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On Equality Upto Constraints over Finite Trees, Context Unification, and OneStep Rewriting
"... We introduce equality upto constraints over finite trees and investigate their expressiveness. Equality upto constraints subsume equality constraints, subtree constraints, and onestep rewriting constraints. ..."
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Cited by 28 (7 self)
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We introduce equality upto constraints over finite trees and investigate their expressiveness. Equality upto constraints subsume equality constraints, subtree constraints, and onestep rewriting constraints.
Word Unification and Transformation of Generalized Equations
 Journal of Automated Reasoning
, 1993
"... Makanin's algorithm [Ma77] shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions [P'e81, Ab87, S ..."
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Cited by 21 (1 self)
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Makanin's algorithm [Ma77] shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions [P'e81, Ab87, Sc90, Ja90]. In this paper we present a version of the algorithm which probably cannot be further simplified without fundamentally new insights which exceed Makanin's original ideas. We give a transformation which is efficient, conceptually simple and applies to arbitrary generalized equations. No further subprocedure is needed for the generation of the search tree. Particular attention is then given to the proof that proper generalized equations are transformed into proper generalized equations. This point, which is important for the termination argument, was treated erroneously in other papers. We also show that a combination of the basic algorithm for stringunification (see [Pl72, Le72, Si75, Si78]...
Lempel–Ziv Factorization Using Less Time & Space
, 1661
"... Abstract. For 30 years the Lempel–Ziv factorization LZx of a string x = x[1..n] has been a fundamental data structure of string processing, especially valuable for string compression and for computing all the repetitions (runs) in x. Traditionally the standard method for computing LZx was based on Θ ..."
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Cited by 11 (2 self)
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Abstract. For 30 years the Lempel–Ziv factorization LZx of a string x = x[1..n] has been a fundamental data structure of string processing, especially valuable for string compression and for computing all the repetitions (runs) in x. Traditionally the standard method for computing LZx was based on Θ(n)time (or, depending on the measure used, O(n log n)time) processing of the suffix tree STx of x. Recently Abouelhoda et al. proposed an efficient Lempel–Ziv factorization algorithm based on an “enhanced ” suffix array – that is, a suffix array SAx together with supporting data structures, principally an “interval tree”. In this paper we introduce a collection of fast spaceefficient algorithms for LZ factorization, also based on suffix arrays, that in theory as well as in many practical circumstances are superior to those previously proposed; one family out of this collection achieves true Θ(n)time alphabetindependent processing in the worst case by avoiding tree structures altogether. Mathematics Subject Classification (2000). 68W05. Keywords. Lempel–Ziv factorization, suffix array, suffix tree, LZ factorization.
On pseudoknot words and their properties
, 2007
"... We study a generalization of the classical notions of bordered and unbordered words, motivated by biomolecular computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in biomolecular computing to encode information. Due to the fact that A is WatsonCrick ( ..."
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We study a generalization of the classical notions of bordered and unbordered words, motivated by biomolecular computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in biomolecular computing to encode information. Due to the fact that A is WatsonCrick (WK) complementary to T and G to C, DNA single strands that are WK complementary can bind to each other or to themselves forming socalled secondary structures. Secondary structures are usually undesirable for biomolecular computational purposes since the strands involved in such structures cannot further interact with other strands. This paper studies pseudoknotbordered words, a mathematical formalization of a common secondary structure, the pseudoknot. We obtain several properties of WKpseudoknotbordered and unbordered words. One of the main results of the paper is that a sufficient condition for a WKpseudoknotunbordered word u to result in all words in u + being WKpseudoknotunbordered is for u not to be primitive word. All our results hold for arbitrary antimorphic involutions, of which the WK complementarity function is a particular case.
On Defect Effect of Biinfinite Words
 In MFCS'98
, 1998
"... We prove the following two variants of the defect theorem. Let X be a finite set of words over a finite alphabet. Then if a nonperiodic biinfinite word w has two Xfactorizations, then the combinatorial rank of X is at most card(X) \Gamma 1, i.e. there exists a set F such that X ` F ..."
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Cited by 6 (6 self)
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We prove the following two variants of the defect theorem. Let X be a finite set of words over a finite alphabet. Then if a nonperiodic biinfinite word w has two Xfactorizations, then the combinatorial rank of X is at most card(X) \Gamma 1, i.e. there exists a set F such that X ` F
Equations on Trees
 Proceedings MFCS'96. Lecture Notes in Computer Science n.1113
, 1996
"... . We introduce the notion of equation on trees, generalizing the corresponding notion for words, and we develop the first steps of a theory of tree equations. The main result of the paper states that, if a pair of trees is the solution of a tree equation with two indeterminates, then the two trees a ..."
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Cited by 4 (4 self)
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. We introduce the notion of equation on trees, generalizing the corresponding notion for words, and we develop the first steps of a theory of tree equations. The main result of the paper states that, if a pair of trees is the solution of a tree equation with two indeterminates, then the two trees are both powers of the same tree. As an application, we show that a tree can be expressed in a unique way as a power of a primitive tree. This extends a basic result of combinatorics on words to trees. Some open problems are finally proposed. 1 Introduction In this paper we are mainly concerned with kary trees whose vertices are labeled by letters of an alphabet A. We look at a labeled kary tree as a generalization of a word, in the sense that words correspond to the particular case of k = 1, i.e. to unary trees. a a b c b a b a c b A binary tree A word Given that combinatorics on words is a well developed theory, with several applications to computer science, it appears to be a natural ...
On Syntactic Groups
, 2003
"... We prove that for any finite prefix code X with n elements, the non special subgroups in the syntactic monoid of X # have degree at most n 1. This implies in particular that the groups in the syntactic monoid of X # are all cyclic when X is a prefix code with three elements. ..."
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Cited by 3 (3 self)
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We prove that for any finite prefix code X with n elements, the non special subgroups in the syntactic monoid of X # have degree at most n 1. This implies in particular that the groups in the syntactic monoid of X # are all cyclic when X is a prefix code with three elements.
Tree codes and Equations
"... Contents 1 Prerequisites in Combinatorics on words 1 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Codes and the defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Stable, left unitary and right unitary monoids . . . . . . . . . ..."
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Cited by 1 (1 self)
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Contents 1 Prerequisites in Combinatorics on words 1 1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Codes and the defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Stable, left unitary and right unitary monoids . . . . . . . . . . . . 4 1.2.2 The defect theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 An algorithm for the code problem . . . . . . . . . . . . . . . . . . 6 1.3 Word Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Generalities on trees 11 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations between kary trees . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Tree codes 25 3.1 Factorizations on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Stable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .