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I understand that most problems are trivial if a halting oracle is available (or, I think equivalently, hyper-computation). However, applying the argument that shows the Halting Problem is impossible for a Turing machine also shows that it is impossible for a Turing+oracle to decide the Halting Problem for a Turing+oracle. Are there any actual, practical, examples of problems unsolvable by a halting oracle?

Note: by "oracle" I mean oracle for a standard Turing machine, not a TM with an oracle itself.

Raphael
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ike
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    There are "arbitrarily undecidable" problems, see e.g. here. I don't know about "practical" examples (which also does not match the title you chose); what qualifies as "practical" for you? – Raphael Jun 02 '14 at 21:07
  • That aren't contrived simply to answer this question. I acknowledged that the next-level halting problem still applies. – ike Jun 03 '14 at 10:41
  • In addition, all languages that are not recursively enumerable are not be reducible to HALT. Examples include FINITE, EMPTY, whether two CFG's derive the same language, etc. –  Jun 15 '14 at 20:20

2 Answers2

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Just take a problem whose Turing degree is above $0'$, which is the degree of The Halting Oracle. In terms of the arithmetical hierarchy you want problems which are above $\Sigma^0_1$. Examples of such problems (where $\phi_n$ is the $n$-th partial computable function and $W_n = \{k \in \mathbb{N} \mid \text{$\phi_n(k)$ is defined}\}$ is the $n$-th computably enumerable set):

  • $\{n \in \mathbb{N} \mid \text{$\varphi_n$ terminates for finitely many inputs}\}$ is $\Sigma^0_2$-complete.
  • $\{n \in \mathbb{N} \mid \text{$\varphi_n$ is a total function}\}$ is $\Pi^0_2$-complete.
  • $\{n \in \mathbb{N} \mid \text{$W_n$ is a computable set}\}$ is $\Sigma^0_3$-complete.

None of these can be solved even if you have a Halting Oracle. For instance, consider the second example, "is $\varphi_n$ total?" Given $n$ how would the Halting Oracle help us decide whether the Turing machine encoded by $n$ halts on every input?

[Added 2014-06-03] For a "practical" aspect of all of this, consider the problem: a programmer has written a function void charge_credit_card(int card_number, int amount) and we would like to know whether the function terminates on all inputs. It is impossible to write a compiler which can automatically check this in general. Moreover, even if we allow the compiler to ask us questions of the form "does charge_credit_card terminate when given input (k,m)?", it is still impossible.

Andrej Bauer
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  • Can you give a scenario where someone might actually ask such a question? Also, I don't really understand the example. – ike Jun 03 '14 at 10:43
  • @ike You may have to learn to live with the fact that not all of (computer) science, and arguably even less of TCS/mathematics has practical applications now. – Raphael Jun 03 '14 at 11:03
  • Sure, I added a "practical" example. – Andrej Bauer Jun 03 '14 at 11:20
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    Sayng "I do not understand the example" without explaining what confuses you is not productive. Did you read the relevant Wikipedia pages I pointed to? Those are directly related to your question, so the first thing you should od is familiarize yourself with the basic concepts involved. – Andrej Bauer Jun 03 '14 at 11:26
  • In the credit card example, write a new program that runs the charge_credit_card function for all possible inputs. (There are a finite number of ints in most languages.) Iff this new program halts (which can be ascertained by one call to a halting oracle), then the charge_credit_card function halts on all inputs. – ike Jun 09 '14 at 14:44
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    @ike, the example was meant to have an infinite amount of int, quite obviously. Do you really need me to write BigInt or some such, or will you then complain that computer memory is finite? – Andrej Bauer Jun 09 '14 at 20:43
  • Computer memory is finite. When you're trying to give a "practical" example, you can't allow arbitrarily large inputs. – ike Jun 10 '14 at 17:14
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    Whatever. I told you what the answer to your question was. If you don't want to understand it in good faith, then don't bother us with questions. – Andrej Bauer Jun 11 '14 at 07:48
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    A practical example is, $\overline{HALT}$, the compliment of halt. This is ${<M,w>: \text{M doesn't halt on w} }$ Given an arbitrary program and input to the program, determine if the program doesn't halt. This problem, along with every other non-recursively enumerable language, doesn't reduce to HALT. –  Jun 15 '14 at 21:18
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    @tAllan: you should post that as an answer. It beats me what the OP considers "practical", but your example is certainly better than mine. – Andrej Bauer Jun 15 '14 at 21:26
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Predicting a random oracle is not solvable using, possibly, any kind of hypercomputation.

Anixx
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