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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.
Solvability of context equations with two context variables is decidable
 THE JOURNAL OF SYMBOLIC COMPUTATION
, 1999
"... Context unification is a natural variant of second order unification that represents a generalization of word unification at the same time. While second order unification is wellknown to be undecidable and word unification is decidable it is currently open if solvability of context equations is deci ..."
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Cited by 25 (2 self)
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Context unification is a natural variant of second order unification that represents a generalization of word unification at the same time. While second order unification is wellknown to be undecidable and word unification is decidable it is currently open if solvability of context equations is decidable. We show that solvability of systems of context equations with two context variables is decidable. The context variables may have an arbitrary number of occurrences, and the equations may contain an arbitrary number of individual variables as well. The result holds under the assumption that the first order background signature is finite.
A decision algorithm for stratified context unification
 FACHBEREICH INFORMATIK, J.W. GOETHEUNIVERSITAT
, 1999
"... Context unification is a variant of second order unification and also a generalization of string unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification. Recently, it turned out that stratified context ..."
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Cited by 17 (1 self)
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Context unification is a variant of second order unification and also a generalization of string unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification. Recently, it turned out that stratified context unification and onestep rewrite constraints are equivalent. This paper contains a description of a decision algorithm SCU for stratified context unification, which shows decidability of stratified context unification as well as of satisfiability of onestep rewrite constraints.
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
Dominance Constraints in Context Unification
, 1998
"... Tree descriptions based on dominance constraints are popular in several areas of computational linguistics including syntax, semantics, and discourse. Tree descriptions in the language of context unification have attracted some interest in unification and rewriting theory. Recently, dominance constr ..."
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Cited by 14 (10 self)
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Tree descriptions based on dominance constraints are popular in several areas of computational linguistics including syntax, semantics, and discourse. Tree descriptions in the language of context unification have attracted some interest in unification and rewriting theory. Recently, dominance constraints and context unification have both been used in different underspecified approaches to the semantics of scope, parallelism, and their interaction. This raises the question whether both description languages are related. In this paper, we show for a first time that dominance constraints can be expressed in context unification. We also prove that dominance constraints extended with parallelism constraints are equal in expressive power to context unification.
BetaReduction As Unification
, 1996
"... this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of terms in . Quite apart from the new light it sheds on fireduction, such an analysis turns out to have several othe ..."
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Cited by 13 (9 self)
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this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of terms in . Quite apart from the new light it sheds on fireduction, such an analysis turns out to have several other benefits
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 6 (3 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.
Decidability of Bounded Second Order Unification
 FB INFORMATIK, J.W. GOETHEUNIVERSITAT FRANKFURT AM MAIN
, 1999
"... It is wellknown that first order unification is decidable, whereas second order (and higherorder) unification is undecidable. Bounded second order unification (BSOU) is second order unification under the restriction that only a bounded number of holes in the instantiating terms for second order va ..."
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Cited by 6 (2 self)
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It is wellknown that first order unification is decidable, whereas second order (and higherorder) unification is undecidable. Bounded second order unification (BSOU) is second order unification under the restriction that only a bounded number of holes in the instantiating terms for second order variables is permitted, however, the size of the instantiation is not restricted. In this paper, a decision algorithm for bounded second order unification is described. This is the first nontrivial decidability result for second order unification, where the (finite) signature is not restricted and there are no restrictions on the occurrences of variables. We show that the monadic second order unification (MSOU), a specialization of BSOU is in \Sigma p 2. Since MSOU is related to word unification, this is compares favourably to the best known upper bound NEXPTIME (and also to the announced upper bound PSPACE) for word unification. This supports the claim that bounded second order unification is easier than context unification, whose decidability is currently an open question.
An Optimized Decision Algorithm for Stratified Context Unification
, 2000
"... Context unification is a variant of second order unification. It can also be seen as a generalization of string unification to tree unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification, which is ..."
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Context unification is a variant of second order unification. It can also be seen as a generalization of string unification to tree unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification, which is decidable. However, the previous algorithm has a very bad worst case complexity. Recently it turned out that stratified context unification is equivalent to satisfiability of onestep rewrite constraints. This paper contains an optimized algorithm for stratified context unification exploiting sharing and power expressions. We prove that the complexity is determined mainly by the maximal depth of SOcycles. Two observations are used: i. For every ambiguous SOcycle, there is a context variable that can be instantiated with a ground context of main depth O(c d), where c is the number of context variables, and d is the depth of the SOcycle. ii. the exponent of periodicity is O(2 ...