UNARY LANGUAGE OPERATIONS, STATE COMPLEXITY AND JACOBSTHAL'S FUNCTION

In this paper we give the cost, in terms of states, of some basic operations (union, intersection, concatenation, and Kleene star) on regular languages in the unary case (where the alphabet contains only one symbol). These costs are given by explicitly determining the number of states in the noncyclic and cyclic parts of the resulting automata. Furthermore, we prove that our bounds are optimal. We also present an interesting connection to Jacobsthal's function from number theory.

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On the state complexity of intersection of regular languages

ACM SIGACT News ◽

10.1145/126537.126543 ◽

1991 ◽

Vol 22 (3) ◽

pp. 52-54 ◽

Cited By ~ 14

Author(s):

Sheng Yu ◽

Qingyu Zhuang

Keyword(s):

The State ◽

Regular Languages ◽

State Complexity

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

Journal of Logic and Computation ◽

10.1093/logcom/exaa007 ◽

2020 ◽

Vol 30 (1) ◽

pp. 175-192

Author(s):

NathanaËl Fijalkow

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Prime Numbers ◽

Regular Languages ◽

Automata Theory ◽

Seminal Paper ◽

Probabilistic Automata ◽

State Complexity ◽

Probabilistic Languages ◽

Alternating Automata

Abstract 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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