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Chapter 7: Q10E (page 323)

Show that ALLDFAisinP.

Short Answer

Expert verified

The ALLDFAisin showed.

Step by step solution

01

Step 1:To Randomized the Class - P

On such a randomized single-tap Turing machine, P is indeed a class of languages that can be answered in polynomial time.

ALLDFA={<A>AisaDFAthatrecognies\mathop∑*}

EDFA={<A>AisaDFAandLA=∅}

is determined by a Turing – machine (TM)

Let E be the Turing machine that determinesEDFA

Let R be the Turing machine that determines ALLDFA

02

The R – Algorithm Steps

The algorithm of R is as follows:

:R=''Oninput<A>,whereAisaDFA

1. Construct a DFA Bthat recognizesL(A¯), by swapping accept and non – accepting states

2. Run the TMEon input<B>, where E determinesEDFA.

3. If E accepts, then accept

4. If E rejects, then reject.”

Clearly theTMR determinesALLDFA in polynomial time.Therefore, ALLDFAis in P.

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Most popular questions from this chapter

Let A⊆1* be any unary language. Show that if A is NP-complete, then P = NP. (Hint: Consider a polynomial time reduction f from SATto A. For a formula ϕ, let ϕ0100be the reduced formula where variables x1, x2, x3, and x4 inϕ are set to the values 0, 1, 0, and 0, respectively. What happens when you apply f to all of these exponentially many reduced formulas?)

Is the following formula satisfiable?

(x∨y)∧(x∨y)∧(x∨y)∧(x∨y).

Let DOUBLE-SAT={}has at least two satisfying assignments}. Show thatDOUBLE-SATisNP- complete

The difference hierarchyDiPis defined recursively as

  1. role="math" localid="1664206013824" D1P∈NPand
  2. DiP={A|A=B\CforBinNPandCinDi-1P}.

(Here BC=B∩C.) For example, a language in D2P is the difference of two NP languages. Sometimes D2P is called DP (and may be written DP). Let Z={(G1,k1,G2,k2)|G1hasak1-cliqueandG2doesn'thaveak2-clique}

.Show that Z is complete for DP. In other words, show that Z is in DP and every language in DP is polynomial time reducible to Z.

Call graphsGandH isomorphic if the nodes of Gmay be reordered so that it is identical to H.

Let ISO=hG,Hi|GandHareisomorphicgraphs.Show that ISO∈NP.
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