriting: from syntax to semantics
Processes, Terms and Cycles
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Lambda theories of effective lambda models
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
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A model of the untyped lambda calculus univocally induces a lambda theory (i.e., a congruence relation on λ-terms closed under α- and β-conversion) through the kernel congruence relation of the interpretation function. A semantics of lambda calculus is (equationally) incomplete if there exists a lambda theory that is not induced by any model in the semantics. In this article, we introduce a new technique to prove in a uniform way the incompleteness of all denotational semantics of lambda calculus that have been proposed so far, including the strongly stable one, whose incompleteness had been conjectured by Bastonero, Gouy and Berline. We apply this technique to prove the incompleteness of any semantics of lambda calculus given in terms of partially ordered models with a bottom element. This incompleteness removes the belief that partial orderings with a bottom element are intrinsic to models of the lambda calculus, and that the incompleteness of a semantics is only due to the richness of the structure of representable functions. Instead, the incompleteness is also due to the richness of the structure of lambda theories. Further results of the article are: (i) an incompleteness theorem for partially ordered models with finitely many connected components (= minimal upward and downward closed sets); (ii) an incompleteness theorem for topological models whose topology satisfies a suitable property of connectedness; (iii) a completeness theorem for topological models whose topology is non-trivial and metrizable.