Handbook of Automata Theory

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

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Language and Automata Theory and Applications

10.1007/978-3-030-68195-1 ◽

2021 ◽

Keyword(s):

Automata Theory

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Automata theory motivated by problem solving

ACM SIGACT News ◽

10.1145/1008304.1008307 ◽

1974 ◽

Vol 6 (2) ◽

pp. 18-23

Author(s):

Ivan M. Havel

Keyword(s):

Problem Solving ◽

Automata Theory

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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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