D. A. Wright. Reduction Types and Intensionality in the Lambda-calculus. PhD thesis, University of Tasmania, 1992. 21  Home/Search   Document Not in Database   Summary   Related Articles   Check

....arises by the Curry Howard isomorphism from such a relevance logic (though see [BBdP95] op allows unrestricted weakening and contraction, but the strictness translation prevents weakening being used to introduce variables of unlifted type. The idea of relevance also lies behind Wright s work [Wri92] on neededness analysis and the connection with relevance logic has been made more explicit by Baker Finch [BF92] Their work is concerned only with analysis and formulates correctness in terms of the syntactic notion of neededness , which is de ned via a labelled reduction system which tracks ....

D. A. Wright. Reduction Types and Intensionality in the Lambda-calculus. PhD thesis, University of Tasmania, 1992. 21

....concretely given function either will or will not depend on its argument, but in the presence of existential variables we still need the ability to remain uncommitted. Therefore our calculus also contains the full function space A u B. A similar calculus has been independently investigated in [Wri92, BF93]: for a comparison see Section 4.1. Labels k : 1 j 0 j u Types A : a j A 1 k A 2 Terms M : c j x j x k :A: M j (M 1 M 2 ) k Note that there are three di erent forms of abstractions and applications, where the latter are distinguished by di erent labels on the argument. It is not ....

....non standard types to represent these intensional information about functions (see [Jen91] for a comparison of these two techniques) However, earlier work as [TMM89] used non standard primitive type to distinguish strict or non strict terms, closed only under intuitionistic implication. Wright [Wri92] seems the rst one to have extended the Curry Howard isomorphism to (the implicational fragment of) relevance logic and explicitly connected the two areas, although both [Bel74] and [Hel77] had previously recognized the link between strictness and relevance 3 . 3 Note that we have became ....

D. A. Wright. Reduction types and intensionality in the lambda-calculus. PhD thesis, University of Tasmania, September 1992.

....of higher order functions and existential variables we still need the ability to remain uncommitted. Therefore our calculus also contains the full function space A u B. We rst concentrate on a version without existential variables. A similar calculus has been independently investigated by Wright [1992] and Baker Finch [1993] for a comparison see the end of Section 4. Labels k : 1 j 0 j u Types A : a j A 1 k A 2 Terms M : c j x j x k :A: M j M 1 M k 2 Note that there are three di erent forms of abstractions and applications, where the latter are distinguished by di erent ....

....non standard primitive type to distinguish strict or non strict terms, closed only under unrestricted function space. In the setting of functional programming, various di erent notions of strictness emerged. However, the absence of recursion and e ects in our setting admits fewer distinctions. Wright [1992] seems to be the rst to have extended the Curry Howard isomorphism to the implicational fragment of relevance logic and explicitly connected the two areas, although both [Belnap 1974] and [Helmann 1977] had previously recognized the link between strictness and relevance. Baker Finch [1993] ....

Wright, D. A. 1992. Reduction types and intensionality in the lambda-calculus. Ph.D. thesis, University of Tasmania.

....lambda calculus [14, 26] does not solve our problems. The polymorphic type for twice is ff:x ff :f ff ff :f(f x) so f will still be required to have the same demand on its argument as its result. Intersection types [5] offer a way out of this bind. Wright took this approach in his thesis [30]. An intersection type for twice is ( ff fi) fi fl) ff fl thus admitting a different type for each occurrence of f . Now for the twice swap x example above we can successfully deduce a demand of Str Theta Abs on x. Inferring intersection types is generally known to be undecidable but ....

David Wright. Reduction Types and Intensionality in Lambda Calculus. PhD thesis, University of Tasmania, 1993.

....interpretation with projection analysis, and Jensen [17] and Benton [5] investigating links between abstract interpretation and the typing approach of Kuo and Mishra. The main contribution of this paper is to demonstrate a close correspondence between the annotated types approach of Wright [25] [26] and Baker Finch [3] 4] 27] and projection analysis. This also suggests a clean extension of projection analysis to the higher order case. The treatment of data structures and recursive types in the type system is new and hence of interest in its own right. The paper is structured as follows. ....

D. A. Wright. Reduction Types and Intensionality in Lambda Calculus. PhD thesis, University of Tasmania, 1993.

....for a comparison of these two techniques) However, earlier work as [TMM89] used non standard primitive type to distinguish strict or non strict terms, closed only under intuitionistic implication. Not forgetting Wadler s early paper [Wad90] on using linear logic for sharing analysis, Wright [Wri91, Wri92] seems the first one to have extended the Curry Howard isomorphism to (the implicational fragment of) relevance logic and explicitly connected the two areas, although both [Bel74] and [Hel77] had previously recognized the link between strictness and relevance 3 . In [BF93] the author summarizes ....

.... compared to the hassle of implementing a strict calculus, only for the purpose of allowing full clause complementation; consider for example the lack so far of a crucial ingredient such as the type reconstruction algorithm (although one for a related calculus is presented in Wright s dissertation [Wri92]) Moreover, for every signature, the definition of the predicate vacuous is completely trivial, while the one for strict is type directed and can be automatically inferred, in the style of Miller s copy clause [Mil89b] On the other hand, this approach is less workable when lifting the ....

D. A. Wright. Reduction types and intensionality in the lambda-calculus. PhD thesis, University of Tasmania, September 1992.

....this way we can go on as long as necessary to calculate the fixpoint. Finally we are allowed to iterate n Gamma q times more to get the property oe n for the fixpoint. Example 1 We can infer fix x Int . x Int : b Int which is more precise than the information Int obtained by [15] in [16] it can be done. In the systems of [2, 6, 7, 16] one can infer the property Int for the term fix x Int .7 whereas we can infer the more precise property n Int . In this system (as well as those of [12, 2, 6, 7, 16] it is possible to infer the property (b Int Int Int b Int ....

....to calculate the fixpoint. Finally we are allowed to iterate n Gamma q times more to get the property oe n for the fixpoint. Example 1 We can infer fix x Int .x Int : b Int which is more precise than the information Int obtained by [15] in [16] it can be done. In the systems of [2, 6, 7, 16] one can infer the property Int for the term fix x Int .7 whereas we can infer the more precise property n Int . In this system (as well as those of [12, 2, 6, 7, 16] it is possible to infer the property (b Int Int Int b Int ) Int b Int Int b Int ) 8 ....

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David A. Wright. Reduction Types and Intensionality in the LambdaCalculus. PhD thesis, University of Tasmania, 1992. 17

....about functions (see[Hug91] for a comparison) However, earlier works, as [TMM89] used non standard primitive type to distinguish strict or non strict terms, closed only under intuitionistic implication. Not forgetting Wadler s early paper [Wad90] on using linear logic for sharing analysis, Wright [Wri91,Wri92] (see also [BF93] seems the first one to have extended the Curry Howard isomorphism to (the implicational fragment of) relevant logic and connected the two areas. Insert a comparison of those systems to ours. 3 Preliminaries In this section we treat the important special case fully applied ....

D. A. Wright. Reduction types and intensionality in the lambda-calculus. PhD thesis, University of Tasmania, September 1992. 46

....can characterize run time properties of programs. For instance intersection types, see [11] and also [1] in their full generality, provide a characterization of normalization. Type systems tailored to specific analysis, such as strictness, totality, binding time etc. have been introduced, see [18, 16, 2, 12, 15, 22, 25, 13]. In this perspective types represent program properties and their inference systems are systems for reasoning formally about them. In this paper we keep a clear distinction between the type structure of the language (types in the usual sense) and the annotated types ( non standard types) which ....

D. A. Wright. Reduction Types and Intensionality in the Lambda-Calculus. PhD thesis, University of Tasmania, 1992. 23

....concretely given function either will or will not depend on its argument, but in the presence of existential variables we still need the ability to remain uncommitted. Therefore our calculus also contains the full function space A u B. A similar calculus have been independently investigated in [16] where the Curry Howard connection with relevant logic is explained. Labels k : 1 j 0 j u Types A : P j A 1 k A 2 Terms M : c j x j x k :A: M j M 1 M 2 k Contexts Gamma : Delta j Gamma; x:A Note that there are three different forms of abstractions and applications, where the ....

D. A. Wright. Reduction types and intensionality in the lambda-calculus. PhD thesis, University of Tasmania, Sept. 1992.

....interpretation with projection analysis, and Jensen [Jen91] and Benton [Ben93] investigating links between abstract interpretation and the typing approach of Kuo and Mishra. The main contribution of this paper is to demonstrate a close correspondence between the annotated types approach of Wright [Wri91, Wri93] and Baker Finch [BF92, BF93, WBF93] and projection analysis. This also suggests a clean extension of projection analysis to the higher order case. The treatment of data structures and recursive types in the type system is new and hence of interest in its own right. The paper is structured as ....

D. A. Wright. Reduction Types and Intensionality in Lambda Calculus. PhD thesis, University of Tasmania, 1993.

....arises by the Curry Howard isomorphism from such a relevance logic (though see [BBdP95] op allows unrestricted weakening and contraction, but the strictness translation prevents weakening being used to introduce variables of unlifted type. The idea of relevance also lies behind Wright s work [Wri92] on neededness analysis and the connection with relevance logic has been made more explicit by Baker Finch [BF92] Their work is concerned only with analysis and formulates correctness in terms of the syntactic notion of neededness , which is defined via a labelled reduction system which tracks ....

D. A. Wright. Reduction Types and Intensionality in the Lambda-calculus. PhD thesis, University of Tasmania, 1992.

....concretely given function either will or will not depend on its argument, but in the presence of existential variables we still need the ability to remain uncommitted. Therefore our calculus also contains the full function space A u B. Similar calculi have been independently investigated in [20, 1]. Labels k : 1 j 0 j u Types A : P j A 1 k A 2 Terms M : c j x j x k :A: M j M 1 M 2 k Contexts Gamma : Delta j Gamma; x:A Note that there are three different forms of abstractions and applications, where the latter are distinguished by different labels on the argument. It ....

D. A. Wright. Reduction types and intensionality in the lambda-calculus. PhD thesis, University of Tasmania, September 1992.

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