Lecture 15 Naveen Z Quazilbash Ambiguity

Overview S-Grammars Ambiguity in Grammars Ambiguous grammars and Unambiguous Grammars

S-grammars Simple grammars a.k.a s-grammars. Rules for s-grammars: Productions of the form A  sB 1 B 2 B 3 ….B n,where s is a symbol and B 1 B 2 B 3 ….B n are variables. For each variable A and symbol s, there is at most one production of the form: A  sB 1 B 2 B 3 ….B n

S-grammars For example, we can not have A  aBC A  aCD Because we must only have one “A  a…” production.

S-grammars Example of a simple grammar: S  aA|bBC A  aA|bb B  bB|aC C  cC|a w=bbacca Parsing: S=>bBC=>bbBC=>bbaC=>bbacC=>bbaccC=>bbacca

S-grammars We use leftmost derivation. At each step we know which production to apply, by looking at the next symbol in w. Remark: Thus, the derivation takes linear time; that is, the time is proportional to |w|.

Ambiguity

Some important notes Applications of CFG’s (like Parsers, DTDs and markup languages) often rely on the grammar to provide the structure of files. Tacit Assumption: A grammar uniquely determines a structure for each string in its language. Problem: However, that’s not always possible. Solution: Redesign the grammar!!

solution Options: 1. Redesign possibility 2. Redesign impossibility ~ Inherently ambiguous (beyond our scope) So, first, lets try to understand the problem now!

Ambiguity Example 1: S  SA | A A  aAb | ε w=aabb Give it lm and rm derivations and also draw parse trees for both lm and rm derivations.

Ambiguity lm: S=>SA=>AA=>aAbA=>aaAbbA=>aabbA=>aabb rm: S=>SA=>SaAb=>SaaAbb=>Saabb=>Aaabb=>aabb

Ambiguity Example 2: E  I E  E+E E  E*E E  (E) I  a|b|Ia|Ib|I0|I1 w=E+E*E Give it lm and rm derivations and also draw parse trees for both lm and rm derivations.

Ambiguity rm: E=>E+E=>E+E*E lm: E=>E*E=>E+E*E

Ambiguity in Grammars There is a significant difference between the two derivations. What’s that? Hence, this grammar is not a good one for providing unique structures. In particular, while it can give correct groupings to arithmetic expressions, it also gives them incorrect groupings.

Ambiguity in Grammars Lets have a look at another very important factor, i-e “what causes ambiguity?” Multiplicity of derivations? //discussion

Ambiguous Grammars and Unambiguous Grammars A CFG, G={V, T, P, S} is ambiguous if there is at least one string w in T* for which we can find two different parse trees, each with root labeled S and yield w. If each string has at most one parse tree, then the grammar is unambiguous.

Unambiguous Grammars - Avoiding ambiguity Next class