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15
Conjunctive Queries over Trees
, 2004
"... We study the complexity and expressive power of conjunctive queries over unranked labeled trees, where the tree structures are represented using "axis relations" such as "child", "descendant", and "following" (we consider a superset of the XPath axes) as well ..."
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Cited by 69 (7 self)
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We study the complexity and expressive power of conjunctive queries over unranked labeled trees, where the tree structures are represented using "axis relations" such as "child", "descendant", and "following" (we consider a superset of the XPath axes) as well as unary relations for node labels. (Cyclic) conjunctive queries over trees occur in a wide range of data management scenarios related to XML, the Web, and computational linguistics. We establish a framework for characterizing structures representing trees for which conjunctive queries can be evaluated e# ciently. Then we completely chart the tractability frontier of the problem for our axis relations, i.e., we find all subsetmaximal sets of axes for which query evaluation is in polynomial time. All polynomialtime results are obtained immediately using the proof techniques from our framework. Finally, we study the expressiveness of conjunctive queries over trees and compare it to the expressive power of fragments of XPath. We show that for each conjunctive query, there is an equivalent acyclic positive query (i.e., a set of acyclic conjunctive queries), but that in general this query is not of polynomial size.
Context Sequence Matching for XML
, 2005
"... Context and sequence variables allow matching to explore termtrees both in depth and in breadth. It makes context sequence matching a suitable computational mechanism for a rulebased language to query and transform XML, or to specify and verify web sites. Such a language would have advantages of b ..."
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Cited by 10 (5 self)
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Context and sequence variables allow matching to explore termtrees both in depth and in breadth. It makes context sequence matching a suitable computational mechanism for a rulebased language to query and transform XML, or to specify and verify web sites. Such a language would have advantages of both pathbased and patternbased languages. We develop a context sequence matching algorithm and its extension for regular expression matching, and prove their soundness, termination and completeness properties.
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 8 (2 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
Matching with Regular Constraints
 SUTCLIFFE G., VORONKOV A., Eds., Proceedings of LPAR’05
, 2005
"... We describe a sound, terminating, and complete matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. Context and sequence variables allow matching to move in term trees to arbitrary depth and breadth, respectively. The ..."
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Cited by 7 (7 self)
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We describe a sound, terminating, and complete matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. Context and sequence variables allow matching to move in term trees to arbitrary depth and breadth, respectively. The values of variables can be constrained by regular expressions which are not necessarily linear. We describe heuristics for optimization, and discuss applications.
Flat matching
 Journal of Symbolic Computation
"... We study matching in flat theories both from theoretical and practical points of view. A flat theory is defined by the axiom f(x, f(y), z). = f(x, y, z) that indicates that nested occurrences of the function symbol f can be flattened out. From the theoretical side, we design a procedure to solve a s ..."
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Cited by 5 (3 self)
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We study matching in flat theories both from theoretical and practical points of view. A flat theory is defined by the axiom f(x, f(y), z). = f(x, y, z) that indicates that nested occurrences of the function symbol f can be flattened out. From the theoretical side, we design a procedure to solve a system of flat matching equations and prove its soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite. The procedure enumerates this set and stops if it is finite. We identify a class of problems on which the procedure stops. From the practical point of view, we look into restrictions of the procedure that give an incomplete terminating algorithm. From this perspective, we give a set of rules that, in our opinion, describes the precise semantics for the flat matching algorithm implemented in the Mathematica system. 1.
Sequence unification through currying
 IN: PROCEEDINGS OF THE 18TH INTERNATIONAL CONFERENCE ON REWRITING TECHNIQUES AND APPLICATIONS, RTA’07, LNCS
, 2007
"... Sequence variables play an interesting role in unification and matching when dealing with terms in an unranked signature. Sequence Unification generalizes Word Unification and seems to be appealing for information extraction in XML documents, program transformation, and rulebased programming. In th ..."
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Cited by 4 (3 self)
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Sequence variables play an interesting role in unification and matching when dealing with terms in an unranked signature. Sequence Unification generalizes Word Unification and seems to be appealing for information extraction in XML documents, program transformation, and rulebased programming. In this work we study a relation between Sequence Unification and another generalization of Word Unification: Context Unification. We introduce a variant of Context Unification, called LeftHole Context Unification that
On the complexity of higherorder matching in the linear λcalculus
 in « International Conference on Rewriting Techniques and Applications  RTA’2003, Valencia, Spain », series Lecture notes in Computer Science
, 2003
"... Abstract. We prove that linear secondorder matching in the linear λcalculus with linear occurrences of the unknowns is NPcomplete. This result shows that context matching and secondorder matching in the linear λcalculus are, in fact, two different problems. 1 ..."
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Cited by 2 (0 self)
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Abstract. We prove that linear secondorder matching in the linear λcalculus with linear occurrences of the unknowns is NPcomplete. This result shows that context matching and secondorder matching in the linear λcalculus are, in fact, two different problems. 1
Solving, Reasoning, and Programming in Common Logic
"... Abstract. Common Logic (CL) is a recent ISO standard for exchanging logicbased information between disparate computer systems. Sharing and reasoning upon knowledge represented in CL require equation solving over terms of this language. We study computationally wellbehaved fragments of such solving ..."
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Cited by 1 (1 self)
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Abstract. Common Logic (CL) is a recent ISO standard for exchanging logicbased information between disparate computer systems. Sharing and reasoning upon knowledge represented in CL require equation solving over terms of this language. We study computationally wellbehaved fragments of such solving problems and show how they can influence reasoning in CL and transformations of CL expressions. 1
On the Relation Between Context and Sequence Unification
"... Both Sequence and Context Unification generalize the same problem: Word Unification. Besides that, Sequence Unification solves equations between unranked terms involving sequence variables, and seems to be appealing for information extraction in XML documents, program transformation, knowledge repre ..."
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Both Sequence and Context Unification generalize the same problem: Word Unification. Besides that, Sequence Unification solves equations between unranked terms involving sequence variables, and seems to be appealing for information extraction in XML documents, program transformation, knowledge representation, and rulebased programming. It is decidable. Context Unification deals with the same problem for ranked terms involving context variables, and has applications in computational linguistics and program transformation. Its decidability is a longstanding open question. In this work we study a relation between these two problems. We introduce a variant (restriction) of Context Unification, called LeftHole Context Unification (LHCU), to which Sequence Unification is Preduced: We define a partial currying procedure to translate sequence unification problems into lefthole context unification problems, and prove soundness of the translation. Furthermore, a precise characterization of the shape of the unifiers allows us to easily reduce LeftHole Context Unification to (the decidable problem of) Word Unification with Regular Constraints, obtaining then a new decidability proof for Sequence Unification. Finally, we define an extension of Sequence Unification (ESU) and, closing the circle, prove the inter Preducibility of LHCU and ESU.