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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.
Linear SecondOrder Unification and Context Unification with TreeRegular Constraints
 Proc. of the 11th Int. Conference on Rewriting Techniques and Applications (RTA’2000), volume 1833 of LNCS
, 2000
"... Linear SecondOrder Unification and Context Unification are closely related problems. However, their equivalence was never formally proved. Context unification is a restriction of linear secondorder unification. Here we prove that linear secondorder unification can be reduced to context unificatio ..."
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Cited by 12 (3 self)
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Linear SecondOrder Unification and Context Unification are closely related problems. However, their equivalence was never formally proved. Context unification is a restriction of linear secondorder unification. Here we prove that linear secondorder unification can be reduced to context unification with treeregular constraints. Decidability of context unification is still an open question. We comment on the possibility that linear secondorder unification is decidable, if context unification is, and how to get rid of the treeregular constraints. This is done by reducing rankbound treeregular constraints to wordregular constraints.
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...
Context unification and traversal equations
 In: Proc. of the 12th International Conference on Rewriting Techniques and Applications (RTA’01
, 2001
"... Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing firstorder variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are secon ..."
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Cited by 8 (7 self)
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Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing firstorder variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are secondorder variables that are restricted to be instantiated by linear terms (a linear term is a λexpression λx1 ···λxn.t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rankbound conjecture is true, then the context unification problem is decidable. This is done reducing context unification to solvability of traversal equations (a kind of word unification modulo certain permutations) and then, reducing traversal equations to word equations with regular constraints. 1
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
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.
Rigid Reachability
 Proc. 4th Asian Computing Science Conference, LNCS 1538
, 1998
"... . We show that rigid reachability, the nonsymmetric form of rigid Eunification, is undecidable already in the case of a single constraint. From this we infer the undecidability of a new rather restricted kind of secondorder unification. We also show that certain decidable subclasses of the proble ..."
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Cited by 5 (4 self)
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. We show that rigid reachability, the nonsymmetric form of rigid Eunification, is undecidable already in the case of a single constraint. From this we infer the undecidability of a new rather restricted kind of secondorder unification. We also show that certain decidable subclasses of the problem which are Pcomplete in the equational case become EXPTIMEcomplete when symmetry is absent. By applying automatatheoretic methods, simultaneous monadic rigid reachability with ground rules is shown to be in EXPTIME. 1 Introduction Rigid reachability is the problem, given a rewrite system R and two terms s and t, whether there exists a ground substitution # such that s# rewrites in some number of steps via R# into t#. The term "rigid" stems from the fact that for no rule more than one instance can be used in the rewriting process. Simultaneous rigid reachability is the problem in which a substitution is sought which simultaneously solves each member of a system of reachability constra...
THE COMPLEXITY OF MONADIC SECONDORDER UNIFICATION ∗
, 1113
"... Abstract. Monadic secondorder unification is secondorder unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NPcomplete, where we use the technique of compressing solutions us ..."
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Cited by 4 (1 self)
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Abstract. Monadic secondorder unification is secondorder unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NPcomplete, where we use the technique of compressing solutions using singleton contextfree grammars. We prove that monadic secondorder matching is also NPcomplete.
Simplifying the signature in secondorder unification
, 2009
"... SecondOrder Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all symbols are monadic, then the problem is NPcomplete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce SecondO ..."
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SecondOrder Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all symbols are monadic, then the problem is NPcomplete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce SecondOrder Unification to SecondOrder Unification with a signature that contains just one binary function symbol and constants. The reduction is based on partially currying the equations by using the binary function symbol for explicit application @. Our work simplifies the study of SecondOrder Unification and some of its variants, like Context Unification and Bounded SecondOrder Unification.