A Survey on Operational State Complexity (original) (raw)

A Review on State Complexity of Individual Operations

Abstract. The state complexity of a regular language is the number of states of its minimal determinitisc finite automaton. The complexity of a language operation is the complexity of the resulting language seen as a function of the complexities of the operation arguments. In this report we review some of the results of state complexity of individual operations for regular and some subregular languages.

Incomplete operational transition complexity of regular languages

Information and Computation, 2015

The state complexity of basic operations on regular languages considering complete deterministic finite automata (DFA) has been extensively studied in the literature. But, if incomplete DFAs are considered, transition complexity is also an significant measure. In this paper we study the incomplete (deterministic) state and transition complexity of some operations for regular and finite languages. For regular languages we give a new tight upper bound for the transition complexity of the union, which refutes the conjecture presented by Y. Gao et al.. For finite languages, we correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right operand is larger than the left one. We also present some experimental results to test the behaviour of those operations on the average case, and we conjecture that for many operations and in practical applications the worst-case complexity is seldom reached.

State complexity of some operations on binary regular languages

Theoretical Computer Science, 2005

We investigate the state complexity of some operations on binary regular languages. In particular, we consider the concatenation of languages represented by deterministic finite automata, and the reversal and complementation of languages represented by nondeterministic finite automata. We prove that the upper bounds on the state complexity of these operations, which were known to be tight for larger alphabets, are tight also for binary alphabets.

State Complexity of Basic Operations on Non-returning Regular Languages

Lecture Notes in Computer Science, 2013

We consider the state complexity of basic operations on non-returning regular languages. For a non-returning minimal DFA, the start state does not have any in-transitions. We establish the precise state complexity of four Boolean operations (union, intersection, difference, symmetric difference), catenation, reverse, and Kleene-star for non-returning regular languages. Our results are usually smaller than the state complexities for general regular languages and larger than the state complexities for suffix-free regular languages. In the case of catenation and reversal, we define witness languages over a ternary alphabet. Then we provide lower bounds for a binary alphabet. For every operation, we also study the unary case.

The Operational Incomplete Transition Complexity on Finite Languages

2013

The state complexity of basic operations on finite languages (considering complete DFAs) has been extensively studied in the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both descriptional measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using family languages with a variable alphabet size. In general the operational complexities depend not only on the complexities of the operands but also on other refined measures.

Estimation of state complexity of combined operations

Theoretical Computer Science, 2009

It appears that the state complexity of each combined operation has its own special features. Thus, it is important and practical to obtain good estimates for some commonly used general cases. In this paper, we consider the state complexity of combined Boolean operations and give an exact bound for all of them in the case when the alphabet is not fixed. Moreover, we show that for any fixed alphabet, this bound can be reached in infinitely many cases. We also consider the state complexity of multiple catenations. The state complexities are obtained in the cases of the catenations of three and four languages. An estimate for the catenation of an arbitrary number of languages is given, which is very close to the state complexities in the three and four languages cases.

Incomplete Transition Complexity of Basic Operations on Finite Languages

Lecture Notes in Computer Science, 2013

The state complexity of basic operations on finite languages (considering complete DFAs) has been in studied the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both descriptional measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using family languages with a variable alphabet size. In general the operational complexities depend not only on the complexities of the operands but also on other refined measures.

On a structural property in the state complexity of projected regular languages

Theoretical Computer Science, 2012

A transition is unobservable if it is labeled by a symbol removed by a projection. The present paper investigates a new structural property of incomplete deterministic finite automata -a number of states incident with an unobservable transition -and its effect on the state complexity of projected regular languages. We show that the known upper bound can be met only by automata with one unobservable transition (up to unobservable multi-transitions). We improve this upper bound by taking into consideration the structural property of minimal incomplete automata, and prove the tightness of new upper bounds. Special attention is focused on the case of finite languages. The paper also presents and discusses several fundamental problems which are still open. A typical automaton model of a real-world system usually consists of a huge number of states. Therefore, the simplification of the system plays an important role in many fields of computer science and engineering, such as compositional verification, fault diagnoses, or supervisory control . Projections, also called natural projections because they can be seen as natural transformations of category theory, are one of the forms of abstraction methods that are used for such a simplification. Given a regular language L and a projection P, it is well-known that the minimal deterministic finite automaton (dfa) accepting the language P(L) can be of exponential size in comparison with the dfa accepting the language L. However, from the practical point of view, only those projections which ensure that the automaton for the projected language is significantly smaller than the automaton of the original language are of interest. In this paper, we summarize the known results on this topic, improve the known upper bounds of the projected regular languages, and formulate several open problems.

State complexity of basic language operations combined with reversal

Information and Computation, 2008

We study the state complexity of combined operations on regular languages. Each of the combined operations is a basic operation combined with reversal. We show that their state complexities are all very different from the compositions of state complexities of individual operations.

On the State Complexity of Operations on Two-Way Finite Automata

Developments in Language Theory

The paper investigates the effect of basic language-theoretic operations on the number of states in two-way deterministic finite automata (2DFAs). If m and n are the number of states in the 2DFAs recognizing the arguments of the following operations, then their result requires the following number of states: at least m + n − o(m + n) and at most 4m + n + const for union; at least m + n − o(m + n) and at most m + n + 1 for intersection; at least (m n) + 2 (n) log m and at most 2m m+1 • 2 n n+1 for concatenation; at least 1 n 2 n 2 −1 and at most 2 O (n n+1) for Kleene star, square and projections; between n + 1 and n + 2 for reversal; exactly 2n for inverse homomorphisms. All results are obtained by first establishing high lower bounds on the number of states in any 1DFAs recognizing these languages, and then using these bounds to reason about the size of any equivalent 2DFAs.