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

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FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453
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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?
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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)
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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?)
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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?
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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?
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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
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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?
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