Context-Free Grammars – Derivations

Lecture 15Section 2.1

Mon, Sep 24, 2007

Context-Free Grammars

• A context-free grammar is a grammar whose rules have the following properties:• The left-hand is a single variable.• The right-hand side is any string of

variables and terminals.

Example

• Let V and be• V = {E}, = {a, b, c, +, *, (, )}.

• Let R be• E E + E | E * E | (E) | a | b | c

Examples

• Find CFGs for the following languages.• {anbn | n 0}• {anbm | n m 0}• {anbm | m n 0}

Equal No. of a’s and b’s

• Find CFGs for the language{w | w contains an equal number of

a’s and b’s}

Equal No. of a’s and b’s

• Let w be a string with an equal number of a’s and b’s.

• There are four cases:• w = aza• w = azb• w = bza• w = bzb.

Equal No. of a’s and b’s

• In each case, what do we know about z?

Equal No. of a’s and b’s

• In the cases azb and bza, z is again a string with an equal number of a’s and b’s.

Equal No. of a’s and b’s

• In the cases aza and bzb, the string w must necessarily split into a concatenation of strings w1w2, where w1 and w2 each have an equal number of a’s and b’s.

• Why?

Derivations

• A string u yields a string v if we can apply a grammar rule to u and get v.

• We write u v.

Derivations

• We say that u derives v if there is a sequence u1, u2, …, uk (k 0), where u = u1, v = uk, and

u1 u2 … uk.

• We write u * v.• The sequence is called a

derivation.

Leftmost and Rightmost Derivations

• A leftmost derivation is a derivation in which, at each step, the leftmost variable is replaced with a string.

• Similarly for a rightmost derivation.

Example

• Using the grammar S SS | aSb | bSa |

find leftmost and rightmost derivations of abab.

Ambiguity

• Some grammars provide more than one way to derive a string.

• For example, abab can be derived in two different ways using the grammar rules

S SS | aSb | bSa | • (Typically the leftmost and

rightmost derivations are different.)

Ambiguity

• If a language contains a string that has more than one leftmost derivation, then the grammar is ambiguous.

• If every grammar for a language is ambiguous, then the language is inherently ambiguous.

Ambiguity

• Consider again the grammar rulesS SS | aSb | bSa |

• Suppose that, for every string, we need to associate with each a a unique b.

• Does the grammar show us how to do this?

Ambiguity

• Consider the grammarE E + E | E * E | (E) | a | b | c

• Derive the string a + b * c in two different ways.

• Is this grammar ambiguous?• Is this language inherently

ambiguous?

Ambiguity

• See Example 2.4 on p. 103 an unambiguous grammar for this language.