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Polymorphic P Systems

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Polymorphic P Systems Hiroshima University. Higashi-Hiroshima, Japan Artiom Alhazov Sergiu Ivanov Yurii Rogozhin Chi in u, Moldova Institute of Mathematics and ... – PowerPoint PPT presentation

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Title: Polymorphic P Systems


1
Polymorphic P Systems
Hiroshima University. Higashi-Hiroshima, Japan
  • Artiom Alhazov
  • Sergiu Ivanov
  • Yurii Rogozhin

Chisinau, Moldova
Institute of Mathematics and Computer
Science Academy of Sciences of Moldova artiom,
sivanov,rogozhin_at_math.md
Technical University of Moldova
if(STATE0)INSTR"increment else
INSTR"decrement" repeat N times INSTR A
(code not using STATE)
repeat N times if(STATE0)increment A else
decrement A (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
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
  • R2an?b, 1n??n/2?

4
Definition polymorphic P systems
  • ?(O,T,µ,ws,w1L,w1R,,wmL,wmR,?,iout),
  • Hs,1L,1R,,mL,mR,sskin,parent(iL)parent(iR),
  • No rules, only features. In this paper ?H?Tar
  • Example notation OPk(polymd(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 iwiL?(wiR,?(i))?Rparent(iL)
  • -d Regions iL are never empty
  • Computing Input ??O in iin.
  • D deterministic (for every input)
  • Deciding Tyes,no, confluent.
  • Generating - N(?), accepting Na(?), deciding
    Nd(?), computing a partial function in the
    deterministic case f(?)

6
Superexponential growth example
  • ?1(a,a,µ,a,a,a,a,a,a,aa,?,1),
  • µ1L2L3L3R2R1Rs, ??here.
  • In the rest of the talk graphical notation.

Initial rules
R2Ra?aa R1Ra?a Rsa?a skin a
  • Multisets defining rules are changing
  • Use old contents, i.e.
  • compute all, then update

7
Superexponential - continued
?n
R2Ra?aa R1Ra?a Rsa?a
R2Ra?aa R1Ra?aa Rsa?a
R2Ra?aa R1Ra?aaaa Rsa?aa
n10
R2Ra?aa R1Ra?a1024 Rsa?a35184372088832 sk
in 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 iu?v becomes uiLviR
  • total of 47 membranes.
  • Focus efficiency of computations
  • e.g., generating/deciding factorials by
  • constant-time multiplication of variable factors.

11
Targets. No cooperation
  • n!nkn?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
    nk1
  • To generate exactly factorials
  • We need to stop the multiplication when we stop
    the increment
  • Seems impossible without cooperative rules.

13
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 rule1ac?an is actually
    neverapplied used to stop the computation.
  • n! are generated in n1 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 nn1 objects, in one step we can obtain
    at most n2n1 objects.

15
Lower bound Superpowers
,,2,,,4,,,16,,,256,,,65536,,,4294967296,,,18446744
073709551616,,,
  • 2(2n)n?0?NOP15(polym-d(ncoo),tar)
  • iterated squaring
  • (b,s)?2(a,s)(b,1R)
  • ak ?(1a?bk) bkk
  • rule 4 cleanup
  • Stopped by rule 3 making 1c?bk.
  • Numbers 2(2n) generated in 3n2 steps.
  • No cooperation! Reminder 15 membranes.

16
Deterministic computing
  • ( n?2(2n) )?DfOP15(polymd(coo),tar)
  • Similar to the previous system
  • a in 1L powers one squaring
  • Input dn in skin
  • cd ?3 ca in 1L

17
Deterministic computing remarks
  • Disabling rules may be avoided d ? -d
  • as 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, nk1steps, OP13(polym-d(ncoo),tar)
  • Gen. n! in n1steps, OP9(polym-d(coo),tar)
  • Generating 2(2n) in 3n2 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)

gt
gt
22
On one slide
  • Strong universality in OP47(polym-d(coo))
  • Superexp. growth in DOP7(polym-d(ncoo))
  • Gen. n!nk, nk1steps, OP13(polym-d(ncoo),tar)
  • Gen. n! in n1steps, OP9(polym-d(coo),tar)
  • Generating 2(2n) in 3n2 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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