Multifunction algebras and the provability of PH↓ Original Research Article

We introduce multifunction algebras View the MathML source where τ is a set of 0 or 1-ary terms used to bound recursion lengths. We show that if for all ℓ∈τ we have View the MathML source then View the MathML source, those multifunctions computable in polynomial time with at most View the MathML source queries to a View the MathML source witness oracle for ℓ∈τ and p a polynomial. We use our algebras to obtain independence results in bounded arithmetic. In particular, we show if View the MathML source proves View the MathML source for some j⩾i then View the MathML source. This implies if View the MathML source then View the MathML source does not prove the polynomial hierarchy collapses. We then consider a subtheory, Z, of the well-studied bounded arithmetic theory View the MathML source. 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 View the MathML source-consequences of View the MathML source cannot prove the hierarchy collapses. (2) If View the MathML source for any i then the polynomial hierarchy collapses. (3) If Z proves the polynomial hierarchy is infinite then for all i, View the MathML source.