A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. J. ACM, 41(2):368--398, Mar. 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... side of rule (g) above (which we call generalised redex) 23] uses and (and calls the combination ) to show that the perpetual reduction strategy nds the longest reduction path when the term is Strongly Normalizing (SN) 27] also introduces reductions similar to those of [23] Furthermore, [13] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi uni cation. 24] uses a reduction which has some common themes with where ( x : y :x)a)b) is transformed into k : x : y :kx)b) a) 3] identi ed the extra power of the CPS transformations of ....

A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. ACM, 41(2):368-398, 1994.

.... is called extended redex in the paper) Reg 94] uses and fl (and calls the combination oe) to show that the perpetual reduction strategy finds the longest reduction path when the term is Strongly Normalising (SN) Vid 89] also introduces reductions similar to those of [Reg 94] Furthermore, KTU 94] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi unification. SF 92] uses a reduction which has some common themes to . Nederpelt 73] and [dG 93] use whereas [KW 95a] uses fl to reduce the problem of strong normalisation for fi reduction to the ....

Kfoury, A.J., Tiuryn, J. and Urzyczyn, P., An analysis of ML typability, J. ACM 41(2), 368-398, 1994.

....type systems whereas Regnier studies Curry style type systems. Reg 94] uses and fl (and calls the combination oe) to show that the perpetual reduction strategy finds the longest reduction path when the term is SN. Vid 89] also introduces reductions similar to those of [Reg 94] Furthermore, KTU 94] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi unification. SF 92] uses a reduction which has some common themes to . dG 93] uses a restricted version of and [KW 95a] uses fl to reduce the problem of strong normalisation for fi reduction to the ....

Kfoury, A.J., Tiuryn, J. and Urzyczyn, P. (1994), An analysis of ML typability, J. ACM 41(2), 368-398. 34

....type systems whereas Regnier studies Curry style type systems. Reg 94] uses and fl (and calls the combination oe) to show that the perpetual reduction strategy finds the longest reduction path when the term is SN. Vid 89] also introduces reductions similar to those of [Reg 94] Furthermore, KTU 94] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi unification. SF 92] uses a reduction which has some common themes to . dG 93] uses a restricted version of and [KW 95a] uses fl to reduce the problem of strong normalisation for fi reduction to the ....

Kfoury, A.J., Tiuryn, J. and Urzyczyn, P., An analysis of ML typability, J. ACM 41(2), 368-398, 1994.

.... the perpetual reduction strategy finds the longest reduction path when the term is strongly normalizing (SN) Reg94] Vidal also introduced similar reductions [Vid89] Kfoury, Tiuryn, and Urzyczyn used (and other reductions) to show that typability in ML is equivalent to acyclic semi unification [KTU94]. Sabry and Felleisen described a relationship between a reduction similar to and a particular style of CPS [SF93] De Groote [dG93] used and Kfoury and Wells [KW95b] used fl to reduce the problem of fi strong normalization to the problem of weak normalization (WN) for related reductions. ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. Journal of the ACM, 41:368--398, March 1994.

.... in the rule (g) above (which we call generalised redex) 18] uses and (and calls the combination ) to show that the perpetual reduction strategy nds the longest reduction path when the term is Strongly Normalizing (SN) 21] also introduces reductions similar to those of [18] Furthermore, [9] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi uni cation. 19] uses a reduction which has some common themes with . 16] and [4] use whereas [12] uses to reduce the problem of strong normalization to the problem of weak normalization (WN) for ....

A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. ACM, 41(2):368-398, 1994.

....scheme [26] Given a monomorphic type system based on a constraint system X, the authors give a generic construction of HM(X) i.e. type inference for ML style (i.e. Hindley Milner) polymorphic constrained types. Type inference for the polymorphic system remains DEXPTIMEcomplete, of course [16, 17]. In the remainder of the introduction we summarize the main idea of the type system for first class messages and of the constraint system OF. 1.1. The Type System The type system contains types for objects and messages and explains what type of messages can be sent to an object of a given ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. Journal of the Association for Computing Machinery, 41(2):368--398, Mar. 1994.

....in I 2 is DEXPTIME complete. Proof: The equivalence of I 2 and 2 typability follows from Theorems 12 and 26. Kfoury and Tiuryn [12] show that 2 typability is polynomial time equivalent to ML typability. ML typability was shown to be DEXPTIMEcomplete independently by Kfoury et al. [15] and by Mairson [22] 2 This equivalence has been shown independently by Yokouchi [35] 20 3 Type inference for I We present the type inference algorithm for I 2 and a proof that it infers principal pairs. The algorithm is not new: it was described briefly in Leivant s original paper [21] ....

.... recursion: rec poly) where 2 S(1) Example 59 When extended by (rec poly) both ML and 2 can type the following terms: w: xyz:z) w 3) w true) 8t:t t; x:xx) 8t:t: Neither is typable with the rule (rec simple) Other examples are given by Mycroft [25] and Kfoury et al. [13, 15], who introduced (rec poly) independently. Unfortunately, type inference for 2 or ML extended by (rec poly) is undecidable [14, 9] so (rec simple) is used in practice. 6.2 Recursive definitions in I 2 The rule (rec simple) is one way of typing recursive definitions in intersection type ....

A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. Journal of the ACM, 41(2), March 1994.

....in I 2 is DEXPTIME complete. Proof: The equivalence of I 2 and 2 typability follows from Theorems 12 and 26. Kfoury and Tiuryn [12] show that 2 typability is polynomial time equivalent to ML typability. ML typability was shown to be DEXPTIMEcomplete independently by Kfoury et al. [15] and by Mairson [22] 2 The equivalence of Theorem 27 has been shown independently by Yokouchi [35] 20 3 Type inference for I We present the type inference algorithm for I 2 and a proof that it infers principal pairs. The algorithm is not new: it was described briefly in Leivant s original ....

.... recursion: rec poly) where 2 S(1) Example 60 When extended by (rec poly) both ML and 2 can type the following terms: w: xyz:z) w 3) w true) 8t:t t; x:xx) 8t:t: Neither is typable with the rule (rec simple) Other examples are given by Mycroft [25] and Kfoury et al. [13, 15], who introduced (rec poly) independently. Unfortunately, type inference for 2 or ML extended by (rec poly) is undecidable [14, 9] so (rec simple) is used in practice. 43 6.2 Recursive definitions in I 2 The rule (rec simple) is one way of typing recursive definitions in intersection type ....

A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. Journal of the ACM, 41(2), March 1994.

....take the following program: Procedure P n 1 (x) Procedure P n (x) P n 1 (y) P n 1 (z) Procedure P 1 (x) P 2 (y) P 2 (z) P 1 (0) The expanded version of this program will contain 2 n 1 Gamma 1 procedures. In ML there are similar problems with types of exponential size [3, 4]. In both cases it can be argued that the examples are highly artificial and that we will not encounter this behavior in practice. A fully implemented prototype of the Turbo Pascal tool can be found at http: www.daimi.aau.dk hougaard itp. This site contains the source code written in Turbo ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. Journal of the ACM, 41:368--398, 1994.

....given below (meta variables x and range, resp. over a countably in nite set of variables and a countably in nite set of type variables) 1 Simple Types : j 0 Types : 8 : j Expressions e : x j x: e j e e 0 j let x = e in e 0 An equivalent version of ML type system [Hen93, KTU93, KTU94, CDDK86] is presented in Fig.1. Predicate inst( 8( i ) i=1: n : 0 ; holds when = 0 [ 1 = 1 ; n = n ] for some 1 ; n . We use 8( i ) i=1: n : as abbreviation for 8 1 : 8 n : where n 0. We sometimes drop the superscripts (i = 1: n) and write 8 i : ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An Analysis of ML Typability. Journal of the ACM, 41(2):368-398, 1994.

....type annotations find if it is typable using the ML (TLC) typing rules. For TLC this is an efficiently solvable problem in linear time [48] by reduction to first order unification [43, 15] but it is also PTIME complete. For Core ML it is EXPTIME complete in the size of the program typed, see [29, 30]. The proof sketch we present in Section 4.2 is based on [22] Fixed Functionality: In [24, 25] we continued the investigation of expressibility and type reconstruction, but for low functionality orders. Fragments of Core ML, based on the order of functionalities used, can be fairly expressive. ....

....one reduction of (x: xx) z: z) namely (z: z) z: z) 5 Note that, Core ML has all the properties of TLC = i.e. Church Rosser, Strong Normalization, Principal Type and Type Reconstruction. There is only one difference: Type reconstruction is no longer in linear time but EXPTIMEcomplete [29, 30]. 2.3 Functionality Order For a finer analysis of expressibility we partition types according to functionality order. The order of a type measures the higher order functionality of a term of that type: o (t) 0 for a type variable t and o ( 0 00 ) max (1 o ( 0 ) o ( 00 ) We ....

A. Kfoury, J. Tiuryn, and P. Urzyczyn. An Analysis of ML Typability. In Proceedings 17th Colloquium on Trees, Algebra and Programming, pp. 206--220. LNCS 431, Springer Verlag, 1990.

....types. For all these programs there are PTIME reduction strategies. 2) In Section 5 we investigate the flexibility of MLtyping. We show that for fixed order functionalities MLtype inference is PTIME in the size of programs. In general, type inference is EXPTIME complete in the size of programs [26, 27]. Thus, in our MLI languages type inference is provably efficient. These languages do simplify our calculations. For example, PTIME queries MLI = 1 is provable without any of the type laundering techniques of [21] 3) In Section 6 we present the main analytic results of this paper. These ....

....expressibility, principal type and type inference properties hold for core ML = where constants and their equality are added as in TLC = Order of functionality is defined in the same way. There are two differences: 1) Type inference is no longer in linear time but EXPTIME complete [26, 27]. 2) Arbitrary order core ML, core ML = TLC, and TLC = all have the same expressive power, but for fixed order type inference allows more core ML than TLC programs to be typed, so it might provide more expressibility. 2.3 Elementary Recursion via List Iteration We briefly review how list ....

A. Kfoury, J. Tiuryn, and P. Urzyczyn. An Analysis of ML Typability. In Proceedings 17th Colloquium on Trees, Algebra and Programming, pp. 206--220. Lecture Notes in Computer Science 431, Springer Verlag, 1990.

....given below (meta variables x and range, resp. over a countably in nite set of variables and a countably in nite set of type variables) 1 Simple Types : j 0 Types : 8 : j Expressions e : x j x: e j e e 0 j let x = e in e 0 An equivalent version of ML type system [Hen93, KTU93, KTU94] is presented in Fig.1. Predicate inst( 8( i ) i=1: n : 0 ; holds when = 0 [ 1 = 1 ; n = n ] for some 1 ; n . We use 8( i ) i=1: n : as abbreviation for 8 1 : 8 n : where n 0. We sometimes drop the superscripts (i = 1: n) and write 8 i : ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An Analysis of ML Typability. Journal of the ACM, 41(2):368-398, 1994.

....given below (meta variables x and range, resp. over a countably in nite set of variables and a countably in nite set of type variables) 1 Simple Types : j 0 Types : 8 : j Expressions e : x j x: e j e e 0 j let x = e in e 0 An equivalent version of ML type system [Hen93, KTU93, KTU94, CDDK86] is presented in Fig.1. For simplicity, we distinguish between lambda bound variables and let bound variables and use meta variable u to denote a lambda bound variable and meta variable x to denote either a lambda bound or a let bound variable, when the distinction is clear from the context or is ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An Analysis of ML Typability. Journal of the ACM, 41(2):368-398, 1994.

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. J. ACM, 41(2):368--398, Mar. 1994.

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A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. In Arnold, editor, 15th Colloquium on Trees in AlgebraandProgramming, CAAP 90, LNCS 431, pages 206 -- 220. Springer Verlag, 1990.

No context found.

A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. In Arnold, editor, 15th Colloquium on Trees in Algebra and Programming, CAAP 90, LNCS 431, pages 206 -- 220. Springer Verlag, 1990.

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A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ml typability.In Arnold, editor, 15th Colloquium on Trees in Algebra and Programming, CAAP 90, LNCS 431, pages 206 -- 220. Springer Verlag, 1990. 24

....using the acyclic restriction of regular semi unification. The problem of inferring rank 2 recursive types is polynomial time equivalent to finding regular solutions for instances of acyclic semi unification. Many of the ideas already used to handle the finite case of acyclic semi unification [KTU94] are used again in the regular case of the same problem. As a result, whether an instance of acyclic semi unification has a regular solution (and, therefore, whether a program is typable with rank 2 recursive types) is DEXPTIME complete. The details of this translation are not yet polished and, ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. J. ACM, 41(2):368--398, Mar. 1994.

.... as type inference for the simply typed # calculus, which can be made to run in linear time and is therefore very efficient in practice [23, 8] Just like first order unification, it is PTIME complete [3] Type inference in ML may require exponential time only in the presence of polymorphic (Let) [11, 14] and this happens only in the case of programs that are arguably pathological [17] Towards filling the huge gap between efficient typeinference with (Monorec) and undecidable type inference with (Polyrec) one of our research goals is to formulate typing rules strictly more powerful that ....

....We prove that the problem of inferring rank 2 recursive types is polynomial time equivalent to finding regular solutions for instances of acyclic semi unification, for which we have an alwaysterminating algorithm. Many of the ideas already used to handle the finite case of acyclic semi unification [14] are used again in the regular case of the same problem. As a result, whether an instance of acyclic semi unification has a regular solution (and, therefore, whether a program is typable with rank 2 recursive types) is DEXPTIME complete. For every k # 3, type inference with rank k recursive ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. J. ACM, 41(2):368--398, Mar. 1994.

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A. J. Kfoury, J. Tiuryn, P. Urzyczyn. An analysis of ML typability. J. ACM, 41(2), 1994. Supersedes [19].

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Kfoury, A. J., J. Tiuryn, and P. Urzyczyn: 1994, `An analysis of ML typability'. Journal of the ACM 41(2), 368--398.

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A. J. Kfoury, J. Tiuryn, P. Urzyczyn. An analysis of ML typability. J. ACM, 41(2), 1994.

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A. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. ACM, 41(2):368--398, 1994.

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