hw7sol - CS 181 — Winter 2008 Formal Languages and...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CS 181 — Winter 2008 Formal Languages and Automata Theory Problem Set #7 Solutions Problem 7.1. (10 points) A useless state in a Turing machine is one that is never entered on any input string. Consider the problem of determining whether a Turing machine has any useless states. Formulate this problem as a language and show that it is undecidable. Solution We formulate this problem as the following language: US TM = {h M i | M is a Turing machine with a useless state } . We will show that US TM is undecidable via a reduction from the language E TM . Recall that E TM = {h M i | M is a Turing machine with L ( M ) = ∅} . Assume, by way of contradiction, that US TM is decidable and let D be a decider for it. Now suppose we wish to decide whether or not some Turing machine M has an empty language (i.e. whether or not h M i ∈ E TM ). Notice that M has an empty language if and only if its accept state q accept is useless. However, we can only use D to find if any of the states (not just q accept ) are useless. Thus, we will need to construct a Turing machine M whose language is exactly that of M , but guarantees that all non-accepting states are not useless. We construct M by starting with M and adding transitions. Let ♠ be a tape symbol not used by M . Then for each state q ∈ Q where q 6 = q accept , M has the ability to go from q to q by writing ♠ to the tape (regardless of what tape symbol is read) and keeping the head in the same position. Since ♠ did not appear in M , no accepting run of M will use these transitions. However, the existence of these transitions guarantee that every state, other than q accept , is not useless....
View Full Document

This homework help was uploaded on 04/02/2008 for the course COM SCI 181 taught by Professor Griebach during the Spring '08 term at UCLA.

Page1 / 3

hw7sol - CS 181 — Winter 2008 Formal Languages and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online