Computational complexity with experiments as oracles

Abstract

We discuss combining physical experiments with machine computations and introduce a form of analogue-digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

Membrane Systems and Hypercomputation

“…-Can HP be implemented via a physical system (an oracle [Tur39,BCLT08]) that is not itself computational?…”

Computational complexity with experiments as oracles. II. Upper bounds

“…In this paper, we answer questions left open in Beggs et al (2008a) by proving upper bounds, which enable us to complete the classification of the power of the three kinds of scatter machines. Combining upper and lower bounds, we formulate our new results as equivalence theorems.…”

“…Since the scatter machine can compute more than the Turing machine, it was an interesting experiment to choose as an oracle. In Beggs et al (2008a), the hybrid machines made by combining the SME and Turing machines were termed analogue-digital (AD) scatter machines. The case studies revealed some concepts and mathematical tools useful for a general theory of Turing machines with experimental oracles.…”

“…In Beggs et al (2008a), we introduced the idea of coupling a Turing machine to an abstract physical experiment as a new form of oracle. We focused on the design and operation of these computational models and the question: What can be computed by Turing machines with such physical oracles?…”

“…We focused on the design and operation of these computational models and the question: What can be computed by Turing machines with such physical oracles? In Beggs et al (2008a), we chose a particular experiment and studied the hybrid machines made by combining Turing machines with the experiment.…”