Context free languages

1. Equivalence of context free grammars2. Normal forms

Context-free grammars In a context free grammar, all

productions are of the formA -> w,where A is a nonterminal or the start symbol S, and w is a string from(N T)*

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 productionA -> Awis called left-recursive

The productionA -> wAis called right recursive

Repeated sentential forms

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

Leftmost derivationsMinimal 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

Weak equivalence Two context-free grammars G1

and G2 are called weakly equivalent ifL(G1) = L(G2)

Example of weak equivalence G1: S -> S01; S -> 1

L(G1) = { 1(01)* } G2: S -> S0S; S -> 1

L(G2) = { 1(01)* }

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.

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*

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.

Example: Strong equivalence G1: S->A; A->1B; A->1; B->0A

L(G1) = { (10)*1 }

G2: S->B; B->A1; B->1; A->B0L(G2) = { 1(01)* }

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

productions removal of left recursive

productions

Substitution If G has the A-rule, A->uBv,

and all the B-rules are:B->w1, B->w2, . . . , B->wk, then

1. Remove the A-rule A->uBv2. Add the A-rules: A->uw1v, A->uw2v, . . . , A->uwkv3. Keep all the other rules of G, including the B-rules

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;

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.

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.

Strong equivalence after expansion If G is context free, and G’ is

obtained from G by expansion, then G and G’ are strongly equivalent.

Useful production A production A->w of a cfg G is useful

if there is a string x from T* such thatS-> . . -> 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

Removing useless productions T-marking S-marking Productions that are both T-

marked and S-marked are useful. All other productions can be removed.

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]

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.

Example: T/S-marking Rule T mark S mark

1. S->H 2 12. H->AB3. H->aH 2 24. H->a 1 25. B->Hb 26. C->aC

Thus only 1,3,4 are useful

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.

Removing non-generative productions

Removing left-recursive rules Let all the X-rules of grammar G be:

X->u1 | u2 | . . . | uk

X->Xw1 | Xw2 | . . . | Xwh

Then these rules may be replaced by the following:X->u1 | u2 | . . . | uk

X->u1Z | u2Z | . . . | ukZZ->w1 | w2 | . . . | wh

Z->w1Z | w2Z | . . . | whZwhere Z is a new non-terminal symbol

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 | xEy T->F;

T->FH;H->cF | dF;H->cFH | dFH;

F->n | xEy

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.

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->AA->wwhere A N and w (N+T)* - N and each production is useful.

Example of well-formed grammars Parenthesis grammar

S->A;A->AA;A->(A);A->();

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->AA->BCA->awhere A,B,C N and a T.

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;

Chomsky Normal Form Theorem From any context free grammar,

one can construct a strongly equivalent grammar in Chomsky normal form.

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->AA->awwhere A N, a T, and w (N+T)*.

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;

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;

Standard Form Theorem From any context free grammar,

one can construct a strongly equivalent grammar in standard form (Greibach normal form).

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, thenw = xuyvz, with uv and y nonempty and xukyvkz L for all k 0.