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31
Unification: A multidisciplinary survey
 ACM Computing Surveys
, 1989
"... The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the ..."
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Cited by 106 (0 self)
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The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the paper include examples that highlight particular uses
Partial polymorphic type inference and higherorder unification
 IN PROCEEDINGS OF THE 1988 ACM CONFERENCE ON LISP AND FUNCTIONAL PROGRAMMING, ACM
, 1988
"... We show that the problem of partial type inference in the nthborder polymorphic Xcalculus is equivalent to nthorder unification. On the one hand, this means that partial type inference in polymorphic Xcalculi of order 2 or higher is undecidable. On the other hand, higherorder unification is oft ..."
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Cited by 83 (8 self)
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We show that the problem of partial type inference in the nthborder polymorphic Xcalculus is equivalent to nthorder unification. On the one hand, this means that partial type inference in polymorphic Xcalculi of order 2 or higher is undecidable. On the other hand, higherorder unification is often tractable in practice, and our translation entails a very useful algorithm for partial type inference in the worder polymorphic Xcalculus. We present an implementation in AProlog in full.
On the Undecidability of SecondOrder Unification
 INFORMATION AND COMPUTATION
, 2000
"... ... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most t ..."
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Cited by 33 (16 self)
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... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two secondorder variables; (2) there is only one secondorder variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all secondorder variables are ground terms of size <n, (ii) the arity of all secondorder variables is <n, (iii) the number of occurrences of secondorder variables is #5, (iv) there is either a single secondorder variable or there are two secondorder variables and no firstorder variables.
Birewrite systems
, 1996
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations ..."
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Cited by 29 (9 self)
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→
Extensions and Applications of Higherorder Unification
, 1990
"... ... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The ric ..."
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Cited by 25 (1 self)
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... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The rich structure of a typedcalculus,asopposedtotraditional,rst generalideaistouseacalculusasametalanguageforrepresentingvariousotherlanguages thelattercase,thealgorithmisincomplete,thoughstillquiteusefulinpractice. Thelastpartofthedissertationprovidesexamplesoftheusefulnessofthealgorithms.The algorithmrstfordependentproduct()types,andsecondforimplicitpolymorphism.In involvessignicantcomplicationsnotarisingHuet'scorrespondingalgorithmforthesimply orderabstractsyntaxtrees,allowsustoexpressrules,e.g.,programtransformationand typedcalculus,primarilybecauseitmustdealwithilltypedterms.Wethenextendthis Wecanthenuseunicationinthemetalanguagetomechanizeapplicationoftheserules.
Decidable higherorder unification problems
 AUTOMATED DEDUCTION  CADE12. SPRINGER LNAI 814
, 1994
"... Secondorder unification is undecidable in general. Miller showed that unification of socalled higherorder patterns is decidable and unitary. Weshow that the unification of a linear higherorder pattern s with an arbitrary secondorder term that shares no variables with s is decidable and finitar ..."
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Cited by 17 (4 self)
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Secondorder unification is undecidable in general. Miller showed that unification of socalled higherorder patterns is decidable and unitary. Weshow that the unification of a linear higherorder pattern s with an arbitrary secondorder term that shares no variables with s is decidable and finitary. A few extensions of this unification problem are still decidable: unifying two secondorder terms, where one term is linear, is undecidable if the terms contain bound variables but decidable if they don't.
Decidable and undecidable secondorder unification problems
 In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications (RTA’98), volume 1379 of LNCS
, 1998
"... Abstract. There is a close relationship between word unification and secondorder unification. This similarity has been exploited for instance for proving decidability of monadic secondorder unification. Word unification can be easily decided by transformation rules (similar to the ones applied in ..."
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Cited by 15 (9 self)
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Abstract. There is a close relationship between word unification and secondorder unification. This similarity has been exploited for instance for proving decidability of monadic secondorder unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higherorder unification procedures) when variables are restricted to occur at most twice. Hence a wellknown open question was the decidability of secondorder unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid Eunification to secondorder unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid Eunification is decidable. This happens, for example, for onevariable problems where the variable occurs at most twice (because rigid Eunification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable secondorder unification problems with variable occurrence restrictions. 1
Tractable and Intractable SecondOrder Matching Problems
 In Proc. 5th Ann. Int. Computing and Combinatorics Conference (COCOON'99), LNCS 1627
, 1999
"... . The secondorder matching problem is the problem of determining, for a finite set {#t i , s i #  i # I} of pairs of a secondorder term t i and a firstorder closed term s i , called a matching expression, whether or not there exists a substitution # such that t i # = s i for each i # I ..."
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Cited by 10 (2 self)
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. The secondorder matching problem is the problem of determining, for a finite set {#t i , s i #  i # I} of pairs of a secondorder term t i and a firstorder closed term s i , called a matching expression, whether or not there exists a substitution # such that t i # = s i for each i # I . It is wellknown that the secondorder matching problem is NPcomplete. In this paper, we introduce the following restrictions of a matching expression: kary, kfv , predicate, ground , and functionfree. Then, we show that the secondorder matching problem is NPcomplete for a unary predicate, a unary ground, a ternary functionfree predicate, a binary functionfree ground, and an 1fv predicate matching expressions, while it is solvable in polynomial time for a binary functionfree predicate, a unary functionfree, a kfv functionfree (k # 0), and a ground predicate matching expressions. 1 Introduction The unification problem is the problem of determining whether or not any two ter...
On Unification Problems in Restricted SecondOrder Languages
 In Annual Conf. of the European Ass. of Computer Science Logic (CSL98
, 1998
"... We review known results and improve known boundaries between the decidable and the undecidable cases of secondorder unification with various restrictions on secondorder variables. As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous ..."
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Cited by 7 (4 self)
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We review known results and improve known boundaries between the decidable and the undecidable cases of secondorder unification with various restrictions on secondorder variables. As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous rigid Eunification.
Monadic secondorder unification is NPcomplete
 In RTA’04, volume 3091 of LNCS
, 2004
"... Abstract. Bounded SecondOrder Unification is the problem of deciding, for a given secondorder equation t? = u and a positive integer m, whether there exists a unifier σ such that, for every secondorder variable F, the terms instantiated for F have at most m occurrences of every bound variable. I ..."
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Cited by 7 (5 self)
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Abstract. Bounded SecondOrder Unification is the problem of deciding, for a given secondorder equation t? = u and a positive integer m, whether there exists a unifier σ such that, for every secondorder variable F, the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded SecondOrder Unification is decidable and NPhard, whereas general SecondOrder Unification is undecidable. We prove that Bounded SecondOrder Unification is NPcomplete, provided that m is given in unary encoding, by proving that a sizeminimal solution can be represented in polynomial space, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. 1