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Context matching for compressed terms
 In LICS 2008
, 2008
"... This paper is an investigation of the matching problem for term equations s = t where s contains context variables, and both terms s and t are given using some kind of compressed representation. In this setting, term representation with dags, but also with the more general formalism of singleton t ..."
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Cited by 8 (6 self)
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This paper is an investigation of the matching problem for term equations s = t where s contains context variables, and both terms s and t are given using some kind of compressed representation. In this setting, term representation with dags, but also with the more general formalism of singleton tree grammars, are considered. The main result is a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. NPcompleteness is obtained when the terms are represented using singleton tree grammars. The special cases of firstorder matching and also unification with STGs are shown to be decidable in PTIME. 1
Dominance Constraints in Stratified Context Unification
 INFORMATION PROCESSING LETTERS
, 2007
"... We express dominance constraints in the onceonly nesting fragment of stratified context unification, which therefore is NPcomplete. ..."
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Cited by 5 (0 self)
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We express dominance constraints in the onceonly nesting fragment of stratified context unification, which therefore is NPcomplete.
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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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.
Congruence Closure of Compressed Terms in Polynomial Time
, 2011
"... The wordproblem for a finite set of equational axioms between ground terms is the question whether for terms s, t the equation s = t is a consequence. We consider this problem under grammar based compression of terms, in particular compression with singleton tree grammars (STGs) and with directed ..."
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The wordproblem for a finite set of equational axioms between ground terms is the question whether for terms s, t the equation s = t is a consequence. We consider this problem under grammar based compression of terms, in particular compression with singleton tree grammars (STGs) and with directed acyclic graphs (DAGs) as a special case. We show that given a DAGcompressed ground and reduced term rewriting system T, the Tnormal form of an STGcompressed term s can be computed in polynomial time, and hence the Tword problem can be solved in polynomial time. This implies that the word problem of STGcompressed terms w.r.t. a set of DAGcompressed ground equations can be decided in polynomial time. If the ground term rewriting system (gTRS) T is STGcompressed, we show NPhardness of Tnormalform computation. For compressed, reduced gTRSs we show a PSPACE upper bound on the complexity of the normal form computation of STGcompressed terms. Also special cases are considered and a prototypical implementation is presented.
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.
On the complexity of Bounded SecondOrder Unification and Stratified Context Unification
"... Bounded SecondOrder Unification is a decidable variant of undecidable SecondOrder Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a longstanding open problem. This paper is a join of two separate previous, preliminary papers on ..."
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Bounded SecondOrder Unification is a decidable variant of undecidable SecondOrder Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a longstanding open problem. This paper is a join of two separate previous, preliminary papers on NPcompleteness of Bounded SecondOrder Unification and Stratified Context Unification. It clarifies some omissions in these papers, joins the algorithmic parts that construct a minimal solution, and gives a clear account of a method of using singleton tree grammars for compression that may have potential usage for other algorithmic questions in related areas.
On the Limits of SecondOrder Unification
"... SecondOrder Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Tw ..."
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SecondOrder Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Two variants of the problem, Bounded SecondOrder Unification and Linear SecondOrder Unification –where the use of bound variables in the instantiations is restricted–, have been extensively studied in the last two decades. In this paper we summarize some decidability/undecidability/complexity results, trying to focus on those that could be more interesting for a wider audience, and involving less technical details. 1