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Chapter 4: Q3E (page 193)

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

Short Answer

Expert verified

It can be shown that ALLDFAis decidable.

Step by step solution

01

Explain decidability

The automata and the grammars helps decide the outcome of the undecidable problems. The Deterministic Finite Automata determines the state of the string given as input.

02

Show that ALLDFA is decidable

Consider the given context-free language, ALLDFA=A|A鈥�鈥�is a鈥�DFA鈥�that鈥�鈥�recognizes鈥�鈭�*. Assume that the following Turing Machine Mdecides ALLDFA.

M=''On鈥�input鈥�A鈥�where鈥�A鈥�鈥�is鈥�a鈥�鈥�DFA:

  1. Construct DFA鈥�鈥�Erecognizes the complete situation, the start state A, until new states are marked by the transition, if no accept state is marked 鈥�补肠肠别辫迟鈥�, otherwise, 鈥�谤别箩别肠迟鈥�.
  2. Run the Turing Machine T, on input B, whereT decides EDFA.
  3. IfT accepts, 鈥�补肠肠别辫迟鈥�.
  4. IfT rejects, 鈥湴鹇岜鸪Τ兮��.

Therefore, the ALLDFAis decidable.

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