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Context-free Grammars

Learn about context-free grammars, which are more powerful than regular languages. Originally developed by linguists, they play an important role in the compilation of programming languages.

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Context-free Grammars

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

  2. [Section 2.1] Context-free Grammars • 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

  3. [Section 2.1] Context-free Grammars • 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 ŧ = ; • 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 u1,u2,…,uk such that • u => u1 => u2 => … uk => v • The language of the grammar is { w2§* | S=>*w }.

  4. [Section 2.1] Context-free Grammars • Examples: give context-free grammars for the following languages: • { aibjck | i=k, i,j,k ¸ 0 } • { aibjck | 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

  5. [Section 2.1] Ambiguity 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.

  6. [Section 2.1] Ambiguity 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 { aibjck | i=j or j=k } ). These are called inherently ambiguous. Example: give an unambigous CFG for the language of arithmetic expressions over { +, x, a }

  7. [Section 2.1] Closure properties of CFLs Thm: The class of context-free languages (CFLs) is closed under union. Thm: The class of CFLs is closed under concatenation. Thm: The class of CFLs is closed under star.

  8. [not in the book?] Closure properties of CFLs Thm: The class of CFLs is not closed under intersection. Thm: The class of CFLs is closed under intersection with regular languages. Thm: Every regular language is context-free.

  9. [Section 2.1] Chomsky Normal Form 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 ?

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

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