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12
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 12 (8 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
Stratified context unification is npcomplete
 In Proc. of the 3rd International Joint Conference on Automated Reasoning, IJCAR’06
, 2006
"... Abstract. Context Unification is the problem to decide for a given set of secondorder equations E where all secondorder variables are unary, whether there exists a unifier, such that for every secondorder variable X, theabstractionλx.r instantiated for X has exactly one occurrence of the bound va ..."
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Cited by 11 (3 self)
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Abstract. Context Unification is the problem to decide for a given set of secondorder equations E where all secondorder variables are unary, whether there exists a unifier, such that for every secondorder variable X, theabstractionλx.r instantiated for X has exactly one occurrence of the bound variable x in r. Stratified Context Unification is a specialization where the nesting of secondorder variables in E is restricted. It is already known that Stratified Context Unification is decidable, NPhard, and in PSPACE, whereas the decidability and the complexity of Context Unification is unknown. We prove that Stratified Context Unification is in NP by proving that a sizeminimal solution can be represented in a singleton tree grammar of polynomial size, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. This also demonstrates the high potential of singleton tree grammars for optimizing programs maintaining large terms. A corollary of our result is that solvability of rewrite constraints is NPcomplete. 1
Polynomial equality testing for terms with shared substructures. Frank report 21, Institut für Informatik
 FB Informatik und Mathematik. J. W. GoetheUniversität Frankfurt am Main
, 2005
"... Abstract. Sharing of substructures like subterms and subcontexts in terms is a common method for spaceefficient representation of terms, which allows for example to represent exponentially large terms in polynomial space, or to represent terms with iterated substructures in a compact form. We prese ..."
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Cited by 11 (4 self)
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Abstract. Sharing of substructures like subterms and subcontexts in terms is a common method for spaceefficient representation of terms, which allows for example to represent exponentially large terms in polynomial space, or to represent terms with iterated substructures in a compact form. We present singleton tree grammars as a general formalism for the treatment of sharing in terms. Singleton tree grammars (STG) are recursionfree contextfree tree grammars without alternatives for nonterminals and at most unary secondorder nonterminals. STGs generalize Plandowski’s singleton context free grammars to terms (trees). We show that the test, whether two different nonterminals in an STG generate the same term can be done in polynomial time, which implies that the equality test of terms with shared terms and contexts, where composition of contexts is permitted, can be done in polynomial time in the size of the representation. This will allow polynomialtime algorithms for terms exploiting sharing. We hope that this technique will lead to improved upper complexity bounds for variants of second order unification algorithms, in particular for variants of context unification and bounded second order unification.
Parameter reduction in grammarcompressed trees
 In 12th FoSSaCS, volume 5504 of LNCS
, 2009
"... Abstract. Trees can be conveniently compressed with linear straightline contextfree tree grammars. Such grammars generalize straightline contextfree string grammars which are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression) ..."
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Abstract. Trees can be conveniently compressed with linear straightline contextfree tree grammars. Such grammars generalize straightline contextfree string grammars which are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression). It is shown that every linear straightline contextfree tree grammar can be transformed in polynomial time into a monadic (and linear) one. A tree grammar is monadic if each nonterminal uses at most one context parameter. Based on this result, a polynomial time algorithm is presented for testing whether a given nondeterministic tree automaton with sibling constraints accepts a tree given by a linear straightline contextfree tree grammar. It is shown that if tree grammars are nondeterministic or nonlinear, then reducing their numbers of parameters cannot be done without an exponential blowup in grammar size. 1
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.
Pattern Matching of Compressed Terms and Contexts and Polynomial Rewriting
, 2011
"... A generalization of the compressed string pattern match that applies to terms with variables is investigated: Given terms s and t compressed by singleton tree grammars, the task is to find an instance of s that occurs as a subterm in t. We show that this problem is in NP and that the task can be pe ..."
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Cited by 3 (1 self)
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A generalization of the compressed string pattern match that applies to terms with variables is investigated: Given terms s and t compressed by singleton tree grammars, the task is to find an instance of s that occurs as a subterm in t. We show that this problem is in NP and that the task can be performed in time O(n cVar(s) ), including the construction of the compressed substitution, and a representation of all occurrences. We show that the special case where s is uncompressed can be performed in polynomial time. As a nice application we show that for an equational deduction of t to t ′ by an equality axiom l = r (a rewrite) a single step can be performed in polynomial time in the size of compression of t and l, r if the number of variables is fixed in l. We also show that n rewriting steps can be performed in polynomial time, if the equational axioms are compressed and assumed to be constant for the rewriting sequence. Another potential application are querying mechanisms on compressed XMLdata bases.
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