Consider the language RTM, where RTM = {(M, w) | M is a Turing Machine that rejects the string w}. 1. How is RTM different from the complement of ATM? 2. Show that RTM is Turing-recognizable. 3. Show that RTM is undecidable.

Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .

Let P be a property of the language of a Turing machine. We say P is non-trivial if it fulfills the following 2 conditions:  - Some, but not all Turing machines satisfy the property P, and - P is a property of the language of the machines – i.e., if M1 and M2 are two Turing machines and L(M1) =L(M2), then they either both satisfy property P or they both do not satisfy property P. ---------------------------------------------------------------------- Note:AT M,HALTT M, and LCF are all problems related to non-trivialproperties. Rice’s Theorem:LetPbe any non-trivial property of the language of aTuring machine. DefineLPas follows: LP={〈M〉 |M is a Turing machine satisfying property P}. ThenLPis undecidable. ---------------------------------------------------------------------- In this problem, you will prove Rice’s Theorem. You may assume that (i) ifL(M) =∅, then〈M〉/∈LP, and (ii) MP is a Turing machine such that〈MP〉 ∈ LP. (a) Use the two assumptions above to create a machine M2 that…

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