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CSCI 2670 Introduction to Theory of Computing

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Theorem: If A m B and B is Turing-recognizable, then A is Turing-recognizable. ... If A T B and B is decidable, then A is decidable ... – PowerPoint PPT presentation

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Title: CSCI 2670 Introduction to Theory of Computing


1
CSCI 2670 Introduction to Theory of Computing
November 3, 2005
2
Agenda
  • Yesterday
  • Reductions (Section 5.3)
  • Today
  • More on reductions
  • Section 6.3
  • We will not be covering section 6.4
  • I will discuss some basic issues of this section
    when covering chapter 7

3
Announcement
  • Remember to let me know if you want to come to my
    pizza party next Wednesday

4
Mapping reducibility
  • Definition Language A is mapping reducible to
    language B, written A?mB, if there is a
    computable function f? ? ?, where for every w,
  • w ? A iff f(w) ? B

5
Why m?
  • In some sense, if A m B, then A is less
    powerful than B
  • For example, we can map from CFGs to decidable
    languages that arent CF, but not vice versa
  • Example
  • We can easily map any CFG C to ACFG
  • f(w) ltC,wgt
  • w ? C iff ltC,wgt ? ACFG
  • You can use this example to help you remember how
    to use reductions

6
Mapping reductions decidability
  • Theorem If A ?m B and B is decidable, then A is
    decidable.
  • Proof Let M be a decider for B and let f be a
    reduction from A to B.
  • Consider the following TM, N
  • N On input w
  • Compute f(w)
  • Run M on f(w) and report Ms output
  • Then N decides A

7
Example
  • Let EV ltAgt A is a DFA all strings in L(A)
    have an even number of 1s
  • How can we prove EV is decidable using a mapping
    reduction?
  • Consider the following DFA B

L(B) w ?? w has an even number of 1s
8
Mapping reduction of L
  • Use EQDFA ltA,Bgt A and B are DFAs with L(A)
    L(B)
  • Mapping from EV to EQDFA
  • f(ltAgt) ltA,A?Bgt
  • A has an even number of 1s if and only if L(A)
    L(A?B)
  • I.e., A ? EV iff f(A) ? EQDFA
  • Since we know EQDFA is decidable and EV m EQDFA,
    we now know EV is decidable

9
Mapping reductions undecidability
  • Corollary If A ?m B and A is undecidable, then
    B is undecidable.
  • We have been using this corollary implicitly
    already

10
Example
  • We showed that HALTTM is undecidable by
    contradiction
  • Now lets show its undecidable using mapping
    reduction
  • Need a function that takes input ltM,wgt
  • Halts if M accepts w, and loops if not

11
Mapping from ATM to HALTTM
  • F On input x
  • If x ? ltM,wgt for some TM M, output x
  • Otherwise, construct the following TM
  • M On input x
  • 1. Run M on x
  • 2. If M accepts, accept
  • 3. If M rejects, enter a loop
  • 2. Output ltM,wgt
  • If M accepts w M halts on w, otherwise M loops
    or generates a string not in HALTTM
  • I.e., ltM,wgt?ATM iff ltM,wgt?HALTTM

12
HALTTM is undecidable
  • We just showed that ATM m HALTTM
  • Since we know ATM is undecidable, we can conclude
    that HALTTM is undecidable

13
Reductions TM-recognizability
  • Theorem If A ?m B and B is Turing-recognizable,
    then A is Turing-recognizable.
  • Proof (same as decidable proof) Let M be a
    recognizer for B and let f be a reduction from A
    to B.
  • Consider the following TM, N
  • N On input w
  • Compute f(w)
  • Run M on f(w) and report Ms output
  • Then N recognizes A

14
Reductions non-TM-recognizability
  • Corollary If A ?m B and A is not
    Turing-recognizable, then B is not
    Turing-recognizable.
  • Question Which language have we seen that is
    not Turing-recognizable?
  • Answer ATM

15
Proving non-Turing-recognizability
  • Question If A ?m U is A ?m U?
  • Answer Yes since x ? A iff f(x) ? U
  • How can we use this to prove non-Turing-recognizab
    ility?
  • Prove ATM ?m U or prove ATM ?m U

16
Is mapping reducibility enough?
  • Mapping reducibility does not completely capture
    our intuition about reductions
  • Example ATM and ATM are not mapping reducible
  • ATM is Turing-recognizable and ATM isnt
  • A solution to ATM would also provide a solution
    to ATM

17
Oracle
  • An oracle for a language B is an external device
    that is capable of reporting whether any string w
    is a member of B
  • We are not concerned how the oracle determines
    membership
  • An oracle Turing machine is a Turing machine that
    can query an oracle
  • The machine MB can query an oracle for the
    language B

18
Example
  • An oracle Turing machine with an oracle for EQTM
    can decide ETM
  • TEQ-TM On input ltMgt
  • Create TM M1 such that L(M1) ?
  • M1 has a transition from start state to reject
    state for every element of ?
  • Call the EQTM oracle on input ltM,M1gt
  • If it accepts, accept if it rejects, reject
  • TEQ-TM decides ETM
  • ETM is decidable relative to EQTM

19
Turing reducibility
  • Definition A language A is Turing reducible to a
    language B, written
  • A ?T B, if A is decidable relative to B
  • Previous slide shows ETM is Turing reducible to
    EQTM
  • Whenever A is mapping reducible to B, then A is
    Turing reducible to B
  • The function in the mapping reducibility could be
    replaced by an oracle

20
Applications
  • If A ?T B and B is decidable, then A is decidable
  • If A ?T B and A is undecidable, then B is
    undecidable
  • If A ?T B and B is Turing-recognizable, then A is
    Turing-recognizable
  • If A ?T B and A is non-Turing-recognizable, then
    B is non-Turing-recognizable

21
Have a great weekend!
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