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

Super-Turing or Non-Turing? Outline Super-Turing I. The Limits of Turing Computation or A. Models & Frames of Relevance Non-Turing? B. The Frame of Turing Computability II. New Computational Models Bruce MacLennan A. Natural Computation B. Field Computation Dept. of Computer Science C. Nanocomputation University of Tennessee, Knoxville III. Summary of Issues www.cs.utk.edu/~mclennan 2006-09-11 Super-Turing or Non-Turing? 2 Models & Simplifying Dangers of Going Outside Assumptions Frame of Relevance • Turing computation is a model of computation • Simplifying assumptions make sense within • A model is like its subject in relevant ways its FoR ⇒ gives good answers • Unlike it in irrelevant ways • Outside FoR, answers may reflect • A model is suited to pose & answer certain simplifying assumptions more than modeled classes of questions system • Thus every model exists in a frame of • Sometimes models give good answers relevance (FoR) outside intended FoR • FoR defines domain of reliable use of model 2006-09-11 Super-Turing or Non-Turing? 3 2006-09-11 Super-Turing or Non-Turing? 4 1

Super-Turing or Non-Turing? The FoR of Turing Computation Idealizing Assumptions • Historical roots : issues of formal • Finite but unbounded resources calculability & provability in axiomatic • Discreteness & definiteness mathematics; hence: • Sequential time – finite number of steps & finite but unlimited • Computational task = evaluation of well- resources defined function – computation viewed as function evaluation – discreteness assumptions 2006-09-11 Super-Turing or Non-Turing? 5 2006-09-11 Super-Turing or Non-Turing? 6 Natural Computation Alternative Frames of Relevance • Natural computation = computation occurring in nature or inspired by it • Is TC the only kind of computation? • Occurs in nervous systems, DNA, microorganisms, animal groups • Natural computation • Good models for robust, efficient & • Field computation effective artificial systems (autonomous • Nanocomputation robots etc.) • Different issues are relevant 2006-09-11 Super-Turing or Non-Turing? 7 2006-09-11 Super-Turing or Non-Turing? 8 2

Super-Turing or Non-Turing? Relevant Issues Outside TC FoR Real-Time Control • Real-time control • Real-time (RT) response constraints • Continuous computation • Asymptotic complexity is usually irrelevant – Input size typically constant or of limited • Robustness variability • Generality, flexibility & adaptability – Computational resources are bounded • Non-functional computation • Relevant: relation of RT response rate to RT rates of its components 2006-09-11 Super-Turing or Non-Turing? 9 2006-09-11 Super-Turing or Non-Turing? 10 Continuous Computation “Metaphysics” of Reals • Inputs & outputs often: • “Metaphysical issues”: – Are continuous quantities – Turing-computable reals vs. standard reals – Vary continuously in real time – Standard reals vs. non-standard reals • Results depend on “metaphysical issues” ⇒ • Computational processes often continuous outside FoR of model • Obviously can be approximated by discrete quantities varying at discrete times, • Naïve real analysis is sufficient for models but … of natural computation 2006-09-11 Super-Turing or Non-Turing? 11 2006-09-11 Super-Turing or Non-Turing? 12 3

Super-Turing or Non-Turing? Within-Frame Comparison Cross-Frame Comparisons • Can we compare models with different Comparison OK FoRs? • Yes: can translate one to other’s FoR • Typically make incompatible simplifying assumptions Model Model 1 2 • Results may depend on specifics of translation Common • E.g., how are continuous quantities Frame of Relevance represented in TC? 2006-09-11 Super-Turing or Non-Turing? 13 2006-09-11 Super-Turing or Non-Turing? 14 Translated Comparison Cross-Frame Comparison Meaningful Comparison, Meaningless Comparison But Relevant? trans- lation Model Model Model Model Model 1 ′ 1 2 1 2 FoR FoR FoR FoR A B A B 2006-09-11 Super-Turing or Non-Turing? 15 2006-09-11 Super-Turing or Non-Turing? 16 4

Super-Turing or Non-Turing? Translation to Third Frame Field Computation Relevant? • Field computation = computation of continuous distributions of continuous data trans- trans- • Examples: lation lation Model Model Model Model – Optical computing 1 ′ 2 ′ 1 2 – Kirkhoff machines – Cortical maps FoR FoR FoR A C B – Very dense cellular automata 2006-09-11 Super-Turing or Non-Turing? 17 2006-09-11 Super-Turing or Non-Turing? 18 Nanocomputation Relevant Issues Outside TC FoR • Computation involving sub-micron devices • Error, noise & uncertainty are unavoidable & arrangements of information – must be part of model of computation – may be used productively • Examples: • Microscopic reversibility may occur – Molecular computation (e.g., DNA) – e.g., reversible chemical reactions – Quantum computation – want statistical or macroscopic progress – Nanoscale electronic logic gates (SETs etc.) • Computation proceeds asynchronously in continuous-time parallelism 2006-09-11 Super-Turing or Non-Turing? 19 2006-09-11 Super-Turing or Non-Turing? 20 5

Super-Turing or Non-Turing? Super-Turing and Non-Turing • Notion of Super-Turing computation is relative to FoR of Turing computation Conclusions • Super-Turing computation is important, but so is Non -Turing computation 2006-09-11 Super-Turing or Non-Turing? 21 2006-09-11 Super-Turing or Non-Turing? 22 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? 2006-09-11 Super-Turing or Non-Turing? 23 6