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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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Cited by 23 (15 self)
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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.
Compositional Characterizations of λterms using Intersection Types (Extended Abstract)
, 2000
"... We show how to characterize compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and weak head normalization. We consider also the ..."
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Cited by 17 (6 self)
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We show how to characterize compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and weak head normalization. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterization results are new, to our knowledge, or else they streamline, strengthen, or generalize earlier results in the literature. The completeness parts of the characterizations are proved uniformly for all the properties, using a settheoretical semantics of intersection types over suitable kinds of stable sets. This technique generalizes Krivine 's and Mitchell's methods for strong normalization to other evaluation properties.
A Continuum of Theories of Lambda Calculus Without Semantics
 16TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS 2001), IEEE COMPUTER
, 2001
"... In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we ..."
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Cited by 16 (11 self)
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In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we positively solve the conjecture, stated by BastoneroGouy [6, 7] and by Berline [10], that the strongly stable semantics is incomplete. 1
Simple easy terms
 Intersection Types and Related Systems, volume 70 of Electronic Notes in Computer Science
, 2002
"... Dipartimento di Informatica Universit`a di Venezia ..."
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Cited by 12 (4 self)
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Dipartimento di Informatica Universit`a di Venezia
The sensible graph theories of lambda calculus
 IN: 19TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS’04), IEEE COMPUTER
, 2004
"... Sensible λtheories are equational extensions of the untyped lambda calculus that equate all the unsolvable λterms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a nonsyntactic model whose equational theory is the least sensible λtheory H (g ..."
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Cited by 12 (8 self)
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Sensible λtheories are equational extensions of the untyped lambda calculus that equate all the unsolvable λterms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a nonsyntactic model whose equational theory is the least sensible λtheory H (generated by equating all the unsolvable terms). A related question is whether, given a class of models, there exist a minimal and maximal sensible λtheory represented by it. In this paper we give a positive answer to this question for the semantics of lambda calculus given in terms of graph models. We conjecture that the least sensible graph theory, where “graph theory ” means “λtheory of a graph model”, is equal to H, while in the main result of the paper we characterize the greatest sensible graph theory as the λtheory B generated by equating λterms with the same Böhm tree. This result is a consequence of the fact that all the equations between solvable λterms, which have different Böhm trees, fail in every sensible graph model. Further results of the paper are: (i) the existence of a continuum of different sensible graph theories strictly included in B (this result positively answers Question 2 in [7, Section 6.3]); (ii) the nonexistence of a graph model whose equational theory is exactly the minimal lambda theory λβ (this result negatively answers Question 1 in [7, Section 6.2] for the restricted class of graph models).
A Complete Characterization of Complete IntersectionType Theories (Extended Abstract)
 ACM TOCL
, 2000
"... M. DEZANICIANCAGLINI Universita di Torino, Italy F. HONSELL Universita di Udine, Italy F. ALESSI Universita di Udine, Italy Abstract We characterize those intersectiontype theories which yield complete intersectiontype assignment systems for lcalculi, with respect to the three canonical ..."
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Cited by 12 (5 self)
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M. DEZANICIANCAGLINI Universita di Torino, Italy F. HONSELL Universita di Udine, Italy F. ALESSI Universita di Udine, Italy Abstract We characterize those intersectiontype theories which yield complete intersectiontype assignment systems for lcalculi, with respect to the three canonical settheoretical semantics for intersectiontypes: the inference semantics, the simple semantics and the Fsemantics. Keywords Lambda Calculus, Intersection Types, Semantic Completeness, Filter Structures. 1 Introduction Intersectiontypes disciplines originated in [6] to overcome the limitations of Curry 's type assignment system and to provide a characterization of strongly normalizing terms of the lcalculus. But very early on, the issue of completeness became crucial. Intersectiontype theories and filter lmodels have been introduced, in [5], precisely to achieve the completeness for the type assignment system l" BCD W , with respect to Scott's simple semantics. And this result, ...
Intersection Types and Lambda Models
, 2005
"... Invariance of interpretation by #conversion is one of the minimal requirements for any standard model for the #calculus. With the intersection type systems being a general framework for the study of semantic domains for the #calculus, the present paper provides a (syntactic) characterisation of t ..."
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Cited by 11 (1 self)
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Invariance of interpretation by #conversion is one of the minimal requirements for any standard model for the #calculus. With the intersection type systems being a general framework for the study of semantic domains for the #calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.
Boolean algebras for lambda calculus
 21TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS 2006), IEEE COMPUTER
, 2006
"... In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combin ..."
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Cited by 10 (8 self)
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In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory T, there is a continuum of lambda theories including T which are omitted by the indecomposable semantics.
The Minimal Graph Model of Lambda Calculus
"... A longstanding open problem in lambdacalculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambdacalculus whose theory is exactly the betatheory or the betaetatheory. A related question, raised recently by C.Berline, is whether, given a class of lambdamode ..."
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Cited by 9 (8 self)
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A longstanding open problem in lambdacalculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambdacalculus whose theory is exactly the betatheory or the betaetatheory. A related question, raised recently by C.Berline, is whether, given a class of lambdamodels, there is a minimal equational theory represented by it.
Lambda theories of effective lambda models
 In 16th EACSL Annual Conference on Computer Science and Logic (CSL’07), LNCS
, 2007
"... Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recu ..."
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Cited by 9 (5 self)
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Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λcalculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem.