Computable Functions


Can a Turing Machine give more output than only accept or reject?

	yes, the symbols left on the TM tape can be treated as output

How can a Turing Machine compute a function?

	the TM starts with the input to the function on the tape
	the TM halts with the output of the function on the tape

What's a Computable Function?

	if some TM M, on every input w, halts with f(w) on its tape
	then f is a computable function

















5.3 Mapping Reducibility


What does it mean to say language A is mapping-reducible
(or many-one reducible) to language B?

	if there is a computable function f,
	where for every w, w is in A if and only if f(w) is in B
	then language A is mapping-reducible to language B


What notation is used to indicate A mapping-reduces to B?

















If A mapping-reduces to B and B is Decidable,
what do you know about A?

How can you use a mapping reduction to show a problem is Decidable?

	give a mapping reduction TO a Known-Decidable problem

How do you build a decider for A,
given a decider for B and a reduction from A to B?

	M is a decider for B
	f is a reduction from A to B
	N is a decider for A

	N = "On input w:
	  1. Compute f(w).
	  2. Run M on input f(w) and output what M outputs."

















If A mapping-reduces to B and A is Undecidable,
what do you know about B?


How can you use a mapping reduction to show a problem is Undecidable?


	give a mapping reduction FROM a Known-Undecidable problem

















The Halting Problem


What's the Halting Problem for Turing Machines?

	HALT[TM] = { <M,w> | M is a TM and M halts on input w }

How do you show A[TM] mapping-reduces to HALT[TM]?

	describe a function with input <M,w> and output <M2,w2>
	such that <M,w> is in A[TM] if and only if <M2,w2> in HALT[TM]


Describe a machine that computes a reduction from A[TM] to HALT[TM].

	F = "On input <M,w>:
	  1. Construct TM M2
	     M2 = "On input x:
		1. Run M on x.           (if M loops, M2 will loop)
		2. If M accepts, accept. (M2 could reject, as long as it halts)
		3. If M rejects, loop."
	  2. Output <M2,w>."

If <M,w> is in A[TM], how do you show <M2,w> is in HALT[TM]?

	if <M,w> is in A[TM], M will accept,
	so M2 will halt, and <M2,w> is in HALT[TM].

If <M,w> is not in A[TM], how do you show <M2,w> is not in HALT[TM]?

	if <M,w> is not in A[TM], M will either loop or reject.
	if M loops, M2 will not halt.
	if M rejects, M2 will not halt.
	In either case <M2,w> is not in HALT[TM].


How do you write the proof?
What's the format to use for the proof?
Suppose you are proving B is Undecidable using a reduction from A.

	1. State that you will prove B is undeciable
	   by showing A mapping-reduces to B.
	2. Describe a function that maps any w in A to f(w) in B.
	   Explain why if w is in A, f(w) must be in B.
	   Explain why if w is not in A, f(w) must be not in B.
	3. State: A is Undecidable, A reduces to B, so B is Undecidable.


Write the proof that HALT[TM] is Undecidable.

	The TM F reduces A[TM] to HALT[TM].

	F = "On input <M,w>:
	  1. Construct TM M2
	     M2 = "On input x:
		1. Run M on x.           (if M loops, M2 will loop)
		2. If M accepts, accept. (M2 could reject, as long as it halts)
		3. If M rejects, loop."
	  2. Output <M2,w>."

	if <M,w> is in A[TM], M will accept,
	so M2 will halt, and <M2,w> is in HALT[TM].

	if <M,w> is not in A[TM], M will either loop or reject.
	if M loops, M2 will not halt.
	if M rejects, M2 will not halt.
	In either case <M2,w> is not in HALT[TM].

	A[TM] is Undecidable, A[TM] reduces to HALT[TM],
	so HALT[TM] is Undecidable.


How is a proof using mapping reduction
different from previous proofs using reduction?

















Classwork
You may work with a partner.

Use a mapping reduction to prove the language is Undecidable.


	L2 = { <M> | M is a TM and M accepts the string 1001
	       (note that M could accept other strings as well) }