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In Search Of Most Complex Regular Languages

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brzozowski-s.pdf (253.66 KB)

Date

2013-09-01

Authors

Brzozowski, Janusz

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Publisher

World Scientific Publishing

Abstract

Sequences (L-n vertical bar n >= k), called streams, of regular languages L-n are considered, where k is some small positive integer, n is the state complexity of L-n, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: (1) the state complexity n of L-n, that is the number of left quotients of L-n (used as a reference); (2) the state complexities of the left, quotients of L-n; (3) the number of atoms of L-n; (4) the state complexities of the atoms of L-n; (5) the size of the syntactic semigroup of L; and the state complexities of the following operations: (6) the reverse of L-n; (7) the star; (8) union, intersection, difference and symmetric difference of and L-n; and the concatenation of L-m and L-n. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (U-n (a, b, c) vertical bar n >= 3 is defined by the deterministic finite automaton with state set {0, 1, ..., n-1}, initial state 0, set {n-1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the;a states, b transposes states 0 and 1, and c maps state n - 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use U-n (a, b, c) and U-n(a, b, c), where the roles of a and b are interchanged in the second language. In this sense, U-n (a, b, c) is a universal witness This witness and its extensions also apply to a large number of combined regular operations.

Description

Electronic version of an article published as International Journal of Foundations of Computer Science, 24(06), 2013, 691–708. http://dx.doi.org/10.1142/S0129054113400133 © World Scientific Publishing Company http://www.worldscientific.com/

Keywords

Atom, complexity of operation, finite automaton, quotient complexity, regular language, state complexity, syntactic semigroup, Witness

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http://dx.doi.org/10.1142/S0129054113400133
 http://hdl.handle.net/10012/12514

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Waterloo Research
Computer Science