Narrow Results

We say a family of strings over an alphabet is an UMFF if every string has a unique maximal factorization over . Foundational work by Chen, Fox and Lyndon established properties of the Lyndon circ-UMFF, which is based on lexicographic ordering. Commencing with the circ-UMFF related to V-order, we then proved analogous factorization families for a further 32 Block-like binary orders. Here we distinguish between UMFFs and circ-UMFFs, and then study the structural properties of circ-UMFFs. These properties give rise to the complete construction of any circ-UMFF. We prove that any circ-UMFF is a totally ordered set and a factorization over it must be monotonic. We define atom words and initiate a study of u, v-atoms. Applications of circ-UMFFs arise in string algorithmics.

Sequences (L_n | 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_n; and the state complexities of the following operations: (6) the reverse of L_n; (7) the star of L_n; (8) union, intersection, difference and symmetric difference of L_m and L_n; and (9) 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) | 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 U_n(a, b, c) and U_n(b, a, 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.

A regular language $L$ 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 $n\ge 4$, there exists a ternary witness of state complexity $n$ 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 ${(n-1)}^n$ elements and requires at least $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ 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.