## Applying Universal Algebra to Lambda Calculus (2007)

Citations: | 2 - 2 self |

### BibTeX

@MISC{Manzonetto07applyinguniversal,

author = {Giulio Manzonetto and Antonino Salibra},

title = {Applying Universal Algebra to Lambda Calculus },

year = {2007}

}

### OpenURL

### Abstract

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 λ-theories ( = equational extensions of untyped λ-calculus) 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 long-standing open questions concerning the representability of λ-theories 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 λ-abstraction algebras. In every combinatory and λ-abstraction 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 λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other non-trivial 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 λ-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.

### Citations

1118 |
The Lambda Calculus: Its Syntax and Semantics
- Barendregt
(Show Context)
Citation Context ...iderot, and by LIX Laboratoire d’Informatique de l’Ecole Polytechnique (Palaiseau, France).Lambda calculus has been originally investigated by using mainly syntactical methods (see Barendregt’s book =-=[4]-=-). At the beginning researchers have focused their interest on a limited number of equational extensions of lambda calculus, called λ-theories. They arise by syntactical or semantic considerations. In... |

321 |
The Calculi of lambda-conversion
- Church
- 1941
(Show Context)
Citation Context ...omputational formalisms which have been introduced, the lambda calculus plays an important role as a bridge between logic and computer science. The lambda calculus was originally introduced by Church =-=[20, 21]-=- as a foundation for logic, where functions, instead of sets, were primitive, and it turned out to be consistent and successful as a tool for formalizing all computable functions. The rise of computer... |

274 | A course in universal algebra
- Burris, Sankappanavar
- 1981
(Show Context)
Citation Context ..., if a variety has factorable congruences and every member of the variety can be represented as a Boolean product of directly indecomposable algebras, then the variety is a discriminator variety (see =-=[22]-=- for the terminology). Discriminator varieties satisfy very strong algebraic properties, in particular they are congruence permutable (i.e., in each algebra the join of two congruences is just their c... |

229 | Domain theory in logical form
- Abramsky
- 1991
(Show Context)
Citation Context ...r of complementary factor congruences. Conversely, given a pair (φ, φ) of complementary factor congruences, the map f defined by: is a decomposition operation. f(x, y) = u if, and only if, x φ u φ y, =-=(1)-=- Notice that if (φ, φ) is a pair of complementary factor congruences, then for all x and y there is just one element u such that x φ u φ y. 2.10 Boolean factor congruences and Boolean products The pro... |

215 |
A filter lambda model and the completeness of type assignment
- Barendregt, Coppo, et al.
- 1984
(Show Context)
Citation Context ...dels, which were isolated in the seventies by Plotkin, Scott and Engeler [33, 61, 69], and the class of filter models, which were isolated at the beginning of eighties by Barendregt, Coppo and Dezani =-=[5]-=- after the introduction of intersection-type discipline at the end of seventies by Coppo and Dezani [26]. Filter models were investigated by Coppo, Dezani, Barendregt et al. in a series of papers and ... |

212 |
Data types as lattices
- Scott
- 1976
(Show Context)
Citation Context ...complete partial orders and morphisms are Scott continuous functions. Scott continuous semantics includes the class of graph models, which were isolated in the seventies by Plotkin, Scott and Engeler =-=[33, 61, 69]-=-, and the class of filter models, which were isolated at the beginning of eighties by Barendregt, Coppo and Dezani [5] after the introduction of intersection-type discipline at the end of seventies by... |

175 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ... similarity type is: (i) A variety if it is closed under subalgebras, homomorphic images and direct products; (ii) An equational class if it is axiomatizable by a set of equations. Birkhoff proved in =-=[14]-=- (see also [54, Thm. 4.131]) that conditions (i) and (ii) are equivalent. A variety K of algebras is generated by an algebra A ∈ K if every equation satisfied by A is also satisfied by every algebra i... |

124 | Continuous lattices - Scott |

101 |
A set of postulates for the foundation of logic
- Church
- 1932
(Show Context)
Citation Context ...omputational formalisms which have been introduced, the lambda calculus plays an important role as a bridge between logic and computer science. The lambda calculus was originally introduced by Church =-=[20, 21]-=- as a foundation for logic, where functions, instead of sets, were primitive, and it turned out to be consistent and successful as a tool for formalizing all computable functions. The rise of computer... |

88 |
An extension of the basic functionality theory for the λ-calculus
- Coppo, Dezani-Ciancaglini
- 1980
(Show Context)
Citation Context ... filter models, which were isolated at the beginning of eighties by Barendregt, Coppo and Dezani [5] after the introduction of intersection-type discipline at the end of seventies by Coppo and Dezani =-=[26]-=-. Filter models were investigated by Coppo, Dezani, Barendregt et al. in a series of papers and are perhaps the most established and studied semantics of lambda calculus (see e. g. [27, 5, 47]). Other... |

79 | Über die Bausteine der mathematischen Logik. Mathematische Annalen, 92:305–316 - Schönfinkel - 1924 |

76 |
spaces
- Stone
- 1982
(Show Context)
Citation Context ...to other varieties of algebras (see [22, Ch. IV]). Actually, this construction has been presented for several years as “the algebra of global sections of sheaves of algebras over Boolean spaces” (see =-=[25, 40]-=-); however, these notions were unnecessarily complex and we prefer to adopt here the following equivalent presentation (see [23]). We recall that a Boolean space is a compact, Hausdorff and totally di... |

69 | Commutator theory for congruence modular varieties
- Freese, McKenzie
- 1987
(Show Context)
Citation Context ...mutator to algebras other than groups is due to the pioneering papers of Smith [73] and Hagemann-Hermann [37]. The commutator is very well behaved in congruence modular varieties (see Freese-McKenzie =-=[34]-=- and Gumm [35]). However, in [64] it was shown that LAA is not congruence modular. As a consequence, it is not possible to apply to LAA the nice theory of commutator developed for congruence modular v... |

68 |
Stable models of typed lambda-calculi
- Berry
- 1978
(Show Context)
Citation Context ...arendregt et al. in a series of papers and are perhaps the most established and studied semantics of lambda calculus (see e. g. [27, 5, 47]). Other semantics of lambda calculus were isolated by Berry =-=[12]-=- and Bucciarelli-Ehrhard [16]: Berry’s stable semantics and Bucciarelli-Ehrhard’s strongly stable semantics are refinements of the continuous semantics introduced to capture the notion of “sequential”... |

52 |
Sequentiality and strong stability
- Bucciarelli, Ehrhard
- 1991
(Show Context)
Citation Context ...of papers and are perhaps the most established and studied semantics of lambda calculus (see e. g. [27, 5, 47]). Other semantics of lambda calculus were isolated by Berry [12] and Bucciarelli-Ehrhard =-=[16]-=-: Berry’s stable semantics and Bucciarelli-Ehrhard’s strongly stable semantics are refinements of the continuous semantics introduced to capture the notion of “sequential” Scott continuous function. A... |

52 |
Extended Type Structures and Filter Lambda Models", Logic Colloquium '82
- Coppo, Dezani-Ciancaglini, et al.
- 1984
(Show Context)
Citation Context ...ppo and Dezani [26]. Filter models were investigated by Coppo, Dezani, Barendregt et al. in a series of papers and are perhaps the most established and studied semantics of lambda calculus (see e. g. =-=[27, 5, 47]-=-). Other semantics of lambda calculus were isolated by Berry [12] and Bucciarelli-Ehrhard [16]: Berry’s stable semantics and Bucciarelli-Ehrhard’s strongly stable semantics are refinements of the cont... |

46 |
What is a Model of the Lambda Calculus
- Meyer
- 1982
(Show Context)
Citation Context .... At the end of the seventies, researchers were able to provide a general algebraic characterization of the models of lambda calculus as an elementary subclass of combinatory algebras called λ-models =-=[55, 70]-=-. Definition 22. An environment with values in C is a total function ρ : Na → C, where Na is the set of names of λ-calculus. We denote by EnvC the set of all environments with values in C. For every a... |

40 | Set-theoretical and other elementary models of the lambda-calculus
- Plotkin
- 1993
(Show Context)
Citation Context ... of mathematical models for lambda calculus have been introduced in various categories of domains and were classified into semantics according to the nature of their representable functions, see e.g. =-=[4, 8, 61]-=-. Scott continuous semantics [70] is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. Scott continuous semantics includes the class of grap... |

35 |
Lambda calculus: Some models, some philosophy
- Scott
- 1980
(Show Context)
Citation Context ...ulus have been introduced in various categories of domains and were classified into semantics according to the nature of their representable functions, see e.g. [4, 8, 61]. Scott continuous semantics =-=[70]-=- is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. Scott continuous semantics includes the class of graph models, which were isolated in ... |

34 |
From computation to foundations via functions and application: the λcalculus and its webbed models
- Berline
(Show Context)
Citation Context ...ent λ-theory. Although researchers have mainly focused their interest on a limited number of them, the lattice of λ-theories, hereafter denoted by λT , has a very rich and complex structure (see e.g. =-=[4, 8, 9]-=-). The lambda calculus, although its axioms are all in the form of equations, is not a genuine equational theory since the variable-binding properties of lambda abstraction prevent “variables” in lamb... |

26 |
Geometrical methods in congruence modular algebras
- Gumm
- 1983
(Show Context)
Citation Context ...ebras other than groups is due to the pioneering papers of Smith [73] and Hagemann-Hermann [37]. The commutator is very well behaved in congruence modular varieties (see Freese-McKenzie [34] and Gumm =-=[35]-=-). However, in [64] it was shown that LAA is not congruence modular. As a consequence, it is not possible to apply to LAA the nice theory of commutator developed for congruence modular varieties. Lipp... |

26 |
Vries. Infinitary lambda calculus
- Kennaway, Klop, et al.
- 1997
(Show Context)
Citation Context ...his makes clear the connection existing between lambda calculus and combinatory logic. Infinitary lambda calculus. Various infinitary versions of λ-calculus have been introduced by several authors in =-=[43, 7, 29]-=-. Here, as an application of Thm. 5, we recall from [62] the completeness theorem for the infinitary λ-calculus. Let Λ⊥ be the similarity type obtained from the similarity type Λ of lambda calculus by... |

24 |
Ronchi Della Rocca S., “An Approximation Theorem for Topological Lambda Models and the Topological Incompleteness of Lambda Calculus
- Honsell
- 1992
(Show Context)
Citation Context ...f there exists a λ-theory which is not the theory of any model in the semantics. In the nineties the problem of the equational incom-pleteness was positively solved by Honsell and Ronchi della Rocca =-=[39]-=- for Scott’s continuous semantics, and by Bastonero and Gouy for Berry’s stable semantics [6]. The proofs of the above results are syntactical and very difficult. In [65, 66] it was shown the equation... |

23 | Topological incompleteness and order incompleteness of the lambda calculus
- Salibra
- 2003
(Show Context)
Citation Context ...onsell and Ronchi della Rocca [39] for Scott’s continuous semantics, and by Bastonero and Gouy for Berry’s stable semantics [6]. The proofs of the above results are syntactical and very difficult. In =-=[65, 66]-=- it was shown the equational incompleteness of all semantics of lambda calculus that involve monotonicity with respect to some partial order and have a bottom element (including the incompleteness of ... |

22 | Uncountable limits and the lambda calculus
- Gianantonio, Honsell, et al.
- 1995
(Show Context)
Citation Context ...Ronchi della Rocca [39]: (P14) Is there a “non-syntactical” model of the untyped lambda calculus whose theory is exactly the least (extensional) λ-theory λβ (λβη)? Di Gianantonio, Honsell and Plotkin =-=[31]-=- have shown that there exists an extensional λ-theory which is minimal among those represented by Scott continuous semantics. Graph models and other classes of models. Graph semantics is the semantics... |

21 |
Alcune Proprieta’ delle forme βη-normali nel λK-calcolo”, Pubblicazioni dell’Istituto per le Applicazioni del Calcolo
- Böhm
- 1968
(Show Context)
Citation Context ...all unsolvable λ-terms) were given in Barendregt’s 1971 thesis [3], while one of the most significant λ-theories is connected with the study of the infinite normal forms of λ-terms through Böhm trees =-=[15, 4]-=-. The set of λ-theories is naturally equipped with a structure of complete lattice (see [4, Chapter 4]). The bottom element of this lattice is the least λ-theory λβ, while the top element is the incon... |

21 |
A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity
- Hagemann, Herrmann
- 1979
(Show Context)
Citation Context ...n be defined in terms of the commutator operation on normal subgroups. The extension of the commutator to algebras other than groups is due to the pioneering papers of Smith [73] and Hagemann-Hermann =-=[37]-=-. The commutator is very well behaved in congruence modular varieties (see Freese-McKenzie [34] and Gumm [35]). However, in [64] it was shown that LAA is not congruence modular. As a consequence, it i... |

20 | Strong stability and the incompleteness of stable models of λ-calculus
- Bastonero, Gouy
- 1999
(Show Context)
Citation Context ...es the problem of the equational incom-pleteness was positively solved by Honsell and Ronchi della Rocca [39] for Scott’s continuous semantics, and by Bastonero and Gouy for Berry’s stable semantics =-=[6]-=-. The proofs of the above results are syntactical and very difficult. In [65, 66] it was shown the equational incompleteness of all semantics of lambda calculus that involve monotonicity with respect ... |

20 | On the algebraic models of lambda calculus
- Salibra
(Show Context)
Citation Context ... the problems we handle, for example in order to investigate the structure of the lattice of λ-theories (see [4, Chapter 4] and [8, 9]) in itself and in connections with the theory of models. Salibra =-=[51, 66, 63]-=- has launched at the end of the nineties a research program for exploring lambda calculus and combinatory logic using techniques of universal algebra. The remark that the lattice of λ-theories is isom... |

19 | Isomorphism and equational equivalence of continuous lambda models - Kerth - 1998 |

19 | The lattice of lambda theories
- Lusin, Salibra
(Show Context)
Citation Context ... the problems we handle, for example in order to investigate the structure of the lattice of λ-theories (see [4, Chapter 4] and [8, 9]) in itself and in connections with the theory of models. Salibra =-=[51, 66, 63]-=- has launched at the end of the nineties a research program for exploring lambda calculus and combinatory logic using techniques of universal algebra. The remark that the lattice of λ-theories is isom... |

18 | The relationship between two commutators
- Kearnes, Szendrei
- 1998
(Show Context)
Citation Context ...t LAA is not congruence modular. As a consequence, it is not possible to apply to LAA the nice theory of commutator developed for congruence modular varieties. Lipparini [49, 50] and Kearnes-Szendrei =-=[41]-=- have recently shown that under very weak hypotheses the commutator proves also useful in studying algebras without congruence modularity. However, in [51] Lusin and Salibra have shown that a lattice ... |

18 | domain theory and theoretical computer science - Topology - 1998 |

18 |
Modules over commutative regular rings
- Pierce
(Show Context)
Citation Context ...representation theorem for Boolean rings (the observation that Boolean algebras could be regarded as rings is due to Stone) admits a generalization, due to Pierce, to commutative rings with unit (see =-=[57]-=- and [40, Ch. V]). To help the reader to get familiar with the argument, we now outline Pierce’s construction. Let A = (A, +, ·, 0, 1) be a commutative ring with unit, and let E(A) = {a ∈ A : a · a = ... |

16 | A continuum of theories of lambda calculus without semantics
- Salibra
- 2001
(Show Context)
Citation Context ...onsell and Ronchi della Rocca [39] for Scott’s continuous semantics, and by Bastonero and Gouy for Berry’s stable semantics [6]. The proofs of the above results are syntactical and very difficult. In =-=[65, 66]-=- it was shown the equational incompleteness of all semantics of lambda calculus that involve monotonicity with respect to some partial order and have a bottom element (including the incompleteness of ... |

16 | Order-incompleteness and finite lambda reduction models
- Selinger
(Show Context)
Citation Context ...calculus can be restated in terms of algebraic properties of varieties of λ-abstraction algebras. For example, the open problem of the order-incompleteness of lambda calculus, raised by Selinger (see =-=[71]-=-), asks for the existence of a λ-theory not arising as the equational theory of a non-trivially partially ordered model of lambda calculus. A partial answer to the order-incompleteness problem was obt... |

15 |
Graph models of λ-calculus at work, and variations
- Berline
(Show Context)
Citation Context ...ntinuous and, furthermore, commutes with “infs of compatible elements”. A strongly stable function between dI-domains with coherence, is a stable function preserving coherence. We refer the reader to =-=[8, 9]-=- for a more detailed description of these semantics. All these semantics are structurally and equationally rich: in particular, in each of them it is possible to build up 2 ℵ0 models having pairwise d... |

15 | Etude des théories équationnelles et des propriétés algébriques des modèles stables du λ-calcul - Gouy - 1995 |

14 |
Algebras and combinators. Algebra universalis
- Engeler
- 1981
(Show Context)
Citation Context ...complete partial orders and morphisms are Scott continuous functions. Scott continuous semantics includes the class of graph models, which were isolated in the seventies by Plotkin, Scott and Engeler =-=[33, 61, 69]-=-, and the class of filter models, which were isolated at the beginning of eighties by Barendregt, Coppo and Dezani [5] after the introduction of intersection-type discipline at the end of seventies by... |

14 |
A finite equational axiomatization of the functional algebras for the lambda calculus
- Salibra, Goldblatt
- 1999
(Show Context)
Citation Context ...(iii). Definition 27. Any algebra isomorphic to a subalgebra of a high-order expansion of a λ-model is called a functional λ-abstraction algebra. The class of all these algebras is denoted by FLA. In =-=[62]-=- it was shown the following representation theorem: Theorem 5. (Goldblatt-Salibra [62]) LAA = FLA. In other words, any λ-abstraction algebra is isomorphic to a subalgebra of a high-order expansion of ... |

13 |
Omega can be anything it should not be
- Baeten, Boerboom
- 1979
(Show Context)
Citation Context ...straction algebra and t(x1, . . . , xn) = u(x1, . . . , xn) be an identity between Λ-terms. Then there exist two λ-terms Mt and Mu such that A |= t(x1, . . . , xn) = u(x1, . . . , xn) ⇔ A |= Mt = Mu. =-=(2)-=- We remark that the proof of (2) is not trivial, because λ-abstraction algebras may admit elements which depend on all the names in Na. This is obviously not true for the term algebra of a λtheory bec... |

13 |
On the construction of stable models of λ-calculus
- Kerth
(Show Context)
Citation Context ...gly stable semantics are refinements of the continuous semantics introduced to capture the notion of “sequential” Scott continuous function. All these semantics are structurally and equationally rich =-=[10, 44, 46]-=- in the sense that it is possible to build up 2 ℵ0 λ-models in each of them inducing, pairwise distinct λ-theories. Nevertheless, the above denotational semantics do not match all possible operational... |

12 |
Lambda abstraction algebras: coordinatizing models of lambda calculus
- Pigozzi, Salibra
- 1998
(Show Context)
Citation Context ...ed by the term algebra of λβ is axiomatized by the finite schema of identities characterizing λ-abstraction algebras. The equational theory of λabstraction algebras, introduced by Pigozzi and Salibra =-=[59, 60]-=-, constitutes a purely algebraic theory of the untyped lambda calculus in the same spirit that cylindric and polyadic (Boolean) algebras constitute an algebraic theory of the first-order predicate log... |

11 | The sensible graph theories of lambda calculus
- Bucciarelli, Salibra
- 2004
(Show Context)
Citation Context ...intersection, so it is not a sublattice of λT . Proof. Let ξ = φ ∧ ψ. By Lemma 6, [Ω]ξ is a non-trivial central element of Λξ. It follows that ξ /∈ T h(C).We recall that the graph models (see, e.g., =-=[9, 18]-=-) and the filter models (see, e.g., [5]) are classes of λ-models within the Scott-continuous semantics. Corollary 3. Let C be one of the following semantics: graph semantics, filter semantics, Scottco... |

11 |
Isomorphisme et équivalence équationnelle entre modèles du calcul
- Kerth
- 1995
(Show Context)
Citation Context ...gly stable semantics are refinements of the continuous semantics introduced to capture the notion of “sequential” Scott continuous function. All these semantics are structurally and equationally rich =-=[10, 44, 46]-=- in the sense that it is possible to build up 2 ℵ0 λ-models in each of them inducing, pairwise distinct λ-theories. Nevertheless, the above denotational semantics do not match all possible operational... |

11 |
Lambda abstraction algebras: representation theorems
- Pigozzi, Salibra
- 1995
(Show Context)
Citation Context ...ed by the term algebra of λβ is axiomatized by the finite schema of identities characterizing λ-abstraction algebras. The equational theory of λabstraction algebras, introduced by Pigozzi and Salibra =-=[59, 60]-=-, constitutes a purely algebraic theory of the untyped lambda calculus in the same spirit that cylindric and polyadic (Boolean) algebras constitute an algebraic theory of the first-order predicate log... |

10 |
Some extensional term models for combinatory logics and -calculi
- Barendregt
- 1971
(Show Context)
Citation Context ...ce relation of the interpretation function. Syntactical proofs of consistency of remarkable λ-theories (for example, the theory equating all unsolvable λ-terms) were given in Barendregt’s 1971 thesis =-=[3]-=-, while one of the most significant λ-theories is connected with the study of the infinite normal forms of λ-terms through Böhm trees [15, 4]. The set of λ-theories is naturally equipped with a struct... |

10 | Boolean algebras of factor congruences - Bigelow, Burris - 1990 |

10 | A characterization of varieties with a difference term
- Lipparini
(Show Context)
Citation Context ...ever, in [64] it was shown that LAA is not congruence modular. As a consequence, it is not possible to apply to LAA the nice theory of commutator developed for congruence modular varieties. Lipparini =-=[49, 50]-=- and Kearnes-Szendrei [41] have recently shown that under very weak hypotheses the commutator proves also useful in studying algebras without congruence modularity. However, in [51] Lusin and Salibra ... |

10 | Boolean algebras for lambda calculus - Manzonetto, Salibra |