Course Notes on Formal Languages and Compilers

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5. REGULAR EXPRESSIONS
These expressions denote sets and are defined as follows:
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(i)
(ii)
(iii)
(iv)
∅ is a regular expression:
{ε} is a regular expression
{a} is a regular expression where "a" is a terminal symbol.
If R and S are regular expressions then the following are also regular
expressions.
(a) R|S
(b) RS
(c) R* (also S*)
- the union
- the concatenation
- the transitive closures
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In practice we will write ε and not {ε}, "a" and not {a}. Of course the set
{a,b,c,d} is then written (a|b|c|d) and of course {a,b} ∪ {c,d} would be written
(a|b) | (c|d) as regular expressions.
Incidentally, ∅* = ε in regular expressions.
Regular expressions are useful for defining the structure of reserved words,
identifiers, numbers occuring in programming languages but are not suitable
for defining the structure of blocks, expressions, statements etc.
Examples of Regular Expressions:
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1) Unsigned Integers:
(0|1|2|...|9)(0|1|2...|9)*
2) Integers with an optional sign
(ε|+|-)(0|1|2|...|9)(0|1|2|...9)*
3) Identifiers:
(a|b|c|...|z)(a|b|c|...|z|0|1|2...|9)*
Exercises: 1) Write a regular expression defining the structure of a real
number which may include a sign, a decimal point and an
exponent which are optional.
2) Change the definition of the first example above so as to
exclude numbers with unneccessary leading zeroes, i.e. to
exclude numbers such as 000, 00156, etc.
Equivalence between regular expressions and regular grammars
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We shall now show that the sets defined by regular expressions and regular
grammars are equivalent.