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Advances in Unconventional Computing

Volume 22 of the series Emergence, Complexity and Computation pp 73-115

Date:

An Analogue-Digital Model of Computation: Turing Machines with Physical Oracles

  • Tânia AmbaramAffiliated withDepartment of Mathematics, Instituto Superior Técnico, Universidade de Lisboa Email author 
  • , Edwin BeggsAffiliated withCollege of Science, Swansea University
  • , José Félix CostaAffiliated withDepartment of Mathematics, Instituto Superior Técnico, Universidade de Lisboa
  • , Diogo PoçasAffiliated withDepartment of Mathematics and Statistics, McMaster University
  • , John V. TuckerAffiliated withCollege of Science, Swansea University

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Abstract

We introduce an abstract analogue-digital model of computation that couples Turing machines to oracles that are physical processes. Since any oracle has the potential to boost the computational power of a Turing machine, the effect on the power of the Turing machine of adding a physical process raises interesting questions. Do physical processes add significantly to the power of Turing machines; can they break the Turing Barrier? Does the power of the Turing machine vary with different physical processes? Specifically, here, we take a physical oracle to be a physical experiment, controlled by the Turing machine, that measures some physical quantity. There are three protocols of communication between the Turing machine and the oracle that simulate the types of error propagation common to analogue-digital devices, namely: infinite precision, unbounded precision, and fixed precision. These three types of precision introduce three variants of the physical oracle model. On fixing one archetypal experiment, we show how to classify the computational power of the three models by establishing the lower and upper bounds. Using new techniques and ideas about timing, we give a complete classification.