Download presentation

Presentation is loading. Please wait.

Published byVernon Waithe Modified over 3 years ago

1
Context free languages 1. Equivalence of context free grammars 2. Normal forms

2
Context-free grammars In a context free grammar, all productions are of the form A -> w, where A is a nonterminal or the start symbol S, and w is a string from (N T)*

3
Handles, recursive productions In the production A -> xw, prefix x, if a single symbol, is called the handle of the production, whether x is in N or T A production A -> Aw is called left-recursive The production A -> wA is called right recursive

4
Repeated sentential forms In a derivation, the sentential form wAx S-> … -> wAx -> … -> wAx -> … is called a repeated sentential form. All the intervening steps are wasted steps.

5
Leftmost derivations Minimal leftmost derivations A derivation is a leftmost derivation if at each step only the leftmost nonterminal symbol is replaced using some rule of the grammar. A leftmost derivation is called minimal if no sentential form is repeated in the derivation

6
Weak equivalence Two context-free grammars G1 and G2 are called weakly equivalent if L(G1) = L(G2)

7
Example of weak equivalence G1: S -> S01; S -> 1 L(G1) = { 1(01)* } G2: S -> S0S; S -> 1 L(G2) = { 1(01)* }

8
Strong equivalence Two CFGs G1 and G2 are called strongly equivalent if they are weakly equivalent, and for each string w of terminals in L(G1) = L(G2), and the minimal left-most derivations of w in G1 and the minimal left-most derivations of w in G2 are exactly the same in number, and so can be put into one-to-one correspondence.

9
Strong equivalence Thus G1 and G2 must both be unambiguous, or must both be ambiguous in exactly the same number of ways, for each string w in T*

10
Weakly equivalent but not strongly equivalent G1: Grammar of expressions S: S -> T | S + T; T -> F | T * F; F -> a | ( S ); G2: Grammar of expressions S: S -> E; E -> E + E | E * E | (E) | a; L(G1) = L(G2) = valid expressions using a, +, *, (, and ). G1 has operator precedence.

11
Example: Strong equivalence G1: S->A; A->1B; A->1; B->0A L(G1) = { (10)*1 } G2: S->B; B->A1; B->1; A->B0 L(G2) = { 1(01)* }

12
Elementary transformations of context free grammars substitution expansion removal of useless productions removal of non-generative productions removal of left recursive productions

13
Substitution If G has the A-rule, A->uBv, and all the B-rules are: B->w 1, B->w 2,..., B->w k, then 1. Remove the A-rule A->uBv 2. Add the A-rules: A->uw 1 v, A->uw 2 v,..., A->uw k v 3. Keep all the other rules of G, including the B-rules

14
Example of substitution G1: S->H; H->TT; T->S; T->aSb; T->c G2: S->H; H->ST; H->aSbT; H->cT; T->S; T->aSb; T->c;

15
Strong equivalence after substitution The grammar G, and the grammar G’ obtained by substitution of B into the A- rule, are strongly equivalent if steps 2 and 3 do not introduce duplicate rules.

16
Expansion If a grammar has the A-rule, A->uv Remove this A-rule, and replace it with the two rules A->Xv; X->u; or with A->uY; Y->v where X (or Y) is a new non-terminal symbol of the grammar.

17
Strong equivalence after expansion If G is context free, and G’ is obtained from G by expansion, then G and G’ are strongly equivalent.

18
Useful production A production A->w of a cfg G is useful if there is a string x from T* such that S->.. -> uAv -> uwv ->.. -> x Otherwise the production, A->w is useless Thus, a production that is never used to derive a string of terminals is useless

19
Removing useless productions T-marking S-marking Productions that are both T-marked and S-marked are useful. All other productions can be removed.

20
T-marking Construct a sequence P0, P1, P2,..., of subsets of P, and a sequence N0, N1, N2,... of subsets of N as follows: P0 = empty, N0 = empty, j = 0 P[j+1] = { A->w|w in (N[j] + T)* } N[j+1] = { A in N | P[j+1] contains a rule A->w } Continue until P[j] = P[j+1] = P[T]

21
S-marking Construct a sequence Q1, Q2, Q3,... of subsets of P[T] as follows: Q1 = {S->w in P[T]} Q[j+1] = Q[j] + {A->w in P[T] | Q[j] contains a rule B->uAv } Continue until Q[j] = Q[j+1] = P[S] P[S] are now the useful productions.

22
Example: T/S-marking RuleT markS mark 1. S->H21 2. H->AB 3. H->aH22 4. H->a12 5. B->Hb2 6. C->aC Thus only 1,3,4 are useful

23
Strong equivalence after removal of useless productions If grammar G’ is obtained from grammar G after removal of useless productions of grammar G, then G and G’ are strongly equivalent.

24
Removing non-generative productions

25
Removing left-recursive rules Let all the X-rules of grammar G be: X->u 1 | u 2 |... | u k X->Xw 1 | Xw 2 |... | Xw h Then these rules may be replaced by the following: X->u 1 | u 2 |... | u k X->u 1 Z | u 2 Z |... | u k Z Z->w 1 | w 2 |... | w h Z->w 1 Z | w 2 Z |... | w h Z where Z is a new non-terminal symbol

26
Example: Removing left- recursive rules S->E;S->E; E->T | aT | bT;E->T | aT | bT; E->EaT | EbT;E->TG | aTG | bTG; T->F;G->aT | bT; T->TcF | TdF;G->aTG | bTG; F->n | xEyT->F; T->FH; H->cF | dF; H->cFH | dFH; F->n | xEy

27
Strong equivalence after removal of left-recursive rules If grammar G’ is obtained from grammar G by replacing the left- recursive rules of G by right recursive rules to get G’, then G and G’ are strongly equivalent.

28
Well-formed grammars A context free grammar G=(N,T,P,S) is well-formed if each production has one of the forms: S-> S->A A->w where A N and w (N+T)* - N and each production is useful.

29
Example of well-formed grammars Parenthesis grammar S->A; A->AA; A->(A); A->();

30
Chomsky Normal form A context free grammar G=(N,T,P,S) is in normal form (Chomsky normal form) if each production has one of the forms: S-> S->A A->BC A->a where A,B,C N and a T.

31
Example of Chomsky normal form grammar Parenthesis grammar S->A;S->A; A->AA;A->AA; A->(A);A->BC; B-> (; C->AD; D->); A->();A->BD;

32
Chomsky Normal Form Theorem From any context free grammar, one can construct a strongly equivalent grammar in Chomsky normal form.

33
Greibach normal form (standard form) A context free grammar G=(N,T,P,S) is in standard form (Greibach normal form) if each production has one of the forms: S-> S->A A->aw where A N, a T, and w (N+T)*.

34
Example: converting to Greibach standard form First remove left-recursive rules: S->E;S->E; E->T;E->T; E->EaT;E->TF; T->n;F->aT; T->xEy;F->aTF; T->n; T->xEy;

35
Converting to Greibach: then substitute to get nonterminal handles S->E;S->E; E->T;E->n | xEy; E->TF;E->nF | xEyF; F->aT;F->aT; F->aTF;F->aTF; T->n;T->n; T->xEy;T->xEy;

36
Standard Form Theorem From any context free grammar, one can construct a strongly equivalent grammar in standard form (Greibach normal form).

37
Pumping Lemma for context free languages If L is a context free language, then there exists a positive integer p such that: if w L and |w| > p, then w = xuyvz, with uv and y nonempty and xu k yv k z L for all k 0.

Similar presentations

OK

1 Simplification of Context-Free Grammars Some useful substitution rules. Removing useless productions. Removing -productions. Removing unit-productions.

1 Simplification of Context-Free Grammars Some useful substitution rules. Removing useless productions. Removing -productions. Removing unit-productions.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on different dance forms of india Ppt on international chamber of commerce Ppt on afforestation in india Ppt on faculty development programme iim Ppt on principles of object-oriented programming in java Ppt on holographic technology video Ppt on cross-sectional study define Ppt on articles of association for non-profit Ppt on dry drunk syndrome Ppt on the road not taken meaning