High Level description of Turing Machines

Question

How can create a Turing machine that checks whether or not an input string is a well-defined regular expression? For example, it recognizes a language that consists of string over {0,1} and the symbols used to write down REs (U,*,+, ...)

Answer 1

Using a fixed alphabet $\Sigma$, one can define regular expressions over $\Sigma$ inductively:

• $\varnothing$ and each $a \in \Sigma$ are regular expressions.
• if $r$ and $e$ are regular expressions then so are $(r + e)$, $(r \cdot e)$, and $r^\ast$.

From there it is easy to define a context-free grammar for all regular expressions over $\Sigma$ with the following productions:

• $S \to \varnothing \mid a$ for all $a \in \Sigma$
• $S \to (S + S) \mid (S \cdot S) \mid S^\ast$

Hence, a Turing machine that accepts exactly these expressions only need to simulate a pushdown automaton that accepts the language of the grammar, which is essentially just checking whether there is a closing ) for each ( and there are never more ) than ( in each prefix, which is a standard exercise in each course or textbook on formal languages.