A Hierarchy for $$ BPP //\log \!\star $$ B P P / / log ⋆ Based on Counting Calls to an Oracle

Abstract

Algorithms whose computations involve making physical measurements can be modelled by Turing machines with oracles that are physical systems and oracle queries that obtain data from observation and measurement. The computational power of many of these physical oracles has been established using non-uniform complexity classes; in particular, for large classes of deterministic physical oracles, with fixed error margins constraining the exchange of data between algorithm and oracle, the computational power has been shown to be the non-uniform class BPP // log. In this paper, we consider non-deterministic oracles that can be modelled by random walks on the line. We show how to classify computations within BPP // log by making an infinite non-collapsing hierarchy between BPP // log and BPP. The hierarchy rests on the theorem that the number of calls to the physical oracle correlates with the size of the responses to queries.

A Survey on Analog Models of Computation

“…In the fixed precision case, ε is a constant of the experiment and cannot be changed. More recently, the authors have developed a hierarchy for BPP//log⋆ based on counting calls to a non-deterministic physical oracle modeled by a random walk on a line [44].…”

East-West paths to unconventional computing

“…The time complexity of a measurement reduces the computational power of dynamic systems with self-advice from their internal parameters. According with [82], this reduction of super-Turing capabilities can be so great that the real numbers add no further power, even assuming that the reals exist beyond the discrete nature of matter and energy. In the best scenario, we are still waiting for some evidence that refutes the following conjecture: No reasonable physical measurement has an associated measurement map performable in polynomial time.…”