Lecture11x - 15-453 FORMAL LANGUAGES, AUTOMATA AND...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A TM = { (M,w) | M is a TM that accepts string w } HALT TM = { (M,w) | M is a TM that halts on string w } E TM = { M | M is a TM and L(M) = } REG TM = { M | M is a TM and L(M) is regular} ALL PDA = { P | P is a PDA and L(P) = Σ* } EQ TM = {( M, N) | M, N are TMs and L(M) =L(N)} ALL UNDECIDABLE Use Reductions to Prove Which are SEMI-DECIDABLE?
Background image of page 2
A B f f Let f : Σ* Σ* be a computable function such that w A f(w) B Say: A is mapping reducible to B; Write: A m B Σ* Σ* Also, ¬ A ¬ B , why?
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Theorem: If A m B and B is ( semi ) decidable, then A is ( semi ) decidable Proof: Let M decide B and let f be a reduction from A to B We build a machine N that decides A as follows: On input w: 1. Compute f (w) 2. Run M on f (w)
Background image of page 4
CLAIM: A TM m HALT TM f: (M,w) (M’, w) where M’( w) = M(w) if M(w) accepts Loops otherwise So, (M, w ) A TM (M’, w) HALT TM A TM = { (M,w) | M is a TM that accepts string w } HALT TM = { (M,w) | M is a TM that halts on string w } CONSTRUCT f : Σ* Σ* So HALT TM is NOT DECIDABLE, but it is SEMI- DECIDABLE (Why?)
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
CLAIM: A TM m ¬ E TM f: (M,w) M w where M w (s) = M(w) if s = w Loops otherwise So, (M, w ) A TM M w ¬ E TM A TM = { (M,w) | M is a TM that accepts string w } E TM = { M | M is a TM and L(M) = } CONSTRUCT f : Σ* Σ* So, M(w) accepts L (M w ) ¬ A TM m E TM So ¬ E TM is NOT DECIDABLE, but it is SEMI- DECIDABLE ( why?) Is E TM SEMI-DECIDABLE?
Background image of page 6
CLAIM: A TM m REG TM f: (M,w) M’ w where M’ w (s) = accept if s = 0 n 1 n M(w) otherwise So, (M, w ) A TM M’ w REG TM A TM = { (M,w) | M is a TM that accepts string w } REG TM = { M | M is a TM and L(M) is regular} CONSTRUCT f : Σ* Σ* So, L (M’ w ) = Σ* if M(w) accepts { 0 n 1 n } if not So REG TM is UNDECIDABLE Is REG SEMI-DECIDABLE?
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
CLAIM: ¬ A TM m REG TM f: (M,w) M” w where M” w (s) = accept if s = 0 n 1 n and M(w) accepts Loop otherwise So, (M, w ) A TM M” w REG TM A TM = { (M,w) | M is a TM that accepts string w } REG TM = { M | M is a TM and L(M) is regular} CONSTRUCT f : Σ* Σ* So, L (M’ w ) = { 0 n 1 n } if M(w) accepts if not So, REG NOT SEMI-DECIDABLE So, REG NOT SEMI-DECIDABLE
Background image of page 8
CLAIM: E TM m EQ TM f: M (M, M ) where M (s) = Loops So, M E TM (M, M ) EQ TM E TM = { M | M is a TM and L(M) = } EQ TM = {( M, N) | M, N are TMs and L(M) =L(N)} CONSTRUCT f : Σ* Σ* So EQ TM is UNDECIDABLE Is EQ TM SEMI-DECIDABLE?
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 52

Lecture11x - 15-453 FORMAL LANGUAGES, AUTOMATA AND...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online