Non-confluent P systemswith active membranes

Antonio E. Porrecaporreca@disco.unimib.it

Dipartimento di Informatica, Sistemistica e ComunicazioneUniversita degli Studi di Milano-Bicocca, Italy

February 26, 2009

P systems with active membranes(without division)

I A membrane structure µ (rooted unorderedtree) with labels Λ and polarizations {+, 0,−}

I A multiset of objects in Γ inside each regionI A set R of rules of the following types:

I Evolution: [a→ w ]αhI Communication: [a]αh → [ ]βh bI Communication: a [ ]αh → [b]βhI Dissolution: [a]αh → b

I No membrane division rules

I Maximally parallel, nondeterministic application

Recogniser P systems

I The alphabet Γ contains objects yes and no

I Every computation is halting, and exactly oneof yes, no is sent out from the skin to signalthe result

I When all computations agree on the result, theP system is said to be confluent

I If this is not necessarily the case, it is said tobe non-confluent and the global result isacceptance iff there exists an acceptingcomputation (like NDTMs)

Solving decision problems semi-uniformly

I Let L ⊆ Σ? be a language

I To each string x ∈ Σ? we associate a P systemΠx deciding whether x ∈ L

I The mapping x 7→ Πx is polynomial-timecomputable via a Turing machine M

I The set Π = {Πx | x ∈ Σ?} is called apolynomially semi-uniform family of P systems

I We say that Π decides L; in symbols L = L(Π)

Time complexity classes for P systems

I L ∈ PMC?NAM iff L is decided by Π whereI Π is a polynomially semi-uniform family of

P systems with active membranes without divisionI Each Πx is confluentI There exists a polynomial p such that, for each

x ∈ Σ?, every computation of Πx halts withinp(|x |) computation steps

I L ∈ NPMC?NAM iff L is decided by Π as above,except that

I Each Πx is non-confluent

I Other classes are defined in an analogous way

Solving the Boolean formula value problem

I Given a Boolean formula φ ∈ Φ0, where

Φ0 → 0 | 1 | (Φ0 ∧Φ0) | (Φ0 ∨Φ0) | ¬Φ0

I . . . decide whether φ evaluates to 1

I (Of course this problem is in P)

I Let us solve it by using P systems

Parse trees of formulae

I A formula can be described by its parse tree

I For instance 1 ∧ ¬0 has the following parsetree:

∧

1 ¬

0

I Formulae are trees, membrane structures aretrees. . .

Representing formulae with membranes

φ ≡ 1 ∧ ¬0 =⇒ 11 ∧2 ¬304

2

1 3

4 1

4

3

2

Evaluating formulae

I We put an object 0 or 1 inside elementarymembranes representing such constants

I We send out from each membrane h an objectrepresenting the value of the subformula havingas root the connective represented by h

I The value are computed by usingcommunication rules and polarizations

I Membrane structures are unordered, but ∧ and∨ are commutative ⇒ no problem

Evaluating formulae (example)

I If φ ≡ 11 ∧2 ¬304, the membrane structure is

2

1 3

41

4

3

2

I The output of each membrane must be

Membrane Output

1 12 13 14 0

Evaluating subformulae

I If h represents a Boolean constant (0 or 1)

[0]0h → [ ]−h 0 [1]0h → [ ]−h 1

I If h represents a conjunction (disj. is dual)

[0]0h → [ ]−h 0 [1]0h → [ ]+h #

[0]+h → [ ]−h 0 [1]+h → [ ]−h 1

I If h represents a negation

[0]0h → [ ]−h 1 [1]0h → [ ]−h 0

Evaluating formulae (conclusions)

I The skin membrane transforms 1, 0 in yes, no

[1]00 → [ ]−0 yes [0]00 → [ ]− no

I The formula φ is accepted iff it evaluates to 1

I Evaluation is performed in linear time

I The P systems can be constructed(semi-uniformly) in polynomial time

I Thus the Boolean formula value problem is inPMC?NAM

PMC?NAM contains P. . .

I Let L ∈ P

I We describe a polynomially semi-uniform familyΠ of P systems with active membranes and nodivision operating in constant time anddeciding L

I The TM M constructing Π, on input x :I First decides if x ∈ L (we have the time to do that)I Then constructs Πyes or Πno accordingly

I Πyes and Πno are trivial P systems which simplyoutput yes or no respectively during their firstcomputation step

I Thus L ∈ PMC?NAM

. . . but also PMC?NAM ⊆ P!

I This is the so-called Milano theorem: maximalparallelism is not enough; without divisionrules, P systems with active membranesoperating in polynomial time can not solve“interesting” problems (unless P = NP)

I (We don’t discuss the proof here; we’ll prove anondeterministic version later)

I Thus PMC?NAM and P are the same

Boolean satisfiability problem

I Given a formula φ ∈ Φ with

Φ→ xi | (Φ ∧Φ) | (Φ ∨Φ) | ¬Φ

I . . . deciding whether φ has a satisfyingassignment is NP-complete

I The usual nondeterministic algorithm isI Guess a truth assignment vI Evaluate φ under v and accept iff v |= φ

I We already know how to evaluate formulae viaP systems

Generating a truth assignment

I Consider the (indexed) formula φ

I Inside the skin membrane 0 we put an objectfor each variable occurring in φ

I A pair of conflicting evolution rules for eachvariable x , choosing “true” or “false”

[x → fx ]00 [x → tx ]00

I Then generate one value-object for eachoccurrence i1, . . . , ik of x in φ

[fx → 0i1 · · · 0ik ]00 [tx → 1i1 · · · 1ik ]00

Moving the objects towards the leaves

I The indexed zeros and ones must be moved tothe elementary membranes representingvariables (e.g. 0h must be sent to membrane h)

I There is only one path in a tree

I If k is on the path 0→ h (and k 6= h) then

0h [ ]αk → [0h]αk 1h [ ]αk → [1h]αk α ∈ {0,+}

I When the value-object 0h (or 1h) reaches h thesubscript is deleted

0h [ ]0h → [0]0h 1h [ ]0h → [1]0h

Evaluation phase

I Now the problem is reduced to the previousone!

I Maybe not all value-objects reach their leavessimultaneously

I However, the new rules do not interfere withthe old ones (the objects are different, thepolarizations do not change)

Solving SAT (conclusions)

I The family of non-confluent P systems solvingSAT can be constructed (semi-uniformly) inpolynomial time O(n2)

I Each P system halts in O(n2) steps

I An accepting computation exists iff a satisfyingassignment exists

I Thus SAT ∈ NPMC?NAM

Simulating non-confluent P systems

I We proved NP ⊆ NPMC?NAMI Can we solve even harder problems efficiently

by using this kind of P systems?

I It turns out the answer is no

I To prove that, we describe an efficientsimulation of non-confluent divisionlessP system by using a nondeterministic Turingmachine

Data structures

I A membrane structure is represented as a tree

I Each membrane (node of the tree) hasassociated data: label, multiset, polarization

I Multisets are represented as vectors ofnonnegative integers

I The developmental rules are represented asrecords with the appropriate fields

Simulation overview

I We maintain a copy of the currentconfiguration C (membrane structure &multisets) of the simulated P system

I We also use an environment multiset (tocollect the objects sent out from the skin)

I Finally let R be the set of rules of the P system

I Choose the rules from R nondeterministicallyand apply them, simulating maximal parallelism

I . . . until yes or no is found in the environment

Simulation algorithm (1)

A1 If yes (no) is found, halt and accept (reject)

A2 Remove from R any rule involving membranesnot existing in C; set C ′ ← C and R ′ ← R

A3 Until R ′ is empty, pick a rule r ∈ R ′

A4 If r = [a→ w ]αh and it can be applied, choose thenumber k of a to rewrite into w , remove theseoccurrences from both C and C ′ and add k timesw to C; remove r from R ′

A5 If r = [a]αh → [ ]βh b and it can be applied, removea from both C and C ′, add b and changepolarization in C; remove r from R ′ together withany communication or division rule involving h

A6,A7 . . . and so on (dissolution is deferred)

Simulation algorithm (2)

A8 If parallelism is not maximal (useful objectsremain) abort the simulation by rejecting

A9 Dissolve all the membranes marked fordissolution (adjusting multisets and membranestructure)

A10 Return to A1

Correctness and time complexity

I The only tricky part is ensuring that maximalparallelism is simulated correctly

I Any violation is due to object evolution rulesI But it is detected and handled in step A8I “Membrane rules” are always applied in a

maximally parallel way

I Complexity analysis is boring; it suffices to saythat the algorithm requires O(p(m, t)) time,i.e. polynomial time with respect to the size mof the P system and the number of steps t tosimulate

I The polynomial is something like O(n3t2). . .

A non-confluent Milano theorem

I Let L ∈ NPMC?NAM and L = L(Π) with Πconstructed by M

I We define a NDTM N deciding L in polynomialtime

I On input x , with |x | = n, construct Πx

I Simulate Πx and accept/reject accordingly

I Πx has size q(n) and runs for r(n) steps (forsome polynomials q, r)

I The total time is then O(p(q(n), r(n))),i.e. O(s(n)) for some polynomial s

I Then L ∈ NP

Conclusions

I We proved NPMC?NAM = NP

I That is, semi-uniform families of non-confluentP systems with active membranes but withoutdivision solve exactly the problems in NP inpolynomial time

I (The deterministic version of the simulationalgorithm can be used to prove the originalMilano theorem)

Open problems

I What happens when we allow the use of bothnon-confluence and division rules? (Conjecture:EXP or even NEXP)

I What about space complexity?

Bibliography (1)

G. Paun,

Computing with membranes: Attacking NP-completeproblems,

in: I. Antoniou, C. Calude, M. J. Dinneen (Eds.),Unconventional Models of Computation, UMC’2K,Proceedings of the Second International Conference,Brussel, Belgium, 13-16 December 2000, Springer, 2001,pp. 94–115.

P. Sosık,

The computational power of cell division in P systems:Beating down parallel computers?,

Natural Computing 2 (3) (2003) 287–298.

Bibliography (2)

P. Sosık, A. Rodrıguez-Paton,

Membrane computing and complexity theory: Acharacterization of PSPACE,

Journal of Computer and System Sciences 73 (1) (2007)137–152.

C. Zandron, C. Ferretti, G. Mauri,

Solving NP-complete problems using P systems withactive membranes,

in: I. Antoniou, C. Calude, M. J. Dinneen (Eds.),Unconventional Models of Computation, UMC’2K,Proceedings of the Second International Conference,Brussel, Belgium, 13–16 December 2000, Springer, 2001,pp. 289–301.