### Alternation Trading Proofs and Their Limitations [chapter]

Sam Buss
2013 Lecture Notes in Computer Science
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 o(1) space, alternation trading proofs can show that deterministic algorithms for Satisfiability require time greater than n cn for c < 2 cos(π/7) (as shown by Williams [21, 19] ), and that
more » ... o-nondeterministic algorithms require time greater than n cn 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). A central open problem in computer science is the question of whether nondeterministic polynomial time (NP) is more powerful than ostensibly weaker computational classes such as polynomial time (P) or logarithmic space (Logspace). These are famously important and difficult questions, and unfortunately, in spite of over 40 years of concerted efforts to prove that NP = P or NP = Logspace, it is generally felt that minimal progress has been made on resolving them. Alternation trading proofs are a method aimed at separating NP from smaller complexity classes, by using "indirect" diagonalization to prove separations. A typical alternation trading proof begins with a simulation assumption, for instance the assumption that the NP-complete problem of Satisfiability (SAT) can be recognized by an algorithm which uses time n c and space n o(1) . Iterated ⋆