Band Monoid Languages Revisited

Polynomial time multiplication and normal forms in free bands

“…in the size of the alphabet and linear in the size of the representations of the elements; see below for a formal description of the complexity. This representation is similar to the "admissible maps" of [20,21] and the "factor automata" of [23].…”

“…This paper is about efficient computational solutions to the problems of checking equality, performing multiplication, and computing minimal representatives of elements of free bands. This paper arose from that of and Rytter [23], and is related to the papers of Neto and Sezinando [20,21]. We will discuss in more detail below the interactions of [20,21,23] and the current paper.…”

“…This paper arose from that of and Rytter [23], and is related to the papers of Neto and Sezinando [20,21]. We will discuss in more detail below the interactions of [20,21,23] and the current paper.…”

“…The starting point for the development of the representation of free band elements by transducers (and the other representations in [20,21,23]), is the solution to the word problem given by Green and Rees in [11]; we will restate this description in Section 2. Recall that we denoted by ∼ the least congruence on A + containing {(x 2 , x) : x ∈ A + }.…”

“…A similar recursive approach was found by Gerhard and Petrich [9] to solve the word problem in relatively free bands (the free objects in sub-varieties of bands). Suppose that A is any fixed alphabet and that u, v ∈ A + are such that |u| + |v| = k for some k ∈ N. Neto and Sezinando [20] estimate the time complexity of Gerhard and Petrich's approach for the free band to be O(2 k ), which is worse than the naïve approach using the Green-Rees Theorem. However, for other relatively free bands, Neto and Sezinando [20] estimate the time complexity of Gerhard and Petrich's approach to be O((k + 1) n−2 ) where n ∈ N is the height of the relatively free band in the lattice of all varieties of bands.…”