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On the Number of Distinct Languages Accepted by Finite Automata with n States
 Journal of Automata, Languages and Combinatorics
, 2001
"... We give asymptotic estimates and some explicit computations for both the number of distinct languages and the number of distinct nite languages over a kletter alphabet that are accepted by deterministic nite automata (resp. nondeterministic nite automata) with n states. ..."
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We give asymptotic estimates and some explicit computations for both the number of distinct languages and the number of distinct nite languages over a kletter alphabet that are accepted by deterministic nite automata (resp. nondeterministic nite automata) with n states.
Sequences With Grouped Factors
 Developments in Language Theory III, Publications of Aristotle University of Thessaloniki
, 1998
"... We define a new class of sequences, sequences with grouped factors, in which all factors of a given length occur consecutively at some position. This class contains periodic sequences, Sturmian sequences, but also sequences with maximal complexity. As a particular case, we obtain a new characteristi ..."
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Cited by 14 (0 self)
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We define a new class of sequences, sequences with grouped factors, in which all factors of a given length occur consecutively at some position. This class contains periodic sequences, Sturmian sequences, but also sequences with maximal complexity. As a particular case, we obtain a new characteristic property of Sturmian words. 1 Introduction The recurrence function R(n) is a classical tool associated to symbolic sequences [6, 8]
Automaticity III: Polynomial Automaticity And ContextFree Languages
 Computational Complexity
, 1996
"... . If L is a formal language, we define AL (n) to be the number of states in the smallest deterministic finite automaton that accepts a language that agrees with L on all inputs of length n. This measure is called automaticity. In this paper, we first study the closure properties of the class DPA of ..."
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Cited by 5 (3 self)
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. If L is a formal language, we define AL (n) to be the number of states in the smallest deterministic finite automaton that accepts a language that agrees with L on all inputs of length n. This measure is called automaticity. In this paper, we first study the closure properties of the class DPA of languages of deterministic polynomial automaticity, i.e., those languages L for which there exists k such that AL (n) = O(n k ). Next, we discuss similar results for a nondeterministic analogue of automaticity, introducing the classes NPA (languages of nondeterministic polynomial automaticity) and NPLA (languages of nondeterministic polylog automaticity). We conclude by showing how to construct a contextfree language of automaticity arbitrarily close to the maximum possible. Key words. automaticity, finite automata, nondeterminism Subject classifications. Primary 68Q68; Secondary 68Q75 68Q45. 1. Introduction. In two previous papers (Shallit & Breitbart (1996) , Pomerance et al. (199...
Automaticity And Rationality
, 2000
"... Automaticity is a measure of descriptional complexity for formal languages L, and measures how closely L can be approximated by regular languages. I survey some of the known results and open problems on automaticity. I also discuss a measure which I call "rationality", and explain how it generalizes ..."
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Automaticity is a measure of descriptional complexity for formal languages L, and measures how closely L can be approximated by regular languages. I survey some of the known results and open problems on automaticity. I also discuss a measure which I call "rationality", and explain how it generalizes the wellknown concept of linear complexity. Keywords: Regular language, finite automata, formal power series, rational series, linear complexity, linear span, linear recurrence, linear feedback shift register sequence, automaticity, rationality, continued fraction. 1. Introduction Let L be a formal language, that is, a subset of \Sigma , where \Sigma is a finite alphabet. We say L is regular if L is accepted by some finite automaton. Of course, not every language is regular, but we can approximate any language arbitrarily closely with regular languages. We say a language L 0 is an n'th order approximation to a language L if L and L 0 agree on all strings of length n, that is, if...
Bounded Error Probabilistic Finite State Automata
"... What power does randomness confer to computing devices? In this article, we focus on this question in what is perhaps its simplest form, namely when the computing device is a finite state automaton. Some of the oldest studies of probabilistic computations, dating as far back as the 40's, implicitly ..."
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What power does randomness confer to computing devices? In this article, we focus on this question in what is perhaps its simplest form, namely when the computing device is a finite state automaton. Some of the oldest studies of probabilistic computations, dating as far back as the 40's, implicitly concern probabilistic finite state devices. A beautiful theory of probabilistic finite state automata was developed starting in the early 60's [Rabin, 1963, Paz, 1971]. This work primarily concerned automata with 1way heads on the input tape, where an input w is considered to be accepted if the probability of reaching the accept state from the initial con guration is greater than some threshold, say 1/2. The class of languages thus accepted is known as the stochastic languages. A pfa for a stochastic language may err by rejecting inputs in the language with probability that approaches 1/2 as the input size increases. It is natural to cons...
Automaticity IV: Sequences, Sets, and Diversity
 J. Th'eorie Nombres Bordeaux
, 1996
"... This paper studies the descriptional complexity of (i) sequences over a finite alphabet; and (ii) subsets of N (the natural numbers). If (s(i)) i0 is a sequence over a finite alphabet \Delta, then we define the kautomaticity of s, A k s (n), to be the smallest possible number of states in any det ..."
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This paper studies the descriptional complexity of (i) sequences over a finite alphabet; and (ii) subsets of N (the natural numbers). If (s(i)) i0 is a sequence over a finite alphabet \Delta, then we define the kautomaticity of s, A k s (n), to be the smallest possible number of states in any deterministic finite automaton that, for all i with 0 i n, takes i expressed in basek as input and computes s(i). We give examples of sequences that have high automaticity in all bases k; for example, we show that the characteristic sequence of the primes has k automaticity A k s (n) = \Omega\Gamma n 1=43 ) for all k 2, thus making quantitative the classical theorem of Minsky and Papert that the set of primes expressed in base2 is not regular. We give examples of sequences with low automaticity in all bases k, and low automaticity in some bases and high in others. We also obtain bounds on the automaticity of certain sequences that are fixed points of homomorphisms, such as the Fibonac...
Number Theory And Formal Languages
 Emerging Applications of Number Theory, IMA Volumes in Mathematics and Applications
, 1999
"... . I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorp ..."
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. I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorphisms, automaticity, and kregular sequences. Key words. finite automata, automatic sequences, transcendence, automaticity AMS(MOS) subject classifications. Primary 11B85, Secondary 11A63 11A55 11J81 1. Introduction. In this paper, I survey some interesting connections between number theory and the theory of formal languages. This is a very large and rapidly growing area, and I focus on a few areas that interest me, rather than attempting to be comprehensive. (An earlier survey of this area, written in French, is [1].) I also give a number of open questions. Number theory deals with the properties of integers, and formal language theory deals with the properties of strings. At the interse...
On a conjecture of J. Shallit
 In Proc. 24th Int'l. Conf. on Automata, Languages, and Programming (ICALP
, 1997
"... We solve a conjecture of J. Shallit related to the automaticity function of a unary language, or equivalently to the first occurrence function in a symbolic sequence. The answer is negative: the conjecture is false, but it can be corrected by changing the constant involved. The proof is based on a s ..."
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We solve a conjecture of J. Shallit related to the automaticity function of a unary language, or equivalently to the first occurrence function in a symbolic sequence. The answer is negative: the conjecture is false, but it can be corrected by changing the constant involved. The proof is based on a study of paths in the Rauzy graphs associated to the sequence.
Juhani Karhumaki
, 1997
"... We study the Shallit's conjecture which states that an infinite word ! is ultimately periodic if and only if lim n!1 inf ju n (!)j n ? 3 \Gamma p 5 2 ; where u n (!) is the longest suffix of length n prefix of ! which occurs also inside length n \Gamma 1 prefix of !. We prove that a weaker co ..."
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We study the Shallit's conjecture which states that an infinite word ! is ultimately periodic if and only if lim n!1 inf ju n (!)j n ? 3 \Gamma p 5 2 ; where u n (!) is the longest suffix of length n prefix of ! which occurs also inside length n \Gamma 1 prefix of !. We prove that a weaker condition holds, namely that the conjecture is true if the constant 3\Gamma p 5 2 is replaced by 13\Gamma p 69 10 . TUCS Research Group Mathematical Structures of Computer Science 1 Introduction One of the most fundamental notions of words is that of the periodicity. In addition to classical results, such as Periodicity Lemma of Fine and Wilf and Critical Factorization Theorem, cf. [5, 8], a very interesting periodicity property was introduced in [11, 10]. Namely that an infinite word ! is ultimately periodic if and only if each long enough prefix of ! contains as a suffix a repetition of order 1 + OE, where OE is the golden ratio 1+ p 5 2 . A related but different characteriza...