John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to rework these results using an axiomatization of equality for call by value semantics such as that given by Riecke [11] but it would clearly be preferable to find a single formulation that can be used for both cases. Another promising approach would be to use ideas from category theory as in [10] for a language with intersection types and subtyping, and in [5] for a system of type classes. One of the main attractions of the categorical approach from the theoretical standpoint is the increased generality resulting from a higher level of abstraction. The main benefit for practical work is ....

J.C. Reynolds. The coherence of languages with intersection types. In Theoretical aspects of computer software. Springer Verlag LNCS 526, 1991.

....this property. Consider for example the binary type constructor direct sum # defined for Abelian groups. One could coerce A into A # B via a coercion # 1 and B into A # B via a coercion # 2 . But then the image of A in A # A depends on the choice of the coercion function Following [3] and [18] we will call a type system coherent, if the coercions are independent of the way they are deduced in the system. 1 In the following we will look at different kinds of coercions which occur and we will state some conditions which will yield the coherence of the system. Besides the technical ....

J. C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, Theoretical Aspects of Computer Software --- International Conference TACS '91, volume 526 of Lecture Notes in Computer Science, pages 675--700, Sendai, Japan, Sept. 1991. Springer-Verlag.

....programming languages, in particular in connection with record types and object oriented programming, but such systems are not expressive enough for our purposes. More promising are enhancements of simple subtypes with intersection types [CDCV81] which have been applied to programming languages [Rey91] and recently also in type theory [Hay91] General decidability of type checking or inference in such calculi is problematic, but under certain restrictions type checking is decidable and principal types exist [Rey88, FP91, CG92] In this paper we tie together ideas from these threads of research ....

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

....(see section 4.6) that transforms an integer into a real. Note that the on the left is the sum for integers, while the one on the right is the sum for reals. In other words, the only form of overloading that is definable in this framework is the coherent overloading (see also [Pie90, Rey88, Rey91] for a complete study on coherent overloading) This restriction has many important consequences. In the context of this thesis, the most remarkable one is that it does not allow full object orientation: even if, in the cited papers, the specialization of a branch is correctly identified with the ....

J.C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, Theoretical Aspects of Computer Software (Sendai, Japan), number 526 in Lecture Notes in Computer Science, pages 675--700. Springer, September 1991.

....that combines intersection types ( 25] and bounded polymorphism ( 7] The system presented here is an implicitly typed version. More information about syntactic, semantic, and pragmatic details can be found in [21] Information about intersection types or bounded polymorphism can be found in [4, 22, 25, 26] or [6, 7, 8, 9, 10, 12, 20] The set of F types is defined by: oe : j ns j oe 1 oe 2 j 8 oe 1 : oe 2 j oe 1 oe 2 Here is the binary intersection operator, ns denotes the null intersection, and 8 oe 1 : oe 2 is for bounded polymorphism. We denote 8 ns: oe 2 as 8 : oe 2 . A type context is a ....

John C. Reynolds. The coherence of languages with intersection types. In Theoretical Aspects of Computer Science, September 1991. To appear.

....U , and this is the case for our overloaded types too. In the case of intersection, though, types, a coherence condition is imposed too, which means, essentially, that when a value can play different roles, we are free to choose any of these roles, without affecting the result of the computation [Rey91] In our context, this is not the case; a programmer may define an overloaded function foo of type fInt Theta Int Int; Real Theta Real Realg which sums two integers but multiplies two real numbers, while in a coherent intersection type discipline an overloaded function with that type ....

J.C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, Theoretical Aspects of Computer Software (Sendai, Japan), number 526 in Lecture Notes in Computer Science, pages 675--700. Springer-Verlag, September 1991.

....programming languages, in particular in connection with record types and object oriented programming, but such systems are not expressive enough for our purposes. More promising are enhancements of simple subtypes with intersection types [CDCV81] which have been applied to programming languages [Rey91] and recently also in type theory [Hay91] General decidability of type checking or inference in such calculi is problematic, but under certain restrictions type checking is decidable and principal types exist [Rey88, FP91, CG92] In this paper we tie together ideas from these threads of research ....

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

....the binary type constructor direct sum Phi defined for Abelian groups. One could coerce A into A Phi B via a coercion OE 1 and B into A Phi B via a coercion OE 2 . But then the image of A in A Phi A depends on the choice of the coercion function Following (Breazu Tannen et al. 1991) and (Reynolds, 1991) we will call a type system coherent, if the coercions are independent of the way they are deduced in the system. y In the following we will look at different kinds of coercions which occur and we will state some conditions which will yield the coherence of the system. Besides the technical ....

Reynolds, J. C. (1991). The coherence of languages with intersection types. In Ito, T. and Meyer, A. R., editors, Theoretical Aspects of Computer Software --- International Conference TACS '91, volume 526 of Lecture Notes in Computer Science, pages 675--700, Sendai, Japan. Springer-Verlag.

....Consider for example the binary type constructor direct sum Phi defined for Abelian groups. One could coerce A into A Phi B via a coercion OE 1 and B into A Phi B via a coercion OE 2 . But then the image of A in A Phi A depends on the choice of the coercion function Following [3] and [18] we will call a type system coherent, if the coercions are independent of the way they are deduced in the system. 1 In the following we will look at different kinds of coercions which occur and we will state some conditions which will yield the coherence of the system. Besides the technical ....

J. C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, Theoretical Aspects of Computer Software --- International Conference TACS '91, volume 526 of Lecture Notes in Computer Science, pages 675--700, Sendai, Japan, Sept. 1991. Springer-Verlag.

....to establish coherence with respect to the definition of provable equality in Section 5 but we have not attempted to prove this. 8. 2 Related work on coherence Coherence results have been established for a number of different systems including those with subtyping [2, 3] intersection types [20], scaling [22] and type classes [1, 6, 10] One standard approach used to establish results of this kind is to define a system of reduction rules on typing derivations. The required coherence property can then be established by showing that the reduction rules are strongly normalizing and preserve ....

J.C. Reynolds. The coherence of languages with intersection types. In Theoretical aspects of computer software. Springer Verlag LNCS 526, 1991.

....i.e. we would have to prove that any translation e 1 and e 2 of a term e given by derivations S j A e ; e 1 : and S j A e ; e 2 : are semantically equivalent. Coherence results for systems with subtyping were given in [15, 28] and for systems with intersection types were given in [83]. We do not continue the development of this semantics and let it rather as a further research. 3.3.3 Untyped Semantics Curry style systems are often called type assignment systems. The interpretation of a term e is some element m of an untyped model U , given by a semantical function [ Gamma] ....

J. C. Reynolds. The coherence of languages with intersection types. In Theoretical Aspects of Computer Software, volume 526 of Springer LNCS, 1991.

.... completeness problem for such systems has been recently settled by Yokouchi in [94] Intersection and union types turn out to be powerful enough to discriminate L evy Longo trees [45, 46] Systems of intersection types a la Church are not trivial to define: they have been extensively studied in [81, 82, 83, 72, 73, 25, 26] and applied to type disciplines for object oriented languages in [27] Lastly we mention that intersection and union types have been extensively used to prove properties of programs, like strictness analysis, detection of dead code, etc. Pioneers in this field were the PhD thesis of Benton [19] ....

J.C. Reynolds, "The Coherence of Language with Intersection Types", TACS'91, LNCS 526, Springer-Verlag, Berlin 1991, 675--700.

....(see section 4.6) that transforms an integer into a real. Note that the on the left is the sum for integers, while the one on the right is the sum for reals. In other words, the only form of overloading that is definable in this framework is the coherent overloading (see also [Pie90, Rey88, Rey91] for a complete study on coherent overloading) This restriction has many important consequences. In the context of this thesis, the most remarkable one is that it does not allow full object orientation: even if, in the cited papers, the specialization of a branch is correctly identified with the ....

J.C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, Theoretical Aspects of Computer Software (Sendai, Japan), number 526 in Lecture Notes in Computer Science, pages 675--700. Springer, September 1991.

....yields a useful soundness theorem relating the semantics to the F type system: valid F typing derivations are translated to well typed (and hence well behaved) F Theta terms. However, this construction has not been shown to be coherent [3] because the standard proof of this property for [32] relies on the existence of least upper bounds in the subtype relation a property that F does not share. An equational theory of provable equivalences between terms of pure F can be shown to be sound for both the untyped and the translation semantics. 8 Future Work A primary practical ....

John C. Reynolds. The coherence of languages with intersection types. In Ito and Meyer [21], pages 675--700.

.... at Torino and many others (Barendregt et al. 1983; Cardone Coppo, 1990; Coppo DezaniCiancaglini, 1980; Coppo et al. 1983; Coppo et al. 1980; Coppo et al. 1981; Coppo et al. 1987; Dezani Ciancaglini Margaria, 1984; Dezani Ciancaglini Margaria, 1986; Hindley, 1982; Reynolds, 1988; Reynolds, 1991; Ronchi della Rocca, 1988; Ronchi della Rocca Venneri, 1984; van Bakel, 1991; van Bakel, 1992) Various extensions of the original intersection type discipline have also been explored, including the notion of infinite intersections (Leivant, 1990) and the dual notion of union types (Barbanera ....

....can always be conjoined to form a single least upper bound, so the nonexistence of lubs is equivalent to the nonexistence of complete finite sets of upper bounds for ordinary types. The most immediate implication of the nonexistence of least upper bounds is that standard techniques developed by Reynolds (1991) for constructing and analyzing models of first order intersection types will not generalize straightforwardly to F . Reynolds s model construction proceeds as follows. First, the set of canonical type expressions is defined as the limit of a series formed by beginning with the primitives and, at ....

Reynolds, J. C. 1991. The Coherence of Languages with Intersection Types. In: (Ito & Meyer, 1991).

....Since we also identify ff convertible terms, this does not essentially restrict the inference rules below. We will call A B the intersection of A and B, but refer to A and B as its conjuncts. Our system is more restrictive than customary formulations of intersection types (see, for example, [1, 19, 20]) The validity judgments below introduce a distinction between proper types and sorts. Proper types behave essentially like simple types and do not contain intersections. Sorts further refine proper types by enabling a more precise classification of terms, but sorts can only be intersected or ....

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

....any base subsorting relation which satisfies the some appropriate consistency conditions. The subsorting judgement has the form: R S Sort R is a sub sort of S, where R and S must be compatible. The subsorting rules present a minor variation for other systems with intersection types such as [Rey91] cnjs R s a sub base R s a sub reflex R R R 1 R 2 R 2 R 3 sub trans R 1 R 3 sub conjL1 R S R sub conjL2 R S S R S R S 0 sub conjR R S S 0 R 0 R S S 0 sub arrow R S R 0 S 0 sub dist (R S) R S 0 ) R (S S 0 ) If R S and S R then ....

....of each type modulo sort equivalence, which can be proved by induction on types. 2.4 Sorting The sorting assignment judgement has the form: Delta M . R Term M has sort R in context Delta. The sorting rules are very similar to those for a system with general intersection types (e.g. see [Rey91] Here each abstraction and fixed point includes the type assigned to the variable during type inference, so the choice of the sort for the variable is restricted to refinements of this type. x.R in Delta srt var Delta x . R Delta; x.R M . S R : A srt lam Delta x:A: M . R S ....

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

....we allow any base subsorting relation which satisfies some appropriate consistency conditions. The subsorting judgement has the form: R S Sort R is a subsort of S, where R and S must be compatible. The subsorting rules present a minor variation for other systems with intersection types such as [Rey91] cnjs R s a sub base R s a sub reflex R R R 1 R 2 R 2 R 3 sub trans R 1 R 3 sub conjL1 R S R sub conjL2 R S S R S R S 0 sub conjR R S S 0 R 0 R S S 0 sub arrow R S R 0 S 0 sub dist (R S) R S 0 ) R (S S 0 ) If R S and S R then we ....

....of each type modulo sort equivalence, which can be proved by induction on types. 6.4 Sorting The sort assignment judgement has the form: Delta M . R Term M has sort R in context Delta. The sorting rules are very similar to those for a system with general intersection types (e.g. see [Rey91] Here each abstraction and fixed point includes the type assigned to the variable during type inference, so the choice of the sort for the variable is restricted to refinements of this type. x.R in Delta srt var Delta x . R Delta; x.R M . S R : A srt lam Delta x:A: M . R S ....

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

....2 . A demonstration that the semantics is still well defined was given in an invited talk at the Logic in Computer Science Symposium [19] but was never written up. A proof that the semantics is coherent, i.e. that different proofs of the same typing do not lead to different meanings, was given in [20]. The functor category semantics is the basis of a scheme for generating intermediate code for Algol like language [21] 15. Conclusions and Future Research There are a number of directions in which it would be desirable to extend Forsythe, providing such extensions do not impact the uniformity ....

Reynolds, J. C. The Coherence of Languages with Intersection Types. in: Theoretical Aspects of Computer Software, International Conference TACS '91, Proceedings, Sendai, Japan, September 24--27, 1991, edited by T. Ito and A. R. Meyer. Lecture Notes in Computer Science, vol. 526, Springer-Verlag, Berlin, 1991, pp. 675--700.

No context found.

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

No context found.

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

No context found.

J.C. Reynolds, "The Coherence of Language with Intersection Types", TACS'91, LNCS 526, Springer-Verlag, Berlin 1991, 675--700.

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J. Reynolds, "The Coherence of Languages with Intersection Types", LNCS 526, 1991, 675-700.

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J.C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, Theoretical Aspects of Computer Software (Sendai, Japan), number 526 in LNCS, pages 675--700. Springer-Verlag, September

No context found.

John C. Reynolds. The coherence of languages with intersection types. In T. Ito and A. R. Meyer, editors, International Conference on Theoretical Aspects of Computer Software, pages 675--700, Sendai, Japan, September 1991. Springer-Verlag LNCS 526.

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