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24
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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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.
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.
Infinite rewriting: from syntax to semantics
 In Processes, Terms and Cycles: Steps on the Road to Infinity: Essays Dedicated to Jan Willem Klop on the Occasion of His 60th Birthday
, 2005
"... Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriti ..."
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Cited by 3 (1 self)
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Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriting has also
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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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 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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Cited by 1 (1 self)
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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.
From lambda calculus to universal algebra and back
 33rd International Symposium on Mathematical Foundations of Computer Science, LNCS
, 2008
"... We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λtheories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest ..."
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We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λtheories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest of a systematic study of the lattice λT of λtheories grows out of several open problems on lambda calculus. For example, the failure of certain lattice identities in λT would imply that the problem of the orderincompleteness of lambda calculus raised by Selinger has a negative answer. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit and the term algebras of all λtheories) to model the ifthenelse instruction of programming and to extend some properties of Boolean algebras to general universal algebras. The interest of Church algebras is that each has a Boolean algebra of central elements, which play the role of the idempotent elements in rings. Central elements are the key tool to represent any Church algebra as a weak Boolean product of directly indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easy λterm and prove that any Church algebra with an “easy set ” of cardinality n admits (at the top) a lattice interval of congruences isomorphic to the free Boolean algebra with n generators. This theorem has the following consequence for λT: for every recursively enumerable λtheory φ and each n, there is a λtheory φn ≥ φ such that {ψ: ψ ≥ φn} “is ” the Boolean lattice with 2 n elements. 1.
Nominal Algebra and the HSP Theorem
"... Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. ..."
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Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitelysupported permutation action); previous work proved soundness and completeness. The HSP theorem characterises the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras, and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to socalled freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this paper we give the constructions which show that, after all, a ‘nominal ’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products, and an atomsabstraction construction specific to nominalstyle semantics. Keywords: universal algebra, equational logic, nominal algebra, HSP or Birkhoff’s theorem, nominal sets, nominal terms 1