1. Let A and B be languages. We say that AmB if there is a computable
function f such that, for every string x:
x 2 A ) f(x) 2 B
and
x 62 A ) f(x) 62 B:
(This is a very important relation on languages.)
Show that m is a transitive relation. That is, show that if AmB and BmC,
then AmC. If AmB and B is recursive, is A recursive? If B is r.e., is A
recursively-enumerable?
If AmB and A is recursive, is B recursive? If A is r.e., is B recursivelyenumerable?
Justify your answers.
2. Explain why m is not a partial order. (As part of this exercise, you might
need to do some searching, to find out what a partial order is.)
3. Let A and B be languages. We say that AmB if AmB and BmA.
Show that m is an equivalence relation.
4. Let HP denote the halting problem. Consider the language:
A = f1x : x 2 HPg [ f0x : x 62 HPg:
Show HPmA and HPmA. Is A recursively-enumerable? Justify your answer.
5. Show that A is decidable if and only if A can be recursively enumerated in
lexicographic order.

Homework due Wednesday, Sept. 141. LetAandBbe languages. We say thatA≤mBif there is a computablefunctionfsuch that, for every stringx:x∈A⇒f(x)∈Bandx6∈A⇒f(x)6∈B.(This is a very important relation on languages.)Show that≤mis a transitive relation. That is, show that ifA≤mBandB≤mC,thenA≤mC.IfA≤mBandBis recursive, isArecursive? IfBis r.e., isArecursively-enumerable?IfA≤mBandAis recursive, isBrecursive? IfAis r.e., isBrecursively-enumerable? Justify your answers.2. Explain why≤misnota partial order. (As part of this exercise, you mightneed to do some searching, to find out what a partial order is.)3. LetAandBbe languages. We say thatA≡mBifA≤mBandB≤mA.Show that≡mis an equivalence relation.4. LetHPdenote the halting problem. Consider the language:A={1x:x∈HP}∪{0x:x6∈HP}.ShowHP≤mAandHP≤mA.IsArecursively-enumerable? Justify your answer.5. Show thatAis decidable if and only ifAcan be recursively enumeratedinlexicographic order.1

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