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23
State Complexity of Regular Languages
 Journal of Automata, Languages and Combinatorics
, 2000
"... State complexity is a descriptive complexity measure for regular languages. We investigate the problems related to the state complexity of regular languages and their operations. In particular, we compare the state complexity results on regular languages with those on finite languages. ..."
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Cited by 34 (6 self)
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State complexity is a descriptive complexity measure for regular languages. We investigate the problems related to the state complexity of regular languages and their operations. In particular, we compare the state complexity results on regular languages with those on finite languages.
Tight Lower Bound for the State Complexity of Shuffle of Regular Languages
"... The upper bound for the state complexity of the shuffle of two regular languages is 2^mn  1. We prove that this bound can be reached for some (not necessarily complete) deterministic finite automata with, respectively, m and n states. Our construction uses an alphabet of size 5. ..."
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Cited by 16 (9 self)
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The upper bound for the state complexity of the shuffle of two regular languages is 2^mn  1. We prove that this bound can be reached for some (not necessarily complete) deterministic finite automata with, respectively, m and n states. Our construction uses an alphabet of size 5.
Descriptional Complexity of Machines with Limited Resources
 J. UNIVERSAL COMPUTER SCI
, 2002
"... Over the last 30 years or so many results have appeared on the descriptional complexity of machines with limited resources. Since these results have appeared in a variety of different contexts, o rgo4 here is to pro vide a survey o these results. Particular emphasis is put o limiting reso rces (e.g. ..."
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Cited by 9 (3 self)
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Over the last 30 years or so many results have appeared on the descriptional complexity of machines with limited resources. Since these results have appeared in a variety of different contexts, o rgo4 here is to pro vide a survey o these results. Particular emphasis is put o limiting reso rces (e.g., no ndeterminism, ambiguity,lo o ahead, etc.) fo vario s types o finite state machines, pushdo wn auto mata, parsers and cellular auto mata ando n the e#ect it haso n their descriptio nal co mplexity. We also address the questio no f how descriptional complexity might help in the future to solve practical issues, such as software reliability.
Time granularities and ultimately periodic automata
 In Proc. of the 9th European Conference on Logics in Artificial Intelligence (JELIA) volume 3229 of Lecture Notes in Computer Science
, 2004
"... Abstract. The relevance of the problem of managing periodic phenomena is widely recognized in the area of knowledge representation and reasoning. One of the most effective attempts at dealing with this problem has been the addition of a notion of time granularity to knowledge representation systems. ..."
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Cited by 7 (1 self)
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Abstract. The relevance of the problem of managing periodic phenomena is widely recognized in the area of knowledge representation and reasoning. One of the most effective attempts at dealing with this problem has been the addition of a notion of time granularity to knowledge representation systems. Different formalizations of such a notion have been proposed in the literature, following algebraic, logical, stringbased, and automatonbased approaches. In this paper, we focus our attention on the automatonbased one, which allows one to represent a large class of granularities in a compact and suitable to algorithmic manipulation form. We further develop such an approach to make it possible to deal with (possibly infinite) sets of granularities instead of single ones. We define a new class of automata, called Ultimately Periodic Automata, we give a characterization of their expressiveness, and we show how they can be used to encode and to solve a number of fundamental problems, such as the membership problem, the equivalence problem, and the problem of granularity comparison. Moreover, we give an example of their application to a concrete problem taken from clinical medicine. 1
State complexity of basic operations on suffixfree regular languages
, 2007
"... We investigate the state complexity of basic operations for suffixfree regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worstcase for the minimal deterministic finitestate automaton that accepts the lan ..."
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Cited by 7 (2 self)
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We investigate the state complexity of basic operations for suffixfree regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worstcase for the minimal deterministic finitestate automaton that accepts the language obtained from the operation. We establish the precise state complexity of catenation, Kleene star, reversal and the Boolean operations for suffixfree regular languages.
State complexity of prefixfree regular languages
 IN: PROCEEDINGS OF DCFS’06
, 2006
"... We investigate the state complexities of basic operations for prefixfree regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worstcase for the minimal deterministic finitestate automaton (DFA) that accepts ..."
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Cited by 6 (4 self)
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We investigate the state complexities of basic operations for prefixfree regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worstcase for the minimal deterministic finitestate automaton (DFA) that accepts the language obtained from the operation. We know that a regular language is prefixfree if and only if its minimal DFA has only one final state and the final state has no outtransitions whose target state is not a sink state. Based on this observation, we reduce the state complexities for prefixfree regular languages compared with the state complexities for (general) regular languages. For both catenation and Kleene star operations of (general) regular languages, the state complexities are exponential in the size of given minimal DFAs. On the other hand, if both regular languages are prefixfree, then the state complexities are at most linear. We also demonstrate that we can reduce the state complexities of intersection and union operations based on the structural properties of prefixfree minimal DFAs.
THE FROBENIUS PROBLEM IN A FREE MONOID
, 2008
"... The classical Frobenius problem over N is to compute the largest integer g not representable as a nonnegative integer linear combination of nonnegative integers x1, x2,..., xk, where gcd(x1, x2,..., xk) = 1. In this paper we consider novel generalizations of the Frobenius problem to the noncommu ..."
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Cited by 6 (3 self)
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The classical Frobenius problem over N is to compute the largest integer g not representable as a nonnegative integer linear combination of nonnegative integers x1, x2,..., xk, where gcd(x1, x2,..., xk) = 1. In this paper we consider novel generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for several analogues of g, with the precise bound depending on the particular measure chosen.
State complexity of union and intersection of finite languages
 In Proceedings of DLT’07, Lecture Notes in Computer Science 4588
, 2007
"... Abstract. We investigate the state complexity of union and intersection for finite languages. Note that the problem of obtaining the tight bounds for both operations was open. We compute the upper bounds based on the structural properties of minimal deterministic finitestate automata (DFAs) for fin ..."
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Cited by 4 (2 self)
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Abstract. We investigate the state complexity of union and intersection for finite languages. Note that the problem of obtaining the tight bounds for both operations was open. We compute the upper bounds based on the structural properties of minimal deterministic finitestate automata (DFAs) for finite languages. Then, we show that the upper bounds are tight if we have a variable sized alphabet that can depend on the size of input DFAs. In addition, we prove that the upper bounds are unreachable for any fixed sized alphabet. 1
On the Average State and Transition Complexity of Finite Languages
, 2007
"... We investigate the averagecase state and transition complexity of deterministic and nondeterministic finite automata, when choosing a finite language of a certain “size” n uniformly at random from all finite languages of that particular size. Here size means that all words of the language are eithe ..."
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Cited by 4 (2 self)
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We investigate the averagecase state and transition complexity of deterministic and nondeterministic finite automata, when choosing a finite language of a certain “size” n uniformly at random from all finite languages of that particular size. Here size means that all words of the language are either of length n, or of length at most n. It is shown that almost all deterministic finite automata accepting finite languages over a binary input alphabet have state complexity Θ ( 2n n), while nondeterministic finite automata are shown to perform better, namely the nondeterministic state complexity is in Θ ( √ 2n). Interestingly, in both cases the aforementioned bounds are asymptotically like in the worstcase. However, the nondeterministic transition complexity is shown to be again Θ ( 2n n). The case of unary finite languages is also considered. Moreover, we develop a framework that allows us to investigate the averagecase complexity of operations like, e.g., union, intersection, complementation, and reversal, on finite languages in this setup.
Shuffle Quotient and Decompositions
 Proceedings of the Fifth International Conference on Developments in Language Theory, DLT2001
"... We introduce a right congruence relation that is the analogy of the Nerode congruence when catenation is replaced by shuffle. Using this relation we show that for certain subclasses of regular languages the shuffle decomposition problem is decidable. We show that shuffle decomposition is undecidable ..."
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Cited by 3 (1 self)
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We introduce a right congruence relation that is the analogy of the Nerode congruence when catenation is replaced by shuffle. Using this relation we show that for certain subclasses of regular languages the shuffle decomposition problem is decidable. We show that shuffle decomposition is undecidable for contextfree languages. 1