Outline Super-Turing I. The Limits of Turing Computation or A. - - PDF document

Super-Turing or Non-Turing? 1

Super-Turing

• r

Non-Turing?

Bruce MacLennan

• Dept. of Computer Science

University of Tennessee, Knoxville www.cs.utk.edu/~mclennan

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Outline

I. The Limits of Turing Computation

A. Models & Frames of Relevance B. The Frame of Turing Computability

II. New Computational Models

A. Natural Computation B. Field Computation C. Nanocomputation

• III. Summary of Issues

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Models & Simplifying Assumptions

• Turing computation is a model of

computation

• A model is like its subject in relevant ways
• Unlike it in irrelevant ways
• A model is suited to pose & answer certain

classes of questions

• Thus every model exists in a frame of

relevance (FoR)

• FoR defines domain of reliable use of model

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Dangers of Going Outside Frame of Relevance

• Simplifying assumptions make sense within

its FoR ⇒ gives good answers

• Outside FoR, answers may reflect

simplifying assumptions more than modeled system

• Sometimes models give good answers
• utside intended FoR

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The FoR of Turing Computation

• Historical roots: issues of formal

calculability & provability in axiomatic mathematics; hence:

– finite number of steps & finite but unlimited resources – computation viewed as function evaluation – discreteness assumptions

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Idealizing Assumptions

• Finite but unbounded resources
• Discreteness & definiteness
• Sequential time
• Computational task = evaluation of well-

defined function

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Alternative Frames of Relevance

• Is TC the only kind of computation?
• Natural computation
• Field computation
• Nanocomputation

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Natural Computation

• Natural computation = computation
• ccurring in nature or inspired by it
• Occurs in nervous systems, DNA,

microorganisms, animal groups

• Good models for robust, efficient &

effective artificial systems (autonomous robots etc.)

• Different issues are relevant

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Relevant Issues Outside TC FoR

• Real-time control
• Continuous computation
• Robustness
• Generality, flexibility & adaptability
• Non-functional computation

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Real-Time Control

• Real-time (RT) response constraints
• Asymptotic complexity is usually irrelevant

– Input size typically constant or of limited variability – Computational resources are bounded

• Relevant: relation of RT response rate to RT

rates of its components

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Continuous Computation

• Inputs & outputs often:

– Are continuous quantities – Vary continuously in real time

• Computational processes often continuous
• Obviously can be approximated by discrete

quantities varying at discrete times, but …

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“Metaphysics” of Reals

• “Metaphysical issues”:

– Turing-computable reals vs. standard reals – Standard reals vs. non-standard reals

• Results depend on “metaphysical issues” ⇒
• utside FoR of model
• Naïve real analysis is sufficient for models
• f natural computation

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Cross-Frame Comparisons

• Can we compare models with different

FoRs?

• Yes: can translate one to other’s FoR
• Typically make incompatible simplifying

assumptions

• Results may depend on specifics of

translation

• E.g., how are continuous quantities

represented in TC?

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Within-Frame Comparison

Common Frame of Relevance Model 1 Model 2 Comparison OK

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FoR B

Cross-Frame Comparison

FoR A Model 1 Model 2 Meaningless Comparison

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FoR B

Translated Comparison

FoR A Model 1 Model 2 Meaningful Comparison, But Relevant? Model 1′ trans- lation

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FoR C FoR B

Translation to Third Frame

FoR A Model 1 Model 2′ Relevant? Model 1′ trans- lation Model 2 trans- lation

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Field Computation

• Field computation = computation of

continuous distributions of continuous data

• Examples:

– Optical computing – Kirkhoff machines – Cortical maps – Very dense cellular automata

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Nanocomputation

• Computation involving sub-micron devices

& arrangements of information

• Examples:

– Molecular computation (e.g., DNA) – Quantum computation – Nanoscale electronic logic gates (SETs etc.)

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Relevant Issues Outside TC FoR

• Error, noise & uncertainty are unavoidable

– must be part of model of computation – may be used productively

• Microscopic reversibility may occur

– e.g., reversible chemical reactions – want statistical or macroscopic progress

• Computation proceeds asynchronously in

continuous-time parallelism

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Conclusions

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Super-Turing and Non-Turing

• Notion of Super-Turing computation is

relative to FoR of Turing computation

• Super-Turing computation is important, but

so is Non-Turing computation

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Some Issues in Non-Turing Computation

• What is computation in broad sense?
• What FoRs are appropriate for non-Turing

computation?

• Models of non-Turing computation
• How fundamentally to incorporate error,

uncertainty, imperfection, reversibility?

• How systematically to exploit new physical

processes?