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 PPP; 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
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

gollark: (because it's bad, and won't do that automatically)
gollark: (technically it also has some code to force it to respond to an instant-lose/instant-win situation)
gollark: It is funny that people keep losing to a fairly trivial piece of code which just decides how good a move is by playing 100 *entirely random games* starting from it and seeing how many it wins.
gollark: Okay, I am now decreasing my estimate of your programming competence.
gollark: I don't know if there's a general strategy. The main thing to exploit is that the AI can't really respond to two threats at once.

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
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