Let Li be the set of strings accepted by the finite-state...

Let $L_{i}$ be the set of strings accepted by the finite-state automaton $A_{i}=\left(\mathcal{I}, \mathcal{S}_{i}, f_{i}, \mathcal{A}_{i}, \sigma_{i}\right), i=1,2 .$ Let $$A=\left(\mathcal{I}, \mathcal{S}_{1} \times \mathcal{S}_{2}, f, \mathcal{A}, \sigma\right)$$ where $$\begin{aligned}f\left(\left(S_{1}, S_{2}\right), x\right) &=\left(f_{1}\left(S_{1}, x\right), f_{2}\left(S_{2}, x\right)\right) \\\mathcal{A} &=\left\{\left(A_{1}, A_{2}\right) \mid A_{1} \in \mathcal{A}_{1} \text { or } A_{2} \in \mathcal{A}_{2}\right\} \\\sigma &=\left(\sigma_{1}, \sigma_{2}\right)\end{aligned}$$ Show that $\operatorname{Ac}(A)=L_{1} \cup L_{2}$.

Let $L_{i}$ be the set of strings accepted by the finite-state automaton $A_{i}=\left(\mathcal{I}, \mathcal{S}_{i}, f_{i}, \mathcal{A}_{i}, \sigma_{i}\right), i=1,2 .$ Let
$$A=\left(\mathcal{I}, \mathcal{S}_{1} \times \mathcal{S}_{2}, f, \mathcal{A}, \sigma\right)$$
where
$$\begin{aligned}f\left(\left(S_{1}, S_{2}\right), x\right) &=\left(f_{1}\left(S_{1}, x\right), f_{2}\left(S_{2}, x\right)\right) \\\mathcal{A} &=\left\{\left(A_{1}, A_{2}\right) \mid A_{1} \in \mathcal{A}_{1} \text { or } A_{2} \in \mathcal{A}_{2}\right\} \\\sigma &=\left(\sigma_{1}, \sigma_{2}\right)\end{aligned}$$
Show that $\operatorname{Ac}(A)=L_{1} \cup L_{2}$.

Key Concepts

Product Construction of Automata

Product construction is a technique used to combine two automata into a single automaton whose state set is the Cartesian product of the state sets of the original automata. This method is often used to design automata that simulate the concurrent processing of input by each component automaton, allowing the combined automaton to recognize languages defined by combinations of the original automata.

Transition Functions in Automata

Transition functions determine how an automaton moves from one state to another upon processing an input symbol. In the product automaton, the transition function is defined such that it applies the original transition functions to each component of the state pair, ensuring that the overall state accurately tracks the progression of each individual automaton's computation.

Language Acceptance

The language accepted by a finite-state automaton consists of all strings that, when processed by the automaton starting from the initial state, cause it to transition into one of its accepting states. This concept is fundamental to understanding how automata recognize and classify sequences of symbols within formal language theory.

Finite-State Automaton

A finite-state automaton (FSA) is a mathematical model used to represent computation. It consists of a finite set of states, transitions between those states based on input symbols, an initial state, and a set of accepting states. FSAs are used to recognize patterns and decide whether a given string belongs to a particular language.

Union of Languages

The union of two languages is a set that includes all strings that belong to at least one of the two languages. In the context of automata, constructing an automaton that recognizes the union of two languages involves ensuring that a string is accepted if it meets the criteria of either of the original automata.

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