$\mathbf{Let}{\mathbf{BAL}}_{\mathrm{DFA}}=\left\{\left(\mathbf{M}\right)\right|\mathbf{M}$is a DFA that accepts some string containing an equal number of 0s and 1s}. Show that ${\mathbf{BAL}}_{\mathrm{DFA}}$ is decidable.

A context-free grammar is G. The set of all strings in $\mathrm{\Sigma }*$ that is deduced for start variable S in V is the definition of the language of G:

$\mathbf{L}\mathbf{\left(}\mathbf{G}\mathbf{\right)}\mathbf{=}\mathbf{w}\mathbf{}\mathbf{is}\mathbf{}\mathbf{a}\mathbf{}\mathbf{member}\mathbf{}\mathbf{of}\mathbf{\Sigma }\mathbf{*}\mathbf{:}\mathbf{S}\mathbf{=}\mathbf{>}\mathbf{w}\mathbf{.}$

If a Context Free Grammar $G$ exists such that $L\left(G\right)$ Equals L, then the language L is referred to as a Context Free Language. Context Free Language (CFL) is the collection of all strings that is produced using Context Free Grammar (CFG) (CFL).

A context-free language created by the grammar $\text{S}\to \text{1S0S|0S1S|}\in \text{.}$ is the language of all strings with an equal number of 0s and 1s.

Assume $P$ is the PDA that is understood this language. For $BA{L}_{DFA}$ which works as to construct a TM M. Use B and P to create a new $PDAR$ that understands the intersection of the languages of $B$ and $P$ on input (B), where B is a DFA. Next, check to see if R's language is empty. Reject it if the language is meaningless; accept it otherwise.

Hence, $BA{L}_{DFA}$ is decidable.