Computer Science 500 Theory of Computation Spring, 2000 Feb' 28

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Computer Science 500Theory of ComputationSpring,
2000Feb. 28 1. Reductions! 2. Homework 5 -
1st two problems
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The Universal Turing Machine
Universal Turing Machine MUniversal Input
(M,I) where M is a Turing Machine
Description I is the input to the machine
M Results L(MUniversal) ATM (M,I) M
accepts I) If M accepts I, then MUniversal
accepts (M,I) If M rejects I, then MUniversal
rejects (M,I) If M does not halt on I,
MUniversal does not halt on (M,I)
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Universal Turing Machine(visualization)
Muniversals initial configuration
M (Q, ???????? q0, qaccept, qreject), Input to
M
U U U U ...
M is described in binary
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Universal Turing Machine(visualization)
Muniversals general configuration
q
0 1 0 0 1 0 U U
M (Q, ???????? q0, qaccept, qreject),
Ms state
Ms tape configuration
M is described in binary
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Definition Computable function
A function f is computable iff there is a Turing
machine that, on input x, halts with f(x) on the
tape.
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6 Theorems

• Assume that L1 mapping reduces to L2.
• If L2 is decidable and then L1 is decidable
• If L1 is undecidable and then L1 is undecidable
• If L2 is recognizable and then L1 is
  recognizable
• If L1 is unrecognizable and then L2 is
  unrecognizable
• If L2 is co-recognizable and then L1 is
  co-recognizable
• If L1 is not co-recognizable and then L2 is not
  co-recognizable

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Proof idea -- construct a T.M. For L1 based on
one for L2
M1
w
f(w)
accept
Mf
M2
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Proof

• Let Mf be a Turing Machine that reduces L1 to
  L2
• Let M2 decide/recognize L2

• Define M1. On input w,
• First, run Mf
• Second, run M2 on f(w)
• Accept iff M2 accepts

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Contrapositive
Assume that L1 reduces to L2. If L1 is un...
and then L2 is un...
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Do not try to prove L1 ltm L2if L2 is
recognizable and L1 is not recognizable !!
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Second ExampleProving L is not recognizable by
reducing an unrecognizable problem to L
Let L M M is a T.M. such that M accepts
no inputs Claim L is not recognizable
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Proof (by mapping reduction)
We will show that
Lu ltm L
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Must demonstrate function f 1. f Z? TMs 2.
f is computable 3. f reduces LU to L
j ? LU f(j) ? L
j ? LU ? Mj dna j ? ... ? f(j) ? L j ? LU
? Mj acc j ? ... ? f(j) ? L
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Reduction Idea
OLU
j
accept/reject
f(j) M
Mf
OL
L(M) f if M does not accept j L(M)
0,1?if M does accept j
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• Define reduction f Z TMs
• f(j) M, a machine that
• on input x, M simulates Mj on j.
• M accepts x iff Mj accepts j.
• Note Mf takes input j and outputs M f(j)

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Prove correctness of function f 1. f Z? TMs
(obvious) 2. f is computable (obvious) 3.
f reduces LU to L
j ? LU f(j) ? L
j ? LU ? Mj dna j ? L(M) f ? f(j) ? L j ? LU
? Mj acc j ? L(M) 0,1 ? f(j) ? L
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Example 3 ATM is undecidable Lu ltm ATM
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The Halting Problemexample 4
Theorem Let LH (M,x) where M does not
halt on input x Then LH is not decidable
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Proof (by mapping reduction)
We will show that
Lu ltm LH
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• Define reduction f Z TMs ? inputs
• f(i) a machine/input pair (M,x) such that
• M on any input w,
• M simulates Mi on i.
• If Mi accepts i, then M accepts w
• If Mi rejects i, then M moves to the right
• forever after finishing the simulation
• else, M does not halt its simulation
• x 0

Proof LU ltm LH
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Need to Prove 3 things 1. f Z? TMs x inputs
(obvious) 2. f is computable (obvious, I
hope) 3. f reduces LU to LH
j ? LU f(j) ? LH
Pf j ? LU ? Mj dna j ? M DNH on x ? f(j) ?
LH j ? LU ? Mj acc j ? M halts on x ? f(j) ?
LH
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Enumerator E like a Turing Machine except that
it takes no input and has an output. That output
sequence is the language L(E)
Scratch Work
Printer
A
w1, w2, w3 ...
R
L(E)
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Definition A language L is recursively
enumerable iff L is the output of an enumerator
Theorem
A language L is recursively enumerable iff L is
recognizable
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Proof
( )
Assume L is recursively enumerable by E. Then to
construct a Turing Machine T that recognizes
L On input w, T runs E and accepts if E ever
outputs w
( )
Assume L is recognizable by T. Then to construct
an enumerator E 1) I 1 2) Run T on the
first I inputs for I steps, output those that are
accepted on the Ith step 3) I I1 goto (1)
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5th Example Proving L is not recognizable by
reducing an unrecognizable problem to L
Let L (A,B) T.M.s A and Bare s.t. L(A)
L(B) Then L is unrecognizable
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Must demonstrate function f 1. f Z? TMs x
TMs 2. f is computable 3. f reduces LU to L
j ? LU f(j) ? L
j ? LU ? Mj dna j ? ... ? f(j) ? L j ? LU
? Mj acc j ? ... ? f(j) ? L
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Reduction Idea
OLU
f(j) (A,B)
j
accept/reject
Mf
OL
L(A) f if M does not accept j L(B)
0,1?if M does accept j
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• Define reduction f Z TMs x TMs
• f(j) (A,B), two machines such that
• A on input x, A simulates Mj on j.
• A accepts x iff Mj accepts j.
• B on input y, B rejects
• Note Mf takes input j and outputs (A,B) f(j)

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Prove correctness of function f 1. f Z? TMs
(obvious) 2. f is computable (obvious) 3.
f reduces LU to L
j ? LU f(j) ? L
j ? LU ? Mj dna j ? L(A) f L(B) f ?
f(j) ? L j ? LU ? Mj acc j ? L(A) 0,1 ?
L(B) f ? f(j) ? L
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Example 6 Proof by reduction
Rices theorem. Let P any subset of Turing
Machines such that a) ? TMs A and B s.t. L(A)
L(B), A ? P ? B ? P b) ? machines C and D such
that C ?P and D ?P. Theorem the language P is
undecidable.
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Proof (by mapping reduction) Case 1 ? T.M. Mf ?
P s.t. L(Mf) ? Note this implies that M
L(M) ??? P. Let MnotinP ? P. Then Claim Lu
ltm P
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Must demonstrate function f 1. f Z? TMs 2. f
is computable 3. f reduces LU to L
j ? LU f(j) ? L
j ? LU ? Mj dna j ? ... ? f(j) ? L j ? LU
? Mj acc j ? ... ? f(j) ? L
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• Define reduction f Z TMs
• f(j) M such that
• M on input x, A simulates Mj on j.
• If Mj accepts j, then M simulates MnotinP
  on x.
• If both Mj accepts j and MnotinP accepts x,
  then M accepts x
• If either Mj DNA j or MnotinP DNA x, then M
  DNA x
• Note Mf takes input j and outputs M f(j)

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• Note that
• if Mj does not accept j, then f(j) accepts
  nothing
• if Mj does accept j, then f(j) accepts
  L(MnotinP)

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Prove correctness of function f 1. f Z? TMs
(obvious) 2. f is computable (obvious) 3.
f reduces LU to L
j ? LU f(j) ? L
j ? LU ? Mj dna j ? L(M) f ? M f(j) ? L j
? LU ? Mj acc j ? L(M) L(MnotinP) ? M f(j)
? L
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Proof (by mapping reduction) Case 2 ? T.M. Mf ?
P s.t. L(Mf) ? Note this implies that M
L(M) ??? P. Let MnotinPB ? P. Then Claim
Lu ltm P Proof is very similar to the previous
proof
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Observation
L is a recognizable language There is a
decidable language L2 such that L x ? w such
that (w, x) ? L2
S1
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Observation
L is a co-recognizable language There is a
decidable language L3 such that L y ? z such
that (y,z) ?L3
P1
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(Problems 1 and 2 of) Homework 5Due Wednesday,
March 8
1. Let B be an undecidable language where B ltm
B. A) Prove that B ?S1 ? P1 B) Give an example
of such a language B
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2. Prove
For all A,B,C, Language A ltm B Language B lt m
C Language A lt m C