## Categorical Completeness Results for the Simply-Typed Lambda-Calculus (1995)

Venue: | Proceedings of TLCA '95, Springer LNCS 902 |

Citations: | 9 - 0 self |

### BibTeX

@INPROCEEDINGS{Simpson95categoricalcompleteness,

author = {Alex K. Simpson},

title = {Categorical Completeness Results for the Simply-Typed Lambda-Calculus},

booktitle = {Proceedings of TLCA '95, Springer LNCS 902},

year = {1995},

pages = {414--427},

publisher = {Springer}

}

### OpenURL

### Abstract

. We investigate, in a categorical setting, some completeness properties of beta-eta conversion between closed terms of the simplytyped lambda calculus. A cartesian-closed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the category that distinguishes them. It is said to have a complete interpretation if there is some interpretation that equates only interconvertible terms. We give simple necessary and sufficient conditions on the category for each of the two forms of completeness to hold. The classic completeness results of, e.g., Friedman and Plotkin are immediate consequences. As another application, we derive a syntactic theorem of Statman characterizing beta-eta conversion as a maximum consistent congruence relation satisfying a property known as typical ambiguity. 1 Introduction In 1970 Friedman proved that beta-eta conversion is complete for deriving all equalities between the (simply-typed) lambda-definable...

### Citations

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(Show Context)
Citation Context ... k); (ii) (k; k; k; k + 1); (iii) (k; k; k + 1; k + 1); (iv) (k; k + 1; k; k + 1); (v) (k; k + 1; k + 1; k + 1). However, (ii) and (v) are impossible. We show this for (v). Clearly (v) requires that f=-=(0; 2)-=- = k +2 and, because RA\ThetaA (h1; 0i; h1; 1i; h1; 2i), that f(1; 2) = k + 1. But then, as RA\ThetaA (h0; 2i; h1; 2i; h2; 2i), there is no possible value for f(2; 2). We claim that for the other case... |

449 |
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(Show Context)
Citation Context ...converse implication holds, and related questions. Before considering such completeness questions we consider the categorical formulation of what an interpretation of the lambda-calculus in C is (see =-=[8]). This fo-=-rmulation is in terms of cartesian-closed functors (CC-functors), which are those functors between CCCs that preserve the cartesian-closed structure "on the nose". 1 Let FX be the free carte... |

61 |
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(Show Context)
Citation Context ...lete for deriving all equalities between the (simply-typed) lambda-definable functionals in the categorysSet [5]. (Incidentally, this result was independently discovered by Plotkin [10], published in =-=[11]-=-.) However, in computer science one is often interested in interpretations in other cartesian closed categories (such as the category of complete partial orders and continuous functions). It is natura... |

60 | Equality between functionals - Friedman - 1973 |

50 |
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- 1991
(Show Context)
Citation Context ....2.28]). More recently, Berger and Schwichtenberg used different techniques to show that completeness holds relative to any model capable of faithfully representing certain basic operations on syntax =-=[3]-=-. In this paper we investigate such completeness questions in a categorical setting. As is well known, cartesian-closed categories (CCCs) provide a general notion of model for the simply-typed lambda ... |

43 |
OnMints’ reductions for ccc-Calculus
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(Show Context)
Citation Context ...roducts of arbitrary finite arity, their tuples and projections. We assume that the reader is acquainted with the rules for beta-eta convertibility, = fij , between terms of identical type (see, e.g, =-=[1, 4, 7]-=-). Two classes of terms, the neutral terms and the long-fij normal forms, are defined by mutual induction. A term is neutral if it has one of the following forms: x oe ; or U (V ) where U is neutral a... |

37 | The Virtues of Eta-expansion
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(Show Context)
Citation Context ...roducts of arbitrary finite arity, their tuples and projections. We assume that the reader is acquainted with the rules for beta-eta convertibility, = fij , between terms of identical type (see, e.g, =-=[1, 4, 7]-=-). Two classes of terms, the neutral terms and the long-fij normal forms, are defined by mutual induction. A term is neutral if it has one of the following forms: x oe ; or U (V ) where U is neutral a... |

33 | The partial lambda calculus - Moggi - 1988 |

23 |
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- 1973
(Show Context)
Citation Context ... conversion is complete for deriving all equalities between the (simply-typed) lambda-definable functionals in the categorysSet [5]. (Incidentally, this result was independently discovered by Plotkin =-=[10]-=-, published in [11].) However, in computer science one is often interested in interpretations in other cartesian closed categories (such as the category of complete partial orders and continuous funct... |

13 |
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(Show Context)
Citation Context ...erence is non-trivial for the question of characterizing when an interpretation is complete. It is interesting to compare our work with Statman's own semantic application of his syntactic results. In =-=[17]-=- he states his important 1-Section Theorem giving necessary and sufficient conditions for an interpretation ofs! f0g in a Henkin model to be complete. (See [12] for a detailed discussion and proof of ... |

12 |
Completeness, Invariance and *-definability
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- 1982
(Show Context)
Citation Context ...le that is complete but which has no complete interpretation is the category of finite sets, FinSet. The completeness of FinSet is proved explicitly in [13], but it is closely related to Theorem 2 of =-=[15]-=- (a result originally due to Plotkin [10]), which is basically a finite model property for beta-eta conversion. The non-existence of a complete interpretation in FinSet was essentially observed by Fri... |

8 |
Embedding of a free cartesian closed category into the category of sets
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(Show Context)
Citation Context ...roducts of arbitrary finite arity, their tuples and projections. We assume that the reader is acquainted with the rules for beta-eta convertibility, = fij , between terms of identical type (see, e.g, =-=[1, 4, 7]-=-). Two classes of terms, the neutral terms and the long-fij normal forms, are defined by mutual induction. A term is neutral if it has one of the following forms: x oe ; or U (V ) where U is neutral a... |

8 | On the existence of closed terms in the typed -calculus - Statman - 1980 |

7 |
definable functionals and fij-conversion
- Statman
- 1983
(Show Context)
Citation Context ...ple ones that are easily checked in particular cases. Moreover, they show the failure of completeness to be the exception rather than the rule. As an application, we use Theorem 1 to obtain Statman's =-=[16]-=- characterization of beta-eta convertibility as a maximally consistent congruence relation satisfying typical ambiguity (Theorem 3). Indeed, as will be seen, our work is closely related to, and also h... |

6 |
Statman’s 1-section theorem
- Riecke
- 1995
(Show Context)
Citation Context ... to be the type (0 ! 0 ! 0) ! 0 ! 0. Proposition 2 (Statman) For all M oe ; N oe 2s! f0g , it holds that M = fij N if and only if, for all L oe!? , L(M ) = fij L(N ). A detailed proof can be found in =-=[12]-=-. Incidentally, in [14, Proposition 1], Statman shows that, for each oe, there exists L oe!? , such that M = fij N if and only if L(M ) = fij L(N ), but we do not need this stronger result here. Propo... |

1 |
fij-equality for coproducts. This volume
- Ghani
- 1995
(Show Context)
Citation Context ... categories. The main obstacle in proving such a generalization is to get a good handle on equality in the internal language. It is already difficult to generalize long-fij normal forms (although see =-=[6]-=- for progress on this question), let alone the deep syntactic results of Statman. On the other hand, for recursion theoretic reasons, it is clear that our results do not generalize to cartesian-closed... |

1 |
The category of finite sets and CCCs
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- 1983
(Show Context)
Citation Context ...completeness. The converse is not true. An example that is complete but which has no complete interpretation is the category of finite sets, FinSet. The completeness of FinSet is proved explicitly in =-=[13]-=-, but it is closely related to Theorem 2 of [15] (a result originally due to Plotkin [10]), which is basically a finite model property for beta-eta conversion. The non-existence of a complete interpre... |