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Chapter 0: 187790-4-2E (page 1)

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.

Short Answer

Expert verified

Answer:

The problem expressed as a language as follows,

EQDFA,REX={<F,R>|FisaDFA,RisaregularExpression,L(F)=L(R)}

The above language is decidable.

Step by step solution

01

Explain decidability.

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

02

Express this problem as a language and show that it is decidable.

Consider the given Context-free language

EQDFA,REX={<F,R>|FisaDFA,RisregularExpression,L(F)=L(R)}.Consider the

following Turing Machine M decides EQDFA.REX. Construct the mapping function

f:鈭�*鈫�鈭�*for the inputs,

M="On inputs <A>where A is a DFA

1. f(<F,R>)=<F,FR>, where FRis the Deterministic Finite Automata Constructed from the

regular expression R, such that L(FR)=L(R)

2. L(F) = L (R), if and only if L (F) = =L(FR),

Thus EQDFA,REX鈮�mEQDFA

Therefore, the language

EQDFA,REX={<F,R>|FisaDFA,RisregularExpression,L(F)=L(R)}is decible

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Most popular questions from this chapter

Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set.

  1. {1,3,5,7,...}
  2. {...,-4,-2,0,2,4,...}
  3. {n|n=2mfor someminN}
  4. {n|n=2mfor someminN, andn=3kfor somekinN}
  5. {w|wis a string of0sand1sandwequals the reverse ofw}
  6. {n|nis an integer andn=n+1}

A queue automaton is like a push-down automaton except that the stack is replaced by a queue. A queue is a tape allowing symbols to be written only on the left-hand end and read only at the right-hand end. Each write operation (we鈥檒l call it a push) adds a symbol to the left-hand end of the queue and each read operation (we鈥檒l call it a pull) reads and removes a symbol at the right-hand end. As with a PDA, the input is placed on a separate read-only input tape, and the head on the input tape can move only from left to right. The input tape contains a cell with a blank symbol following the input, so that the end of the input can be detected. A queue automaton accepts its input by entering a special accept state at any time. Show that a language can be recognized by a deterministic queue automaton iff the language is Turing-recognizable.

Give informal English descriptions of PDAs for the languages in Exercise 2.6

Give context-free grammars generating the following languages.

a. The set of strings over the alphabet a,bwith more a's than b's

b. The complement of the language anbnn鈮�0.

c. w#xwRis a substring of x for w,x 鈭�0,1*

d. localid="1662105288591" x1#x2#...#xkk鈮�1,each xilocalid="1662105304877" 鈭�a,b*,and for some i and j ,localid="1662105320570" xi=xjR

Question: Answer all parts for the following DFA and give reasons for your answers.

a.Is<M,0100>鈭�ADFA?b. Is<M,011>鈭�ADFA?c. Is<M>鈭�ADFA?d. Is<M,0100>鈭�AREX?e. Is<M>鈭�EDFA?f. Is<M,M>鈭�EQDFA?

Modify the proof of Theorem 3.16 to obtain Corollary 3.19, showing that a language is decidable if some nondeterministic Turing machine decides it. (You may assume the following theorem about trees. If every node in a tree has finitely many children and every branch of the tree has finitely many nodes, the tree itself has finitely many nodes.)
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