Turing Machine Resisting Isolated Bursts of Faults

1. Turing Machine Resisting Isolated Bursts of Faults IlirCapuni and Peter Gacs Boston University

2. Introduction • A fault: a violation of the transition function • Change the state, the tape symbol and move the head arbitrarily … 0 0 1 1 0 … … 0 0 0 1 0 … step i q q q’ … 0 0 1 1 0 … step i+1 q’ ‘Regular’ transition A faulty transition

3. The main question • Is there a Turing machine that can carry out its computation despite faults (violations of the transition function) that occur independently of each other with small probability?

4. Reduced goal: resistance to isolated bursts of faults 0 1 0 1 0 … 0 0 1 1 0 0 … q

5. Our result illustrated Given: 1) Machine with the alphabet and state set … 0 0 0 0 1 0 1 1 0 0 0 0 … q 2) The size of the bursts We construct: 3) Block code (E, D) of block-sizeQ 1) IntegersV, Q, C, all depending linearly on 2) Machine with the alphabet and state set that does not contain a halting state … a3 a1 a2 a3 b1 b2 b3 a1 a2 a3 b1 b2 b3 a1 a2 a3 …

6. Such that … 0 0 0 0 1 0 1 0 0 0 0 … 0 q0 1 0 … M2 … 0 0 0 0 1 1 0 1 0 0 0 0 … q5 T-1 … 0 0 0 0 1 0 1 1 0 0 0 0 … qf T Burst are separated by at least V fault- free steps qf The result … 0 … p0 1 … M1 … a a a a b b b … p5 t-Q No faults for Q steps … a a a a b b b b a a a a … pt t, t>CT … … q’

7. Sketch of the construction Colonies Info 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 State _ _ _ _ _ q q q q q _ _ _ _ _ … … Addr 0 1 2 3 4 0 1 2 3 4 0 1 2 2 3 4 … 0 0 1 1 0 … Sweep 2 2 2 2 2 g g g g g L L L L L Drift 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q Other fields Head Mode = Normal Addr = 0 Sweep = 2 Other fields

8. Burst -tolerant Tur(n)ing machine 1 q 0 Info 0 0 0 0 0 1 1 1 1 1 1 1 1 1 L State _ _ _ _ _ q q q q _ _ _ _ _ 1 … … Addr • Information:there is an error-correcting code and global repetitions to handle computation errors • Simulation:There is a structure that could be restored locally using a “recovery procedure” that does not restore lost information 0 1 2 3 4 1 2 3 4 0 1 2 3 4 Sweep L L L L g g g g g L L L L L Drift 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Other fields Head Mode = Normal Addr = 0 Sweep = 1 Other fields Computation phase Transferring phase Two lines of defense

9. Islands and stains • Intervals of cells whose structure is altered by faults will be called islands • The cells where Info and State are altered by faults will be called stains

10. An example Recovery procedure Simulation resumes, but a stain can remain until the next decoding-encoding takes placeover this colony

11. Localization with zigging Alarm is called Sweep - 1 Sweep + 1 Sweep + 2 Sweep - 1 Extensive damage Damage localized with zigging

12. Recovery procedure • Opens a constant size recovery interval around the cell where the Alarm is called • In several sweeps it computes majority of Address, Sweep, and Drift fields, and computes the values that need to be filled in the ‘damaged’ area • Then it leaves the head close to the front of the sweep

13. Disturbing the recovery • The burst can leave the machine in some arbitrary state • The recovery procedure can itself be also disturbed by a burst • Our recovery procedure handles provably all these cases Bad news Good news

14. Proofs • Certain proofs are long and require detailed case analysis

15. Extensions 0 1 a 1 0 b 1 0 a 0 1 a 1 0 Prog a Info a a b b b a b a b a a b a b a State _ _ _ _ _ b a b a b _ _ _ _ _ … Addr 0 1 2 3 4 0 1 2 3 4 0 1 2 2 3 4 Sweep 2 2 2 2 2 g g g g g L L L L L Drift 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Other fields • Coding: Store the program and the state of M2 on the tape of M1 and make M1 act as a Universal Turing machine • The slowdown of the machine now becomes quadratic on the burst size Head Mode = Normal Addr = 0 Sweep = 2 Other fields … … 0 0 1 1 0 … q

16. Toward resistance to probabilistic noise • Our noise model was combinatorial • How to deal with low-probability noise combinatorially? • Level 1: Consider first noise that has low frequency: Bursts are b1 long but V1 steps apart from each other • Level 2: Then allow violations of the previous occurring with low frequency: Occur in at most b2 steps separated by at least V2 steps from each other • And so on… We are left with a noise set that is “sparse”

17. From the Tur(n)ing machine to a Turing machine that can withstand probabilistic noise • We use the previous construction as a building block • Some serious additional challenges must be handled to maintain the hierarchy of simulations …

18. 0 1 a 1 0 1 a 1 0 0 1 a 1 0 Prog 0 Info a a b b b a b a b a a b a b a State _ _ _ _ _ a a b b b _ _ _ _ _ … … Addr 0 1 2 3 4 0 1 2 3 4 0 1 2 2 3 4 … 0 0 1 1 0 … Sweep 2 2 2 2 2 g g g g g L L L L L Drift 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q Other fields Head Mode = Normal Addr = 0 Sweep = 2 Other fields 0 1 a 1 0 1 a 1 0 0 1 a 1 0 Prog 0 1 a 1 0 1 a 1 0 0 1 a 1 0 Prog 0 0 Info Info a a b b b a b a b a a b a b a a a b b b 1 b 0 b a a b a b a State State _ _ _ _ _ a a c b b _ _ _ _ _ _ _ _ _ _ a 1 c b b _ _ _ _ _ … … … … Addr Addr 0 1 2 3 4 0 1 2 3 4 0 1 2 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 2 3 4 Sweep Sweep 2 2 2 2 2 g g g g g 2 2 2 2 2 g g g g g L L L L L L L L L L Drift Drift 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Other fields Other fields Head Head Mode = Normal Mode = Normal Addr = 0 Addr = 0 Sweep = 2 Sweep = 2 Other fields Other fields

19. Why do we care? • It is a natural simple question • Just as with cellular automata, it is surprising that the solutions seem to require so much complexity • So far, this is the simplest universal machine that can resist isolated bursts of faults … Q u e s t i o n s ? T h a n k _ y o u ! q P q’ q’’ q q q q f f f l k k k k k k k k