A Survey on Analog Models of Computation Amaury Pouly Joint work with Olivier Bournez Université de Paris, IRIF, CNRS, F-75013 Paris, France 30 june 2020 Survey: https://arxiv.org/abs/1805.05729 1 / 24 Nowadays: “analog” = continuous/opposite of digital orthogonal concepts ) even continuous/discrete unclear: hybrid exists ) The meaning of “analog” Historically: “analog” = by analogy, i.e. same evolution R L z b k V (t) q C m F(t) F = mz¨ + bz˙ + kz V = Lq¨ + Rq˙ + 1 q , C 2 / 24 The meaning of “analog” Historically: “analog” = by analogy, i.e. same evolution R L z b k V (t) q C m F(t) F = mz¨ + bz˙ + kz V = Lq¨ + Rq˙ + 1 q , C Nowadays: “analog” = continuous/opposite of digital orthogonal concepts ) even continuous/discrete unclear: hybrid exists ) 2 / 24 Some analog machines Slide Rule Difference Engine Antikythera mechanism Linear Planimeter 3 / 24 Some analog machines Admiralty Fire Control Table ENIAC Differential Analyzer Kelvin’s Tide Predicter 4 / 24 MathematicalComputabilityNot general purpose model model Analog Circuits Differential Analyzer Planimetery 0 = f (y) AntikytheraContinuousGPAC y 0 Tide= p(y Predicter) DynamicalAFCT System Difference Engine Discrete Turingyn+1 machine= f (yn) Dynamical System ENIAC Commodore Digital Circuits Slide Rule laptop server supercomputer Classifying machines/models time continuous space discrete 5 / 24 MathematicalComputabilityNot general purpose model model Analog Circuits Differential Analyzer Planimetery 0 = f (y) AntikytheraContinuousGPAC y 0 Tide= p(y Predicter) DynamicalAFCT System Difference Engine Discrete Turingyn+1 machine= f (yn) Dynamical System Slide Rule Classifying machines/models time continuous space ENIAC Commodore Digital Circuits laptop server discrete supercomputer 5 / 24 MathematicalComputabilityNot general purpose model model y 0 = f (y) ContinuousGPAC y 0 = p(y) Dynamical System Difference Engine Discrete Turingyn+1 machine= f (yn) Dynamical System Slide Rule Classifying machines/models time continuous Analog Circuits Differential Analyzer Planimeter Antikythera Tide Predicter AFCT space ENIAC Commodore Digital Circuits laptop server discrete supercomputer 5 / 24 MathematicalComputabilityNot general purpose model model y 0 = f (y) ContinuousGPAC y 0 = p(y) Dynamical System Discrete Turingyn+1 machine= f (yn) Dynamical System Classifying machines/models time continuous Analog Circuits Differential Analyzer Planimeter Antikythera Tide Predicter AFCT Difference Engine space ENIAC Commodore Digital Circuits Slide Rule laptop server discrete supercomputer 5 / 24 MathematicalComputability model model y 0 = f (y) ContinuousGPAC y 0 = p(y) Dynamical System Discrete Turingyn+1 machine= f (yn) Dynamical System Classifying machines/models time Not general purpose continuous Analog Circuits Differential Analyzer Planimeter Antikythera Tide Predicter AFCT Difference Engine space ENIAC Commodore Digital Circuits Slide Rule laptop server discrete supercomputer 5 / 24 MathematicalComputabilityNot general purpose model model y 0 = f (y) ContinuousGPAC y 0 = p(y) Dynamical System Discrete Turingyn+1 machine= f (yn) Dynamical System Classifying machines/models time continuous Analog Circuits Differential Analyzer space ENIAC Commodore Digital Circuits laptop server discrete supercomputer 5 / 24 ComputabilityNot general purpose model GPAC y 0 = p(y) Turing machine Classifying machines/models time continuous Mathematical model Analog Circuits Differential Analyzer = ( ) Continuous y 0 f y Dynamical System space Discrete yn+1 = f (yn) Dynamical System ENIAC Commodore Digital Circuits laptop server discrete supercomputer 5 / 24 MathematicalNot general purpose model = ( ) Continuous y 0 f y Dynamical System Discrete yn+1 = f (yn) Dynamical System Classifying machines/models time continuous Computability model Analog Circuits Differential Analyzer GPAC y 0 = p(y) space Turing machine ENIAC Commodore Digital Circuits laptop server discrete supercomputer 5 / 24 R recursive functions − Large population protocols Hybrid Systems Timed automata Physarum computing Boolean difference equation models Shannon’s GPAC Black hole models Hopfield’s neural networks Reaction-Diffusion Systems Chemical reaction networks Continuous Automata Deep learning models Neural networks Blum Shub Smale machines Hybrid systems Natural computing influence dynamics Signal machines The many many models time continuous space Population protocols Finite state automata Turing machines Petri nets Lambda calculus Recursive functions discrete Post systems Cellular automata Chemical reaction networks 6 / 24 R recursive functions − Large population protocols Hybrid Systems Timed automata Physarum computing Boolean difference equation models Shannon’s GPAC Black hole models Hopfield’s neural networks Reaction-Diffusion Systems Chemical reaction networks The many many models time continuous space Population protocols Continuous Automata Finite state automata Deep learning models Turing machines Neural networks Petri nets Lambda calculus Blum Shub Smale machines Recursive functions Hybrid systems discrete Post systems Natural computing influence dynamics Cellular automata Signal machines Chemical reaction networks 6 / 24 Boolean difference equation models The many many models time R recursive functions continuous − Large population protocols Hybrid Systems Timed automata Physarum computing Shannon’s GPAC Black hole models Hopfield’s neural networks Reaction-Diffusion Systems Chemical reaction networks space Population protocols Continuous Automata Finite state automata Deep learning models Turing machines Neural networks Petri nets Lambda calculus Blum Shub Smale machines Recursive functions Hybrid systems discrete Post systems Natural computing influence dynamics Cellular automata Signal machines Chemical reaction networks 6 / 24 The many many models time R recursive functions continuous − Large population protocols Hybrid Systems Timed automata Physarum computing Boolean difference equation models Shannon’s GPAC Black hole models Hopfield’s neural networks Reaction-Diffusion Systems Chemical reaction networks space Population protocols Continuous Automata Finite state automata Deep learning models Turing machines Neural networks Petri nets Lambda calculus Blum Shub Smale machines Recursive functions Hybrid systems discrete Post systems Natural computing influence dynamics Cellular automata Signal machines Chemical reaction networks 6 / 24 ? quantum analog Making sense of all these models logic boolean circuits discrete recursive Turing lambda functions machines calculus continuous “Church” thesis All discrete models are Turing machine-computable. 7 / 24 ? analog Making sense of all these models logic boolean circuits discrete recursive Turing lambda functions machines calculus continuous quantum “Church” thesis All discrete models are Turing machine-computable. 7 / 24 Making sense of all these models logic boolean circuits discrete recursive Turing lambda functions machines calculus ? continuous quantum analog “Church” thesis ? All models are Turing machine-computable. Clearly wrong: a single real number (⌦ of Chaitin) is super-Turing pow- erful. 7 / 24 Making sense of all these models logic boolean circuits discrete recursive Turing lambda functions machines calculus ? continuous quantum analog “Church” thesis ? All physical machine-based models are Turing machine-computable. Several issues with that statement. 7 / 24 abstraction mathematical model ? proof physical interpretation computability truth results I mathematical model = abstraction of a system I properties of model = properties of system 6 I conclusion might be quantitatively or qualitatively wrong Machine vs mathematical model physical machine 8 / 24 ? proof physical interpretation computability truth results I properties of model = properties of system 6 I conclusion might be quantitatively or qualitatively wrong Machine vs mathematical model physical abstraction mathematical machine model I mathematical model = abstraction of a system 8 / 24 ? physical interpretation truth I conclusion might be quantitatively or qualitatively wrong Machine vs mathematical model physical abstraction mathematical machine model proof computability results I mathematical model = abstraction of a system I properties of model = properties of system 6 8 / 24 Machine vs mathematical model physical abstraction mathematical machine model ? proof physical interpretation computability truth results I mathematical model = abstraction of a system I properties of model = properties of system 6 I conclusion might be quantitatively or qualitatively wrong 8 / 24 Informal theorem If slowly rotating Kerr black holes exists, one can check consistency of ZFC or solve the Turing halting problem in finite time. I conclusion: hypercomputations are possible ? Common occurrence in analog models: non-computable reals, Zeno phenomena, ... Black hole model and hypercomputations I machine: the universe I model: general relativity 9 / 24 Common occurrence in analog models: non-computable reals, Zeno phenomena, ... Black hole model and hypercomputations I machine: the universe I model: general relativity Informal theorem If slowly rotating Kerr black holes exists, one can check consistency of ZFC or solve the Turing halting problem in finite time. I conclusion: hypercomputations are possible ? 9 / 24 Black hole model and hypercomputations I machine: the universe I model: general relativity Informal theorem If slowly rotating Kerr black holes exists, one can check consistency of ZFC or solve the Turing halting