Automaticity IV : sequences, sets, and diversity

Shallit, Jeffrey

doi:10.5802/jtnb.173

Cited by 19 publications

(22 citation statements)

References 22 publications

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“…Their result is unconditional, and makes use of Dirichlet's theorem on arithmetic progressions of prime numbers. A related and stronger result has been proved by Shallit [19], which says that the deterministic automaticity of the prime numbers is not subexponential.…”

Section: The Alternating State Complexity Of Prime Numbersmentioning

confidence: 86%

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Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

Fijalkow

2020

Journal of Logic and Computation

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This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp.Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity.We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

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Section: The Alternating State Complexity Of Prime Numbersmentioning

confidence: 86%

“…Automaticity was defined by Shallit and Breitbart and studied in depth in a series of four papers [16,17,18,19]. The conceptual difference is that automaticity is a non-uniform notion, since there is a finite automaton for each n, whereas state complexity is uniform, since it considers one infinite automaton.…”

Section: Related Workmentioning

confidence: 99%

Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

Fijalkow

2020

Journal of Logic and Computation

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“…Additional results on automaticity can befound in Shallit 1996Shallit . cc 7 1998 As usual, we de ne a nite automaton M to bea5-tuple, Q; ; ; q 0 ; F , where Q is a nite set of states, is a nite input alphabet, q 0 is the start state, and F is a set of nal states.…”

Section: Introductionmentioning

confidence: 99%

Automaticity III: Polynomial automaticity and context-free languages

Glaister¹,

Shallit²

1998

Computational Complexity

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If L is a formal language, we de ne A L n t o b e t h e n umber of states in the smallest deterministic nite automaton that accepts a language which agrees with L on all inputs of length no more than n. This measure is called automaticity. In this paper, we rst 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 A L 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 poly-log automaticity. We conclude by showing how to construct a context-free language of automaticity arbitrarily close to the maximum possible.

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“…This problem of bounding the length of the common prefix of f and g is related to the concept of b-automaticity of infinite sequences [9], which measures the minimum number of states of a base-b automaton that computes the length-n prefix of the sequence. In particular, we are able to get a lower bound on the b-automaticity of an a-automatic sequence.…”

mentioning

confidence: 99%

“…The problem of determining the maximum length of a common prefix of a b-automatic sequence and a Sturmian sequence was examined by Shallit [9]. Upper bounds on the length of the common prefix can be deduced from the automaticity results given by Shallit.…”

mentioning

confidence: 99%