lecture 19 - iitkhome.iitk.ac.in/~akasha/mth401/lectures/lecture19.pdfcfg pdm akash anand Ø let l =...
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MTH401ATheory of Computation
Lecture 19
CFG and PDM
Akash Anand
Equivalence of Context Free Languages and Pushdown
Machines
Conversion of CFG to PDMConversion of PDM to CFG
CFG ➜ PDM
Akash Anand
Ø Let L = L(G)Ø Construct PDM P such that N(P) = L
CFG ➜ PDM
Akash Anand
Ø Let L = L(G)Ø Construct PDM P such that N(P) = L
Ø P has :q One state qq Input symbols = terminals of Gq Stack symbols = all symbols of Gq Start symbol = start symbol of Gq Transitions ??
CFG ➜ PDM
Akash Anand
Ø P has :q Transitions :
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then
δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
Example : Grammar, G : S → 0S0 | 1S1 | ε.
Pushdown Machine, P :States, Q = {q}Input Symbols, Σ= {0, 1}Stack Symbols, Γ= {0,1, S}Start State, q0 = qFinal State, F = {q}
CFG ➜ PDM
Akash Anand
Example : Grammar, G : S → 0S0 | 1S1 | ε.
Pushdown Machine, P :Start Symbol, Z0 = S
CFG ➜ PDM
Akash Anand
Example : Grammar, G : S → 0S0 | 1S1 | ε.
Pushdown Machine, P :Start Symbol, Z0 = STransition Function, δ :
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
Example : Grammar, G : S → 0S0 | 1S1 | ε.
Pushdown Machine, P :Start Symbol, Z0 = STransition Function, δ :δ(q, 0, 0) = {(q, ε)}δ(q, 1, 1) = {(q, ε)}δ(q, ε, S) = {(q, 0S0), (q, 1S1), (q, ε)}
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1 1 0 1
q
CFG ➜ PDM
Akash Anand
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1 1 0 1
q
δ(q, ε, S) = {(q,0S0), (q,1S1), (q, ε)}
CFG ➜ PDM
Akash Anand
1
S
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1 1 0 1
q
CFG ➜ PDM
Akash Anand
1
S
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1 1 0 1
q
δ(q, 1, 1) = {(q, ε)}
CFG ➜ PDM
Akash Anand
1
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
0 1 1 0 1
q
CFG ➜ PDM
Akash Anand
1
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
0 1 1 0 1
q
δ(q, ε, S) = {(q,0S0), (q,1S1), (q, ε)}
CFG ➜ PDM
Akash Anand
1
0
S
0
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
0 1 1 0 1
q
CFG ➜ PDM
Akash Anand
1
0
S
0
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
0 1 1 0 1
q
δ(q, 0, 0) = {(q, ε)}
CFG ➜ PDM
Akash Anand
1
0
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 1 0 1
q
CFG ➜ PDM
Akash Anand
1
0
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 1 0 1
q
δ(q, ε, S) = {(q,0S0), (q,1S1), (q, ε)}
CFG ➜ PDM
Akash Anand
1
0
1
S
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 1 0 1
q
CFG ➜ PDM
Akash Anand
1
0
1
S
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 1 0 1
q
δ(q, 1, 1) = {(q, ε)}
CFG ➜ PDM
Akash Anand
1
0
1
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1
q
CFG ➜ PDM
Akash Anand
1
0
1
S
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1
q
δ(q, ε, S) = {(q,0S0), (q,1S1), (q, ε)}
CFG ➜ PDM
Akash Anand
1
0
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1
q
CFG ➜ PDM
Akash Anand
1
0
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1 0 1
q
δ(q, 1, 1) = {(q, ε)}
CFG ➜ PDM
Akash Anand
1
0
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
0 1
q
CFG ➜ PDM
Akash Anand
1
0
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
0 1
q
δ(q, 0, 0) = {(q, ε)}
CFG ➜ PDM
Akash Anand
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1
q
CFG ➜ PDM
Akash Anand
1
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
1
q
δ(q, 1, 1) = {(q, ε)}
CFG ➜ PDM
Akash Anand
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
q
CFG ➜ PDM
Akash Anand
Example : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
q
Accept !
CFG ➜ PDM
Akash Anand
Example (continued) : 101101 : S⇒1S1⇒10S01⇒101S101⇒101101
(q, 101101, S) ⊦ (q, 101101, 1S1) ⊦(q, 01101, S1) ⊦ (q, 01101, 0S01) ⊦(q, 1101, S01) ⊦ (q, 1101, 1S101) ⊦(q, 101, S101) ⊦ (q, 101, 101) ⊦(q, 01, 01) ⊦ (q, 1, 1) ⊦ (q, ε, ε).
CFG ➜ PDM
Akash Anand
Ø Formally :
(q,wx,S) ⊦* (q,x,α) ⇔ S ⇒* wα
CFG ➜ PDM
Akash Anand
Ø Formally :
(q,wx,S) ⊦* (q,x,α) ⇔ S ⇒* wα
( ⇒ ) is an induction on the number of steps made by P.
CFG ➜ PDM
Akash Anand
Ø Formally :
(q,wx,S) ⊦* (q,x,α) ⇔ S ⇒* wα
( ⇒ ) is an induction on the number of steps made by P.
Basis : (0 steps)Then, (q,wx,S) ⊦* (q,x,α) andS ⇒* wα are true with α=S, w=ε.
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
Induction : Consider n moves of P (q,wx,S) ⊦* (q,x,α) and assume the IH for n-1 moves.
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
Induction : Consider n moves of P (q,wx,S) ⊦* (q,x,α) and assume the IH for n-1 moves.
There are two cases, depending on whether the last move uses a type 1 or type 2 rule.
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
Induction : Consider n moves of P (q,wx,S) ⊦* (q,x,α) and assume the IH for n-1 moves.
There are two cases, depending on whether the last move uses a type 1 or type 2 rule.
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
1. The move sequence must be of the form (q,yax,S) ⊦* (q,ax,aα) ⊦(q,x,α), where w = ya.
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
1. The move sequence must be of the form (q,yax,S) ⊦* (q,ax,aα) ⊦(q,x,α), where w = ya.By IH (applied to first n-1 steps), S ⇒* yaα but w = ya so S ⇒* wα.
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
2. The move sequence must be of the form (q,wx,S) ⊦* (q,x,Aβ) ⊦(q,x,γβ), where A → γ is a production and α = γβ.
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
( ⇒ ) continued …
2. The move sequence must be of the form (q,wx,S) ⊦* (q,x,Aβ) ⊦(q,x,γβ), where A → γ is a production and α = γβ.By IH (applied to first n-1 steps), S ⇒* wAβ. Thus S ⇒* wγβ = wα.
1. δ(q, a, a) = {(q, ε)}.2. If A → α is a production of G then δ(q, ε, A) contains (q, α).
CFG ➜ PDM
Akash Anand
Ø (q,wx,S) ⊦* (q,x,α) ⇔ S ⇒* wα
(⇐) is an induction on the number of steps in the leftmost derivation.
CFG ➜ PDM
Akash Anand
Ø (q,wx,S) ⊦* (q,x,α) ⇔ S ⇒* wα
(⇐) is an induction on the number of steps in the leftmost derivation.
Similar idea! Complete the proof.
CFG ➜ PDM
Akash Anand
Ø (q,wx,S) ⊦* (q,x,α) ⇔ S ⇒* wα
(⇐) is an induction on the number of steps in the leftmost derivation.
Similar idea! Complete the proof.ü If it helps, you can assume that
the grammar is in Chomsky Normal form.
PDM ➜ GFG
Akash Anand
PDM ➜ GFG
Akash Anand
Ø Let L = N(P)Ø Construct CFG G such that L(G) = L
PDM ➜ GFG
Akash Anand
Ø Let L = N(P)Ø Construct CFG G such that L(G) = L
Ø If P accepts w by (p,w,X) ⊦* (q,ε,ε) then we need to have a variable that generates w and its productions :q Let’s call this variable [pXq]q [pXq] generates all and only those
strings w such that (p,w,X) ⊦* (q,ε,ε)
PDM ➜ GFG
Akash Anand
X
Simplest Case :
p
δ(p, a, X) = {…, (q, ε), …}
a
PDM ➜ GFG
Akash Anand
Simplest Case :
q
PDM ➜ GFG
Akash Anand
Simplest Case :
Then the production is : [pXq] → a
q
PDM ➜ GFG
Akash Anand
Next Simplest Case :
δ(p, a, X) = {…, (r, Y), …}
αn
:
α2
α1
X
p
a w1 w2… wn
PDM ➜ GFG
Akash Anand
αn
:
α2
α1
Y
Next Simplest Case :
r
w1 w2 … wn
PDM ➜ GFG
Akash Anand
αn
:
α2
α1
Y
Next Simplest Case :
Then the production is : [pXq] → a [rYq]
r
w1 w2 … wn