Toda's theorem

Toda's theorem is a result in computational complexity theory that was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy"[1] and was given the 1998 Gödel Prize.

Statement

The theorem states that the entire polynomial hierarchy PH is contained in P^PP; this implies a closely related statement, that PH is contained in P^#P.

Definitions

#P is the problem of exactly counting the number of solutions to a polynomially-verifiable question (that is, to a question in NP), while loosely speaking, PP is the problem of giving an answer that is correct more than half the time. The class P^#P consists of all the problems that can be solved in polynomial time if you have access to instantaneous answers to any counting problem in #P (polynomial time relative to a #P oracle). Thus Toda's theorem implies that for any problem in the polynomial hierarchy there is a deterministic polynomial-time Turing reduction to a counting problem.[2]

An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real Turing machines) was proved by Saugata Basu and Thierry Zell in 2009[3] and a complex analogue of Toda's theorem was proved by Saugata Basu in 2011.[4]

Proof

The proof is broken into two parts.

• First, it is established that

${\displaystyle \Sigma ^{P}\cdot {\mathsf {BP}}\cdot \oplus {\mathsf {P}}\subseteq {\mathsf {BP}}\cdot \oplus {\mathsf {P}}}$

The proof uses a variation of Valiant–Vazirani theorem. Because ${\displaystyle {\mathsf {BP}}\cdot \oplus {\mathsf {P}}}$ contains ${\displaystyle {\mathsf {P}}}$ and is closed under complement, it follows by induction that ${\displaystyle {\mathsf {PH}}\subseteq {\mathsf {BP}}\cdot \oplus {\mathsf {P}}}$.

• Second, it is established that

${\displaystyle {\mathsf {BP}}\cdot \oplus {\mathsf {P}}\subseteq {\mathsf {P}}^{\sharp P}}$

Together, the two parts imply

${\displaystyle {\mathsf {PH}}\subseteq {\mathsf {BP}}\cdot \oplus {\mathsf {P}}\subseteq {\mathsf {P}}\cdot \oplus {\mathsf {P}}\subseteq {\mathsf {P}}^{\sharp P}}$

References

1. Toda, Seinosuke (October 1991). "PP is as Hard as the Polynomial-Time Hierarchy". SIAM Journal on Computing. 20 (5): 865–877. CiteSeerX 10.1.1.121.1246. doi:10.1137/0220053. ISSN 0097-5397.
2. 1998 Gödel Prize. Seinosuke Toda
3. Saugata Basu and Thierry Zell (2009); Polynomial Hierarchy, Betti Numbers and a Real Analogue of Toda's Theorem, in Foundations of Computational Mathematics
4. Saugata Basu (2011); A Complex Analogue of Toda's Theorem, in Foundations of Computational Mathematics