. Bruno Durand LIRMM CNRS Universit de Montpellier II November26 - PowerPoint PPT Presentation

Faulty Universality . Bruno Durand LIRMM – CNRS – Université de Montpellier II November 26 th 2011 . .

. 1. introduction . .

. One can compute in a faulty medium One can compute in a faulty medium . An informal statement derived from real theorems The author excepted: The author excepted: • only one person (Lawrence Gray) • only one person (Lawrence Gray) Peter Gàcs Peter Gàcs . . says he understands the proofs says he understands the proofs . There exists a Universal cellular fault tol- There exists a Universal cellular fault tol- . . • a few people say they think the • a few people say they think the . erant cellular automaton in erant cellular automaton in . THM . . . . proofs are correct proofs are correct . • 3D + time – easy • 3D + time – easy • nobody is willing to explain the • nobody is willing to explain the • 2D + time – less easy • 2D + time – less easy proofs proofs • 1D + time – difficult • 1D + time – difficult We (B.D., A. Romashchenko, A. Shen) can reconstruct a proof for 2D by using A set of theorems with deep conse- A set of theorems with deep conse- . . several powerful techniques combined. . . quences (e.g. ergodicity) quences (e.g. ergodicity) . 1. introduction 3/13 .

A model for universality Several possibilities : Several possibilities : . . . . Turing universality Turing universality . intrinsic universality among CA intrinsic universality among CA Several models are possible Problem: if the computation model is . “For almost all fault sequence…” too complex, then one can cheat. . . . Models are needed Models are needed . A model for faults We alternate We alternate . . 1. an iteration of the Cellular 1. an iteration of the Cellular . . . . automaton automaton . 2. a perturbation with a small 2. a perturbation with a small probability probability . 1. introduction 4/13 .

A model for universality Several possibilities : Several possibilities : . . . . Turing universality Turing universality . intrinsic universality among CA intrinsic universality among CA . Problem: if the computation model is too complex, then one can cheat. . . Models are needed Models are needed . A model for faults We alternate We alternate . . 1. an iteration of the Cellular 1. an iteration of the Cellular . . . . automaton automaton . 2. a perturbation with a small 2. a perturbation with a small probability probability Several models are possible . “For almost all fault sequence…” . 1. introduction 4/13 .

Problem: if the computation model is too complex, then one can cheat. . . Models are needed Models are needed . A model for faults We alternate We alternate A model for universality . . 1. an iteration of the Cellular 1. an iteration of the Cellular . Several possibilities : Several possibilities : . . . automaton automaton . . . . . • Turing universality • Turing universality 2. a perturbation with a small 2. a perturbation with a small . probability probability • intrinsic universality among CA • intrinsic universality among CA Several models are possible . . “For almost all fault sequence…” . 1. introduction 4/13 .

. Models are needed Models are needed . A model for faults We alternate We alternate A model for universality . . 1. an iteration of the Cellular 1. an iteration of the Cellular . Several possibilities : Several possibilities : . . . automaton automaton . . . . . • Turing universality • Turing universality 2. a perturbation with a small 2. a perturbation with a small . probability probability • intrinsic universality among CA • intrinsic universality among CA Several models are possible . . Problem: if the computation model is “For almost all fault sequence…” too complex, then one can cheat. . . 1. introduction 4/13 .

the encoding computes (instead the encoding computes (instead of our CA) of our CA) . THM . . . . the halting condition computes the halting condition computes the decoding computes the decoding computes . What we would like to explain in this talk What we would like to explain in this talk . . If the computation model allows too Fault tolerance implies a complex com- complex encoding, halting, or decod- . . putation model (necessary condition in- ing, then one can cheat. dependent of proofs) Examples : . • an encoding fonction • an encoding fonction . . . . • a halting condition • a halting condition . • a decoding fonction • a decoding fonction 1. introduction 5/13 .

. What we would like to explain in this talk What we would like to explain in this talk . If the computation model allows too Fault tolerance implies a complex com- complex encoding, halting, or decod- . . putation model (necessary condition in- ing, then one can cheat. dependent of proofs) Examples : . • an encoding fonction • an encoding fonction • the encoding computes (instead • the encoding computes (instead . of our CA) of our CA) . . . • a halting condition • a halting condition . . THM . . . . • the halting condition computes • the halting condition computes • a decoding fonction • a decoding fonction • the decoding computes • the decoding computes . 1. introduction 5/13 .

. The situation without faults is much more simple The situation without faults is much more simple . Example : A. Gajardo, E. Goles, A. Moreira A. Gajardo, E. Goles, A. Moreira . . . . THM . • an encoding function maps a The Langton ant in the plane is Turing- The Langton ant in the plane is Turing- • an encoding function maps a . . . finite object into a finite zone universal universal finite object into a finite zone • a finitary halting condition : • a finitary halting condition : Many others : . appearance of a state or bounded . . J. Conway J. Conway appearance of a state or bounded . THM . . . . . pattern . The Game of Life is Turing-universal The Game of Life is Turing-universal pattern • a decoding fonction that reads a • a decoding fonction that reads a But more and more complex models are finite word in the medium finite word in the medium needed (Damien Woods’ talk). See N. Ollinger Universalities in Cellular Automata . . 1. introduction 6/13 .

. 2. remembering one bit forever . .

. Toom’s rule Toom’s rule . Finite patterns disappear : . . A cellular automaton A cellular automaton • binary alphabet • binary alphabet . . . . DEF . • in the plane • in the plane • majority of center, top, right • majority of center, top, right . 2. remembering one bit forever 8/13 .

. Toom’s rule is fault tolerant Toom’s rule is fault tolerant . Our technique (with A. Romashchenko): a hierarchy of islands of errors in the space-time diagram . . Easy to be convinced Not trivial to prove Does not work in 1D . 2. remembering one bit forever 9/13 .

Alexandro’s solution : “The measure …ergodicity …conver- “The measure …ergodicity …conver- . . . . . gence…” gence…” True but not constructive enough . A.R. Anahi’s solution : B.D. A.R. THM EXT . This hierarchical construction can be used to prove theorems in per- . . “Let’s put there an ant and see the limit “Let’s put there an ant and see the limit . Any constructive asymptotic solution is OK . . . colation theory . frequency of what it observes” frequency of what it observes” Much better ! . . . . . . . Reading the conserved bit Reading the conserved bit . . Toom’s game : • Martin chooses x = 0 or x = 1 . . • Marcos fills the plane with x • Ivan alternates Toom/faults/Toom/faults/… as many times as he wants Eric would like to find x with probability 1 . 2. remembering one bit forever 10/13 .

A.R. Anahi’s solution : B.D. A.R. THM EXT . This hierarchical construction can be used to prove theorems in per- . . “Let’s put there an ant and see the limit “Let’s put there an ant and see the limit . Any constructive asymptotic solution is OK . . . colation theory . frequency of what it observes” frequency of what it observes” Much better ! . . . . . . . . Reading the conserved bit Reading the conserved bit . Alexandro’s solution : “The measure …ergodicity …conver- “The measure …ergodicity …conver- . . . Toom’s game : . . gence…” gence…” • Martin chooses x = 0 or x = 1 . . True but not constructive enough . • Marcos fills the plane with x • Ivan alternates Toom/faults/Toom/faults/… as many times as he wants Eric would like to find x with probability 1 . 2. remembering one bit forever 10/13 .