polymorphic p systems

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Hiroshima University . Higashi-Hiroshima , Japan. Chi ş in ă u , Moldova. Institute of Mathematics and Computer Science Academy of Sciences of Moldova { artiom , sivanov , rogozhin } @math.md. Polymorphic P Systems. Technical University of Moldova. Artiom Alhazov Sergiu Ivanov - PowerPoint PPT Presentation

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  • Polymorphic P SystemsArtiom AlhazovSergiu IvanovYurii RogozhinHiroshima University. Higashi-Hiroshima, Japan

    Chiinu, Moldova

    Institute of Mathematics and Computer ScienceAcademy of Sciences of Moldova{artiom, sivanov,rogozhin}@math.mdTechnical University of Moldovarepeat N times{if(STATE=0)increment A else decrement A; (code not using STATE)}if(STATE=0)INSTR="increment else INSTR="decrement";repeat N times{INSTR A; (code not using STATE)}

  • Three motives for yet another extensionPractical problems need more than just computational completenessDeterminism; both input & outputProper internal data representationEfficiency; complicated data structuresVon Neumann architectureProgram is dataWhat is a Nucleus in P systems?It is insideIt describes the rules

  • Main ideaMost papers changing the active rulessubsets of a predefined finite setWhat is data?Multisets in regionsMost natural way to specify rules as dataInterpret a pair of regions as a ruleContents left/right side of the rule

    Input: byte (in unary).Output: number of bits 1R={2:anb, 1:nn/2}Try it (time and descriptionally efficiently) with P systems studied so far

  • Definition: polymorphic P systems=(O,T,,ws,w1L,w1R,,wmL,wmR,,iout),H={s,1L,1R,,mL,mR},s=skin,parent(iL)=parent(iR),No rules, only features. In this paper :HTar

    Example notation: OPk(polym+d(coo),tar)+d: disabling rules allowed (by left side=)coo: cooperative rulestar: targets allowed (here,inj,out).k: membrane bound. (thus #rules(k-1)/2)

  • More definitionsInitial rules are i:wiL(wiR,(i))Rparent(iL)-d: Regions iL are never emptyComputing: Input O in iin.D: deterministic (for every input)Deciding: T={yes,no}, confluent.Generating - N(), accepting Na(), deciding Nd(), computing a partial function in the deterministic case f()

  • Superexponential growth example1=({a},{a},,a,a,a,a,a,a,aa,,1),=[[]1L[[]2L[[]3L[]3R]2R]1R]s, here.In the rest of the talk graphical notation.R2R={aaa}R1R={aa}Rs={aa}skin: aInitial rulesMultisets defining rules are changingUse old contents, i.e.compute all, then update

  • Superexponential - continued2

    1

    1 1R2R={aaa}R1R={aa}Rs={aa}R2R={aaa}R1R={aaa}Rs={aa}R2R={aaa}R1R={aaaaa}Rs={aaa}nR2R={aaa}R1R={aa1024}Rs={aa35184372088832}skin:a1329227995784915872903807060280344576n=10

  • Maximal GrowthI: initial number of objectsc: maximum right side sizen: number of stepsd: membrane structure depthNon-polymorphic systems: IcnPolymorphic, no targets: Icp(n), deg(p)=d-1

  • ResultsUniversality with 47 membranesGenerating without cooperation and without disablingFactorials with cooperationGenerating even faster with targetsComputing functionsStay tuned

  • NOP47(polym-d(coo))=NRE[AlhazovVerlan2008]: strongly universal P system with 1 membrane and 23 rulesEach rule i:uv becomes [u]iL[v]iRtotal of 47 membranes.Focus: efficiency of computationse.g., generating/deciding factorials byconstant-time multiplication of variable factors.

  • Targets. No cooperation{n!nk|n1,k0}NOP13(polym-d(ncoo),tar)

    b produces copies of derased non-deterministicallyd enters 1R as a, increasing n in 1: aanThe number of objects a in skin is multiplied by nUntil rule 3 changes rule 1 to can. Non-det.If b is erased too soon, multiplication continues without growing n.

  • RemarksIf multiplication stops while n still grows, a factorial of a smaller number is generatedThe shortest computation generating n!nk is only n+k+1To generate exactly factorialsWe need to stop the multiplication when we stop the incrementSeems impossible without cooperative rules.

  • Exactly factorials{n!|n1}NOP9(polym-d(coo),tar)Similar to the previous systemRule 3 stops both incre- ment and multiplicationA non-cooperative rule 1:acan is actually never applied; used to stop the computation.n! are generated in n+1 step.

  • Yet faster growthPolymorphic, no targets: exponential of polynomialPolymorphic, targets: exponential of exponential.Upper boundThe fastest growth is by squaringHaving n+n+1 objects, in one step we can obtain at most n2+n+1 objects.

  • Lower bound: Superpowers{2^(2n)|n0}NOP15(polym-d(ncoo),tar) iterated squaring (b,s)2(a,s)(b,1R) ak (1:abk) bkk rule 4: cleanupStopped by rule 3 making 1:cbk.Numbers 2^(2n) generated in 3n+2 steps.No cooperation! Reminder: 15 membranes.,,2,,,4,,,16,,,256,,,65536,,,4294967296,,,18446744073709551616,,,

  • Deterministic computing( n2^(2n) )DfOP15(polym+d(coo),tar)Similar to the previous systema in 1L powers one squaringInput dn in skincd 3 ca in 1L

  • Deterministic computing: remarksDisabling rules may be avoided: +d -das appear in skin every 3rd stepNo need to disable rule 1 in the process of computingDeterministic subtraction and appearance checkingc moves into 1L and blocks rule 1n2^(2n) computed in O(n)

  • Deterministic deciding{n!|n1}NdDOP37(polym-d(coo),tar)Deciding is more than acceptingIterated division of the input number4-step cyclesVerifying quotient and remainderA number kn! is decided in at most 4n steps (sublogarithmic w.r.t. k)

  • Summary - 1Polymorphic P systems as a variant of object rewriting model of P systems: rules areNot specified explicitly (only features e.g. targets are)Dynamically inferred from the contents of inner regionsIdea: similar with cell nucleus, but simplerConventional computing: von Neumann architecture VS Harvard architectureUsual P systems cannot grow with factorial speed; polymorphic P systems can deterministically decide factorials of n in O(n)Nice possibilities like constant-time multiplication/divisionExtensions possible

  • Summary - 2Strong universality in OP47(polym-d(coo))Superexp. growth in DOP7(polym-d(ncoo))Gen. {n!nk}, n+k+1steps, OP13(polym-d(ncoo),tar)Gen. n! in n+1steps, OP9(polym-d(coo),tar)Generating 2^(2n) in 3n+2 steps by a P system in OP15(polym-d(coo),tar)Computing n2^(2n) in O(n) steps by a P system in DOP*(polym-d(coo),tar)Deciding factorials in sublogarithmic time by a P system in DOP37(polym-d(coo),tar)

  • Summary - 3Growthpolymorphic with targets (exp of exp)polymorphic without targets (exp of poly)non-polymorphic (exp)There exists infinite sets of numbers that are accepted in time which is sublinear w.r.t. the size of the input in binary representation (without cheating by only examining a part of the input).Selected open questionsCharacterization of restricted classes like OP*(polym-d(ncoo),ntar)Real applications for which non-polymorphic P systems are not suitableCan polymorphic P systems use superexponential growth to attack intractable problems in polytime? (Conjecture: no)

    >>

  • On one slideStrong universality in OP47(polym-d(coo))Superexp. growth in DOP7(polym-d(ncoo))Gen. {n!nk}, n+k+1steps, OP13(polym-d(ncoo),tar)Gen. n! in n+1steps, OP9(polym-d(coo),tar)Generating 2^(2n) in 3n+2 steps by a P system in OP15(polym-d(ncoo),tar)Computing n2^(2n) in O(n) steps by a P system in DOP*(polym-d(coo),tar)Deciding factorials in sublogarithmic time by a P system in DOP37(polym-d(coo),tar)Thank youfor your questions8

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