For languages A and B, let the perfect shufﬂe of A and B be the language {w| w = a1b1 ···akbk, where a1 ···ak ∈ A and b1 ···bk ∈ B, each ai,bi ∈ Σ}. prove that the perfect shuffle of A and B is regular We will create a NFA in the following manner: […]

Let $[k] = \{0,\dots,k-1\}$. Consider the set $F(n,m)$ of functions $f:[n]\times[m]\rightarrow[m]$. The cardinality of $F(n,m)$ is $|F(n,m)| = m^{nm}$. Consider the equivalence relation $f \simeq g$ between functions $f,g \in F(n,m)$ iff there are permutations $\pi:[n]\rightarrow [n]$ and $\tau:[m]\rightarrow [m]$ such that $\tau(f(n,m)) = g(\pi(n),\tau(m))$ (see Harary/Palmer: Enumeration of Finite Automata). Harary/Palmer give an explicit […]

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