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Chapter 0: Q25P (page 1)

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

Short Answer

Expert verified

Turing-Recognizable subset of incompressible strings doesn鈥檛 exist.

Step by step solution

01

Turing Recognizable

A language L is said to be Turing Recognizable if and only if there exist any Turing Machine (TM) which recognize it i.e. Turing Machine halt and accept strings belong to language L and will reject or not halt on the input strings that doesn鈥檛 belong to language L.

02

Proving Turing-Recognizable subset of incompressible strings doesn’t exist

Let鈥檚 prove the above statement by contradiction.

So, let there is infinite subset S of incompressible strings.

Let M be the Turing Machine that recognize S. We will construct a Turing Machine P as follows:

  • Obtain its description 鈥筆鈥�
  • Repeat S, until it finds a string x whose length is greater than 鈥�P,鈭�鈥�i.e.,|x|>|鈥�P,鈭�鈥�|.
  • Output: x .

S must contain strings which are larger than |鈥筆鈥簗 in terms of length because length of each string in S must be infinite. Therefore, description of x must be 鈥�P,鈭�鈥�implies: K(x)鈮�|鈥�P,鈭�鈥�|<|x|.

But it must be noted that x is an incompressible string, so our above assumption is contradict.

Hence, there exist no infinite Turing-Recognizable subset of incompressible strings.

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Most popular questions from this chapter

A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form

未 : Q 脳 螕鈭掆啋Q 脳 螕 脳 {R, RESET}.

If 未(q, a) = (r, b, RESET), when the machine is in state q reading an a, the machine鈥檚 head jumps to the left-hand end of the tape after it writes b on the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages.

Give a formal definition of an enumerator. Consider it to be a type of two-tape Turing machine that uses its second tape as the printer. Include a definition of the enumerated language

Question:Consider the algorithm MINIMIZE, which takes a DFA as input and outputs DFA .

MINIMIZE = 鈥淥n input , where M=(Q,危,未,q0,A) is a DFA:

1.Remove all states of G that are unreachable from the start state.

2. Construct the following undirected graph G whose nodes are the states of .

3. Place an edge in G connecting every accept state with every non accept state. Add additional edges as follows.

4. Repeat until no new edges are added to G :

5. For every pair of distinct states q and r of and every a鈭�危 :

6. Add the edge (q,r) to G if 未q,a,未r,a is an edge of G .

7. For each state q,let[q] be the collection of statesq={r鈭�Q|noedge joins q and r in G }.

8.Form a new DFA M'=Q',危,未',q'0,A'where

Q'={[q]|q鈭�Q}(ifq=r,onlyoneofthemisinQ'),未'(q,a)=[未q,a]foreveryq鈭�Qanda鈭�危,q00=[q0],andA0={[q]|q鈭�A}

9. Output ( M')鈥�

a. Show that M and M' are equivalent.

b. Show that M0 is minimal鈥攖hat is, no DFA with fewer states recognizes the same language. You may use the result of Problem 1.52 without proof.

c. Show that MINIMIZE operates in polynomial time.

Let X be the set {1,2,3,4,5}and Y be the set {6,7,8,9,10}.The unary function f:X鈫�Yand the binary function g:X脳Y鈫�Yare described in the following tables.

g12345f(n)67676 g123456789101010101010789106789106789106789106

a. What is the value of f(2)?
b.What are the range and domain of f?
c. What is the value of g (2, 10) ?
d. What are the range and domain ofg?
e. What is the value ofg(4, f (4))?

Give an example in the spirit of the recursion theorem of a program in a real programming language (or a reasonable approximation thereof) that prints itself out.
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