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Unformatted text preview: CSE 105: Automata and Computability Theory Winter 2011 Homework #5
Due: Tuesday, March 1st, 2011 Problem 1 Let COMPLDFA be the language
A, B A and B are DFAs over the same alphabet Σ and L(A) = L(B ) . (Notice the complementation bar over L(B ) above!) Show that COMPLDFA is decidable.
Problem 2 We say that a string over the alphabet Σ = {0, 1} is sorted if any 0s in it occur
before any 1s. (For example, 111 is sorted, whereas 00110 is not.) We consider the
empty string to be sorted. Let MESSYDFA be the language
A A is a DFA over the alphabet Σ = {0, 1}
and no string in L(A) is sorted . Show that MESSYDFA is decidable.
Hint: Think about intersecting two regular languages.
Problem 3 (Sipser 4.22) A useless state in a pushdown automaton is never entered on any
input string. Consider the problem of determining whether a pushdown automaton has
any useless states. Formulate this problem as a language and show that it is decidable.
Problem 4 NOTSTATE be the language
M , w, q M is a Turing machine, w is a string, and q is a state;
and M , when run on input w, never enters the state q . . Show that NOSTATE is undecidable.
Hint: Assume that NOSTATE is decidable, and use a decider for NOSTATE to decide
the acceptance problem ATM , yielding a contradiction. ...
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This note was uploaded on 08/31/2011 for the course CSE 105 taught by Professor Paturi during the Spring '99 term at UCSD.
 Spring '99
 Paturi

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