In Search of Most Complex Regular Languages

Abstract

Received (Day Month Year) Accepted (Day Month Year) Communicated by (xxxxxxxxxx)Sequences (Ln | n k), called streams, of regular languages Ln are considered, where k is some small positive integer, n is the state complexity of Ln, 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 Ln, that is, the number of left quotients of Ln (used as a reference); 2) the state complexities of the left quotients of Ln; 3) the number of atoms of Ln; 4) the state complexities of the atoms of Ln; 5) the size of the syntactic semigroup of Ln; and the state complexities of the following operations: 6) the reverse of Ln; 7) the star of Ln; 8) union, intersection, difference and symmetric difference of Lm and Ln; and 9) the concatenation of Lm and Ln. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (Un(a, b, c) | 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 n 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 Un(a, b, c) and Un(b, a, c), where the roles of a and b are interchanged in the second language. In this sense, Un(a, b, c) is a universal witness. This witness and its extensions also apply to a large number of combined regular operations.

Descriptional Complexity in Encoded Blum Static Complexity Spaces

“…A similar phenomenon was described by Brzozowski in [5,6], where he shows a series of simple languages as witnesses for the state complexity of several operations on regular languages. It is not surprising that their description is short, while being witnesses for the complexities of various operations.…”

Complexity of Proper Prefix-Convex Regular Languages

“…If L n is a regular language of complexity n, and • is a unary operation, the complexity of • is the maximal value of κ(L • n ), expressed as a function of n, as L n ranges over all languages of complexity n. If L ′ m and L n are regular languages of complexities m and n respectively, and • is a binary operation, the complexity of • is the maximal value of κ(L ′ m • L n ), expressed as a function of m and n, as L ′ m and L n range over all languages of complexities m and n. The complexity of an operation is a lower bound on its time and space complexities. The operations reversal, (Kleene) star, product (concatenation), and binary boolean operations are considered "common", and their complexities are known; see [4,17,18].…”

“…For a binary operation we need two streams. The same stream cannot always be used for both operands, but for all common binary operations the second stream can be a "dialect" of the first, that is it can "differ only slightly" from the first [4].…”

“…If D n is a minimal DFA of L n , then T Dn is isomorphic to the syntactic semigroup T Ln of L n , and we represent elements of T Ln by transformations in T Dn . The size of the syntactic semigroup has been used as a measure of complexity for regular languages [4,10,12,14].…”

“…The number of atoms and their complexities were suggested as possible measures of complexity [4], because all the quotients of a language and the quotients of its atoms are unions of atoms [9]. Most Complex Regular Stream The stream (D n (a, b, c) | n 3) of Definition 1 and Figure 1 will be used as a component in the class of proper prefixconvex languages.…”