A Note on Universal Measures for Weak Implicit Computational Complexity

Abstract

Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate "universal measures" by "dynamic ordinals", and "weak implicit computational complexity" by "bounded arithmetic". Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories.

Generalised dynamic ordinals — universal measures for implicit computational complexity

“…With id we denote the identity function, id(n) = n. The classes FP Σ b i (wit, poly) and FP Σ b i (wit, O(log n)) considered by Krajíček can be expressed as respectively FP Σ b i (wit, | id |) and FP Σ b i (wit, O(| id | 2 )) using the previous definition. The following characterisations of definable multivalued functions by bounded polynomial time WOTMs can be read of the results by Pollett [20], see [20] or [3] for more details. Let us remind that 2 m (x) and |x| m denote the m-fold iterations of the exponentiation function, respectively binary length function.…”

“…We start this section by repeating definitions of and results on dynamic ordinals for some fragments of bounded arithmetic from [4] and [3]. The underlying language will always be the language L BA (α) of relativised bounded arithmetic.…”

“…In [3] we have described the following connections between the dynamic ordinal of some relativised bounded arithmetic theories and the Σ b 2 -definable multivalued functions of their unrelativised companions. For T from the following infinite list of theories Hence, the characterisation of definable multivalued functions of bounded arithmetic theories from the previous section suggests the following definition of generalised dynamic ordinals (see discussion in [3] for more details):…”

“…We will approach the general idea of finding universal measures by doing a case study for a particular framework of weak implicit computational complexity called bounded arithmetic. We already argued in [3] that so-called dynamic ordinals can be viewed as universal measures for some fragments of bounded arithmetic and corresponding bounded witness oracle Turing machine classes. In this paper, we will extend this project by defining and computing generalised dynamic ordinals and indicating their role as universal measures for all bounded arithmetic theories.…”

“…DO(T (α)) is a set of unary number-theoretic functions, which characterises the amount of Π b 1 (α)-orderinduction provable in T (α). In [3], we have described how this fits into our general program on finding universal measure by connecting dynamic ordinals with witness oracle computations. The above mentioned characterisation of definable multivalued functions of higher bounded arithmetic theories suggests the following definition of generalised dynamic ordinals (for more details on this motivation see the discussion in [3]): The i-th generalised dynamic ordinal DO i (T (α)) of a relativised theory of bounded arithmetic T (α) characterises the amount of Π b i (α)-order-induction provable in T (α) (thus, the usual dynamic ordinal is just the first generalised dynamic ordinal).…”