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15
Graph lambda theories
 Journal of Logic and Computation
, 2004
"... Lambda theories are equational extensions of the untyped lambda calculus that are closed under derivation. The set of lambda theories is naturally equipped with a structure of complete lattice, where the meet of a family of lambda theories is their intersection, and the join is the least lambda theo ..."
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Cited by 26 (13 self)
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Lambda theories are equational extensions of the untyped lambda calculus that are closed under derivation. The set of lambda theories is naturally equipped with a structure of complete lattice, where the meet of a family of lambda theories is their intersection, and the join is the least lambda theory containing their union. In this paper we study the structure of the lattice of lambda theories by universal algebraic methods. We show that nontrivial quasiidentities in the language of lattices hold in the lattice of lambda theories, while every nontrivial lattice identity fails in the lattice of lambda theories if the language of lambda calculus is enriched by a suitable finite number of constants. We also show that there exists a sublattice of the lattice of lambda theories which satisfies: (i) a restricted form of distributivity, called meet semidistributivity; and (ii) a nontrivial identity in the language of lattices enriched by the relative product of binary relations.
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 11 (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 11 (6 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.
Applying Universal Algebra to Lambda Calculus
, 2008
"... 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 s ..."
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Cited by 10 (4 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.
Easiness in graph models
 Theoretical Computer Science
, 1993
"... We generalize Baeten and Boerboom’s method of forcing, and apply this to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λterms satisfying the following properties: a) For any countable sequence (gk)k∈ω of continuous functions (of arbitrary arity) on the power set of an arbitrary co ..."
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Cited by 6 (4 self)
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We generalize Baeten and Boerboom’s method of forcing, and apply this to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λterms satisfying the following properties: a) For any countable sequence (gk)k∈ω of continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that (λx.xx)(λx.xx)uk represents gk in the model. b) For any countable sequence (tk)k∈ω of closed λterms there is a graph model that satisfies (λx.xx)(λx.xx)uk = tk for all k. We apply these two results to show the existence of 1. a finitely axiomatized λtheory L such that the interval lattice constituted by the λtheories extending L is distributive; 2. a continuum of pairwise inconsistent graph theories ( = λtheories that can be realized as theories of graph models); 3. a congruence distributive variety of combinatory algebras (lambda
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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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.
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.
Isomorphic Data Encodings in Haskell and their Generalization to Hylomorphisms on Hereditarily Finite Data Types
, 808
"... Abstract. This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, DNA sequences etc.) and ..."
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Abstract. This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, DNA sequences etc.) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides anytoany encodings automatically. Besides applications to experimental mathematics, a few examples of “free algorithms ” obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to selfdelimiting codes, succinct data representations and generation of random instances. The paper covers 47 data types and, through the use of the embedded combinator language, provides 2162 distinct bijective transformations between them. The selfcontained source code of the paper, as generated from a literate Haskell program, is available at
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.