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
The meaning of “analog” Historically: “analog” = by analogy, i.e. same evolution R L z b k V ( t ) q C m F ( t ) q + 1 F = m ¨ z + b ˙ z + kz , V = L ¨ q + R ˙ C q 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 ) q + 1 F = m ¨ z + b ˙ z + kz , V = L ¨ q + R ˙ C q 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
Classifying machines/models time continuous space discrete 5 / 24
Classifying machines/models time continuous space ENIAC Commodore Digital Circuits laptop server supercomputer discrete 5 / 24
Classifying machines/models time continuous Analog Circuits Differential Analyzer Planimeter Antikythera Tide Predicter AFCT space ENIAC Commodore Digital Circuits laptop server supercomputer discrete 5 / 24
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 supercomputer discrete 5 / 24
Classifying machines/models time continuous Not general purpose Analog Circuits Differential Analyzer Planimeter Antikythera Tide Predicter AFCT Difference Engine space ENIAC Commodore Digital Circuits Slide Rule laptop server supercomputer discrete 5 / 24
Classifying machines/models time continuous Analog Circuits Differential Analyzer space ENIAC Commodore Digital Circuits laptop server supercomputer discrete 5 / 24
Classifying machines/models time continuous Mathematical model Analog Circuits Differential Analyzer y 0 = f ( y ) Continuous Dynamical System space y n + 1 = f ( y n ) Discrete Dynamical System ENIAC Commodore Digital Circuits laptop server supercomputer discrete 5 / 24
Classifying machines/models time continuous Computability model Analog Circuits Differential Analyzer y 0 = p ( y ) GPAC space Turing machine ENIAC Commodore Digital Circuits laptop server supercomputer discrete 5 / 24
The many many models time continuous space Population protocols Finite state automata Turing machines Petri nets Lambda calculus Recursive functions Post systems discrete Cellular automata Chemical reaction networks 6 / 24
The many many models time continuous space Population protocols Continuous Automata Deep learning models Finite state automata Turing machines Neural networks Petri nets Lambda calculus Blum Shub Smale machines Hybrid systems Recursive functions Post systems Natural computing influence dynamics discrete Signal machines Cellular automata Chemical reaction networks 6 / 24
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 Deep learning models Finite state automata Turing machines Neural networks Petri nets Lambda calculus Blum Shub Smale machines Hybrid systems Recursive functions Post systems Natural computing influence dynamics discrete Signal machines Cellular automata 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 Deep learning models Finite state automata Turing machines Neural networks Petri nets Lambda calculus Blum Shub Smale machines Hybrid systems Recursive functions Post systems Natural computing influence dynamics discrete Signal machines Cellular automata Chemical reaction networks 6 / 24
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
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 ( Ω 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
Machine vs mathematical model physical machine 8 / 24
Machine vs mathematical model physical mathematical abstraction machine model I mathematical model = abstraction of a system 8 / 24
Machine vs mathematical model physical mathematical abstraction machine model proof computability results I mathematical model = abstraction of a system I properties of model 6 = properties of system 8 / 24
Machine vs mathematical model physical mathematical abstraction machine model proof ? interpretation physical computability truth results I mathematical model = abstraction of a system I properties of model 6 = properties of system I conclusion might be quantitatively or qualitatively wrong 8 / 24
Black hole model and hypercomputations I machine: the universe I model: general relativity 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 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 problem in finite time. I conclusion: hypercomputations are possible ? Common occurrence in analog models: non-computable reals, Zeno phenomena, ... 9 / 24
Back to the Church thesis Distinguish machines from models: Actual Church thesis Every effective computation can be carried out by a Turing machine, and vice versa. ) effective = systematic method in logic/mathematics/CS 10 / 24
Back to the Church thesis Distinguish machines from models: Actual Church thesis Every effective computation can be carried out by a Turing machine, and vice versa. ) effective = systematic method in logic/mathematics/CS Physical Church Turing thesis/Thesis M Whatever can be calculated by a machine (with finite data/instructions) is Turing machine-computable. ) machine that conforms to the physical laws 10 / 24
Back to the Church thesis Distinguish machines from models: Actual Church thesis Every effective computation can be carried out by a Turing machine, and vice versa. ) effective = systematic method in logic/mathematics/CS Physical Church Turing thesis/Thesis M Whatever can be calculated by a machine (with finite data/instructions) is Turing machine-computable. ) machine that conforms to the physical laws Alternative thesis All reasonable models of computations are equivalent to Turing machines. 10 / 24
Chemical Reaction Networks (CRNs) A reaction system is a finite set of I molecular species y 1 , . . . , y n f I reactions of the form P i a i y i � ! P i b i y i ( a i , b i 2 N , f = rate) Example (any resemblance to chemistry is purely coincidental): 2H + O ! H 2 O C + O 2 ! CO 2 11 / 24
Chemical Reaction Networks (CRNs) A reaction system is a finite set of I molecular species y 1 , . . . , y n f I reactions of the form P i a i y i � ! P i b i y i ( a i , b i 2 N , f = rate) Example (any resemblance to chemistry is purely coincidental): 2H + O ! H 2 O C + O 2 ! CO 2 Semantics (assuming law of mass action): I discrete I differential I stochastic 11 / 24
Chemical Reaction Networks (CRNs) A reaction system is a finite set of I molecular species y 1 , . . . , y n f I reactions of the form P i a i y i � ! P i b i y i ( a i , b i 2 N , f = rate) Example (any resemblance to chemistry is purely coincidental): 2H + O ! H 2 O C + O 2 ! CO 2 Semantics (assuming law of mass action): I discrete ! y i = molecule count I differential close to population protocols I stochastic 11 / 24
Recommend
More recommend
