admin ps5 due ps6 out today barrier results relativization barrier results Q. is P vs NP hard to resolve? Q. is P vs NP *possible* to resolve? proof vs truth halting problem Thm: is undecidable Thm: halting problem statments not all provable in any "reasonable" proof system Cor: any "reasonable" proof system is "incomplete" equiv: these statements are *independent* of the axioms Q. could there be natural statements not provable in reasonable systems? A. yes A. no A. mabye parallel postulate euclid's axioms parallel postulate at most one parallel line through a point two lines joinly perpendicular to another line never meet Q. *prove* the parallel postulate? A(Bolyai,Gauss,Beltrami,...): no! sketch create "worlds"/"models" of core axioms where parallel postulate is false elliptical geometry hyperbolic geometry cor: parallel postulate is *independent* of rest of euclidean geometry Q. could P vs NP be independent? Q. could P vs NP be independent of *known* techniques? ie. a barrier result relativization def: oracle TM TIME^A P^A NTIME^A NP^A def: relativization "only simulate" thm: TIME hierarchy relativizes pf thm: BPP in PH relativizes rmk: common set of techniques efficiently simulate one TM with another "black box" enumerate over all TMs can attempt to formalize this further thm: exist oracle A where P^A=NP^A pf: idea: make A "big enough" so P vs NP disappears A=TQBF P^A=PSPACE NP^A=NPSPACE =PSPACE thm: exist oracle B where P^B\ne NP^B pf: idea: create a "world" from scratch idea: the oracle B is an "input" of size 2^n each oracle call is a query to this input use separation between P^{query} and NP^{query} + diagonalization over all TM's define OR^L={1^n: some x\in\bits^n with B(x)=1} lem: OR^L\in NP^L, any L prop: TM M^L running in time t(n) solving OR^L for any L => t(n)-depth decisision tree computing OR pf lem: some B, OR^B\notin P^B pf idea: plug in L where any TM gets OR^L wrong enumerate polytime TMs M_i with clocks n^c+c find n_i where B|_{\bits^{n_i}} not defined yet n^c+c<2^n set B so M_i fails rmk: B is computable any resolution of P vs NP must "notice" lack of oracle most known results in complexity theory relativize eg BPP\subset PH most open questions are known to require non-relativizing techniques arithmetization is non-relativizing exists A, coNP^A\not\subseteq IP coNP\subseteq P^{#SAT}\subseteq IP arithmetization: f:\bits^n\to\bits \mapsto \hat{f}:\F^n\to\F^n \-> would also need to do this to oracle algebraization[AaronsonWigderson] allow access to low-degree extension to oracle most open problems still cannot be resolved next time natural proofs barrier