# Properties of Membrane Systems - CMC12.ppt by hcj

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```									Properties of Membrane Systems

a                       a
aàb                     aàb
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DOP(ncoo)
OP(ncoo)

Artiom Alhazov*
Università degli Studi
di Milano-Bicocca, Milano
and
Institute of Mathematics and Computer Science,
__________
Academy of Sciences of Moldova, Chişinău
*Warning: small text size present. Make sure you sit sufficiently close J
• P systems with symbol objects
– Distributed parallel multiset processing
• Dynamic properties
– as opposed to syntactic ones
• Their effect
– Power, efficiency, descr. complexity, etc.
Brief P systems classes taxonomy
• Derivation mode            •Rewriting obj.-obj. = ncoo, cat,
multi-stable cat, bincoo, fullcoo
– Maxpar, seq, asyn,
•Moving obj.-obj. = targets,
maxstrategy, minpar,
uniport, sym/anti, protons,
cond. uni.
• Syntactic prop.            •Obj.-membr. = active
– Cooperativeness          membranes variants, with|w/o
– Control                  polarizations
•By objects = Promoters,
• Descr. compl.               inhibitors, activators
– #membr./cells, rule       •By structure = Permeab.,
weight, #cat./protons,    polar., dissolution…
alph. size, #rules...     •By appl. ruleset = GP
• Dynamic prop.               systems,
(de-) inhibiting, polymorphism
A Dynamic Property
• Depends on the behaviour of a system
• Cannot be easily derived from the description of
a P system …unlike syntactic properties
• Easily verifiable for any finite computation
• Typically undecidable
– Computations can be infinite
– Non-determinism
– Input or other parameters
• Yields a meaningful and motivated subclass of a
certain class of P systems
– Unlike., e.g., property “always |w|a≠|w|b "a≠b”
Dynamic properties we focus on
• 1. Determinism
– 2. Confluence after each step
– A. Strong determinism •8. Time freeness
• 3. Reversibility           •C. Asynchrony
– B. Strong reversibility •+. Petri-net like,
• 4. Always halting [£1&REC] probabilistic/
stochastic
• 5. Confluence                 - Out of the scope
– 6. Strong confluence
– 7. Ultimate confluence
Effect of Determinism
•   Considered since 2003-2004
•   Sym/anti: still computationally complete
•   Acceptors or IàO
•   Intuitively needs more cooperation/control
– E.g., for fault-free a.c./0test
• Binary cooperation in r/w suffices
– sim. reg. mach.
Effect of Determinism: Separation
Catalytic P systems [power]
– 2 catalysts àcomputational completeness
– Determinism ànot c.c.
• Symport/antiport of size 2 [rule size]
– C.c. even with 2 membr. (under certain conventions)
– Determinism: not c.c. with size 2 (except tissue)
• [cannot increase the total number of objects]
– , but c.c. with size 3
• Evolution-communication [best known
– C.c. with 2 membranes results]
– Determinism: c.c. with 3 membranes
• [det. 0test: ctrl obj. moves r2àelem or r2àskin]
Other roles of determinism
• Practical reasons
– Simplicity of analysis
– Result comes from one computation, not all
• Limited non-determinism
– k-determinism
• Uninteresting computations are easily detected
• Fun trivia: determinism under min. par.
– every membrane evolves sequentially
Determinism: Other models*
• Active membranes with non-cooperative rules
preserving structure AND membrane separation
– Selecting 1 answer: 2s in s+1. Strong confluence
__________
• Non-cooperative rewriting with * AFAIK, no strict
promoters/inhibitors                  separation proof
– DEC by split and “=1” check        has been published
– Ultimate confluence                for these cases
• Alphabet size
– Optimal repres. of states: not incomparable multisets
• Active membranes w/o polarizations
– Representing registers by membranes, operations
keep 1 membrane busy
• Conditional uniport
– Seems to be the most fragile c.c. model in MC
Variants of determinism and
reversibility
• Fixed membrane structure
– Properties not affected by flattening: 1 region
• Det. = no mult. transitions from reachable
• Strong det. = no multiple transitions ever
– Syntactic (1rule if seq.)
• Rev. = no mult. preimages into reachable
• Strong rev. = no multiple preimages ever
– Decidable (syntactic if seq.)
Sequential and maximally parallel
systems
Property     pure   Pri   inh   Pure Pri   inh   pro,inh   U - universal,
ACCEPTING B         U     U     U    U     U     U         C - conjectured N,
deterministic L     U     U     U    U     U     U         B - characterized by
Strongly det. L     U     ?     L    U     ?     U         partially blind
GENERATIVE B        U     U     U    U     U     U         counter machines,
Reversible     N    U     U     C    U     U     U         N - non-universal,
Strongly rev. L     ?     ?     L   C      C     C         L - sublinear

• Determinism: NO, except pure seq.
• Reversibility: NO, except maybe pure.
• Strong determinism: YES in pure, maybe with
inh*., NO otherwise.
• Strong reversibility: maybe YES, but proved for
pure only.
Separation analysis cont’d
• Uncontrolled rev. P systems: non-univ. conjecture
– univ. seems to need 0test or appearance checking
– 1known way with no ctrl: try-and-wait-and-then-check
– not changing state WHEN accessing symbolàno rev.
• Limitations of strong (vs. not-strong) reversibility
– simulation of a reg. mach. needs to satisfy the uniq. of
prev. config. even with multiple state symbols.
• Strong versions of det./rev. may be desired
– decidability and motivation: reflecting microscopic
physical properties.
– System design is more complicated/impossible
– 14 cases of property-mode-control combinations
– separation proved for pure maximally parallel D vs SD
– 4 cases (pure seq., priorities or pro.+inh.): NO
Recall: not strict may be too strict
• 4 slides ago: 5 cases
– Case 1: strong confluence, case 2: ultimate confluence, other
cases: try to prove [-] and try with lookahead
• Det. VS confluent: popular in solving intractable
problems in poly#steps
– Active membranes, tissue with cell div., etc.
– In most cases, confl. improved to det.
• Sometimes with many extra actions (either order à replicate and
parallelize)
• Indistinguishable objects/membranes
• Res. dir ê : descr. compl. penalty of determinism
– E.g., arbitrary VS linear order in generating truth assignments
(simpler rules VS phase end signal)
Time-freeness and asynchrony
• Relaxing the global clock
• Time-freeness
– Not requiring equal rule times of 1 step
– Requiring result is independent on rule times
– Undecidable
• Asynchrony
– Not forcing execution of applicable rule
– Actually a derivation mode
– May be a syntactic property
• aàa "a with modified halting
Time-freeness and asynchrony:
effect
• 1 bi-stable catalyst àuniversality
• Time-freenessà4bi-stable catalysts enough

• Uncontrolled multiset rewriting
– Asynchronous equivalent to sequential
• Control: universality
– De facto sequential simulation of reg. mach.
Conclusions - I
• A number of dynamic properties is
considered
• Gap in computational power
– Yes/conjectured/unknown/no
• Other penalties
– e.g., descriptional complexity parameters
Conclusions – II: Determinism
• Catalytic: gap in power
• Symport/antiport: gap in weight
• Evolution-communication: gap in best known
number of membranes
• Other cases where one might expect a gap
– Membrane separation, non-cooperative controlled
multiset processing, small alphabets, polarizationless
active membranes, conditional uniport
– Seem to heavily rely on non-determinism
– Informal justifying comments were given
Conclusions III – det.&rev. etc.
• Compared to strong and unrestricted
– Depending on the controls
– Open problems commented
• Gap only known in pure cases

• Determinism compared to confluence
• Time-freeness
– Gap in best known no. of bi-stable catalysts
• Asynchrony
– Gap for pure systems
Conclusions - IV
• Survey does not pretend to cover all
classes of P systems corresponding to the
restrictions by dynamic properties
• Aim1: try to give a uniform perspective of
the role of dynamic properties
• Aim2: encourage researchers to formally
prove separation of computational power
by the dynamic properties, e.g., in cases
from slide “Determinism: Other models”.
• Thank you for your attention
• A number of dynamic properties is considered
• Gap in computational power: Yes/conjectured/unknown/no
• Other penalties: e.g., descriptional complexity parameters
Determinism
• Catalytic: gap in power                     Summary
• Symport/antiport: gap in weight
• Evolution-communication: gap in best known number of membranes
• Other cases where one might expect a gap
– Membrane separation, non-cooperative controlled multiset processing,
small alphabets, polarizationless active membranes, conditional uniport
– Seem to heavily rely on non-determinism
– Informally justified            Prop - Pri inh -       Pri inh pro,inh
Determinism and reversibility      ACCB       U    U     U    U   U   U
• Cmp’d to strong and unrestricted det L      U    U     U    U   U   U
– Depending on the controls       Sdet L U    ?     L    U   ?   U
– Open problems commented         GEN B U     U     U    U   U   U
•   Gap only known in pure cases       rev N U     U     C    U   U   U
Srev L ?    ?     L    C   C   C
•   Determinism compared to confluence
•   Time-freeness: Gap in best known no. of bi-stable catalysts
•   Asynchrony: Gap for pure systems

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