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Chapter 4: 13P (page 211)

Let A={R,S|R鈥and鈥S鈥are鈥regular鈥expressions鈥and鈥LR鈯LS}. Show that A is decidable.

Short Answer

Expert verified

A is decidable.

Step by step solution

01

Explain Decidable Language

A problem is decidable if a Turing machine halts in finite amount of time for every input and gives the answers as 鈥測es鈥 or 鈥渘o.鈥滱 decidable problem has an algorithm to determine the answer for a given input.

02

Show that A is decidable

A language is decidable if a Turing machine accepts the input of a language. Consider that Rand Sare regular expressions.

Consider a Turing machine M that decides the language AOn input string W

  1. Consider that the input has two possibilities.
  2. If the input decodes a pair of regular expressions, then it takes finite time.
  3. If it doesn鈥檛 decode a pair of regular expressions, then it stops rejecting.

Therefore, the language given is decidable and has been proved.

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Most popular questions from this chapter

Let 鈭={0,1}.Show that the problem of determining whether a CFG generates some string in 1*is decidable. In other words, show that

{G|G鈥鈥is鈥鈥a鈥鈥CFG鈥over鈥0,1鈥and鈥1*鈭LG鈮鈭}is a decidable language.

Let ALLDFA={A|A鈥鈥is鈥a鈥DFA andLA=鈭*}. Show thatALLDFAis decidable.

Let A={R|Ris a regular expression describing a language containing at least one string w that has 111 as a substring i.e.,w=x111y鈥for鈥some鈥x鈥and鈥y}. Show that A is decidable.

Let E={M|M is a DFA that accepts some string with more 1s than 0s}. Show that E is decidable. (Hint: Theorems about CFLs are helpful here.)

Let S={M|M鈥is鈥鈥a鈥鈥DFA鈥鈥that鈥鈥accepts鈥鈥wR鈥whenever鈥鈥it鈥鈥accepts鈥鈥w} Show that S is decidable.
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