How to define an automata for zig zag concatenation? [duplicate]

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Zigzag concatenation of two languages (1 answer)

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I have two DFAs one for language A and one for language B. I'm asked to make an FDA that is the zig-zag concatenation of letters of A and letters of B. This is described by the following: {w: w = $a_1 b_1$...$a_k b_k$ and $a_1...a_k \in A$ and $b_1 ... b_k \in B$}. With $1 \leq i \leq k$ and $a_i \in \Sigma$ and $b_i \in \Sigma$

This automata should be described as a 5-tuple {Q, $\Sigma$,$\delta$,$q_0$,$F$}.

I simply do not know how I would go about defining the total function $\delta$.

This is what I tried:

$Q = Q_A \cap Q_B$ // We only want the words that contains both a and b

$F = F_A \cap F_B$ // The accepted states should contain both letter from a and from b.

$q_0 = q_A$ //because the word starts with a letter from a.

$\Sigma$ // in this problem we aren't interested in defining the alphabet we just leave the symbol as is in the 5-tuple.

Answer 1

let $Q=Q_A\times Q_B\times \{1,2\}$ (1 means A, 2 means B)

also define $F=F_A\times F_B\times \{2\}$ (we end with B)

and $q_0=(q_A,q_B,1)$ (we start with A)

with the delta function: $\delta ((q_1,q_2, i),\sigma)= \begin{cases} (\delta(q_1,\sigma),q_2,2) &i=1 \\ (q_1, \delta(q_2,\sigma),1) &i=2 \end{cases}$

The idea is to encode both A and B together, with some "index" telling us if we need to check A or B