Complexity of proper prefix-convex regular languages

Abstract

A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ * , whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefixclosed, and prefix-free languages. We study complexity properties of prefix-convex regular languages. In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit most complex prefix-convex languages that meet the complexity bounds for all the measures listed above.Finally, if X = X ′ = ∅ and r ∈ Y ′ \ Y , then distinguish J X,Y and J X ′ ,Y ′ by a word that sends r → n − 2 and Q n \ {r} → {n − 1}. Hence, J X,Y and J X ′ ,Y ′ are distinct in all cases. Therefore, the quotients of A S counted in the upper bound are pairwise distinct and L n,k has maximal atomic complexity.

Complexity of bifix-free regular languages

“…Other kinds of dialects are possible (e.g. [7]), though permutational dialects are the most restricted.…”

“…It may be surprising that such a single witness exists for most of the natural subclasses of regular languages: the class of all regular languages [3], right-, left-, and two-sided ideals [4], and prefixconvex languages [7]. However, there does not exist a single witness for the class of suffix-free languages [9], where two different witnesses must be used.…”

Complexity of Proper Prefix-Convex Regular Languages

“…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. This stream together with some dialects meets the complexity bounds for reversal, star, product, and all binary boolean operations [7,8]. More-over, it has the maximal syntactic semigroup and most complex atoms, making it a most complex regular stream.…”

“…Prefix-Convex Languages We examine the complexity properties of a class of regular languages that has never been studied before: the class of proper prefixconvex languages [7]. Let Σ be a finite alphabet; if w = xy, for x, y ∈ Σ * , then x is a prefix of w. A language L ⊆ Σ * is prefix-convex [1,16] if whenever x and xyz are in L, then so is xy.…”

“…We now turn to the "real" prefix-convex languages that do not belong to any of the three special classes. Omitted proofs can be found in [7].…”