Course Notes on Formal Languages and Compilers

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1. LANGUAGES AND GRAMMARS
We shall now formalize the notion of language and grammar.
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Strings and sets of strings
If V is a set, then V* denotes the set of all finite strings of elements of V
including the empty string which will be denoted by ε.
e.g. {0,1}* = {ε, 0, 1, 00, 01, 10, 11, 000, 001,... }
The set of all non empty strings of elements of V is denoted by V+.
Usually, V+ = V* \ {ε}, but when ε∈V, V+ = V*.
e.g. {0,1}+ = {0, 1, 00, 01, 10, 11, 000, 001,... }
but {ε, 0, 1}+ = {0,1}* = {ε, 0, 1, 00, 01, 10, 11, 000, 001,... }
If x∈V* and y∈V* then xy will denote their concatenation, that is, the string
consisting of x followed by y.
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If x∈V* then x n = xxx.....x
n-times
n≥0
We assume x 0 = ε the empty string.
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e.g. {a}* = {ε, a, a 2 , a 3 , ...a n ,.....} = { a n : n ≥ 0}
{a}+ = {a, a 2 , a 3 , ....a n , ....} = { a n : n ≥ 1}
Similarly, if X, Y are sets of strings, then their concatenation is also denoted
by XY. Of course XY={xy: x∈X and y∈Y}.
Also, X n = XXX.....X n≥0. Of course X 0 = {ε}.
n-times
e.g. {0, 1} {a, b, c} = {0a, 0b, 0c, 1a ,1b, 1c}
{0, 1} 3 = {000, 001, 010, 011, 100, 101, 110, 111}
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If x is a string, then |x| denotes the length of x, and this is the number of
indivisible symbols in x. Of course |ε| = 0.
Exercises
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1) Determine the following sets.
(a) {0,1} {ε, a, ba} (b) {b, aa}*
2) Let V be a set of strings. Does V+ = V V*
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