Product construction for given two finite automata

Question

I need to construct a finite automata which accept a language $L = L_1 \cap L_2$, where $L_1$ and $L_2$ are given below.

$L_1 = \{ w \mid w $ is divisible by 2 }

$L_1 = \{ w \mid w $ is divisible by 3 }

I know how to design a finite automata for a single number but how do I combine the automatons which check divisibility by 2 and 3? I am able to combine the initial states of the two automatons by using ϵ transitions from a new state but how can I combine the final states of the two automatons? Input is in the set of decimal numbers
Σ = {0,1,2,3,4,5,6,7,8,9}

Answer 1

Here is a product construction for finite automata. I am considering you have two finite automata $M_1$ and $M_2$.

Let $M_1$ = $(Q_1,\Sigma ,\delta_1,q_{01},F_1)$ and $M_2$ = $(Q_2,\Sigma ,\delta_2,q_{02},F_2)$ be two finite automata with the same input alphabet. Let us define $M$ such that

$$L(M) = L(M_1) \cap L(M_2) $$

Finite automata for $L(M)$ is defined as $M$ = $(Q,\Sigma ,\delta,q_0,F)$ , where $Q$ is a finite set of states, $\Sigma$ is a input alphabet, $q_0$ is a start state, $\delta$ is a transition function and $F$ is final state of a finite automata $M$.

• $Q$ = $Q_1$ X $Q_2$ (Cartesian product),
• $\delta((q_1,q_2),a) = (\delta_1(q_1,a),\delta_2(q_2,a))$,
• $q_0 = (q_{01},q_{02})$,
• $(q_{1},q_{2}) \in F $ iff $q_1 \in F_1 $ and $q_2 \in F_2$.

Each state in $Q$ is a pair consisting of a state from $Q_1$ and a state from $Q_2$.

Reference : the Product Construction for Automata by Laura Kovács.