Kleene Closure on Regular and Prefix-Free Languages

The Complexity of Languages Resulting from the Concatenation Operation

“…This improves the result from [8], where an alphabet of size growing exponentially with n is used to produce the whole range of complexities for the concatenation operation. Our result complements similar results from [10,14], where a linear alphabet is used to get the whole range of complexities for the reversal and Kleene closure operations.…”

“…The result for reversal and star was improved in [10,14] by showing that a linear alphabet is enough to produce the whole range of complexities.…”

A Survey on Operational State Complexity

“…Jirásková [107] proved that for all integers m and α with either m = 1 and α ∈ [1,2], or m ≥ 2 and α ∈ [1, 2 m−1 + 2 m−2 ], there exists a language L over an alphabet of size 2 m such that sc(L) = m and sc(L ) = α. This result was improved by Jirásková et al [120] by using an alphabet of size atmost 2m. Again, no gaps or magic numbers exist for the Kleene star operation.…”

“…Free languages Table 7 summarizes state complexity results of individual operations on prefix-free languages [82,83,112,22,120,116,102,55]. In the case of state complexity, the results are valid for Boolean operations if m, n ≥ 3; for catenation if m, n ≥ 2; for star if k = 1, then m ≥ 3, if k = 2 then m = 3, and else m ≥ 2; and for reversal if m ≥ 4 and the tight bound cannot be reached if k = 2 [112].…”

Complexity of bifix-free regular languages

“…Here we provide a complete characterization for the state complexity of L * n and show that there are exactly two possibilities for its state complexity: n − 1 and n − 2. This may be compared with prefix-free languages, where there are exactly three possibilities for the state complexity L * n : n, n − 1, and n − 2 [18]. Theorem 6.…”