# 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" (1991) 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 which is correct at least half the time. The class P^{#P} consists of all the problems which 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.^{[1]}

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.^{[2]}

## Proof

The proof is broken into two parts.

- First, it is established that

- The proof uses a variation of Valiant-Vazirani theorem. Because contains and is closed under complement, it follows by induction that .

- Second, it is established that

Together, the two parts imply

## References

- ↑ 1998 Gödel Prize. Seinosuke Toda
- ↑ Saugata Basu and Thierry Zell (2009);
*Polynomial Hierarchy, Betti Numbers and a Real Analogue of Toda's Theorem*, in*Foundations of Computational Mathematics*