Algorithm Design and Analysis 4/8/2008
Lecture 33: More Reductions
Instructor: Mike Kowalczyk
1 Examples: Reducing from Halt to show that a problem is un-
decidable
1.1 Variable attains a certain value during program execution
Suppose we want to solve the problem of figuring out if the variable vever becomes 7 as program
Pruns on input i. Let’s call this the v7 problem. It turns out to be undecidable, meaning there
is no algorithm which returns the correct output for every input. A problem instance consists
of a pair (P, i), and the answer is “yes” if the program Phas a variable vand vbecomes 7 at
some point during execution on input i, and “no” otherwise. We would like to make a reduction
Halt ≤mv7. This would show that v7 is undecidable. Remark: In our discussion of algorithms and
reductions here, we are assuming determinism (in particular, no randomness or nondeterministic
program forking).
reduceHaltToV7(Program P, input i) {
Define program P’’ =
"Same as P, except all instances of variable v are replaced by some unused variable."
Define program P’ =
"1) Run P’’ on i
2) v = 7"
return (P’, i);
}
Let’s prove that this is a reduction from the halting problem to the v7 problem. We need to show
that every instance of the halting problem is mapped to an instance of the v7 problem, and “yes”
and “no” instances of the halting problem map to “yes” and “no” instances of the v7 problem,
respectively. If (P, i) is an instance of the halting problem, that means Phalts on i. Then P00 also
halts on i. Thus, P0reaches line 2, and vgets set to 7 during P0execution on input i. In other
words, (P0, i) is a “yes” instance of the v7 problem. Now suppose that the input (P , i) is a “no”
instance of the halting problem. Then Pdoesn’t halt on input i. Then P00 doesn’t halt on input i
either, and P00 never sets vto 7 on input i(it doesn’t even use the variable at all). Thus, (P0, i)
is a “no” instance of the v7 problem, and our reduction is complete. Halt ≤mv7. In particular,
since Halt is undecidable, we can also conclude that v7 is undecidable.
1.2 The 2-halt problem
Define the 2Halt problem as follows: “On input (P, Q, i), the output should be “yes” if programs
Pand Qboth halt on input i, and “no” otherwise.”. Let’s try writing a reduction from Halt to
show that 2Halt is undecidable.
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