Title: Polymorphic P Systems
1Polymorphic 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)
2Three 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
3Main 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?
4Definition 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)
5More 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(?)
6Superexponential 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
7Superexponential - 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
8Maximal 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
9Results
- Universality with 47 membranes
- Generating without cooperation and without
disabling - Factorials with cooperation
- Generating even faster with targets
- Computing functions
- Stay tuned
10NOP47(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.
11Targets. 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.
12Remarks
- 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.
13Exactly 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.
14Yet 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.
15Lower 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.
16Deterministic 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
17Deterministic 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)
18Deterministic 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)
19Summary - 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
20Summary - 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)
21Summary - 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
22On 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
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