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13
Wellnested context unification
 In CADE 2005, LNCS 3632
"... Abstract. Context unification (CU) is the open problem of solving context equations for trees. We distinguish a new decidable variant of CU– wellnested CU – and present a new unification algorithm that solves wellnested context equations in nondeterministic polynomial time. We show that minimal w ..."
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Cited by 14 (8 self)
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Abstract. Context unification (CU) is the open problem of solving context equations for trees. We distinguish a new decidable variant of CU– wellnested CU – and present a new unification algorithm that solves wellnested context equations in nondeterministic polynomial time. We show that minimal wellnested solutions of context equations can be composed from the material present in the equation (see Theorem 1). This property is wishful when modeling natural language ellipsis in CU. 1
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.
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
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.
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 7 (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
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
Parallelism and Tree Regular Constraints
"... Abstract. Parallelism constraints are logical descriptions of trees. Parallelism constraints subsume dominance constraints and are equal in expressive power to context unification. Parallelism constraints belong to the constraint language for lambda structures (CLLS) which serves for modeling natura ..."
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Cited by 1 (1 self)
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Abstract. Parallelism constraints are logical descriptions of trees. Parallelism constraints subsume dominance constraints and are equal in expressive power to context unification. Parallelism constraints belong to the constraint language for lambda structures (CLLS) which serves for modeling natural language semantics. In this paper, we investigate the extension of parallelism constraints by tree regular constraints. This canonical extension is subsumed by the monadic secondorder logic over parallelism constraints. We analyze the precise expressiveness of this extension on basis of a new relationship between tree automata and logic. Our result is relevant for classifying different extensions of parallelism constraints, as in CLLS. Finally, we prove that parallelism constraints and context unification remain equivalent when extended with tree regular constraints.
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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Cited by 1 (1 self)
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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.
Bounded higherorder unification using regular terms. In EPiC, 2013. To appear. G S Makanin. The problem of solvability of equations in a free semigroup
 Math. USSRSbornik
, 1977
"... We present a procedure for the bounded unification of higherorder terms [22]. The procedure extends G. P. Huet’s preunification procedure [11] with rules for the generation and folding of regular terms. The concise form of the procedure allows the reuse of the preunification correctness proof. Fur ..."
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Cited by 1 (1 self)
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We present a procedure for the bounded unification of higherorder terms [22]. The procedure extends G. P. Huet’s preunification procedure [11] with rules for the generation and folding of regular terms. The concise form of the procedure allows the reuse of the preunification correctness proof. Furthermore, the regular terms can be restricted in order to decide the unifiability problem. Finally, the procedure avoids recomputation of terms in a nondeterministic search which leads to a better performance in practice when compared
Can Context Sequence Matching Be Used for XML Querying?
 the 19th International Workshop on Unification (UNIF’05
, 2005
"... We describe a matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. The algorithm is called a context sequence matching algorithm. Context variables allow matching to descend in termtrees to arbitrary depth. Sequence v ..."
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We describe a matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. The algorithm is called a context sequence matching algorithm. Context variables allow matching to descend in termtrees to arbitrary depth. Sequence variables allow matching to move in termtrees in arbitrary breadth. The ability to explore terms in two orthogonal directions in a uniform way may be useful for querying data available as a large term, like XML documents. We extend the algorithm to process regular constraints and discuss its possible application in XML querying.