## Lambda calculus: models and theories

Venue: | Proceedings of the Third AMAST Workshop on Algebraic Methods in Language Processing (AMiLP-2003), number 21 in TWLT Proceedings, pages 39–54, University of Twente, 2003. Invited Lecture |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Salibra_lambdacalculus:,

author = {Antonino Salibra},

title = {Lambda calculus: models and theories},

booktitle = {Proceedings of the Third AMAST Workshop on Algebraic Methods in Language Processing (AMiLP-2003), number 21 in TWLT Proceedings, pages 39–54, University of Twente, 2003. Invited Lecture},

year = {}

}

### OpenURL

### Abstract

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 λ-theories, the equational incompleteness of lambda calculus semantics, and the λ-theories induced by graph models of lambda calculus.

### Citations

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(Show Context)
Citation Context ...sic Notions and Notation To keep this article self-contained, we summarize some definitions and results that we need in the subsequent part of the paper. Our main reference will be Barendregt’s book (=-=Barendregt, 1984-=-). 2.1 Lambda calculus The two primitive notions of the lambda calculus are application, the operation of applying a function to an argument (expressed as juxtaposition of terms), and lambda (function... |

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67 |
Stable models of typed lambda-calculi
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- 1978
(Show Context)
Citation Context ...; Berline (2000); Plotkin (1993). Continuous semantics (Scott, 1972) is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. Stable semantics (=-=Berry, 1978-=-) and strongly stable semantics (Bucciarelli and Ehrhard, 1991) are a strengthening of the continuous semantics, introduced to capture the “sequential” Scott continuous functions. All these semantics ... |

52 |
Sequentiality and Strong Stability
- Bucciarelli, Ehrhard
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(Show Context)
Citation Context ... semantics (Scott, 1972) is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. Stable semantics (Berry, 1978) and strongly stable semantics (=-=Bucciarelli and Ehrhard, 1991-=-) are a strengthening of the continuous semantics, introduced to capture the “sequential” Scott continuous functions. All these semantics are structurally and equationally rich in the sense that it is... |

52 | Extended type structure and filter lambda models - Coppo, Dezani-Ciancaglini, et al. - 1984 |

46 |
What is a model of the lambda calculus
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(Show Context)
Citation Context ... The axioms of an elementary subclass of combinatory algebras, called λ-models or models of the lambda calculus, were expressly chosen to make coherent the previous definition of interpretation (see (=-=Meyer, 1982-=-), (Scott, 1980), (Barendregt, 1984, Def. 5.2.7)). The Meyer-Scott axiom is the most important axiom in the definition of a λ-model. In the first-order language of combinatory algebras it takes the fo... |

40 | Set-theoretical and other elementary models of the λ-calculus - Plotkin - 1993 |

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34 |
Lambda Calculus: Some Models, Some Philosophy’, in Barwise et al
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(Show Context)
Citation Context ...an elementary subclass of combinatory algebras, called λ-models or models of the lambda calculus, were expressly chosen to make coherent the previous definition of interpretation (see (Meyer, 1982), (=-=Scott, 1980-=-), (Barendregt, 1984, Def. 5.2.7)). The Meyer-Scott axiom is the most important axiom in the definition of a λ-model. In the first-order language of combinatory algebras it takes the following form ∀x... |

32 |
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(Show Context)
Citation Context ...domains and were classified into semantics according to the nature of their representable functions, see e.g. Abramski (1991); Barendregt (1984); Berline (2000); Plotkin (1993). Continuous semantics (=-=Scott, 1972-=-) is given in the category whose objects are complete partial orders and morphisms are Scott continuous functions. Stable semantics (Berry, 1978) and strongly stable semantics (Bucciarelli and Ehrhard... |

23 | Topological incompleteness and order incompleteness of the lambda calculus - Salibra |

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(Show Context)
Citation Context ...ollowing conditions: (i) C is extensional, (ii) ΩΩ = Ω, (iii) λx.Ω(Ωx) = Ω, (iv) Ω �= λx.Ω. The Park stable model Ps satisfies the above conditions (i)-(iv) and equalizes X and Z (see Theorem 4.1 in (=-=Bastonero and Gouy, 1999-=-)). This proves that no continuous model has the theory of Ps. The incompleteness of the stable semantics was proven by Bastonero and Gouy in a similar way. They show that the same λ-terms X and Z are... |

20 | On the algebraic models of lambda calculus - Salibra |

19 |
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(Show Context)
Citation Context ... Scott continuous functions. All these semantics are structurally and equationally rich in the sense that it is possible to build up 2 ℵ0 models in each of them inducing pairwise distinct λ-theories (=-=Kerth, 1998-=-, 2001), Although a rich host of different λ-theories now have a “fully abstract” syntax-free model (i.e., a model which induces precisely those identities which hold in the given theory), the abovesd... |

16 | Further properties of lattices of equational theories. Algebra Universalis 28: 459–486 - Lampe - 1991 |

16 | A continuum of theories of lambda calculus without semantics - Salibra - 2001 |

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

14 | A finite equational axiomatization of the functional algebras for the lambda calculus - Salibra, Goldblatt - 1999 |

13 | Omega can be anything it should not be - Baeten, Boerboom - 1979 |

13 | On the construction of stable models of λ-calculus - Kerth |

12 | Lambda abstraction algebras: coordinatizing models of lambda calculus - Pigozzi, Salibra - 1998 |

11 | Isomorphisme et équivalence équationnelle entre modèles du calcul - Kerth - 1995 |

11 | Lambda abstraction algebras: representation theorems - Pigozzi, Salibra - 1995 |

10 | Topology, domain theory and theoretical computer science. Topology and Its Applications
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(Show Context)
Citation Context ...tt in the category of algebraic lattices and Scott continuous functions. After Scott, many authors tried to find lambda calculus models in Cartesian closed categories of topological spaces. Abramsky (=-=Mislove, 1998-=-, Theorem 5.11) and Plotkin (Mislove, 1998, Theorem 5.14) have shown respectively that there exists no nondegenerate model of the lambda calculus in the category of posets and monotone mappings, and i... |

10 | Nonmodularity results for lambda calculus - Salibra |

9 | The minimal graph model of lambda calculus - Bucciarelli, Salibra - 2003 |

9 | An introduction to lambda abstraction algebras, IX Simposio Latinoamericano de - Pigozzi, Salibra - 1993 |

7 | All compact Hausdorff lambda models are degenerate - Hofmann, Mislove - 1993 |

5 | Computing with Böhm trees - David - 2001 |

3 | Towards lambda calculus order-incompleteness - Salibra - 2001 |

2 | Reasoning about interpretation in qualitative λmodels - Honsell, Rocca, et al. - 1990 |

1 | An Approximation Theorem for Topological Incompleteness of Lambda Calculus - Honsell, Rocca, et al. - 1992 |