The bounded arithmetic hierarchy

Keith Harrow
1978 Information and Control  
The class of bounded arithmetic predicates (BA) is the smallest class containing the polynomial predicates and closed under bounded quantification ((3w)<~ R(x, y, w) or (Vw)~<~ R(x, y, w)). The bounded arithmetic predicates are a small subset of the recursively enumerable, but theY include most of the standard examples from recursive function theory and form a basis for the r.e. sets. BA is closed under Boolean operations, and quantification bounded by a polynomial, but it is not closed under
more » ... antification bounded by x v. In analogy with Kleene's arithmetic hierarchy, there is a bounded arithmetic hierarchy of predicate classes within B-d, based on the number of alternations of bounded quantifiers. The closure properties of these classes are also studied. Although the existence of a strict hierarchy is not established, necessary and sufficient conditions for the hierarchy to be strict are shown. The relationshi p of BA to other known classes of predicates is also discussed.
doi:10.1016/s0019-9958(78)90257-7 fatcat:aslt6dyy7fbk3l6xnzmzva4zk4