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55
Quotient complexity of ideal languages
 In: LATIN 2010. LNCS 6034, SpringerVerlag
, 2010
"... Abstract. We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ ∗ is a right (left) ideal if it satisfies L = LΣ ∗ (L = Σ ∗ L). It is a twosided ideal if L = Σ ∗ LΣ ∗ , and an allsided ideal if L = Σ ∗ L, the shuffle of Σ ∗ with L. We prefer the term ..."
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Cited by 22 (12 self)
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Abstract. We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ ∗ is a right (left) ideal if it satisfies L = LΣ ∗ (L = Σ ∗ L). It is a twosided ideal if L = Σ ∗ LΣ ∗ , and an allsided ideal if L = Σ ∗ L, the shuffle of Σ ∗ with L. We prefer the term “quotient complexity ” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages.
Partial Words for DNA Coding
 In Preliminary Proceedings of 10th International Workshop on DNABased Computers, DNA 2004 (University of MilanoBicocca
, 2004
"... A very basic problem in all DNA computations is nding a good encoding. Apart from the fact that they must provide a solution, the strands involved should not exhibit any undesired behaviour like forming secondary structures. ..."
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Cited by 16 (1 self)
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A very basic problem in all DNA computations is nding a good encoding. Apart from the fact that they must provide a solution, the strands involved should not exhibit any undesired behaviour like forming secondary structures.
Language equations, maximality and errordetection
, 2005
"... We use some ‘natural’ language operations, such as shuffle (scattered insertion) and scattered deletion to model noisy channels, that is, nondeterministic processes transforming words to words. In this spirit, we also introduce the operation of scattered substitution and derive the closure propertie ..."
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Cited by 15 (8 self)
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We use some ‘natural’ language operations, such as shuffle (scattered insertion) and scattered deletion to model noisy channels, that is, nondeterministic processes transforming words to words. In this spirit, we also introduce the operation of scattered substitution and derive the closure properties of the language families in the Chomsky hierarchy under this operation. Moreover, we consider a certain type of language inequations involving language operations and observe that, by varying the parameters of such an inequation, we can define families of codes such as prefix and infix, as well as families of errordetecting languages. Our results on this type of inequations include a characterization of the maximal solutions, which provides a uniform method for deciding whether a given regular code of the type defined by the inequation is maximal.
Watson–Crick conjugate and commutative words, in
 Proc. of DNA 13, in: LNCS
, 2008
"... Abstract. This paper is a theoretical study of notions in combinatorics of words motivated by information being encoded as DNA strands in DNA computing. We generalize the classical notions of conjugacy and commutativity of words to incorporate the notion of an involution function, a formalization of ..."
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Cited by 12 (10 self)
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Abstract. This paper is a theoretical study of notions in combinatorics of words motivated by information being encoded as DNA strands in DNA computing. We generalize the classical notions of conjugacy and commutativity of words to incorporate the notion of an involution function, a formalization of the WatsonCrick complementarity of DNA singlestrands. We define and study properties of WatsonCrick conjugate and commutative words, as well as WatsonCrick palindromes. We obtain, for example, a complete characterization of the set of all words that are not WatsonCrick palindromes. Our results hold for more general functions, such as arbitrary morphic and antimorphic involutions. They generalize classical results in combinatorics of words, while formalizing concepts meaningful for DNA computing experiments. 1
Quotient complexity of bifix, factor, and subwordfree regular languages
, 2011
"... A language L is prefixfree if, whenever words u and v are in L and u is a prefix of v, then u = v. Suffix, factor, and subwordfree languages are defined similarly, where “subword ” means “subsequence”. A language is bifixfree if it is both prefix and suffixfree. We study the quotient complex ..."
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Cited by 11 (7 self)
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A language L is prefixfree if, whenever words u and v are in L and u is a prefix of v, then u = v. Suffix, factor, and subwordfree languages are defined similarly, where “subword ” means “subsequence”. A language is bifixfree if it is both prefix and suffixfree. We study the quotient complexity, more commonly known as state complexity, of operations in the classes of bifix, factor, and subwordfree regular languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.
Conway's Problem and the commutation of languages
 Bulletin of EATCS
, 2001
"... We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The se ..."
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Cited by 10 (5 self)
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We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1
Languages convex with respect to binary relations, and their closure properties. Acta Cybernet
"... A language is prefixconvex if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefixconvex languages include prefixclosed languages at one end of the spectrum, and prefixfree languages, which include prefix c ..."
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Cited by 10 (8 self)
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A language is prefixconvex if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefixconvex languages include prefixclosed languages at one end of the spectrum, and prefixfree languages, which include prefix codes, at the other. In a similar way, we define suffix, bifix, factor, and subwordconvex languages and their closed and free counterparts. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages. We generalize these notions to arbitrary binary relations on the set of all words over a given alphabet, and study the closure properties of such languages.
Maximal and minimal solutions to language equations
 J. Comp. Sys. Sci
, 1996
"... RhX=LhY, where h is a binary word (language) operation, L, R are given constant languages and X, Y are the unknowns. We investigate the existence and uniqueness of maximal and minimal solutions, properties of solutions, and the decidability of the existence of solutions.] 1996 Academic Press, Inc. 1 ..."
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Cited by 9 (3 self)
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RhX=LhY, where h is a binary word (language) operation, L, R are given constant languages and X, Y are the unknowns. We investigate the existence and uniqueness of maximal and minimal solutions, properties of solutions, and the decidability of the existence of solutions.] 1996 Academic Press, Inc. 1.
Involutively bordered words
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... In this paper we study a generalization of the classical notions of bordered and unbordered words, motivated by DNA computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in DNA computing to encode information. Due to the fact that A is WatsonCrick compl ..."
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Cited by 8 (6 self)
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In this paper we study a generalization of the classical notions of bordered and unbordered words, motivated by DNA computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in DNA computing to encode information. Due to the fact that A is WatsonCrick complementary to T and G to C, DNA single strands that are WatsonCrick complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: If some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpinlike structure. This paper studies a mathematical formalization of a particular case of hairpins, the WatsonCrick bordered words. A WatsonCrick bordered word is a word with the property that it has a prefix that is WatsonCrick complementary to its suffix. More generally, we investigate the notion of θbordered words, where θ is a morphic or antimorphic involution. We show that the set of all θbordered words is regular, when θ is an antimorphic involution and the set of all θbordered words is contextsensitive when θ is a morphic involution. We study the properties of θbordered and θunbordered words and also the relation between θbordered and θunbordered words and certain type of involution codes.
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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Cited by 8 (7 self)
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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.