Multifunction algebras and the provability of PH

We introduce multifunction algebras Bτ i where τ is a set of 0 or 1-ary terms used to bound recursion lengths. We show that if for all ` ∈ τ we have ` ∈ O(|x|) then Bτ i = FP p i−1(wit, τ̇), those multifunctions computable in polynomial time with at most O(p(`(x))) queries to a Σpi−1 witness oracle for ` ∈ τ and p a polynomial. We use our algebras to obtain independence results in bounded arithmetic. In particular, we show if Si 2 proves Σ b j = PH for some j ≥ i then Si 2 1B(Σ̂bi+1) S2. This implies if PNP 6= PNP (log) then S1 2 does not prove the polynomial hierarchy collapses. We then consider a subtheory, Z, of the well-studied bounded arithmetic theory S2 = ∪iS 2. Using our algebras (mainly the i = 1 variants of our algebras) we establish the following properties of this theory: (1) Z cannot prove the polynomial hierarchy collapses. In fact, even Z+Π̂0-consequences of S2 cannot prove the hierarchy collapses. (2) If Z ⊆ Si 2 for any i then the polynomial hierarchy collapses. (3) If Z proves the polynomial hierarchy is infinite then for all i, Si 2 ` Σpi 6= Πpi .