Properties of Membrane Systems - CMC12.ppt by hcj


									Properties of Membrane Systems

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                     aàb                     aàb
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                                              Artiom Alhazov*
                                   Università degli Studi
                                di Milano-Bicocca, Milano
                 Institute of Mathematics and Computer Science,
                   Academy of Sciences of Moldova, Chişinău
*Warning: small text size present. Make sure you sit sufficiently close J
      What is this talk about?
• 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,
    lookahead, etc.
                             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/
• 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
  – Lookahead mode
• 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
• 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
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
   – 9 unanswered
   – 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
• 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:
• 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
• 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
• 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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