Alternation Trading Proofs and Their Limitations

Alternation trading proofs are motivated by the goal of separating NP from complexity classes such as Logspace or NL; they have been used to give super-linear runtime bounds for deterministic and conondeterministic sublinear space algorithms which solve the Satisfiability problem. For algorithms which use n space, alternation trading proofs can show that deterministic algorithms for Satisfiability require time greater than ncn for c < 2 cos(π/7) (as shown by Williams [21, 19]), and that co-nondeterministic algorithms require time greater than ncn for c < 3 √ 4 (as shown by Diehl, van Melkebeek and Williams [5]). It is open whether these values of c are optimal, but Buss and Williams [2] have shown that for deterministic algorithms, c < 2 cos(π/7) is the best that can obtained using present-day known techniques of alternation trading. This talk will survey alternation trading proofs, and discuss the optimality of the unlikely value of 2 cos(π/7).