hub:
  repo_type: generator
  builtin: p-systems
  tags:
  - banzhaf-yamamoto
chemistries:
- id: p-systems
  name: P systems (membrane computing)
  intuition: 'Computing with nested membranes. Objects are symbols held in the regions of a membrane tree,
    and rules rewrite multisets of them while targeting the products - keep them here, send them out,
    or push them into an inner membrane - and may dissolve the membrane they fired in. Every rule that
    can fire in a step does, so the parallelism is maximal. The structure computes: halting configurations
    of a fixed small system can yield exactly the square numbers.

    '
  aliases:
  - membrane systems
  - transition P systems
  origin: Paun, 1998
  book: 9.5; multicompartment SSA 4.3; biology 18.3.2
  refs:
  - '[678]'
  - '[679]'
  - '[680]'
  - '[681]'
  - '[683]'
  - '[661]'
  - '[718]'
  - '[150]'
  - doi:10.1006/jcss.1999.1693
  - doi:10.1016/S0304-3975(02)00136-6
  family: rewriting
  kind: formalism
  constructive: false
  S:
    repr: 'symbol objects over an alphabet O, located in the regions of a tree of nested membranes [1
      [2 ]2 [3 ]3 ]1 (1 = skin); multisets written as strings (a^2 b^3 c). Species id = object@region
      (a@2), with env for the environment outside the skin

      '
    definition: explicit
  R:
    definition: explicit
    arity: variable
    scheme: 'multiset rewriting u -> v with target indications on the products, here (default), out (the
      surrounding region, the environment for the skin), in (a non-deterministically chosen inner membrane)
      or in_j, optionally with delta (dissolve the membrane of the region): aab -> a b_out c c_in; d ->
      d delta. Priority relations rho_i order the rules of a region. Chemart syntax: ''r1: 2 a + b ->
      a + b@out + c + c@in'', ''d -> d + delta'', priorities ''r1 > r3''

      '
    notes: 'A rule u -> v in region i becomes reactions u@i -> v@target over compartment-qualified species.
      Catalytic rules ca -> cv are ordinary rules with the catalyst on both sides. Membrane division [a]
      -> [b][c] (P systems with active membranes [679, 683]) is not implemented.

      '
  A:
    reactor:
    - maximally-parallel
    - compartments
    dilution: none
    notes: 'A global clock; in each step every region applies a non-deterministically chosen maximal multiset
      of its rules (no further rule can use the remaining objects), then the products move to their targets
      and dissolved membranes release their contents upwards. A computation is successful when it halts
      (no rule applicable anywhere) and the output region still exists; the result is the number of objects
      in the output region (internal output) or sent to the environment (external output). Priorities
      are strong (a rule is not used when a rule with priority over it is applicable) or weak (the higher
      rule takes as many objects as it can and the lower rule uses the rest). The multicompartmental Gillespie
      algorithm (MGA) [661] and its variants [718] replace this with stochastic kinetics per membrane;
      they are not implemented.

      '
  params:
  - name: system
    type: enum
    default: divisibility
    choices:
    - divisibility
    - n-squared
    - custom
    role: structural
    meaning: 'published system: divisibility (book eq. 9.21 and fig. 9.3: is n a multiple of k? output
      membrane 3 is empty iff it is), n-squared (Paun 2006 fig. 3: generates n^2, n >= 1, sent to the
      environment), custom (membranes, objects, rules, priorities, output)'
  - name: n
    type: int
    default: 7
    min: 1
    max: 1000000
    range: positive; the computation halts after about n/k + 3 steps, so raise max_steps for large n/k
    role: population
    meaning: 'divisibility only: copies of a in membrane 2 (the number tested)'
  - name: k
    type: int
    default: 3
    min: 1
    max: 1000000
    role: population
    meaning: 'divisibility only: copies of c in membrane 2 (the divisor)'
  - name: membranes
    type: str
    default: ''
    range: labelled brackets, e.g. '[1 [2 ]2 [3 ]3 ]1'; labels are letters, digits or _, 'env' and '0'
      are reserved
    role: spatial
    meaning: 'custom only: membrane structure mu'
  - name: objects
    type: dict
    default: {}
    range: 'e.g. {''2'': ''7 a + 3 c + d''} or {''2'': {''a'': 7, ''c'': 3, ''d'': 1}}'
    role: population
    meaning: 'custom only: initial multiset w_i per region'
  - name: rules
    type: dict
    default: {}
    range: 'e.g. {''2'': [''r1: a + c -> c2'', ''r3: d -> d + delta''], ''1'': [''e -> e@out'', ''x ->
      y@in3'']}; targets @here (default), @out, @in, @in<label>; delta or δ dissolves; the optional label
      defaults to r<position>'
    role: structural
    meaning: 'custom only: rule set R_i per region, in the explicit reaction syntax with target suffixes'
  - name: priorities
    type: dict
    default: {}
    range: 'e.g. {''2'': [''r1 > r3'', ''r2 > r3'']}; chains ''r1 > r2 > r3'' allowed; closed transitively,
      cycles rejected'
    role: structural
    meaning: 'custom only: priority relation rho_i per region over its rule labels'
  - name: output
    type: str
    default: ''
    range: a membrane label, or 'env' / '0' for the environment
    role: structural
    meaning: 'custom only: output region i_o'
  - name: priority
    type: enum
    default: strong
    choices:
    - strong
    - weak
    role: structural
    meaning: 'semantics of the priority relations in the computation: strong (a lower rule is not used
      while a higher one is applicable; needed by the book''s divisibility system) or weak (the higher
      rule is used maximally, the lower one on the remaining objects); Paun 2006 sec. 11'
  - name: max_steps
    type: int
    default: 1000
    min: 1
    max: 10000000
    range: a halting n-squared computation with result n^2 takes 2n + 2 steps; non-halting computations
      stop here
    role: population
    meaning: step budget of the maximally parallel computation recorded in extras.analysis (does not change
      the network)
  provides:
  - topology
  - stoichiometry
  - catalysts
  - compartments
  - initial-state
  fidelity: reconstructed
  sources:
  - 'Banzhaf, W. & Yamamoto, L. (2015). Artificial Chemistries, MIT Press, section 9.5: notation, target
    indications, dissolution, eq. 9.20, and the divisibility system eq. 9.21 / fig. 9.3 (primes and the
    in_3 subscript checked on the rendered PDF page 190, because the text extraction drops them).'
  - 'Paun, G. (2006). Introduction to Membrane Computing. In G. Ciobanu, G. Paun, M. J. Perez-Jimenez
    (eds.), Applications of Membrane Computing, Springer, pp. 1-42 (doi:10.1007/3-540-29937-8_1): maximal
    parallelism and the a^5 b^2 c^6 example (sec. 4), target indications and dissolution (sec. 7), the
    formal definition of a transition P system (sec. 8), computations, halting, internal and external
    output and the n^2 system of fig. 3 (sec. 9), strong and weak priorities with the ff -> f / cf ->
    cd delta example (sec. 12). Preprint used: https://cs.ioc.ee/yik/schools/win2007/paun/IntrodMC.pdf
    and http://bioinfo.uib.es/~recerca/BUM/paun.pdf'
  decisions:
  - 'Original papers not reached: Paun''s TUCS Report 208 / JCSS 61 (2000) [678], the 2002 book [680]
    (with the yes/no version of the divisibility system), the Paun-Rozenberg guide [681] and ppage.psystems.eu
    (connection refused) could not be downloaded; the formal definition and semantics come from Paun''s
    2006 introduction, which restates them. The yes/no divisibility system of [680] is therefore not offered.'
  - 'Scope: transition P systems with symbol objects, targets here/out/in/in_j, dissolution and priorities
    (book 9.5). Membrane division and active membranes [679, 683], symport/antiport, promoters/inhibitors,
    string objects and the stochastic MGA variants [661, 718] (book 4.3, 18.3.2) are not implemented:
    the book gives no algorithmic detail or rate constants for them. The v1 A.reactor ''ssa'' is dropped.'
  - 'Priorities: the book states rho_2 = {r1 > r3, r2 > r3} and that r3 fires ''when neither r1 nor r2
    can be fired'', which is Paun''s strong interpretation; under the weak one r3 would fire in the first
    step, as soon as the c are used up, and the computation would give a wrong answer. Default priority
    = strong; weak is available for Paun 2006 sec. 11.'
  - 'Non-determinism: any maximal multiset of rules is allowed by the definition; Chemart samples one
    by repeatedly drawing a still-applicable rule with weight equal to the number of times it can still
    be applied and applying it a uniform number of times in [1, that], until no rule can take the remaining
    objects. An object sent ''in'' goes to a uniformly chosen inner membrane. This is one admissible scheduler,
    not a published distribution.'
  - 'Step semantics (Paun 2006 secs. 7-9): all regions choose their multisets on the configuration at
    the start of the step; products are delivered by the membrane structure at the start of the step (objects
    leaving the skin go to the environment and never return); then every membrane whose region used a
    delta rule dissolves, and its objects and inner membranes join the nearest upper membrane not dissolved
    in the same step; the skin can never dissolve (a delta rule in the skin is rejected). An ''in'' target
    with no (such) inner membrane makes the rule inapplicable.'
  - 'Halting and output: a computation halts when no rule is applicable in any region; the result is the
    number of objects in the output region in the halting configuration, or of objects sent to the environment
    when i_o = env (Paun uses i_o = 0, also accepted). A computation whose output membrane was dissolved,
    or that does not halt within max_steps, has result null. The terminal-object restriction T is not
    modelled (all objects count); catalysts are not declared separately.'
  - 'Network: every rule becomes reactions over object@region species. Because dissolution changes the
    tree, out/in/in_j targets resolve to every region reachable in some structure (a further ancestor
    when the intermediate membranes have delta rules, a deeper membrane when the membranes between can
    dissolve), and an ''in'' with several possible membranes gives one reaction per distribution of the
    objects, so the network is the complete set of reactions the system can perform. Status complete.'
  - 'Facts without a formula: products of a delta rule keep the rule''s own region (d -> d delta is the
    reaction d@2 -> d@2); the dissolution itself and the release of contents are recorded in extras.dissolution,
    priorities in extras.priorities (transitive closure, [higher, lower] pairs) and extras.priority_semantics.
    Reactions have no rate: P systems define none.'
  - 'Book notation: the text extraction of eq. 9.21 loses the primes and the subscript; the rendered page
    reads R_1 = {r4: dcc'' -> a_in3}, R_2 = {r1: ac -> c'', r2: ac'' -> c, r3: d -> d delta}, w_2 = a^n
    c^k d, i_o = 3. The book''s ''adapted from [150, 680]'' system is used as printed.'
  - 'Paun 2006 fig. 3 system (n-squared): i_o is the environment, as the text says; rules r1..r3 of region
    3 are a -> ab, a -> b delta, f -> ff; region 2 has b -> d, d -> de, ff -> f, cf -> cd delta; region
    1 has e -> e_out, f -> f. It has no priorities: computations in which cf -> cd delta is used too early
    never halt because of f -> f.'
  - 'v1 parameters: Sigma (the alphabet is implied by the objects and rules), mu, w_i, R_i (callable),
    rho_i and i_o became the system choice plus the custom JSON/text fields membranes, objects, rules,
    priorities and output; n and k give the divisibility input; priority and max_steps control the recorded
    computation.'
  - 'extras: system, membranes (canonical structure), compartments (per membrane: parent, children, skin,
    dissolvable, rules, species; plus env), rules (id, region, label, text, dissolves, reaction indices),
    reaction_rules (rule id of each reaction), priorities, priority_semantics, dissolution, output_region,
    inapplicable_rules (rules no structure lets fire), and analysis (halted, steps, output, result, final
    structure and configuration, dissolutions, rule applications, a trace of the first 100 steps; divisibility
    adds n_multiple_of_k, n-squared result_is_square).'
  phenomena:
  - 'divisibility (book 9.5, eq. 9.21): r1 and r2 alternately consume k copies of a per step, r3 dissolves
    membrane 2 once a is exhausted, and r4 puts one a into membrane 3 exactly when n is not a multiple
    of k'
  - 'N(Pi) = {n^2 | n >= 1} for the system of Paun 2006 fig. 3: every halting computation sends (n+1)^2
    copies of e to the environment'
  - 'maximal parallelism (Paun 2006 sec. 4): on a^5 b^2 c^6 the rules aab -> abcc and bb -> aac give either
    a^3 b^2 c^10 or a^7 c^7 in one step'
  - 'strong vs weak priority (Paun 2006 sec. 12): with ff -> f > cf -> cd delta on fffc, the strong reading
    uses only ff -> f, the weak reading also uses cf -> cd delta on the remaining fc'
  - 'target indications: aab -> a b_out c c_in is used twice on a^5 b^2 c^6 (Paun 2006 sec. 7)'
  - computational universality, space-for-time trade-off via membrane division or worm objects, and polynomial
    solutions of NP-complete problems are reported in [679, 683] and are not reproduced
  notes: 'Formally Pi = (Sigma, mu, w_1..w_m, (R_1,rho_1)..(R_m,rho_m), i_o). The module also exports
    parse_membranes, build_system, instantiate, select and run, and the published systems divisibility(n,
    k) and N_SQUARED.

    '