Ambiguous Grammars















What's an Ambiguous Grammar?

How can you show that a grammar is ambiguous?

Use the string 2*3+6 to show the grammar is ambiguous.

	E -> E + E
	E -> E - E
	E -> E * E
	E -> E / E
	E -> ( E )
	E -> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

Does this ambiguity cause any problems?

Which is correct based on Operator Precedence?















Use the string 3-4+5 to show the grammar is ambiguous.

	E -> E + E
	E -> E - E
	E -> E * E
	E -> E / E
	E -> ( E )
	E -> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

Does this ambiguity cause any problems?

Which is correct based on Operator Associativity?















Precedence Grammars

Rewrite the grammar to give correct precedence.

	E -> E + E
	E -> E - E
	E -> E * E
	E -> E / E
	E -> ( E )
	E -> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

Draw parse trees for 2*3+6 and 3-4+5 using the new grammar.

Is the grammar still ambiguous?

Does the grammar give correct precedence?

Does the grammar give correct associativity?

Rewrite the grammar to give correct associativity.















How would you write the grammar for an operator
that associates from right to left?

	E -> E + T   gives left to right associativity
	E -> T + E   gives right to left associativity

What kind of associativity is provided by this grammar?

	E -> T + T
	E -> T - T
	E -> T
	T -> F * F
	T -> F / F
	T -> F
	F -> ( E )
	F -> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
















Classwork
You may work with a partner.

The following grammar for boolean expressions is ambiguous.
Rewrite the grammar so that it is unambiguous.
Use the following rules for precedence and associativity.
Order of precedence is 'not' (highest), 'and' (middle), 'or' (lowest).
The binary operators 'and' and 'or' associate left-to-right.
The unary operator 'not' associates right-to-left.

	B -> B and B
	B -> B or B
	B -> not B
	B -> ( B )
	B -> true | false