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Lecture11

# Lecture11 - 15-453 FORMAL LANGUAGES AUTOMATA AND...

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FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453

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ATM = { (M,w) | M is a TM that accepts string w } HALTTM = { (M,w) | M is a TM that halts on string w } ETM = { M | M is a TM and L(M) =  } REGTM = { M | M is a TM and L(M) is regular} ALLPDA = { P | P is a PDA and L(P) = Σ* } EQTM = {( M, N) | M, N are TMs and L(M) =L(N)} ALL UNDECIDABLE Use Reductions to Prove Which are SEMI-DECIDABLE?
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 m B , why?

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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)
CLAIM: ATM m HALTTM f: (M,w)  (M’, w) where M’( w) = M(w) if M(w) accepts Loops otherwise So, (M, w ) ATM  (M’, w)  HALTTM ATM = { (M,w) | M is a TM that accepts string w } HALTTM = { (M,w) | M is a TM that halts on string w } CONSTRUCT f : Σ*  Σ* So HALTTM is NOT DECIDABLE, but it is SEMI-DECIDABLE (Why?)

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CLAIM: ATM m  ETM f: (M,w)  Mw where Mw (s) = M(w) if s = w Loops otherwise So, (M, w ) ATM  Mw  ETM ATM = { (M,w) | M is a TM that accepts string w } ETM = { M | M is a TM and L(M) =  } CONSTRUCT f : Σ*  Σ* So, M(w) accepts  L (Mw)   ATM m ETM So  ETM is NOT DECIDABLE, but it is SEMI-DECIDABLE ( why?) Is ETM SEMI-DECIDABLE?
CLAIM: ATM m REGTM f: (M,w)  M’w where M’w (s) = accept if s = 0n1n M(w) otherwise So, (M, w ) ATM  M’w  REGTM ATM = { (M,w) | M is a TM that accepts string w } REGTM = { M | M is a TM and L(M) is regular} CONSTRUCT f : Σ*  Σ* So, L (M’w) = Σ* if M(w) accepts { 0n1n } if not So REGTM is UNDECIDABLE Is REG SEMI-DECIDABLE?

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CLAIM:  ATM m REGTM f: (M,w)  M”w where M”w (s) = accept if s = 0n1n and M(w) accepts Loop otherwise So, (M, w )  ATM  M”w  REGTM ATM = { (M,w) | M is a TM that accepts string w } REGTM = { M | M is a TM and L(M) is regular} CONSTRUCT f : Σ*  Σ* So, L (M’w) = { 0n1n } if M(w) accepts if not So, REG NOT SEMI-DECIDABLE So, REG NOT SEMI-DECIDABLE
CLAIM: ETM m EQTM f: M  (M, M  ) where M  (s) = Loops So, M E TM  (M, M  ) EQTM ETM = { M | M is a TM and L(M) =  } EQTM = {( M, N) | M, N are TMs and L(M) =L(N)} CONSTRUCT f : Σ*  Σ* So EQTM is UNDECIDABLE Is EQTM SEMI-DECIDABLE?

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Lecture11 - 15-453 FORMAL LANGUAGES AUTOMATA AND...

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