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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.
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 11 (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).
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.
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.
What is a Categorical Model of the Differential and the Resource λCalculi?
"... The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitab ..."
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Cited by 5 (1 self)
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The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows to write the full Taylor expansion of a program. Through this expansion every program can be decomposed into an infinite sum (representing nondeterministic choice) of ‘simpler’ programs that are strictly linear. The aim of this paper is to develop an abstract ‘model theory ’ for the untyped differential λcalculus. In particular, we investigate what should be a general categorical definition of denotational model for this calculus. Starting from the work of Blute, Cockett and Seely on differential categories we provide the notion of Cartesian closed differential category and we prove that linear reflexive objects living in such categories constitute sound models of the untyped differential λcalculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This entails that every model living in such categories equates all programs having the same full Taylor expansion. We then
A general class of models of H
 in "Mathematical Foundations of Computer Science (MFCS’09)", Lecture Notes in Computer Science
"... Abstract. We recently introduced an extensional model of the pure λcalculus living in a cartesian closed category of sets and relations. In this paper, we provide sufficient conditions for categorical models living in arbitrary cpoenriched cartesian closed categories to have H ∗ , the maximal cons ..."
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Cited by 3 (1 self)
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Abstract. We recently introduced an extensional model of the pure λcalculus living in a cartesian closed category of sets and relations. In this paper, we provide sufficient conditions for categorical models living in arbitrary cpoenriched cartesian closed categories to have H ∗ , the maximal consistent sensible λtheory, as their equational theory. Finally, we prove that our relational model fulfils these conditions.
Towards Lambda Calculus OrderIncompleteness
 Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science
"... After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially orde ..."
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Cited by 3 (3 self)
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After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially ordered model (orderincompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is orderincomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the orderincompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).
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.
Applying Universal Algebra to Lambda Calculus
, 2007
"... The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to se ..."
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Cited by 2 (2 self)
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The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the longstanding open questions concerning the representability of λtheories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λabstraction algebras. In every combinatory and λabstraction algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λabstraction algebra as a weak Boolean product of directly indecomposable 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 λtheories. 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.
The Visser topology of lambda calculus
"... A longstanding open problem in lambda calculus is whether there exists a nonsyntactical model of the untyped lambda calculus whose theory is exactly the least λtheory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a m ..."
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Cited by 1 (1 self)
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A longstanding open problem in lambda calculus is whether there exists a nonsyntactical model of the untyped lambda calculus whose theory is exactly the least λtheory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a model can be recursively enumerable (r.e. for brevity). We introduce the notion of an effective model of lambda calculus and prove the following results: (i) The equational theory of an effective model cannot be λβ, λβη; (ii) The order theory of an effective model cannot be r.e.; (iii) 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 the following, where “graph theory ” is a shortcut for “theory of a graph model”: (iv) There exists a minimum order graph theory (for equational graph theories this was proved in [9, 10]). (v) The minimum equational/order graph theory is the theory of an effective graph model. (vi) No order graph theory can be r.e. (vii) Every equational/order graph theory is the theory of a graph model having a countable web. This last result proves that the class of graph models enjoys a kind of (downwards) LöwenheimSkolem theorem, and it answers positively Question 3 in [4, Section 6.3] for the class of graph models.