A new Characterization of Type-2 Feasibility

Abstract

K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is de ned using a generalization of A. Cobham's de nition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equivalent to the familiar poly time functions. We generalize this equivalence to type 2 functionals. This requires a de nition of the notion`poly time in the length of type 1 inputs'. The proof of this equivalence is not a simple generalization of the proof for type 1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation, and requires an analysis of the structure of oracle queries in time bounded computations. Key words. type 2 computability, polynomialtime, notational recursion, oracle Turing machine AMS subject classi cations. 68Q05,68Q15,03D65,03D20 2. A Computational Model for Functionals. Our model for type 2 computability is a generalization of the familiar multi-tape oracle Turing machine (OTM). However, we allow arbitrary type 1 functions as oracles, rather than subsets of N. Note A preliminary version of this paper appeared as 7].

Characterizing polynomial time complexity of stream programs using interpretations

“…This definition is different from the one used in [3] (denoted here by ||.||). Indeed, the size of a function was defined by ||F ||(n) = max |k|≤n |F (k)|, in other words, ||f ||(n) = |f |(2 n − 1).…”

“…This model (uotm) is adapted from the Oracle Turing Machine model used by Kapron and Cook in their characterization of Basic Feasible functionals (bff) [3]. In the following, |x| will denote the size of the binary encoding of x ∈ N, namely ⌈log 2 (x)⌉.…”

“…This is the case since it is a max over a polynomial number of elements of polynomial length. Note that this would not be true with the size function ||.|| as defined in [3] since is not computable in polynomial time. …”

“…This model is well-suited in our framework since it manipulates functions as objects with a well-defined notion of complexity. uotm are a refinement of Oracle Turing Machines (otm) on binary words which correspond exactly to the bff algebra in [3] under polynomial restrictions. uotm are better suited than otm to study stream complexity with a realistic complexity measure, since in the uotm model accessing the n th element costs n transitions whereas it costs log(n) in the otm model.…”

“…The Implicit Computational Complexity (ICC) community has proposed characterizations of otm complexity classes using function algebra [3,4] and type systems [5,6] or recently as a logic [7].…”