Oracle definition

For the oracle definition of the polynomial hierarchy, define

${\displaystyle \Delta _{0}^{\mathsf {P}}:=\Sigma _{0}^{\mathsf {P}}:=\Pi _{0}^{\mathsf {P}}:={\mathsf {P}},}$

where P is the set of decision problems solvable in polynomial time. Then for i ≥ 0 define

${\displaystyle \Delta _{i+1}^{\mathsf {P}}:={\mathsf {P}}^{\Sigma _{i}^{\mathsf {P}}}}$

${\displaystyle \Sigma _{i+1}^{\mathsf {P}}:={\mathsf {NP}}^{\Sigma _{i}^{\mathsf {P}}}}$

${\displaystyle \Pi _{i+1}^{\mathsf {P}}:={\mathsf {coNP}}^{\Sigma _{i}^{\mathsf {P}}}}$

where ${\displaystyle {\mathsf {P}}^{\rm {A}}}$ is the set of decision problems solvable in polynomial time by a Turing machine augmented by an oracle for some complete problem in class A; the classes ${\displaystyle {\mathsf {NP}}^{\rm {A}}}$ and ${\displaystyle {\mathsf {coNP}}^{\rm {A}}}$ are defined analogously. For example, ${\displaystyle \Sigma _{1}^{\mathsf {P}}={\mathsf {NP}},\Pi _{1}^{\mathsf {P}}={\mathsf {coNP}}}$, and ${\displaystyle \Delta _{2}^{\mathsf {P}}={\mathsf {P^{NP}}}}$ is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for some NP-complete problem.^[2]

Quantified boolean formulae definition

For the existential/universal definition of the polynomial hierarchy, let L be a language (i.e. a decision problem, a subset of {0,1}^*), let p be a polynomial, and define

${\displaystyle \exists ^{p}L:=\left\{x\in \{0,1\}^{*}\ \left|\ \left(\exists w\in \{0,1\}^{\leq p(|x|)}\right)\langle x,w\rangle \in L\right.\right\},}$

where ${\displaystyle \langle x,w\rangle \in \{0,1\}^{*}}$ is some standard encoding of the pair of binary strings x and w as a single binary string. The language L represents a set of ordered pairs of strings, where the first string x is a member of ${\displaystyle \exists ^{p}L}$, and the second string w is a "short" (${\displaystyle |w|\leq p(|x|)}$) witness testifying that x is a member of ${\displaystyle \exists ^{p}L}$. In other words, ${\displaystyle x\in \exists ^{p}L}$ if and only if there exists a short witness w such that ${\displaystyle \langle x,w\rangle \in L}$. Similarly, define

${\displaystyle \forall ^{p}L:=\left\{x\in \{0,1\}^{*}\ \left|\ \left(\forall w\in \{0,1\}^{\leq p(|x|)}\right)\langle x,w\rangle \in L\right.\right\}}$

Note that De Morgan's laws hold: ${\displaystyle \left(\exists ^{p}L\right)^{\rm {c}}=\forall ^{p}L^{\rm {c}}}$ and ${\displaystyle \left(\forall ^{p}L\right)^{\rm {c}}=\exists ^{p}L^{\rm {c}}}$, where L^c is the complement of L.

Let C be a class of languages. Extend these operators to work on whole classes of languages by the definition

${\displaystyle \exists ^{\mathsf {P}}{\mathcal {C}}:=\left\{\exists ^{p}L\ |\ p{\text{ is a polynomial and }}L\in {\mathcal {C}}\right\}}$

${\displaystyle \forall ^{\mathsf {P}}{\mathcal {C}}:=\left\{\forall ^{p}L\ |\ p{\text{ is a polynomial and }}L\in {\mathcal {C}}\right\}}$

Again, De Morgan's laws hold: ${\displaystyle {\mathsf {co}}\exists ^{\mathsf {P}}{\mathcal {C}}=\forall ^{\mathsf {P}}{\mathsf {co}}{\mathcal {C}}}$ and ${\displaystyle {\mathsf {co}}\forall ^{\mathsf {P}}{\mathcal {C}}=\exists ^{\mathsf {P}}{\mathsf {co}}{\mathcal {C}}}$, where ${\displaystyle {\mathsf {co}}{\mathcal {C}}=\left\{L^{c}|L\in {\mathcal {C}}\right\}}$.

The classes NP and co-NP can be defined as ${\displaystyle {\mathsf {NP}}=\exists ^{\mathsf {P}}{\mathsf {P}}}$, and ${\displaystyle {\mathsf {coNP}}=\forall ^{\mathsf {P}}{\mathsf {P}}}$, where P is the class of all feasibly (polynomial-time) decidable languages. The polynomial hierarchy can be defined recursively as

${\displaystyle \Sigma _{0}^{\mathsf {P}}:=\Pi _{0}^{\mathsf {P}}:={\mathsf {P}}}$

${\displaystyle \Sigma _{k+1}^{\mathsf {P}}:=\exists ^{\mathsf {P}}\Pi _{k}^{\mathsf {P}}}$

${\displaystyle \Pi _{k+1}^{\mathsf {P}}:=\forall ^{\mathsf {P}}\Sigma _{k}^{\mathsf {P}}}$

Note that ${\displaystyle {\mathsf {NP}}=\Sigma _{1}^{\mathsf {P}}}$, and ${\displaystyle {\mathsf {coNP}}=\Pi _{1}^{\mathsf {P}}}$.

This definition reflects the close connection between the polynomial hierarchy and the arithmetical hierarchy, where R and RE play roles analogous to P and NP, respectively. The analytic hierarchy is also defined in a similar way to give a hierarchy of subsets of the real numbers.

Alternating Turing machines definition

An alternating Turing machine is a non-deterministic Turing machine with non-final states partitioned into existential and universal states. It is eventually accepting from its current configuration if: it is in an existential state and can transition into some eventually accepting configuration; or, it is in a universal state and every transition is into some eventually accepting configuration; or, it is in an accepting state.^[3]

We define ${\displaystyle \Sigma _{k}^{\mathsf {P}}}$ to be the class of languages accepted by an alternating Turing machine in polynomial time such that the initial state is an existential state and every path the machine can take swaps at most k – 1 times between existential and universal states. We define ${\displaystyle \Pi _{k}^{\mathsf {P}}}$ similarly, except that the initial state is a universal state.^[4]

If we omit the requirement of at most k – 1 swaps between the existential and universal states, so that we only require that our alternating Turing machine runs in polynomial time, then we have the definition of the class AP, which is equal to PSPACE.^[5]

General references

1. Arora, Sanjeev; Barak, Boaz (2009). Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4. section 1.4, "Machines as strings and the universal Turing machine" and 1.7, "Proof of theorem 1.9"
2. A. R. Meyer and L. J. Stockmeyer. The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space. In Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pp. 125–129, 1972. The paper that introduced the polynomial hierarchy.
3. L. J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, vol.3, pp. 1–22, 1976.
4. C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994. Chapter 17. Polynomial hierarchy, pp. 409–438.
5. Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. Section 7.2: The Polynomial Hierarchy, pp. 161–167.

Citations