Context-free Grammars [Section 2.1] - more powerful than regular languages - originally developed by linguists - important for compilation of programming languages

Context-free Grammars [Section 2.1] Example:A -> 0A1 A -> B B -> # Terminology: - substitution rules (productions) - variables (including the start variable) – typically upper-case - terminals – typically lower-case, other symbols - derivation, parse tree

Context-free Grammars [Section 2.1] Def 2.2: A context-free grammar is a 4-tuple (V, §,R,S), where - V is a finite set of variables - § is a finite set of terminals, V Å § = ; - R is a finite set of rules, each rule is of the form A -> w where A 2 V and w 2 (V [ § )* - S 2 V is the start variable If A -> w 2 P, then we write uAv => uwv (read “uAv yields uwv”), and we write u =>* v (read “u derives v”) if u=v or if there exists a sequence u 1,u 2,…,u k such that u => u 1 => u 2 => … u k => v The language of the grammar is { w 2 § * | S=>*w }.

Context-free Grammars [Section 2.1] Examples: give context-free grammars for the following languages: - { a i b j c k | i=k, i,j,k ¸ 0 } - { a i b j c k | i=j, i,j,k ¸ 0 } - strings over { (, ) } that are well-parenthesized - strings over { 0,1 } that contain equal number of 0’s and 1’s

Ambiguity [Section 2.1] Example: [EXPR] ->[EXPR] + [EXPR] | [EXPR] x [EXPR] | ( [EXPR] ) | a Give a derivation (and parse trees) for the string a+axa. Notice: for every parse tree there is a unique left-most derivation.

Ambiguity [Section 2.1] Def 2.7: A context-free grammar is called ambiguous if there exists a string that can be generated by two different left-most derivations. Note: Some context-free languages do not have unambigous grammars (e.g { a i b j c k | i=j or j=k } ). These are called inherently ambiguous. Example: give an unambigous CFG for the language of arithmetic expressions over { +, x, a }

Ambiguity [Section 2.1] Def 2.7: A context-free grammar is called ambiguous if there exists a string that can be generated by two different left-most derivations. For every x 2 \Sigma let y… L = {a^ib^jc^k | i,j,k\geq 0}. Example: give an unambigous CFG for the language of arithmetic expressions over { +, x, a }

Chomsky Normal Form [Section 2.1] Def 2.8: A CFG is in Chomsky normal form if every rule is of the form A -> BC or A -> a, where B,C 2 V-{S}, and a 2 §. Thm 2.9: Any CFL can be generated by a CFG in Chomsky normal form. Note: what about CFL that contains ε ? Why a normal form ?

Chomsky Normal Form [Section 2.1] Thm 2.9: Any CFL can be generated by a CFG in Chomsky normal form. “Proof” by example: S -> ASA | aBThings to fix: 1. A -> B | S 2. B -> b | ε 3. 4.