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(?)

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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

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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
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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

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Results

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

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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)

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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)

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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

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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)

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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
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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