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11
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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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.
Reflexive domains are not complete for the extensional lambda calculus
 Proc. of LICS’09, IEEE Computer Society Publications
"... Reflexive domains are not complete for the extensional λcalculus ..."
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Reflexive domains are not complete for the extensional λcalculus
Reflexive Scott domains are not complete for the extensional lambda calculus
"... A longstanding open problem is whether there exists a model of the untyped λcalculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λtheory λβ or the least extensional λtheory λβη. In this paper we analyze the class of reflexive ..."
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A longstanding open problem is whether there exists a model of the untyped λcalculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λtheory λβ or the least extensional λtheory λβη. In this paper we analyze the class of reflexive Scott domains, the models of λcalculus living in the category of Scott domains (a full subcategory of Cpo). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the λβηcalculus, i.e., there are equations not in λβη which hold in all extensional reflexive Scott domains. (ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott’s information systems, called iwebmodels. The class of iwebmodels includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in Cpo. Based on a finegrained study of an “effective” version of Scott’s information systems, we have shown that there are equations not in λβ (resp. λβη) which hold in all (extensional) iwebmodels.
Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators
, 906
"... The symmetric interaction combinators are an equally expressive variant of Lafont’s interaction combinators. They are a graphrewriting model of deterministic computation. We define two notions of observational equivalence for them, analogous to normal form and head normal form equivalence in the la ..."
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The symmetric interaction combinators are an equally expressive variant of Lafont’s interaction combinators. They are a graphrewriting model of deterministic computation. We define two notions of observational equivalence for them, analogous to normal form and head normal form equivalence in the lambdacalculus. Then, we prove a full abstraction result for each of the two equivalences. This is obtained by interpreting nets as certain subsets of the Cantor space, called edifices, which play the same role as Böhm trees in the theory of the lambdacalculus.
Effective λmodels versus recursively enumerable λtheories
"... 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 e ..."
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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 among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem.
Effective λmodels versus recursively enumerable λtheories
"... 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 rec ..."
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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 among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem. key words: λcalculus, effective λmodels, effectively given domains, recursively enumerable λtheories, graph models, LöwenheimSkolem theorem. Contents 1
Graph Lambda Theories
, 2007
"... A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the λβ or the least sensible λtheory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, ther ..."
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A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the λβ or the least sensible λtheory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there are a minimal λtheory and a minimal sensible λtheory represented by it. In this paper, we give a positive answer to this question for the class of graph models à la PlotkinScottEngeler. In particular, we build two graph models whose theories are respectively the set of equations satisfied in any graph model and in any sensible graph model. We conjecture that the least sensible graph theory, where “graph theory ” means “λtheory of a graph model”, is equal to H, while in one of the main results of the paper we show the nonexistence of a graph model whose equational theory is exactly the λβ theory. Another related question is whether, given a class of lambda models, there is a maximal sensible λtheory represented by it. 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 main technical theorem of the paper: all the equations between solvable λterms, which have different Böhm trees, fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in B.