Understanding Context-Free Languages and Their Properties

1. CONTEXT FREE LANGUAGE by: Er. Sukhwinder kaur

2. Topics to be discussed… • Context Free Language • Parse Tree • Chomsky Normal Form (CNF) • Greibech normal form(GNF) • Removing null production • Unit Productions • Pushdown Automata: a preview • Pumping Lemma

3. Context Free Language Facts: 1. each non terminal symbol can derive many different strings. 2. Every string in a derivation is called a sentential form. 3. Every sentential form containing no non terminal symbols is called a sentence. 4. The language L(G) generated by a CFG G is the set of sentences derivable from a distinguished non terminal called the start symbol of G. (eg. <stmt> ) 5. A language is said to be context free (or a context free language (CFL)) if it can be generated by a CFG. A sentence may have many different derivations; a grammar is called unambiguous if this cannot happen (eg: previous grammar is unambiguous)

4. CFGs: a formal definition a CFG is a quadruple G = (N,S,P,S) where N is a finite set (of non terminal symbols) S is a finite set (of terminal symbols) disjoint from N. S  N is the start symbol. P is a a finite subset of N x (N S)* (The productions) Conventions: Non terminals: A,B,C,… terminals: a,b,c,… strings in (N S)* : a,b,g,… Each (A,a)  P is called a production rule and is usually written as: A a. A set of rules with the same LHS: A a1 A a2 A a3 can be abbreviated as A a1| a2 | a3. back

5. S S S ) ( S ) ( S e e Parse Tree Features of the parse tree: 1. The root node is [labeled by] the start symbol: S 2. The left to right traversal of all leaves corresponds to the input string : ( ) ( ). 3. If X is an internal node and Y1 Y2 … YK are an left-to-right listing of all its children in the tree, then X --> Y1Y2… Yk is a rule of G. 4. Every step of derivation corresponds to one-level growth of an internal node

6. Leftmost, Rightmost Derivations Definition. A left-most derivation of a sentential form is one in which rules transforming the left-most nonterminal are always applied Definition. A right-most derivation of a sentential form is one in which rules transforming the right-most nonterminal are always applied

7. Trading Left- & Right-Recursion Left recursion: A  A a Right recursion: A a A Most algorithms have trouble with one, In recursive descent, avoid left recursion. back

8. Chomsky Normal Form (CNF) Let G be a CFG for some L-{} Definition: G is said to be in Chomsky Normal Form if all its productions are in one of the following two forms: A  BC where A,B,C are variables, or A  a where a is a terminal G has no useless symbols G has no unit productions G has no -productions

9. CNF checklist • G1: • E  E+T | T*F | (E) | Ia | Ib | I0 | I1 • T  T*F | (E) | Ia | Ib | I0 | I1 • F  (E) | Ia | Ib | I0 | I1 • I  a | b | Ia | Ib | I0 | I1 Is this grammar in CNF? • Checklist: • G has no -productions • G has no unit productions • G has no useless symbols • But… • the normal form for productions is violated So, the grammar is not in CNF back

10. Greibech normal form(GNF) • A CFG is in Greibach normal form if each rule has one these forms: • A  aA1A2…An • A  a • S   where a   and Ai V – {S} for i = 1, 2,…, n back

11. Removing -Productions Remove all  productions: (1) If there is a rule P Q and Q is nullable, Then: Add the rule P. (2) Delete all rules Q. back

12. Unit Productions A unit production is a rule whose right-hand side consists of a single nonterminal symbol. Example: SX Y X A AB | a Bb YT TY | c

13. Removing Unit Productions • removeUnits(G) = • 1. Let G = G. • 2. Until no unit productions remain in G do: • 2.1 Choose some unit production X Y. • 2.2 Remove it from G. • 2.3 Consider only rules that still remain. For every rule Y , • where V*, do: • Add to G the rule X  unless it is a rule that has • already been removed once. • 3. Return G. SX Y Aa | b Bb T c X a | b Y c Example: SX Y X A AB | a Bb Y T T Y | c back

14. Pushdown Automata: a preview FAs recognize regular languages. What kinds of machines recognize CFLs ? ===> Pushdown automata (PDAs) PDA: Like FAs but with an additional stack as working memory. Actions of a PDA 1. Move right one tape cell (as usual FAs) 2. push a symbol onto stack 3. pop a symbol from the stack. Actions of a PDA depend on 1. current state 2. currently scanned I/P symbol 3. current top stack symbol. A string x is accepted by a PDA if it can enter a final state (or clear all stack symbols) after scanning the entire input. More details defer to later chapters. back

15. Pumping Lemma If a language L is accepted by a DFA M with m states, then any string x in L with |x| > m can be written as x = uvw such that (1) v ≠ε, and (2) uv*w is a subset of L (i.e., for any n> 0, uv w in L).

16. Proof • Consider the path associated with x (|x| > m). x Since |x| > m, # of nodes on the path is At least m+1. Therefore, there is a state Appearing twice.

17. v w u v ≠ ε because M is DFA because there is a path associated with uw from initial state to a final state. uw in L n uv w in L due to the same reason as above back

18. Thank You