An implicit characterisation of the polynomial hierarchy in an unbounded arithmetic

Today, there are countless approaches towards characterising complexity classes via logic. Foremost amongst these lies the proof-theoretic approach, characterising classes as the ‘representable’ functions of some logic or theory. Examples include bounded arithmetic [6] [13] [9], applicative theories [7] [12], intrinsic and ramified theories [16] [4], fragments of linear logic [11] [10] [14] [1] and fragments of intuitionistic logic [15]. To some extent there is a distinction between various notions of ‘representability’, namely between logics that type terms computing functions of a given complexity class, and theories that prove the totality or convergence of programs computing functions in a given complexity class. A somewhat orthogonal distinction is whether the constraints on the logic or theory are implicit or explicit. The former includes constraints such as ramification, type level and substructural considerations, while the latter includes bounded quantification, bounded modalities etc. This distinction is also naturally exhibited in associated function algebras, e.g. Cobham’s limited recursion on notation [8] vs. Bellantoni and Cook’s predicative recursion on notation [3]. While implicit constraints may be preferable since no bounds occur in the characterisation itself per se, explicit bounds are typically far more useful for more fine-grained characterisations of complexity classes. For instance, the polynomial hierarchy, PH, and its levels can be neatly characterised by the theories S 2 of bounded arithmetic, using bounds on quantifiers to control complexity [6]. 1 In this work we improve the situation by using implicit methods in first-order theories to characterise PH. To achieve this we work with a function algebra of Bellantoni from [5] in which to extract programs, and use the witness function method of Buss to extract programs at ground type and preserve quantifier information, necessary to delineate the levels of PH.