W. D. Goldfarb. The undecidability of the secondorder unification problem. Theoretical Computer Science, 13:225--230, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of second order unification and also a generalization of string unification. There are unification procedures for the more general problem of higher order unification (see e.g. Pie73,Hue75,SG89,Pre95] It is well known that higher order unification and second order unification are undecidable [Gol81,Far91,LV00]. String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in EXPSPACE [Gut98] in NEXPTIME [Pla99a] and even in PSPACE [Pla99b] Context unification problems are restricted second order unification problems. Context variables represent ....

W.D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

....# ] # :#.#s:s # (s # ) T ] holds if and only if there exists pure M : # such that the types #[M s] and # # [M s] are equal. Thus, deciding subsignature or equivalence queries in the presence of existentials would be as hard as higherorder unification, which is known to be undecidable [10]. 4.2 Syntactic Principal Signatures It has been argued for reasons related to separate compilation that principal signatures should be expressible in the syntax available to the programmer. This provides the strongest support for separate compilation, because a programmer can break a program at ....

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225-- 230, 1981.

....unification but not always terminating. The decidability of linear second order unification is open but as for context unification, three decidable fragments are known [12] Note also that linear second order unification is a subproblem of second order unification, which is undecidable [7] but only a fragment of higher order unification [19, 8] Ellipses in Natural Language. The motivation of the authors for the investigation of equality up to constraints stems from the area of semantic processing of natural language. This line of research started with higher order unification [5, ....

W. D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

....: # :#.#s:s # (s # ) T ] holds if and only if there exists determinate M : # such that the types # [M s] and # # [M s] are equal. Thus, deciding subsignature or equivalence queries in the presence of existentials would be as hard as higher order unification, which is known to be undecidable [8]. We have explored a variety of alternative formalizations of primitive singletons as well, and none has led to a type system we have been able to prove decidable. 5 Other Issues 5.1 Modules as First Class Values It is desirable for modules to be usable as first class values. This is useful to ....

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

....operators. Theorem proving is undecidable, but it is unfortunate that each resolution step is undecidable. We can recover decidability by restricting unification. Limiting the search gives unpredictable results. Second order matching is decidable [18] though second order unification is not [11]. Perhaps first order unification plus second order matching is a practical compromise. Ketonen s EKL proves theorems using first order unification plus higher order matching. Ketonen claims that higher order matching is decidable, without proof [20] Huet tells me that decidability is an open ....

W. D. Goldfarb, The undecidability of the second-order unification problem, Theoretical Computer Science 13 (1981), pages 225--230.

....# ] # :#.#s:s # (s # ) T ] holds if and only if there exists pure M : # such that the types # [M s] and # # [M s] are equal. Thus, deciding subsignature or equivalence queries in the presence of existentials would be as hard as higher order unification, which is known to be undecidable [9]. We have explored a variety of alternative formalizations of primitive singletons as well, and none has led to a type system we have been able to prove decidable. 4.2 Syntactic Principal Signatures It has been argued for reasons related to separate compilation that principal signatures should ....

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

....# ] # :s.Ps:s s (s # ) T ] holds if and only if there exists pure M : s such that the types t[M s] and t # [M s] are equal. Thus, deciding subsignature or equivalence queries in the presence of existentials would be as hard as higherorder unification, which is known to be undecidable [10]. 4.2 Syntactic Principal Signatures It has been argued for reasons related to separate compilation that principal signatures should be expressible in the syntax available to the programmer. This provides the strongest support for separate compilation, because a programmer can break a program at ....

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225-- 230, 1981.

....Dowek s survey article [4] Higher order unification has first been considered by Huet [11] Huet showed that the general problem is undecidable and gave a semi algorithm (which may not terminate) for its solution. The undecidability result was later strengthened to include second order terms, too [8, 7]. Miller [20] defined higher order patterns, for which the unification problem is decidable and unitary, regardless of the order of the terms. In the monadic case, where all constant symbols are unary, unification is decidable for second order terms [6] but undecidable at higher orders [26] ....

W. D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Comput. Sci., 13(2):225--230, Feb. 1981.

....Recently, Miller and Nadathur [1987] have recasted this idea in a logic programming language called Prolog. To be complete, such a system must incorporate a higher order unification algorithm [Huet, 1975] Unfortunately, higher order unification is undecidable [Huet, 1973; Goldfarb, 1981] and many possible solutions have to be taken into account if two terms are to be unified whose initial symbols are function variables. Besides an ongoing discussion of whether higher order extensions of Prolog are needed [Warren, 1982] this last problem leads us to consider a first order ....

F. Goldfarb. The undecidability of the second-order unification problem. Journal of Theoretical Computer Science, 13:225--230, 1981.

....that we still have to apply a placeholder to the empty list. 2 A free occurrence of a parameter in another parameter type does not generally guarantee a successful elaboration. One reason for this is that argument synthesis is based on unification, which is not decidable in the higher order case [Gol81]. 3 This rules out the inference of the type argument of nil . M. Luther sometimes does not hide arguments at marked positions, even when they are inferable. Both defects result from di#culties to define reduction for a language with forced arguments properly since uniqueness of elaboration ....

W. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

....also explains why we need to define a tree equation in terms of ordered trees. Actually our Open Problem generalizes at the same time both First Order Unification and Word Unification problems and is a particular case of Second Order Unification. The latter problem is in general undecidable (cf. [18]) However we do not know if Second Order Unification becomes decidable for this particular subclass of equations. In fact, the proof of the undecidability of Second Order Unification given in [18] cannot be extended directly to our case. This shows the relevance of the solvability problem of tree ....

....is a particular case of Second Order Unification. The latter problem is in general undecidable (cf. 18] However we do not know if Second Order Unification becomes decidable for this particular subclass of equations. In fact, the proof of the undecidability of Second Order Unification given in [18] cannot be extended directly to our case. This shows the relevance of the solvability problem of tree equations. The Open Problem can be generalized to the one of finding solution over more general sets of graded trees (A; ff) # , as well as to equations with constants. As to concern the search ....

W. Goldfarb. The Undecidability of the Second-Order Unification Problem. Theoretical Computer Science No. 13, pagg. 225--230, 1981.

....(UP) hE=V (E)i for (second order) terms, simply denoted hEi. Intuitively W describes the set of variables substitution is restricted to, whereas W c : V (E) n W contains the protected variables where substitution is not allowed. Second order unification is in general undecidable as shown in [8] 3 , but of course the set of unifiers for a given problem is recursively enumerable. Next we give some basic definitions that are important for finite descriptions of unifiers sets. These definitions are a straightforward generalization for RUP s of the ones given in [24] for ordinary UP s. ....

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

....of expressing preferences between offers. Offers are formulae in , hence the most obvious way of representing preferences between formulae would be as a second order relation in . However, this would mean that would be a higherorder logic, with the associated computational problems of such logics [6]. As a result we prefer to express preferences as a meta language with the following minimum requirements: 1. To represent formulae in as terms in . 2. To express preferences between formulae in . For example, given the sentences , and in , we can express a preference for the first over the ....

W. D. Goldfarb. The undecidability of the second-order unification problem. , 13:225--230, 1981.

....theory is not straightforward. The reason is twofold: the size of logical terms in proofs (with cuts) cannot be bounded by the number of lines n of the proof, and the problem of substitution of terms in proofs can be reduced to the second order unification problem which is known to be undecidable [Gol81]. References related to undecidability questions on the number of lines in proofs are [Bus91, KP88] 3 Distortion in Groups We present a few examples with the aim to illustrate the notion of distortion for finitely presented groups. We take them from [Gro93] where the notion and its Riemannian ....

W.D. Goldfarb. The undecidability of the second-order unification problem. In Theoretical Computer Science, 13:225--230, 1981.

....unification and also a generalization of string unification. There are unification procedures for the more general problem of higher order unification (see e.g. Pie73,Hue75,SG89,Wol93,Pre95] It is well known that general higher order unification and second order unification are undecidable [Gol81,Far91,LV99] and that string unification is decidable [Mak77] Recent upper complexity estimations for string unification are NEXPTIME [Pla99a] and PSPACE [Pla99b] Context unification problems are restricted second order unification problems: context variables represent terms with exactly one hole in ....

W.D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

No context found.

W. D. Goldfarb. The undecidability of the secondorder unification problem. Theoretical Computer Science, 13:225--230, 1981.

No context found.

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

No context found.

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

No context found.

Warren D. Goldfarb. The undecidability of the secondorder unification problem. Theoretical Computer Science, 13:225--230, 1981.

No context found.

Warren D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

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Goldfarb, W. D. The undecidability of the second-order unification problem. Theoretical Computer Science 13 (1981) 225 -- 230.

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Goldfarb, W.D., "The undecidability of the second-order unification problem". Theoretical Computer Science, Vol 13, pp 225-230, 1981.

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W. D. Goldfarb. The undecidability of the secondorder unification problem. Theoretical Computer Science, 13:225--230, 1981.

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W. D. Goldfarb, The undecidability of the second-order unification problem, Theoretical Comput. Sci., 13 (1981), pp. 225--230.

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W.D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225--230, 1981.

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