A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-uni cation problem. Inform. & Comput., 102(1):83-101, Jan. 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for System F. 1.4 What does this article contribute The main contribution of this report is proving the undecidability of Typ and TC for System F. 1. We rst prove that the problem of semi uni cation can be reduced to TC using a simple encoding. Because semi uni cation is undecidable [KTU93], so is TC. 2. We then reduce TC to Typ using a novel method of building terms which simulate arbitrarily chosen type environments. The proof begins by showing that there exists a typable term J such that in every typing of J , its bound variable x is assigned the type : Then, building ....

....A M : 2.5 Semi Uni cation For convenience, we de ne semi uni cation using a rst order signature containing the single in x binary function symbol and for the case where there are only two pairs of terms. The general de nition of semi uni cation is reducible to this special case [Pud88, KTU93] and the proof that semi uni cation is undecidable is actually for this special case [KTU93] This notion is not completely de ned because there may be multiple occurrences of a subterm N M with distinct typings. This does not pose a problem in this report because it is always clear from ....

[Article contains additional citation context not shown here]

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semiuni cation problem. Inf. & Comput., 102(1):83-101, Jan. 1993.

....can only type the same expressions that are typed with rule (FIX M) used in ML. The result of undecidability of type inference for ML (FIX P) was proved independently by Henglein [Hen93] and Kfoury et al. KTU93] It is a corollary of the result of undecidability of the semi uni cation problem [KTU90], obtained by showing that the semi uni cation problem is polynomial time reducible to typability in ML (FIX P) Our rule (FIX 0 ) is based essentially on the same idea of the rule proposed in [Myc84] that uses a kind of transformation from in nitary to nitary polymorphism, in a similar way to ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semiuni cation problem. In Proc. of the 22nd Annual ACM Symp. on Theory of Computation (STOC), pages 468-476. ACM Press, 1990.

....which proceeds by accumulation of constraints, and where term substitutions are replaced by equality constraints. The proof of subject reduction holds independently of the computation domain, under the only assumption that the type of predicates is de nitional generic in the sense of [11] [10]. In section 5 we describe the implementation of our type system. The solving of subtype inequalities is done by an interface to the Wallace constraint handling library [15] developed initially for typing the OCaml programming language. In section 6 we report experimental results on the use of ....

....type to be member : int list(int) list(int) for satisfying the de nitional genericity condition. This notion of de nitional genericity was introduced in [11] for escaping from the undecidability results for inferring the types of predicates inside 5 mutually recursive de nitions, similarly to ML [10]. In the next section we show that it provides also a sucient condition for subject reduction in our context. The following proposition shows that if an expression is well typed in a variable typing U , it remains typable in any instance U . Proposition 1. For any variable typing U , any ....

A.J. Kfoury, J. Tiruyn, and P. Urzyczyn. The undecidability of the semi-unication problem. Technical report, BUCS-89-010, Boston Univ., October 1989.

....i p S( 0 ) is solvable (by some substitution R i ) for every inequality i p 0 2 C. Notice that for each particular index i, the set of inequalities associated with i (S( i p S( 0 ) may be solved by di erent substitutions R i . Semi uni cation is known to be undecidable [KTU93] but a practical semi decision procedure was de ned by Henglein [Hen93] So far, no natural counterexample (i.e. a program that would make the algorithm loop) is known, and the results of the present paper corroborate the practicality of the algorithm. 3.2 Algorithm The core of our constraint ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-unication problem. Information and Computation, 102(1):83-101, January 1993.

....rule for recursion than Milner and shows it to be sound. Independently, Henglein [3] and Kfoury, Tiuryn and Urzyczyn [4] show that type inference with Mycroft s rule for recursion is equivalent to the semi uni cation problem which was later shown undecidable by Kfoury, Tiuryn and Urzyczyn [6]. However, note that the type checking problem for Mycroft s rule is decidable and was used in both HOPE and the Prolog type system above. We say more about this in section 3.3. The rest of this section details these claims and may be skipped by a reader familiar with the above work. Although the ....

Kfoury, A.J., Tiuryn, J. and Urzyczyn, P. \The undecidability of the semi-unication problem", Proc. ACM Symp. on Theory of Computing, 1990.

....cation instance decide whether has a semi uni er. If contains a single pair, then we call it the uniform semi uni cation problem (USUP) The name uniform comes from the fact that it is equivalent to restricting the matching substitutions to be the same substitution. It has been shown in [ Kfoury et al. 1993 ] that SUP is undecidable. USUP, on the other hand, is decidable [ Kapur et al. 1991; Henglein, 1993; Pudlack, 1988 ] with a time complexity of O(n 2 (n) 2 ) where (n) is the inverse of Ackerman s function [ Oliart and Snyder, 1998 ] De nition 7 A substitution is the most general ....

A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semiuni cation problem. Information and Computation, 102(1):83-101, January 1993.

....is a log space computable expression of the form rec x:P such that P has no recusive denition (i.e. no declarations hu = Qi with u 2 fn(Q) and that I is semi uniable if and only if rec x:P is MM typable. Proof for Theorem 7. 1 follows since semi unication has been shown recursively undecidable [21, 36]. We show in fact that typability in the Damas like type inference system D is also undecidable Theorem 7.2 The type inference problem for D [ is undecidable. Proof To prove Theorem 7.2, it is suOEcient to prove that the semi unication problem reduces to typability in D . Suppose ....

A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-unication problem. In Proceedings of the 22nd annual ACM symposium on theory of Computation (STOC), pages 468476, New-York, 1990. ACM.

....for System F. 1.4 What does this article contribute The main contribution of this report is proving the undecidability of Typ and TC for System F. 1. We rst prove that the problem of semi uni cation can be reduced to TC using a simple encoding. Because semi uni cation is undecidable [KTU93], so is TC. 2. We then reduce TC to Typ using a novel method of building terms which simulate arbitrarily chosen type environments. The proof begins by showing that there exists a typable term J such that in every typing of J , its bound variable x is assigned the type : J ....

....is A M : 2.5 Semi Uni cation For convenience, we de ne semi uni cation using a rst order signature containing the single in x binary function symbol and for the case where there are only two pairs of terms. The general de nition of semi uni cation is reducible to this special case [Pud88, KTU93] and the proof that semi uni cation is undecidable is actually for this special case [KTU93] 8 This notion is not completely de ned because there may be multiple occurrences of a subterm N M with distinct typings. This does not pose a problem in this report because it is always clear from ....

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semiuni cation problem. Inf. & Comput., 102(1):83-101, Jan. 1993.

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-uni cation problem. Inform. & Comput., 102(1):83-101, Jan. 1993.

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Kfoury, A.J., Tiuryn, J., and Urzyczyn, P., \The undecidability of the semi-uni cation problem". Information and Computation, Vol 102, no. 1, pp 83-101, 1993.

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-uni cation problem. Inform. & Comput., 102(1):83-101, Jan. 1993.

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-uni cation problem. Inform. & Comput., 102(1):83-101, Jan. 1993.

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-uni cation problem. Inform. & Comput., 102(1):83-101, Jan. 1993.

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A. J. Kfoury, J. Tiuryn, and P. Urzyczyn. The undecidability of the semi-uni cation problem. Inform. & Comput., 102(1):83-101, Jan. 1993.

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