CS Rewriting System - grammars, fa, and PDA

Transcription

1 Restricted version of PDA If (p, γ) δ(q, a, X), a Σ {ε}, p. q Q, X Γ. restrict γ Γ in three ways: i) if γ = YX, (q, ay, Xβ) (p, y, YXβ) push Y Γ, ii) if γ = X, (q, ay, Xβ) (p, y, Xβ) no change on stack, iii) if γ = ε, (q, ay, Xβ) (p, y, β) pop X Γ. We can simulate normal PDA with restricted PDA Extended version of PDA δ: (Q Σ Γ ) 2 (Q Σ Γ ) or δ (Q Σ Γ ) (Q Σ Γ ). if (p, x, β) δ(q, xy, α), then (q, xyz, αγ) (p, yz, βγ). x, y Σ, α, β Γ. (Note that we can not write on the input tape). We can simulate extended PDA with normal(restricted) PDA 12/21/11 Kwang-Moo Choe 1

2 R = (A, P) is a rewriting system on A, if P A A or α β P where α, β A. γαδ γβδ, if α β P where α, β, γ, δ A. We say γαδ is rewritten(derived) to γβδ. A (unrestricted) grammar G = (V, T, P, S) is a rewriting system R = (V T, P) where P (V T) (V T). unrestricted grammar is a rewriting ststem on (V T). S γαδ α β γβδ xyz T. A context-free grammar is a restricted rewriting system on (V T). with A β P where A V and β (V T). S lm xaγ A β xβγ lm xyγ lm xyz T. S rm αaz rm αβz rm αyz rm xyz T. 12/21/11 Kwang-Moo Choe 2

3 Finite automata and regular grammars are also rewriting systems Finite automata is a rewriting system on Q Σ. (q S, xyz) (q A, yz) (q B, z) (q F, ε). (q A, y) (q B, ε) δ. (state, remained input string) Q Σ. δ (δ (q S, x), yz)) = δ (δ(q A, y), z) = δ (q B, z) = q F F. q B δ(q A, y) We may use (q, x) (p, ε) or q xp instead of p δ(q, x). Regular grammar. S xa xyb xyz. A yb P. Regular grammar is a rewriting system on (V {ε}) T. (S, xyz) (A, yz) (B, z) (ε, ε). (A, y) (B, ε) P. (nonterminal if any, remained input string) (V {ε}) T. Consider the difference and similarity of rewriting systems RG on (V {ε}) T, FA on Q Σ, respectively 12/21/11 Kwang-Moo Choe 3

4 Leftmost derivation is a rewriting system on (V T). S lm xaγ lm xβγ lm xyγ lm xyz. Consider the position( ) and remained stack and input string pair. S lm x Aγ lm x βγ lm xy γ lm xyz. (S, xyz) L (Aγ, yz) A β (βγ, yz) L (γ, z) L (ε, ε). Guess and verify(left) parser in PDA L is a rewriting system on (V T) T (stack, input). guess A as β (A, ε) L (β, ε), A β P. verify a (a, a) L (ε, ε), a Σ. 12/21/11 Kwang-Moo Choe 4

5 Rightmost derivationis a rewriting system on (V T). S rm αaz rm αβz rm αyz rm xyz. Consider the rightmost derivation in reversed order. xyz rm αyz rm αβz rm αaz rm S. Consider the position( ) and remained stack and input string pair. xyz rm α yz rm αβ z rm αa z rm S. (ε, xyz) R (α R, yz) R (α R β R, z) β A (α R A, z) R (S, ε). Why reversal( R ) in stack contents? rightmost symbol in the derivation should be on the stack top! Shift and reduce(right) parser in PDA R is a rewriting system shift a (ε, a) R (a, ε), a Σ. reduce β to A (β R, ε) R (A, ε), A β P. 12/21/11 Kwang-Moo Choe 5

6 P = (Γ, Σ,, γ 0, F) is a modified PDA with no state where i) Γ is a set of stack symbols, ii) Σ is a set of input symbols, iii) is a set of rewriting rules on Γ Σ, where a rule, (α, xy) (β, y), derives the rewriting (αγ, xyz) (βγ, yz). iv) γ 0 Γ is an initial stack content, v) F Γ is a set of final stack contents. Language accepted by a PDA P is L(P) = {x Σ (γ 0, x) (φ, ε), φ F} Initial and final stack contents in PDA 12/21/11 Kwang-Moo Choe 6

7 Top-down(Left) and Bottom-up(Right) Parser Let G = (N, T, P, S) be a cfg. Then Top-down and Bottom-up parser of the G is P T = (N T, T, T, S, {ε}) where A α P, (A, ε) T (α, ε) T guess as α, a T, (a, a) T (ε, ε) T verify a. L(P L ) = {x Σ (S, x) T (ε, ε), ε F} Top down parsing and leftmost derivation P B = (N T, T, B, ε, {S}) where a T, (ε, a) B (a, ε) B shift a onto stack, A α P, (α R, ε) B (A, ε) B reduce α to A. L(P R ) = {x Σ (ε, x) B (S, ε), S F} Bottom up parsing and rightmost derivation in reversed order 12/21/11 Kwang-Moo Choe 7

8 Example a+aa Top-down Parsing(Left Parse) (E, a+aa) Guess E E+T (E+T, a+aa) E E+T P Guess E a (a+t, a+aa) E a P Verify a (+T, +aa) Verify + (T, aa) Guess T TF (TF, aa) T TF P Guess T a (af, aa) T a P Verify a (F, a) Verify (F, a) Guess F a (a, a) F a P Verify a (ε, ε) (a+aa, E) E E+T E a T TF T a F a (ε, ε) Left parse a seq. of leftmost derivation(guess actions) 12/21/11 Kwang-Moo Choe 8

9 Bottup-up Parsing(right parse) (ε, a+aa) Shift a onto stack (a, +aa) Reduce a E (E, +aa) E a P Shift + onto stack (+E, aa) Shift a onto stack (a+e, a) Reduce a T (T+E, a) T a P Shift onto stack (T+E, a) Shift a onto stack (at+e, ε) Reduce a F (FT+E, ε) F a P Reduce FT T (T+E, ε) T TF P Reduce T+E E (E, ε) E E+T P (a+aa, ε) E a T a F a T TF E E+T (ε, E) Right parse a seq. of rightmost derivation in reversed order (reduce actions) 12/21/11 Kwang-Moo Choe 9

10 Deterministic Top down and Bottup up parsing P = (Q, Γ, Σ,, γ 0, F) is a modified PDA with state where i) Q is a set of states Let G = (N, T, P, S) be a cfg. Then determinstic version of left and right parser of the G is and P L = (Q T, S, {ε}) and P B = (Q B, ε, {S}) A α P, q, p Q, (q, A, x) T (p, α, x) T, a T, q, p Q, (q, a, a) T (p, ε, ε) T. LL(k) and strong LL(k) grammars. a T, q, p Q, (q, ε, a) B (p, a, ε) B, A α P, q, p Q, (q, α R, x) B (p, A, x) B. LR(k), LALR(k)(in yacc), and SLR(k) grammars. Adding states and lookahead string x ( x =k) in guess and reduce actions 12/21/11 Kwang-Moo Choe 10