The most natural question of reliable computation, in every computation model and noise model,is whether given a certain level of noise, a machine of that model exists that canperform arbitrarily complex computations under noise of that level. This question has positive answers for circuits, cellular automata, and recently for Turing machines.

Here, we raise the question of the existence of a random access machine that---with some moderate slowdown --- can simulate any other random access machine even if the simulator is subjected to constant size bursts of faults separated by a certain minimum number of steps from each other.

We will analyze and spell out the problems and difficulties that need to be addressed in such construction

Capuni, Ilir

Dervishaj, Ervin

Epoka University

21Towards a Universal RAM Machine Resistant toIsolated Bursts of FaultsIlir Çapuni Ervin DervishajComputer Engineering DepartmentEpoka University{icapuni, edervishaj10}@epoka.edu.alAbstract—The most natural question of reliable computa-tion, in every computation model and noise model, is whethergiven a certain level of noise, a machine of that model existsthat can perform arbitrarily complex computations undernoise of that level. This question has positive answers for cir-cuits, cellular automata, and recently for Turing machines [3],[4].Here, we raise the question of the existence of a randomaccess machine that—with some moderate slowdown — cansimulate any other random access machine even if the sim-ulator is subjected to constant size bursts of faults separatedby a certain minimum number of steps from each other.We will analyze and spell out the problems and dicultiesthat need to be addressed in such construction.Index Terms—Random access machines, faults, reliability.I. IntroductionThe problem of constructing fault-proof machines fromcomponents that can fail was rst considered by von Neu-mann in [12], who addressed the problem in the Booleancircuits model. New advances along this path were madein [9], [10]. The question has been considered in uniformmodels of computation as well. A simple rule for two-dimensional cellular automata that keeps one bit forever eventhough each cell can fail with some small probability wasgiven in [11]. A 3-dimensional reliably computing cellularautomaton using Toom’s rule was constructed in [7]. Alas,all simple one-dimensional cellular automata appear to be“ergodic” (forgetting everything about their initial congu-ration in time independent of the size). The rst, complex,nonergodic cellular automaton was constructed in [5], andimproved upon in [6]. Surprisingly, even non-ergodic Turingmachines exists [3], [4].Computing reliably with RAM machines is considered in amyriad of papers with various assumptions on the noise. Asimilar problem but where only the memory is subject to alimited amount of noise is considered in [8]. The cited workinitiated an entire line of research approaching the problemfrom data structures’ and algorithmic perspective. A recentresult in this line is [2].Our approach diers from the papers above for that faultscan perturb the central unit and for that that the noise comesin bursts that are spaced out of each other. The reason forThe rst author’s dedication to this work was made possible by a researchgrant RD 01-2013 of Zanus ltd, Ulqin Montenegrothese assumptions on noise is that — similarly as in [4],[6]— we hope to use this construction as a building blockfor a hierarchically organized RAM that can withstand faultsthat occur independently of each other with some smallprobability.A. The random access machine modelWe view the random-access machine as an extension of theTuring machine in that that the tape consists of cells able tostore an arbitrary integer, and that the head can jump at anycell of the memory M specied by its address.Register PC is a special register called the programcounter. Let Γ = {PC,R0, . . . ,Rk } be the set of registers ofthe control unit. Register R0 will be called the accumulator.The program is a sequence of instructions from the Table I.Denition I.1. A Random Access Machine is a pair (k,Π)where k > 0 is the number of registers, and Π is the program(i.e., a nite sequence of instructions). yThe operation of a RAM machine is described below.1) Initially, the input x is loaded in the rst m cells of thememory M , where m is the length of x . The registersR1, . . . ,Rk , PC and the other memory cells M[m], M[m+1], . . . are initialized to the value 0.2) At each step of the execution, the random accessmachine executes the program line pointed to by theprogram counter PC. After executing the instruction,the value of PC is incremented by 1, unless the in-struction is jump, jzero, or jpos If the instructionis jzero or jpos and the condition is not satised,PC is also incremented by 1.Symbols are encoded by integers, and we assume that theblank symbol is encoded by 0.Example I.2. If Σ = {#,a,b}, then we might assign # = 0,a =1,b = 2. Then the input abba would be represented in therst four cells of the memory as 1,2,2,1. yA conguration of a Random Access Machine (k,Π) is ak + 2-tuple(PC,R0, . . . ,Rk ,M ),where M ∈ ZZ is the memory conguration.ISCIM 2013, pp. 21-23 © 2013 Authors22TABLE IInstructions of a RAM machineInstruction Semantics Descriptionload n R0 ← n Put the value n into R0load Rk R0 ← Rk Put the value of Rk into R0store Rk Rk ← R0 Put the value of R0 into Rkread Rk R0 ← M[Rk ] Copy the value at memory location Rk into R0write Rk M[Rk ]← R0 Write the value of R0 at a memory location Rkadd n R0 ← R0 + n Add the value n to R0add Rk R0 ← R0 + Rk Add the value Rk to R0mult Rk R0 ← R0 ∗ Rk Multiply the value in R0 with Rkjump n PC← n Set the program counter to njzero n if R0 = 0 then PC← n If R0 is zero, then set the PC to njpos n if R0 > 0 then PC← n If R0 is positive, then set the PC to nThe work of the machine can be described as a sequenceof congurations C0,C1,C2, . . . , where Ct is the congurationat time t .The program Π tells us how to compute the next cong-uration from the present one.Denition I.3 (Fault). We say that a fault occurs at timet if conguration Ct+1 is obtained from Ct by violating thetransition function specied by the program. yB. What faults can doBy denition, if a fault occurred at time t , then, at time t+1a random cell and the content of the registers in the controlunit are arbitrary. This means that unlike the eects of burstsin a Turing machine — where bursts produce islands of cellsof diameter β close to the head, in RAM model, these islandsare scattered in a memory in a random way. This is a majordiculty in construction of a fault-tolerant RAM since thismeans that the information stored in the memory decays.For example, consider a computation whose space iss . Suppose further that for time proportional to s2, thecomputation was carried out in only the rst half of thecomputation space. Then, within this time, the second halfof the computation space may be completely ruined by faultsthat occurred during this time period.II. The Desired ResultIn this section we will spell out a desired result. Before weneed the following denition.Denition II.1 (Codes). Let Σ1,Σ2 be two subsets of Z. Ablock code is given by a positive integer Q— called the blocksize — and a pair of functionsψ∗ : Σ2 → ΣQ1 , ψ ∗ : ΣQ1 → Σ2with the property ψ ∗ (ψ∗ (x )) = x . yNow we can spell out the desired statement.Let RAM machine M2 start from an input x with astarting conguration ξ0 and suppose that it haltswithin T steps, writing the result in the memorylocation with address 0. Let S be the amount ofspace that M2 used during its computation.Then, the following can be constructed.1) RAM machine M1 with k registers that doesnot have a halting state.2) Constant Q and a block code (ψ∗,ψ ∗) of blocksize Q .3) Constants T ′ depending on T and S , andconstant k ′ = O (k ).Suppose M1 starts from the initial congurationξ ′0 = ξ′0 (x ), and it operates under noise that consistsof isolated bursts of size at most β .Then, at any time t , t > T ′, the result of M2 can bedecoded from the memory block k ′, . . . ,k ′ +Q − 1.III. Solutions and difficultiesA. Solving the problem by simulating the fault-tolerant TuringmachineIt is natural to ask if we can achieve the needed faulttolerance by just simulating the fault-tolerant Turing ma-chine from [3] or [4]. The answer is alas negative. The fault-tolerant Turing machines constructed in [3] and [4] havean underlying assumption that the information on the tapedoes not decay. However, here faults can ruin portions ofthe memory far from the location of the head of the Turingmachine.B. Redundancy and memory updatesLet us consider a dierent solution. We will encode thecontrol unit of M2 using some error-correcting code intoa xed constant size portion of the memory of M1. Usingthe same code we will encode the memory of M2 onto thememory of M1.Then, similar to the simulations in [3], [4], we will simulateone step of M2 by many steps of M1.Since information “decays” in the memory, we need to nda way to constantly update and check all the parts of thecomputation space in the memory. Since RAM is a sequentialmachine, we need to space out the bursts enough such thatthe machine can “catch up” with the decay of the informationin the memory. However, the minimal distance between twoconsecutive bursts will now depend on the amount of spacethat M2 needs during the computation.23C. A hierarchical organization with digestsIn the previous section we have seen that in order topreserve the information from decaying in the memory, weneed to constantly refresh it by decoding and encoding withan error-correcting code. Doing this for the entire space s ofcomputation may be time consuming.Using the approach of [1] we may restrict doing this for apart of the memory of size O (log s ) which will be consideredmore reliable.Giving a fully edged construction based on this idea andproving the desired result spelled out in Section II will be asubject of our forthcoming research.References[1] Blum M., Evans W., Gemmel P., Kannan S., and Naor M.: Checkingthe Correctness of Memories. In: Algorithmica, 1995, pp. 90-99.[2] Christiano P, Demaine D. E., and Kishore Sh.: Lossless Fault-TolerantData Structures with Additive Overhead. In Proceedings of the 12thinternational conference on Algorithms and data structures (WADS’11),Frank Dehne, John Iacono, and Jörg-Rüdiger Sack (Eds.). Springer-Verlag, Berlin, Heidelberg, 243-254.[3] Çapuni I., Gács, P. : A Turing machine resisting isolated bursts offaults. In: Chicago Journal of Theoretical Computer Science, vol. 2013.[4] Ilir Capuni. 2013. A Fault-Tolerant Turing Machine. Ph.D. Disser-tation. Boston University, Boston, MA, USA. Advisor(s) Peter Gacs.AAI3536949.[5] Gács, P.: Reliable computation with cellular automata. Journal ofComputer System Science, 32/1, (1986) 15-78.[6] Gács, P.: Reliable cellular automata with self-organization. Journal ofStatistical Physics 103/1-2 (2001), 45-267.[7] Gács, P., Reif, J.A: Simple three-dimensional real-time reliable cellulararray. J. Comput. Syst. Sci. 36/2 (1988) 125-147.[8] Finocchi, I., Grandoni, F., Italiano, F., G.: Designing Reliable Algo-rithms in Unreliable Memories. Computer Science Review 1/2 (2007)77-87.[9] Pippenger,N.: On networks of noisy gates. In: Proc. of the 26-th IEEEFOCS Symposium (1985) 30-38.[10] Spielman, D.: Highly fault-tolerant parallel computation. In: Proc. ofthe 37th IEEE FOCS Symposium (1996) 154-163.[11] Toom, A.: Stable and attractive trajectories in multicomponent sys-tems. In Multicomponent Systems (R.L. Dobrushin, ed.), Advances inProbability 6, Dekker, New York, (1980) 549-575.[12] von Neumann, J.: Probabilistic logics and the synthesis of reliableorganisms from unreliable components. In: Automata Studies (C.Shannon and McCarthy eds.), Princeton University Press, Princeton,NJ. (1956)

A Universal RAM Machine Resistant to Isolated Bursts of Faults

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A Universal RAM Machine Resistant to Isolated Bursts of Faults

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