Question

Given any regular expression A, is there a way to to transform it to anouther regular expression B, that accepts all the strings and prefixes of strings that A accepts.

for example if /apple/ is the given regular expression, is there a generalized way to convert it to /a|ap|app|appl|apple/

Answer 1

If you’re talking about formal regular expressions (i.e. regular expressions that describe regular languages), then here’s a procedure to convert a regular expression into one that accepts prefixes.

Any regular expression has a DFA; here’s the DFA for /apple/ (with transitions to failure states left out):

To produce a DFA that matches prefixes of strings accepted by this DFA, convert states to accepting states if they lie along paths that lead to accepting states in the original DFA:

There are several methods for reading a regular expression from a DFA. If we use the state removal technique, we arrive at the following DFA:

This corresponds to the regular expression /a|ap|app|appl|apple|/, plus the empty string (since the empty string is a prefix of any regular expression).

The apple example is trivial, but this same technique can be used for more complicated regular expressions. For example, consider /(00)*1(00|1)*/:

This DFA accepts the string 00100 but doesn’t accept 0010101. After converting the appropriate states to final states and combining two identical states, we have

This is equivalent to

from which we can read the regular expression /(00)*(0?|1(1|00)*0?)/, which includes the empty string.

This regular expression rejects 00101 because it causes the original DFA to transition into a failing state, but accepts ‘0’ and ’00’, because those strings do not cause the original DFA to enter a failure state.