Multiple Concatenation and State Complexity (Extended Abstract)

We describe witnesses for the concatenation of k languages over an alphabet of k + 1 symbols with a significantly simpler proof then that from the literature. Then we use slightly modified Maslov's automata to get witnesses over a k-letter alphabet which solves an open problem stated by Caron et al. [2018, Fund. Inform. 160, 255-279]. We prove that for k = 3, the ternary alphabet is optimal. We also obtain lower bounds n(1) - 1 + (1/2(2k-2))2(n2 + . . . +nk) and (1/2(2k-2))n(1)2(n2 + . . . +nk) in the binary and ternary case, respectively. Finally, we show that an upper bound for unary cyclic languages is n(1)n(k)/d + n(1) + . . . + n(k) - k +1+ d where n(1 )<= . . . <= n(k) and d = gcd(n1, . . . , n(k)).