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Chapter 0: Q19P (page 1)

In the silly Post Correspondence Problem, SPCP, the top string in each pair has the same length as the bottom string. Show that the SPCP is decidable.

Short Answer

Expert verified

It is proved that Silly Post Correspondence Problem (SPCP) is decidable.

Step by step solution

01

 Decidability

A problem is decidable if there Turing Machine exist which will halt in finite amount of time.

02

 Proving SPCP is decidable

Now as the each pairs have equal length at top and at bottom string, so it means total length of string at top and at bottom are equal.

Thus if we construct a Turing Machine, there is only one way by which our TM will be decidable i.e., if top and bottom in the pair are the same.

Let us construct a Turning Machine M such that

M = for each pair

Check if TOP and BOTTOM are equal

  • ·If EQUAL, then check the next pair

If NOT EQUAL, Reject M .

If all the pairs are resolved, Accept M .

Thus, SPCP is decidable.

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Most popular questions from this chapter

Consider the problem of determining whether a DFA and a regular expression are equivalent. Express this problem as a language and show that it is decidable.

Show that the single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.

LetAbe the set{x,y,z}andBbe the set{x,y}.

  1. IsAa subset ofB?
  2. IsBa subset ofA?
  3. What isA∪B?
  4. What isA∩B?
  5. What isA×B?
  6. What is the power set ofB ?

Show that the set of incompressible strings contains no infinite subset that is Turing-recognizable.

Question: Each of the following languages is the intersection of two simpler languages. In each part, construct DFAs for the simpler languages, then combine them using the construction discussed in footnote 3 (page 46) to give the state diagram of a DFA for the language given. In all parts,∑=a,b.

a.{w|w\kern1pt has  at  least  three  a's  and  at  least  two  b's}b.w|w\kern1pt has  exactly  two  a's  and  at  least  two  b'sc.w|w\kern1pt has  even number  of  a's  and  one or  two  b'sd.{w|w\kern1pt has  even number  of  a's  and eachais followed by at least  one b}e.w|w\kern1pt starts  with  an  a  and  has at most one  bf.w|w\kern1pt has an odd number ofa's  and ends with a  bg.w|w\kern1pt has even length and an odd number ofa's
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