Projective frame

In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.

Background

The statement of Post's theorem uses several concepts relating to definabilityTemplate:Dn and recursion theory. This section gives a brief overview of these concepts, which are covered in depth in their respective articles.

The arithmetical hierarchy classifies certain sets of natural numbers that are definable in the language of Peano arithmetic. A formula is said to be ${\mathrm{\Sigma }}_m^0$ if it is an existential statement in prenex normal form (all quantifiers at the front) with $m$ alternations between existential and universal quantifiers applied to a quantifier free formula. Formally a formula $\varphi (s)$ in the language of Peano arithmetic is a ${\mathrm{\Sigma }}_m^0$ formula if it is of the form

$\mathrm{\exists }{n}_1\mathrm{\forall }{n}_2\mathrm{\exists }{n}_3\mathrm{\forall }{n}_4\cdots Q{n}_m\rho (n_1,\dots ,n_m,x_1,\dots ,x_k),$

where ρ is a quantifier free formula and Q is $\mathrm{\forall }$ if m is even and $\mathrm{\exists }$ if m is odd. Note that any formula of the form

$(\mathrm{\exists }{n}_1^1\mathrm{\exists }{n}_2^1\cdots \mathrm{\exists }{n}_{j_1}^1)(\mathrm{\forall }{n}_1^2\cdots \mathrm{\forall }{n}_{j_2}^2)(\mathrm{\exists }{n}_1^3\cdots )\cdots (Q_1{n}_1^m\cdots )\rho (n_1^1,\dots n_{j_m}^m,x_1,\dots ,x_k)$

where $\rho$ contains only bounded quantifiers is provably equivalent to a formula of the above form from the axioms of Peano arithmetic. Thus it isn't uncommon to see ${\mathrm{\Sigma }}_m^0$ formulas defined in this alternative and technically nonequivalent manner since in practice the distinction is rarely important.

A set of natural numbers A is said to be ${\mathrm{\Sigma }}_m^0$ if it is definable by a ${\mathrm{\Sigma }}_m^0$ formula, that is, if there is a ${\mathrm{\Sigma }}_m^0$ formula $\varphi (s)$ such that each number n is in A if and only if $\varphi (n)$ holds. It is known that if a set is ${\mathrm{\Sigma }}_m^0$ then it is ${\mathrm{\Sigma }}_n^0$ for any $n>m$, but for each m there is a ${\mathrm{\Sigma }}_{m+1}^0$ set that is not ${\mathrm{\Sigma }}_m^0$. Thus the number of quantifier alternations required to define a set gives a measure of the complexity of the set.

Post's theorem uses the relativized arithmetical hierarchy as well as the unrelativized hierarchy just defined. A set A of natural numbers is said to be ${\mathrm{\Sigma }}_m^0$ relative to a set B, written ${\mathrm{\Sigma }}_m^{0,B}$, if A is definable by a ${\mathrm{\Sigma }}_m^0$ formula in an extended language that includes a predicate for membership in B.

While the arithmetical hierarchy measures definability of sets of natural numbers, Turing degrees measure the level of uncomputability of sets of natural numbers. A set A is said to be Turing reducible to a set B, written $A{\le }_{T}B$, if there is an oracle Turing machine that, given an oracle for B, computes the characteristic function of A. The Turing jump of a set A is a form of the Halting problem relative to A. Given a set A, the Turing jump $A^{\prime }$ is the set of indices of oracle Turing machines that halt on input 0 when run with oracle A. It is known that every set A is Turing reducible to its Turing jump, but the Turing jump of a set is never Turing reducible to the original set.

Post's theorem uses finitely iterated Turing jumps. For any set A of natural numbers, the notation $A^{(n)}$ indicates the n-fold iterated Turing jump of A. Thus $A^{(0)}$ is just A, and $A^{(n+1)}$ is the Turing jump of $A^{(n)}$.

Post's theorem and corollaries

Post's theorem establishes a close connection between the arithmetical hierarchy and the Turing degrees of the form ${\mathrm{\varnothing }}^{(n)}$, that is, finitely iterated Turing jumps of the empty set. (The empty set could be replaced with any other computable set without changing the truth of the theorem.)

Post's theorem states:

1. A set B is ${\mathrm{\Sigma }}_{n+1}^0$ if and only if $B$ is recursively enumerable by an oracle Turing machine with an oracle for ${\mathrm{\varnothing }}^{(n)}$, that is, if and only if B is ${\mathrm{\Sigma }}_1^{0,{\mathrm{\varnothing }}^{(n)}}$.
2. The set ${\mathrm{\varnothing }}^{(n)}$ is ${\mathrm{\Sigma }}_n^0$ complete for every $n>0$. This means that every ${\mathrm{\Sigma }}_n^0$ set is many-one reducible to ${\mathrm{\varnothing }}^{(n)}$.

Post's theorem has many corollaries that expose additional relationships between the arithmetical hierarchy and the Turing degrees. These include:

1. Fix a set C. A set B is ${\mathrm{\Sigma }}_{n+1}^{0,C}$ if and only if B is ${\mathrm{\Sigma }}_1^{0,C^{(n)}}$. This is the relativization of the first part of Post's theorem to the oracle C.
2. A set B is ${\mathrm{\Delta }}_{n+1}$ if and only if $B{\le }_T{\mathrm{\varnothing }}^{(n)}$. More generally, B is ${\mathrm{\Delta }}_{n+1}^C$ if and only if $B{\le }_T{C}^{(n)}$.
3. A set is defined to be arithmetical if it is ${\mathrm{\Sigma }}_n^0$ for some n. Post's theorem shows that, equivalently, a set is arithmetical if and only if it is Turing reducible to ${\mathrm{\varnothing }}^{(m)}$ for some m.

References

Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1

Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. ISBN 3-540-15299-7