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Chapter 8: Q11P (page 358)

Question:Show that if every NP-hard language is also PSPACE-hard, then PSPACE = NP.

Short Answer

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Step by step solution

01

Step-1: NP-Hard and PSPACE

The NP problems are a class of problems whose solutions are difficult to find but simple to prove, and which are solved in polynomial time by a Non-Deterministic Machine.A problem is NP-hard if all problems in NP can be reduced to it in polynomial time, even if it isn't in NP.

PSPACE is the collection of all decision problems that a Turing machine can answer using a polynomial space in computational complexity theory.

02

 Step-2: the explanantion of PSPACE-hard

If every NP-hard language is also PSPACE-hard, then SAT is also PSPACE-hard, and every PSPACE language can be reduced to SAT in polynomial time. As a result,PSPACE⊆NPbecause SAT∈NPand thereforePSPACE=NPbecause we know.NP⊆PSPACE

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Most popular questions from this chapter

The game of Nim is played with a collection of piles of sticks. In one move, aplayer may remove any nonzero number of sticks from a single pile. The players alternately take turns making moves. The player who removes the very last stick loses. Say that we have a game position in Nim with k piles containing s1,.....,sksticks. Call the position balanced if each column of bits contains an even number of 1s when each of the numbers s , is written in binary, and the binary numbers are written as rows of a matrix aligned at the low order bits. Prove the following two facts.

  1. Starting in an unbalanced position, a single move exists that changes theposition into a balanced one.
  2. Starting in a balanced position, every single move changes the position intoan unbalanced one.

Let NIM={⟨s1,...,sk⟩|each siis a binary number and Player I has a winningstrategy in the Nim game starting at this position}. Use the preceding facts about balanced positions to show that NIM∈Lis missing.

LetMULT={a#b#c|a,b,care binary natural numbers and a×b=c}. Showthat MULT∈L.

Define UPATHto be the counterpart of PATHfor undirected graphs. Show that BIPARTITE________________≤LUPATH. (Note: In fact, we can proveUPATH∈L, and thereforeBIPARTITE∈L, but the algorithm [62] is too difficult to present here.)

A ladderis a sequence of strings s1, s2, . . . ,sk, wherein every string differs from the preceding one by exactly one character. For example, the following is a ladder of English words, starting with “head” and ending with “free”:head, hear, near, fear, bear, beer, deer, deed, feed, feet, fret, free.

Let

LADDERDFA={(M,s,t)|MisaDFAandL(M) containsaladderofstrings,startingwithsandendingwitht}

Show that LADDERDFA is in PSPACE.

Show that for any function f:N→R+where f(n)≥n, the space complexity class isSPACE(f(n))the same whether you define the class by using the single tape TM model or the two-tape read-only input TM model.
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