Structure and definability in general bounded arithmetic theories

A Note on Universal Measures for Weak Implicit Computational Complexity

“…one whose binary length is polynomially bounded in the binary length of the query. These kind of results are extended and generalized by Pollett [16].…”

“…Building on this one can show T i 2 (X) = S i+1 2 (X) [12]. Here, the relativized theories S [16]. Independently of these, and with completely different methods (see below), we have shown separation results for theories of relativized bounded arithmetic in [3,4].…”

“…A natural stratification of WOTMs, called bounded WOTMs, is obtained by bounding the number of oracle queries. Pollett in [16] has given characterization of definable multivalued functions of theories of bounded arithmetic analogous to Krajíček's characterization of the Σ i (wit, O(log n)) is the class of multivalued functions computable by a polynomial time WOTM which on inputs of length n uses fewer than O(log n) witness queries to a Σ b i -oracle. For Φ is a set of formulas, a multivalued function f is called Φ-definable in some theory T if there is a formula ϕ(x, y) in Φ such that ϕ describes the graph of f and T proves the totality of f via ϕ:…”

Characterising Definable Search Problems in Bounded Arithmetic via Proof Notations

“…• Pollett [Pol99] showed that the Σ b j+1 -definable multi-functions in S i 2 for j > i are exactly FP…”

An unexpected separation result in Linearly Bounded Arithmetic

“…The strict classes have been studied in several places, e.g. by Pollett [19] and Beckmann [3], and in particular by Impagliazzo and Krajíček [11], on which the present paper builds.…”