## A Semantics for Static Type Inference (1993)

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Venue: | Information and Computation |

Citations: | 11 - 0 self |

### BibTeX

@ARTICLE{Plotkin93asemantics,

author = {Gordon Plotkin},

title = {A Semantics for Static Type Inference},

journal = {Information and Computation},

year = {1993},

volume = {109},

pages = {256--299}

}

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### Abstract

Curry's system for F-deducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show completeness for Curry's system itself, relative to an extended notion of model that validates reduction but not conversion.

### Citations

1212 |
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(Show Context)
Citation Context ...low Scott and model information by a partial order, we would want the denotation of M to be less than that of N. We will therefore work with an ordered variant of the notion of a syntactical ),-model =-=[2, 5]-=-. To this end, following an idea of Mitchell [29], we first define interpretations, and only then consider models. An ordered lambda interpretation of the )*-calculus is a triple: P=swhere P is a part... |

1008 | A theory of type polymorphism in programming
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Citation Context ... simplest systems for type inference for the untyped k-calculus [3,10,11,18]. It is a decidable system that can be considered as the basis of static type inference algorithms for languages such as ML =-=[27]-=-. These algorithms are static, in that the type of a term is calculated without any reductions being carried out. To this end, the calculation is syntax-directed, in that the types of terms are found ... |

523 |
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Citation Context ...ticipated by Girard in [16]. In the categorical literature, orderenriched categories [23] and the more general 2-categories [32,33] have been considered. In the context of universal algebra, Meseguer =-=[25] considere-=-d categorical, preorder and partial order interpretations of a "rewriting logic" for conditional rewriting modulo a set of equations; he proves soundness and completeness results relative to... |

227 |
A filter lambda model and the completeness of type assignment
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(Show Context)
Citation Context ...tation. A certain equality rule is then valid, that if two terms are interconvertible then they have the same types. Adding this rule 2 yields a proper extension of Curry's system. Hindley and others =-=[4,18,19]-=- proved that completeness indeed holds for the extended system with respect to any of various classes of models; these differ only in how type expressions are interpreted (the simple semantics, the F-... |

158 |
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Citation Context ... order is, per se, hardly new. In the context of combinatory logic, Meyer, Bunder and Powers [26] considered an asymmetric combinatory logic, with such axioms as Kxy_ largely anticipated by Girard in =-=[16]-=-. In the categorical literature, orderenriched categories [23] and the more general 2-categories [32,33] have been considered. In the context of universal algebra, Meseguer [25] considered categorical... |

129 |
An Ideal Model for Recursive Polymorphic Types
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(Show Context)
Citation Context ... the polymorphic type discipline with the intersection type discipline of Coppo et al is of immediate interest, and one can also consider, for example, existential and union types and recursive types =-=[1,6,7,24]-=-. In a different direction, it would be interesting to extend the language to include other features such as, perhaps, a recursion combinator. Continuing in this way, it would be particularly interest... |

106 |
Polymorphic Type Inference and Containment
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(Show Context)
Citation Context ... logic" for conditional rewriting modulo a set of equations; he proves soundness and completeness results relative to all three classes of interpretations. In section 4, following an idea of Mitc=-=hell [28]-=-, we consider a generalisation of the simple notion of type interpretation, for the i We may informally understand a rule to be admissible in a formal system if, for any instantiation of the schematic... |

58 |
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Citation Context ....) transformations, it is easy to see that these two mappings are inverses; they are also clearly monotonic. The relation between filter models and set-theoretic 35 models has also been considered in =-=[8]-=-, where it is shown that the filter model of [4] can be embedded in Engeler's model; use is made there of a function similar to the (.)0 transformation. Since the filter and set-theoretic models consi... |

49 |
Two extension of Curry's type inference system
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(Show Context)
Citation Context ...ssions are used, being those satisfying a certain infinitary condition; consequently, application is not continuous. Finally, in section 6, we consider the intersection type discipline of Coppo et al =-=[6,9]-=-. Here completeness has already been obtained for the simple semantics [4,17] and the Fsemantics [12]. We obtain completeness for two more general classes of type interpretations, where there is a ple... |

49 | A type-inference approach to reduction properties and semantics of polymorphic expressions - Mitchell - 1986 |

42 | Type inference with recursive types: Syntax and semantics - Cardone, Coppo - 1991 |

36 |
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Citation Context ...ssions are used, being those satisfying a certain infinitary condition; consequently, application is not continuous. Finally, in section 6, we consider the intersection type discipline of Coppo et al =-=[6,9]-=-. Here completeness has already been obtained for the simple semantics [4,17] and the Fsemantics [12]. We obtain completeness for two more general classes of type interpretations, where there is a ple... |

25 |
The simple semantics for Coppo-Dezani-Sallé types
- Hindley
- 1982
(Show Context)
Citation Context ...nference model of polymorphic types. The provable sequents of his system are those of the basic system, except that the type expressions occurring are restricted to be Hindley’s “Normal type schemes” =-=[17]-=-. The term model For our first proof of completeness, we use the term model of β-reduction Cβ = < C,⋅, ⋅ (⋅) > whose elements are equivalence classes of (open) terms under β−conversion, with the trivi... |

20 |
Foundations of equational deduction: A categorical treatment of equational proofs and uni algorithms
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(Show Context)
Citation Context ...sidered an asymmetric combinatory logic, with such axioms as Kxy_ largely anticipated by Girard in [16]. In the categorical literature, orderenriched categories [23] and the more general 2-categories =-=[32,33] have been-=- considered. In the context of universal algebra, Meseguer [25] considered categorical, preorder and partial order interpretations of a "rewriting logic" for conditional rewriting modulo a s... |

19 |
Modelling computations: a 2-categorical framework
- Seely
- 1987
(Show Context)
Citation Context ...sidered an asymmetric combinatory logic, with such axioms as Kxy_ largely anticipated by Girard in [16]. In the categorical literature, orderenriched categories [23] and the more general 2-categories =-=[32,33] have been-=- considered. In the context of universal algebra, Meseguer [25] considered categorical, preorder and partial order interpretations of a "rewriting logic" for conditional rewriting modulo a s... |

18 |
Combinatory Logic Vol
- Curry, Feys
- 1958
(Show Context)
Citation Context ...ns to systems with polymorphic or intersection types are also considered. I Introduction Curry's system for F-deducibility is one of the simplest systems for type inference for the untyped k-calculus =-=[3,10,11,18]-=-. It is a decidable system that can be considered as the basis of static type inference algorithms for languages such as ML [27]. These algorithms are static, in that the type of a term is calculated ... |

17 |
Algebras and combinators
- Engeler
- 1981
(Show Context)
Citation Context ...tain completeness for two more general classes of type interpretations, where there is a pleasing correspondence between the type-expression models and the set-theoretic models of Plotkin and Engeler =-=[30, 14]-=-. It would be interesting to have a formal definition of a notion of a static type inference system. Such a system provides the basis for static type inference (or type checking) algorithms. It is des... |

17 |
Filter models with polymorphic types
- Jacobs, Margaria, et al.
- 1992
(Show Context)
Citation Context ...cessitates the use of a wider class of models than those considered by Hindley. We present here one such analysis based on the notion of a model of reduction. This terminology was first introduced in =-=[22]-=-, but what is called a model of reduction here is called a model of expansion there; the reason for such a - prima facie confusing - switch is discussed below. Models of reduction arise if one conside... |

16 |
Type inference problems: A survey
- TIURYN
- 1990
(Show Context)
Citation Context ... would have applications in a more complex setting. 19 5 Polymorphic Type Inference A system for polymorphic type inference is obtained by adding universal quantification vt. to type expressions (see =-=[3,13,15,16,28,34,36]-=- for information on this system and further references). We do not distinguish m-equivalent type expressions - those which differ only in the names of bound variables. There are also evident notions o... |

14 |
A characterisation of F-complete type assignments
- Dezani-Ciancaglini, Margaria
- 1986
(Show Context)
Citation Context ...ot continuous. Finally, in section 6, we consider the intersection type discipline of Coppo et al [6,9]. Here completeness has already been obtained for the simple semantics [4,17] and the Fsemantics =-=[12]-=-. We obtain completeness for two more general classes of type interpretations, where there is a pleasing correspondence between the type-expression models and the set-theoretic models of Plotkin and E... |

14 |
Types with intersection: An introduction
- Hindley
- 1991
(Show Context)
Citation Context ...[[M]](p) (as [[M]](p) satisfies condition (2s)) 31 6. The Intersection Type Discipline Intersection types were independently introduced by Coppo and Dezani-Ciancaglini, by Sails and by Pottinger (see =-=[3,6,9,20,36]-=- for references and further discussion). The surprising feature of such systems is that even though (EQ) is not derivable, it is admissible - provided empty intersections are allowed. We may then use ... |

9 |
Modelling reductions in confluent categories
- Jay
- 1990
(Show Context)
Citation Context ...ogic, Meyer, Bunder and Powers [26] considered an asymmetric combinatory logic, with such axioms as Kxy_ largely anticipated by Girard in [16]. In the categorical literature, orderenriched categories =-=[23] and -=-the more general 2-categories [32,33] have been considered. In the context of universal algebra, Meseguer [25] considered categorical, preorder and partial order interpretations of a "rewriting l... |

7 |
S.R.D.: Type inference: Some results, some problems
- Giannini, Honsell, et al.
- 1993
(Show Context)
Citation Context ...e systems for intersection types, and therefore typability of closed terms is undecidable. In the case of the polymorphic discipline, the decision problem of typability is an outstanding open problem =-=[15]-=-. It would also be of interest to extend the semantic analysis to other type disciplines. The combination of the polymorphic type discipline with the intersection type discipline of Coppo et al is of ... |

7 |
The Completeness Theorem for Typing lambda-Terms
- Hindley
- 1983
(Show Context)
Citation Context ...ns to systems with polymorphic or intersection types are also considered. I Introduction Curry's system for F-deducibility is one of the simplest systems for type inference for the untyped k-calculus =-=[3,10,11,18]-=-. It is a decidable system that can be considered as the basis of static type inference algorithms for languages such as ML [27]. These algorithms are static, in that the type of a term is calculated ... |

4 |
Expansion and conversion models in the lambda calculus from filters with polymorphic types
- Jacobs, Margaria, et al.
- 1989
(Show Context)
Citation Context ...free in M).sThis improves the proof of (Sub) in [2] where the equality version of (v) is also assumed. The possibility of this lemma was suggested to the author by work of Jacobs, Margaria and Zacchi =-=[21]-=-. There is a multiple substitution consequence of (Sub): [[ [ 1I /s]M ]](p) = [[ M ]](p(s:=[[ 1I ]](p))) Here a rightmost convention is adopted to resolve conflicts in the interpretation of p( := ), i... |

4 |
Implementing the fool's model of combinatory logic
- Meyer, Bunder, et al.
- 1991
(Show Context)
Citation Context ...le is indeed modelled, but not necessarily the expansion rule. The idea of modelling reduction by a partial order is, per se, hardly new. In the context of combinatory logic, Meyer, Bunder and Powers =-=[26]-=- considered an asymmetric combinatory logic, with such axioms as Kxy_ largely anticipated by Girard in [16]. In the categorical literature, orderenriched categories [23] and the more general 2-categor... |

2 | Curry's type rules are complete with respect to the Fsemantics too - Hindley - 1983 |

1 |
Intersection and union types, in: Theoretical Aspects of Computer Software
- Barbanera, Dezani-Ciancaglini
- 1991
(Show Context)
Citation Context ... the polymorphic type discipline with the intersection type discipline of Coppo et al is of immediate interest, and one can also consider, for example, existential and union types and recursive types =-=[1,6,7,24]-=-. In a different direction, it would be interesting to extend the language to include other features such as, perhaps, a recursion combinator. Continuing in this way, it would be particularly interest... |

1 | Lambda calculi with types. To appear - Barendregt |

1 | A result for combinators, BCK logics and BCK algebras. Logique et Analyse 109 - Bunder, Meyer - 1985 |

1 |
Polymorphic types, fixedpoint combinators and continuous lambda models
- Dezani-Ciancaglini, Margaria
- 1987
(Show Context)
Citation Context ... would have applications in a more complex setting. 19 5 Polymorphic Type Inference A system for polymorphic type inference is obtained by adding universal quantification vt. to type expressions (see =-=[3,13,15,16,28,34,36]-=- for information on this system and further references). We do not distinguish m-equivalent type expressions - those which differ only in the names of bound variables. There are also evident notions o... |

1 |
The simple semantics for Coppo-Dezani-Sal16 types
- Hindley
- 1982
(Show Context)
Citation Context ...nference model of polymorphic types. The provable sequents of his system are those of the basic system, except that the type expressions occurring are restricted to be Hindley's "Normal type sche=-=mes" [17]-=-. The term model For our first proof of completeness, we use the term model of p-reduction C6=swhose elements are equivalence classes of (open) terms under 6-conversion, with the trivial partial order... |

1 |
A set-theoretical definition of application. Research Memorandum MIP-R-95
- Plotkin
- 1972
(Show Context)
Citation Context ...tain completeness for two more general classes of type interpretations, where there is a pleasing correspondence between the type-expression models and the set-theoretic models of Plotkin and Engeler =-=[30, 14]-=-. It would be interesting to have a formal definition of a notion of a static type inference system. Such a system provides the basis for static type inference (or type checking) algorithms. It is des... |

1 | A semantics for type checking, in: Theoretical Aspects of Computer Software - Plotkin - 1991 |