Consider partial maps ${\Sigma }^{*}$$\to$$ℝ$ with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure...

Consider partial maps ∑* → $ℝ$ with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.

The cyclic shift of a language $L$, defined as $shift\left(L\right)$=$\left\{vu\right|uv\in L\}$, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslov’s pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl. 11 (1970) 1373–1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n-1)! $·$ 2${}^{(n-1)(n-2)}$, which shows that the state complexity...

The cyclic shift of a language L, defined as SHIFT(L) = {vu | uv ∈ L}, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslov's pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl.11 (1970) 1373–1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n-1)! . 2(n-1)(n-2), which...