Unrestricted State Complexity of Binary Operations on Regular Languages

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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Most Complex Non-Returning Regular Languages

International Journal of Foundations of Computer Science ◽

10.1142/s0129054119400239 ◽

2019 ◽

Vol 30 (06n07) ◽

pp. 921-957

Author(s):

Janusz A. Brzozowski ◽

Sylvie Davies

Keyword(s):

Star Product ◽

Upper Bounds ◽

The State ◽

Boolean Operations ◽

Deterministic Finite Automaton ◽

Initial State ◽

Complexity Measures ◽

State Complexity ◽

Complex Sequence ◽

Binary Operations

A regular language [Formula: see text] is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each [Formula: see text], there exists a ternary witness of state complexity [Formula: see text] that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has [Formula: see text] elements and requires at least [Formula: see text] generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

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