A. J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order #-calculus. Inf. & Comput., 98:228--257, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can type fewer terms than ML. But starting with rank 2, the systems can type more terms than ML. Rank 2 of System F, which we call 2 , has received the most study. McCracken [23] proposed a type inference algorithm for 2 based on Leivant s ideas. This algorithm is incorrect. Kfoury and Tiuryn [12] show that the complexity of typability in 2 is identical to that of ML. Kfoury and Wells [16, 17] give a correct type inference algorithm, and show that ranks 3 and higher in System F are undecidable. Leivant s original paper is almost the only work on rank 2 of the intersection type ....

....and oe may be read, oe is more general than . We make an exception in the case of to be consistent with its use in ML [24] We now define 2 , our version of the rank 2 fragment of System F. The superscript s in 2 indicates that the system is syntax directed. See Kfoury and Tiuryn [12] for a definition of 2 , the non syntax directed version. The judgments of the system are defined by the rules of Figure 4. We write 2 . A M : if A M : is derivable from these rules, where types in type environments are restricted to S(1) and derived types are restricted to S (2) ....

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A.J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, June 1992.

.... s) s Figure 2: Types that go beyond rank 2 rank 2, the systems can type more terms than ML. Rank 2 of System F, which we call 2 , has received the most study. McCracken [23] proposed a type inference algorithm for 2 based on Leivant s ideas. This algorithm is incorrect. Kfoury and Tiuryn [12] show that the complexity of typability in 2 is identical to that of ML. Kfoury and Wells [16, 17] give a correct type inference algorithm, and show that ranks 3 and higher in System F are undecidable. Leivant s original paper is almost the only work on rank 2 of the intersection type ....

....and oe may be read, oe is more general than . We make an exception in the case of to be consistent with its use in ML [24] We now define 2 , our version of the rank 2 fragment of System F. The superscript s in 2 indicates that the system is syntax directed. See Kfoury and Tiuryn [12] for a definition of 2 , the non syntax directed version. The judgments of the system are defined by the rules of Figure 4. We write 2 . A M : if A M : is derivable from these rules, where types in type environments are restricted to S(1) and derived types are restricted to S (2) ....

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A.J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, June 1992.

.... be typed) and it does not have principal typings (see [17, 25] System F [23] provides a much more general notion of polymorphism, but lacks principal types, and type inference is undecidable in general (although it is decidable for some subsystems, in particular if we consider types of rank 2 [27]) Intersection type systems [12] are somewhere in the middle with respect to polymorphism, and have principal typings. But type assignment is again undecidable; decidability is recovered if we restrict ourselves to intersection types of finite rank [29] In [24] a system for LC combining ....

....ii) Does it have the principal typing property We answer these questions in the affirmative. The restriction to types of rank 2 of the combined system of polymorphic and intersection types is decidable. This restricted system can be seen as a combination of the systems considered in [6] and [27]. The combination is two fold: not only the type systems of those two papers are combined (resp. intersection and polymorphic types of rank 2) but also their calculi are combined (resp. CuTRS and LC) In our Rank 2 system each typeable term has a principal typing. This is the case also in the ....

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A.J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, 1992.

....the minimum rank typing for M (if one exists) For example: term minrank term minrank I = #x.x) 0 (II)R i 2i 2 R 1 = #w.ww) 2 (#z.z(IR 1 )R 2 ) 4 R i = #y.yR i 1 ) 2i (#z. z(IR 2 )R1 ) 5 IR i 2i 1 (R1R1 ) undefined Our definition of rank is equivalent to others found in the literature [Lei83, Jim96, vB93, KT92], except that we consider the rank of typed terms instead of the rank of derivations, and the rank of a typed term is not affected by unused type assumptions. The restriction of System I to finite rank k, called System I k , is the restriction deriving only judgements of the form A # M, ....

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order #-calculus. Inform. & Comput., 98(2):228--257, June 1992.

.... of derivations in F to finite values, e.g. depth of bound type variable from binding quantifier [GRDR91] the number of generations of instantiation of quantifiers themselves introduced by instantiation [Lei91] and the rank of polymorphic types (introduced in [Lei83] further studied in [McC84, KT92, KW94]) Urzyczyn recently showed Typ to be undecidable for F s powerful extension, the system F [Urz93b] which allows types to contain functions from types to types. The proof, which reduces the halting problem for Turing machines to Typ for F , actually establishes the recursive inseparability of ....

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Inf. & Comput., 98(2):228--257, June 1992.

....expression, no quantified type appears as a constituent in another type, leaving us with effectively only one level typing, that is, a first order system. Our T system is related to a very similar family of systems that have level 2 types constructed by quantification over level one types [Lei91, KT92] Type systems with stratification of types into levels are all essentially predicative. At the other extreme are essentially impredicative type systems, such as the second order lambda calculus [Gir72, Rey74] that allow types with quantification over all types. The T type system is essentially ....

....We differ from systems with recursively defined types [MPS86, CC91] because our dependent types are predicative; either finite or well founded recursions. On the other hand, we are similar to other implicitly typed and first order or predicative systems in the ML family [KTU93a, Hen93, Lei91, KT92, McC84] 4.2.2 Polymorphism and Dependent Types There are many kinds of polymorphism. To simplify our study, we use monomorphic type schemes as in the original work by Hindley [Hin69] By monomorphic we mean that no types are formed using universal quantifiers over type variables. By type ....

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A. J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second order lambda calculus. Information and Computation, 98:228--257, 1992.

....a principal typing expects less of its free variables, and provides more than any other typing judgment. 2.1 Comparison with Rank 2 of System F The system P 2 is closely connected to 2 , the restriction of System F to rank 2 types. Our presentation of 2 is based on that of Kfoury and Tiuryn [15]. The types of System F are defined by the following grammar: t j ( 1 2 ) j (8t ) We consider System F types to be syntactically equal modulo renaming of bound type variables, reordering of adjacent quantifiers, and elimination of unnecessary quantifiers. 7 (var) A x [ fx : oeg x : ....

....for 2 [19] Corollary 5 Typability in P 2 is DEXPTIME complete. The proof of Theorem 4 relies on the principal type property of ML and is given in a separate paper [11] a similar theorem has been shown independently by Yokouchi [32] Corollary 5 follows by the results of Kfoury and Tiuryn [15] ( 2 typability is polynomial time equivalent to ML typability) and Kfoury et al. 18] and Mairson [22] ML typability is DEXPTIME complete) 8 2.2 Subtype satisfaction In order to perform type inference, we must solve subtype satisfaction problems, which generalize unification. Solving ....

A.J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, June 1992.

....ae ffl is of the restricted rank 2. It is easy to establish invertibility Lemma 2.1 (Invertibility) If Gamma ffl M : ae ffl is derivable and ae ffl = Pi v : sort : ae 0 , then Gamma ffl M : ae 0 is derivable too. By this we have a simplified version of a theorem from [KT92] Theorem 2.2 (Calculus minimization) Gamma 1 M : ae iff Gamma ffl M : ae ffl Proof: That lifts to 1 is immediate. For the other direction an easy induction over the length of the derivation tree in 1 gives the result. We highlight a few cases VAR 1 Suppose VAR 1 Gamma; x ....

A.J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, June 1992.

.... is limited (some programs that arise naturally cannot be typed) System F [18] provides a much more general notion of polymorphism, but lacks principal types, and type inference is undecidable in general (although it is decidable for some subsystems, in particular if we consider types of rank 2 [21]) Intersection type systems [10] are somewhere in the middle with respect to polymorphism (they provide less polymorphism than System F but more than ML) and principal types can be constructed for typeable terms. But type assignment is again undecidable; decidability is recovered if we restrict ....

....algebraic rewriting into account. We also answer the first question in the affirmative. The restriction to types of rank 2 of the combined system of polymorphic and intersection types is decidable. This restricted system can be seen as a combination of the systems considered in [4] and [21]. The combination is twofold: not only the type systems of those two papers are combined (resp. intersection and polymorphic types of Rank 2) but also their calculi are combined (resp. CuTRS and LC) In our Rank 2 system each typeable term has a principal type such that every type derivable for ....

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A.J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. I&C 98(2):228--257, 1992.

....ae ffl is of the restricted rank 2. It is easy to establish invertibility Lemma 2.1 (Invertibility) If Gamma ffl M : ae ffl is derivable and ae ffl = Pi v : sort : ae 0 , then Gamma ffl M : ae 0 is derivable too. By this we have a simplified version of a theorem from [KT92] Theorem 2.2 (Calculus minimization) Gamma 1 M : ae iff Gamma ffl M : ae ffl This establishes adequacy of our type system based on instead of 1 to infer rank 2 bounded types. 2.4 Subject reduction In order to establish that this calculus preserves types under ....

A.J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, June 1992.

....of such a result; instead, in Subsection 6.1, we will briefly discuss the fundamental differences between the two systems. To avoid confusion, it is necessary to point out that there also exists a notion of type assignment that is called the Rank 2 Polymorphic Type Assignment System, defined in [25]. This system is an extension of Milner s system, by allowing for the 8 type constructor to occur also on the left hand side of an arrow type, instead of only at top level. It is also a restriction of the Polymorphic Type Discipline [23] where types are restricted to polymorphic types of Rank ....

A.J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order - calculus. Information and Computation, 98(2):228--257, 1992.

....and Okada in [21] in order to take the presence of (higher order) algebraic rewriting into account. Finally, we present a decidable restriction of this combined system, by limiting types to those of Rank 2. This restricted system can be seen as a combination of the systems considered in [5] and [22]. The combination is twofold: not only the type systems of those two papers are combined (resp. intersection and polymorphic types of Rank 2) but also their caluli are combined (resp. CuTRS and LC) In our Rank 2 system, each typeable term has a principal type such that every type derivable for ....

....CuTRS and LC) In our Rank 2 system, each typeable term has a principal type such that every type derivable for the term can be seen as an instance (under suitable operations) of the type. This is the case also in the Rank 2 intersection system of [5] but not in the Rank 2 polymorphic system of [22]. For the latter, a type inference algorithm of the same complexity of that of ML was given in [23] where the problems that occur due to the lack of principal types are discussed in detail. This paper is organised as follows: In Section 1 we define TRS with application, abstraction and ....

A.J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Information and Computation, 98(2):228--257, 1992.

....in less detail. For more on the relationship between Klop s and Nederpelt s technique, see [41, II.4] For more on the relationship between de Groote s and Kfoury and Wells technique, see [37] The notions of reduction discussed in this section have been considered in a number of other contexts [1, 33, 34, 35, 48, 57, 60, 71], see [37] 3.1. The technique by de Groote This subsection presents de Groote s [14] technique to reduce strong normalization for the systems in the cube [3] to weak normalization of related systems. In particular, adopting a version of the Turing Prawitz proof, he proves strong normalization ....

A. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Journal of Information and Computation, 98(2):228--257, 1992.

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A. J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order #-calculus. Inf. & Comput., 98:228--257, 1992.

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A. J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order _ -calculus. Inf. & Comput., 98:228--257, 1992.

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A. J. Kfoury and J. Tiuryn. Type reconstruction in finite rank fragments of the second-order #-calculus. Inf. & Comput., 98:228--257, 1992.

.... the minimum rank typing for M (if one exists) For example: term minrank term minrank I = x:x) 0 (II)R i 2i 2 R 1 = w:ww) 2 (z:z(IR 1 )R 2 ) 4 R i = y:yR i Gamma1 ) 2i (z:z(IR2 )R1 ) 5 IR i 2i 1 (R1R1 ) undefined Our definition of rank is equivalent to others found in the literature [Lei83, Jim96, vB93, KT92], except that we consider the rank of typed terms instead of the rank of derivations, and the rank of a typed term is not affected by unused type assumptions. 5 The restriction of System I to finite rank k, called System I k , is the restriction deriving only judgements of the form A M; M : ....

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Inf. & Comput., 98(2):228--257, June 1992.

.... themselves introduced by instantiation [Lei91] One natural restriction which we consider in this paper results from stratifying F according to the rank of types allowed in the typing of terms and restricting the rank to various finite values (introduced in [Lei83] and further studied in [McC84, KT92]) All of these systems improve on the expressive power of ML. Unfortunately, it is often the case that the more flexible and powerful a particular polymorphic type system is, the more likely that automatic type inference will be infeasible or impossible. As discouraging examples, the problems of ....

....rank 2. The undecidability of type checking at rank 3 can be seen by observing that the proof for the undecidability of type checking in F in [Wel93] requires only rank 3 types. The undecidability of typability at rank 3 results from the fact that the constants c and f defined in section 5 of [KT92] can be encoded using the methods of [Wel93] in System 3 (the rank 3 fragment of F) and from Theorem 30 of [KT92] We give this encoding in this paper. Since it was already known from [KT92] that typability is decidable for System 2 (the rank 2 fragment of F) we know exactly where the boundary ....

[Article contains additional citation context not shown here]

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Inf. Comput., 98(2):228--257, June 1992.

....we are told used something like fl and 1 . Also in 1989, Moggi used 3 along with many other transformations in a paper [Mog89] In 1990, Kfoury, Tiuryn, and Urzyczyn used 1 and 3 together with another transformation as part of the proof that typability in ML is DEXPTIMEcomplete [KTU90, KTU94] A 1992 paper by Kfoury and Tiuryn uses a variant of fl (denoted ( L ) in analyzing the rank 2 restriction of system F [KT92] Also in 1992, Sabry and Felleisen introduced a generalization (denoted fi lift ) of 1 and 3 [SF92] This development by Sabry and Felleisen is especially interesting ....

.... In 1990, Kfoury, Tiuryn, and Urzyczyn used 1 and 3 together with another transformation as part of the proof that typability in ML is DEXPTIMEcomplete [KTU90, KTU94] A 1992 paper by Kfoury and Tiuryn uses a variant of fl (denoted ( L ) in analyzing the rank 2 restriction of system F [KT92] Also in 1992, Sabry and Felleisen introduced a generalization (denoted fi lift ) of 1 and 3 [SF92] This development by Sabry and Felleisen is especially interesting since the fi lift rule was derived from equivalences induced by continuation passing style transformations. In 1993, de ....

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order -calculus. Inf. Comput., 98(2):228--257, June 1992.

.... themselves introduced by instantiation in [Lei91] One natural restriction which we consider in this paper results from stratifying F according to the rank of types allowed in the typing of terms and restricting the rank to various finite values (introduced in [Lei83] and further studied in [McC84, KT92]) All of these systems improve on the expressive power of ML. Unfortunately, it is often the case that the more flexible and powerful a particular polymorphic type system is, the more likely that it will be infeasible to implement. As discouraging examples, the problems of typability and type ....

....2. The undecidability of type checking at rank 3 can be seen by observing that the proof for the undecidability of type checking in F in [Wel93] requires only rank 3 types. 1 The undecidability of typability at rank 3 results from the fact that the constants c and f defined in section 5 of [KT92] can be encoded using the methods of [Wel93] in System 3 (the rank 3 fragment of F) and from Theorem 30 of [KT92] We give this encoding in this paper. Since it was already known from [KT92] that typability is decidable for System 2 (the rank 2 fragment of F) we know exactly where the boundary ....

[Article contains additional citation context not shown here]

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the secondorder -calculus. Inf. Comput., 98(2):228--257, June 1992.

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A. Kfoury. Type reconstruction in finite rank fragments of second-order lambda calculus. Information and Computation, 98(2):228--257, June 1992.

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A. Kfoury. Type reconstruction in finite rank fragments of second-order lambda calculus. Information and Computation, 98(2):228--257, June 1992.

No context found.

A. Kfoury. Type reconstruction in finite rank fragments of second-order lambda calculus. Information and Computation, 98(2):228--257, June 1992.

No context found.

A. J. Kfoury and J. Tiuryn. Type reconstruction in finite-rank fragments of the second-order #-calculus. Inform. & Comput., 98(2):228--257, June 1992.

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AJ Kfoury [June 1992], "Type reconstruction in finite rank fragments of secondorder lambda calculus," Information and Computation98, 228--257.

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