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Polymorphic P Systems

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

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Polymorphic P Systems

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  1. Hiroshima University. Higashi-Hiroshima, Japan Chişinău, Moldova Institute of Mathematics andComputer Science Academy of Sciences of Moldova {artiom, sivanov,rogozhin}@math.md Polymorphic P Systems Technical University of Moldova Artiom Alhazov Sergiu Ivanov Yurii Rogozhin if(STATE=0)INSTR="increment“ elseINSTR="decrement"; repeat N times {INSTRA; (code not using STATE) } repeat N times {if(STATE=0)incrementA else decrementA; (code not using STATE) }

  2. Three motives for yet another extension • Practical problems need more than just computational completeness • Determinism; both input & output • Proper internal data representation • Efficiency; complicated data structures • Von Neumann architecture • Program is data • What is a Nucleus in P systems? • It is inside • It describes the rules

  3. Input over a a128 aa a 1L 1R b 2L 2R s Main idea • Most papers changing the active rules • subsets of a predefined finite set • What is data? • Multisets in regions • Most natural way to specify rules as data • Interpret a pair of regions as a rule • Contents left/right side of the rule Try it (time and descriptionally efficiently) with P systems studied so far • Input: byte (in unary). • Output: number of bits 1 • R={2:anb, 1:nn/2}

  4. Definition: polymorphic P systems • =(O,T,μ,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 :HTar • Example notation: OPk(polym+d(coo),tar) • +d: disabling rules allowed (by left side=) • coo: cooperative rules • tar: targets allowed (here,inj,out). • k: membrane bound. (thus #rules(k-1)/2)

  5. More definitions • Initial rules are i:wiL(wiR,(i))Rparent(iL) • -d: Regions iL are never empty • Computing: 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()

  6. a a a a aa 3L 3R a 2L 2R a 1L 1R s Superexponential growth example • 1=({a},{a},μ,a,a,a,a,a,a,aa,,1), • μ=[[]1L[[]2L[[]3L[]3R]2R]1R]s,here. • In the rest of the talk – graphical notation. Initial rules R2R={aaa} R1R={aa} Rs={aa} skin: a • Multisets defining rules are changing • Use “old contents”, i.e. • compute all, then update

  7. a2^(n(n-1)(n-2)/6) a2^(n(n-1)/2) a2^n a aa 3L 3R a 2L 2R a 1L 1R a a s a a aa Exponential of polynomial 3L 3R a 2L 2R a 1L 1R 2 2 1 1 2 4 2 1 2 2n 2(n(n-1)/2) 2(n(n-1)(n-2)/6) s … x x x … x x x … x x x n Superexponential - continued ⇒n R2R={aaa} R1R={aa} Rs={aa} R2R={aaa} R1R={aaa} Rs={aa} R2R={aaa} R1R={aaaaa} Rs={aaa} n=10 R2R={aaa} R1R={aa1024} Rs={aa35184372088832} skin: a1329227995784915872903807060280344576 2 1 1 1

  8. Maximal Growth • I: initial number of objects • c: maximum right side size • n: number of steps • d: membrane structure depth • Non-polymorphic systems: Icn • Polymorphic, no targets: Icp(n), deg(p)=d-1

  9. Results Universality with 47 membranes Generating without cooperation and without disabling Factorials with cooperation Generating even faster with targets Computing functions Stay tuned

  10. NOP47(polym-d(coo))=NRE • [AlhazovVerlan2008]: strongly universal P system with 1 membrane and 23 rules • Each rule i:uv becomes [u]iL[v]iR • total of 47 membranes. • Focus: efficiency of computations • e.g., generating/deciding factorials by constant-time multiplication of variable factors.

  11. ab 4:bbd 5:b 6:d(a,in1R) ab a 2:aa 3:ac a b bd a a 4L 4R 2L 2R b  a c 5L 5R 3L 3R d a 6L 6R a a 1L 1R 1L 1R s s Targets. No cooperation • {n!nk|n1,k0}NOP13(polym-d(ncoo),tar) • b produces copies of d • erased non-deterministically • d enters 1R as a, increasing n in 1: aan • The number of objects a in skin is multiplied by n • Until rule 3 changes rule 1 to can. Non-det. • If b is erased too soon, multiplication continues without growing n.

  12. Remarks • If multiplication stops while n still grows, a factorial of a smaller number is generated • The shortest computation generating n!nk is only n+k+1 • To generate exactly factorials • We need to stop the multiplication when we stop the increment • Seems impossible without cooperative rules.

  13. 2:bbd 3:b(c,in1L) 4:d(a,in1R) ab a a 1L 1R s Exactly factorials • {n!|n1}NOP9(polym-d(coo),tar) • Similar to the previous system • Rule 3 stops both incre-ment and multiplication • A non-cooperative rule1:acan is actually neverapplied; used to stop the computation. • n! are generated in n+1 step.

  14. Yet faster growth • Polymorphic, no targets: exponential of polynomial • Polymorphic, targets: exponential of exponential. • Upper bound • The fastest growth is by squaring • Having n+n+1 objects, in one step we can obtain at most n2+n+1 objects.

  15. 5:ba’b’ 6:a’a 7:b’(b,in1R) bb 2:aa 3:ac a 4:b  1L 1R s Lower bound: Superpowers ,,2,,,4,,,16,,,256,,,65536,,,4294967296,,,18446744073709551616,,, … • {2^(2n)|n0}NOP15(polym-d(ncoo),tar) iterated squaring (b,s)⇒2(a,s)(b,1R) ak⇒(1:abk) bkk rule 4: cleanup • Stopped by rule 3 making 1:cbk. • Numbers 2^(2n) generated in 3n+2 steps. • No cooperation! Reminder: 15 membranes.

  16. Deterministic computing • ( n2^(2n) )DfOP15(polym+d(coo),tar) • Similar to the previous system • a in 1L powers one squaring • Input dn in skin • cd⇒3ca in 1L

  17. Deterministic computing: remarks • Disabling rules may be avoided: +d  -d • a‘s appear in skin every 3rd step • No need to disable rule 1 in the process of computing • Deterministic subtraction and appearance checking • c moves into 1L and blocks rule 1 • n2^(2n) computed in O(n)

  18. Deterministic deciding • {n!|n1}NdDOP37(polym-d(coo),tar) • Deciding is more than accepting • Iterated division of the input number • 4-step cycles • Verifying quotient and remainder • A number kn! is decided in at most 4n steps (sublogarithmic w.r.t. k)

  19. Summary - 1 • Polymorphic P systems as a variant of object rewriting model of P systems: rules are • Not specified explicitly (only features e.g. targets are) • Dynamically inferred from the contents of inner regions • Idea: similar with cell nucleus, but simpler • Conventional computing: von Neumann architecture VS Harvard architecture • Usual 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/division • Extensions possible

  20. Summary - 2 • Strong 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 n2^(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)

  21. Summary - 3 > • Growth • polymorphic 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 questions • Characterization of restricted classeslike OP*(polym-d(ncoo),ntar) • “Real” applications for which non-polymorphic P systems are not suitable • Can polymorphic P systems use superexponential growth to attack intractable problems in polytime? (Conjecture: no) >

  22. a85 b4 a a128 aa aa a a 1L 1L 1R 1R b b 2L 2L 2R 2R s s On one slide • Strong 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 n2^(2n) in O(n)steps by a P system inDOP*(polym-d(coo),tar) • Deciding factorials in sublogarithmic time by a P system in DOP37(polym-d(coo),tar) ⇒8 for your questions Thank you

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