While trying to understand the concept of polynomial hierarchy, I noticed that there are several ways to define it. And the most confusing thing about the situation is to see the equivalence between those definitions intuitively. What I want to know for now is the relation between oracle machines and existential/universal quantifiers from the viewpoint of polynomial hierarchy.

By Wiki, the polynomial hierarchy is the generalization of the classes P, NP, and co-NP to oracle machines. The addition of oracles (not sure this is a suitable expression) results in the increase of the hierarchy.

On the other hand, another way to define the polynomial hierarchy is to use the quantifiers $\forall$ and $\exists$. According to this lecture note (http://www.cs.cornell.edu/courses/cs6810/2009sp/scribe/lecture5.pdf), a language $L$ is in $\Sigma_i$ iff $\exists$ a polynomial time relation $R$ s.t. $x\in L$ iff \begin{equation} \exists y_1\forall y_2\cdots Q_iy_iR(x,y_1,\cdots y_i), \end{equation} where $Q_j=\forall$ if $j$ is even and $Q_j=\exists$ if $j$ is odd. And $L$ is in $\Pi_i$ iff its complementary $\bar{L}$ is in $\Sigma_i$.

From the above definitions, it seems that there exist a sort of one-to-one correspondence between the inclusion of oracle and the addition of quantifiers.

Can anyone help me understand how the oracles in a Turing machine plays the role of quantifiers in predicates?


1 Answer 1


The oracles do not play the role of quantifiers, they play the role of everything but the first quantifiers.

First, observe how NP relates to $\exists yR(x,y)$ with $R$ in P. That means that NP allows existential nondeterminism followed by computation in P.

Similarly, $\Pi_1$ "is" $\forall yR(x,y)$ with $R$ in P. Now for $\Sigma_2$ we have the same with an additional $\exists$ at the front: $\exists y_1\forall y_2R(x,y_1,y_2)$ with $R$ in P. Or, in other words, $\exists y_1 R'(x,y_1)$ with $R'$ in $\Pi_1$. So we have existential nondeterminism followed by computation in $\Pi_1$. And that is more or less the same as NP with a $\Pi_1$ oracle.

In general, the first quantifier specifies whether we are in the existential or universal case. The rest of the formula then describes a lower level of the hierarchy which we use as an oracle.

  • $\begingroup$ Thank you for the answer. So the oracle is not directly related to the concept of quantifiers themselves. It seems that I understand better now about how the increase of quantifiers is realized with oralcles. $\endgroup$
    – Beom.Jean
    Mar 23, 2018 at 2:35

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