Space complexity for P systems

Antonio E. Porrecaporreca@disco.unimib.it

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

BWMC7 – February 5, 2009

Size of a configuration (1)

Suppose we manage to build a real P system withactive membranes. How large is it?

DefinitionLet Π be a P system with active membranes. Thesize |C| of a configuration C of Π is the sum of thenumber of membranes and the total number ofobjects inside its regions.

Size of a configuration (2)

. . . or maybe we just want to simulate P systemsin silico?

DefinitionLet Π be a P system with active membranes. Thesize |C| of a configuration C of Π is the sum of thenumber of membranes and the number of bitsrequired to store the number of objects inside eachregion.

Size of a configuration (3)

ProblemWhich definition do we choose? Does our choicechange the theory in any important way, i.e. is thenumber of objects as important as the number ofmembranes?

On the number of membranes and objects

I Let Π be a recogniser P system with activemembranes

I Let m be the maximum number of membranesin Π (considering all computations and allconfigurations)

I Let o be the maximum number of objectsinside Π

ProblemCan we always modify Π (without changing itsbehaviour) in such a way that o ≤ poly(m)?

Size of a computation

Meanwhile, assume the first definition of “size of aconfiguration”.

DefinitionLet Π be a P system with active membranes, andlet ~C = (C0, . . . , Cn) be a halting computation of Π.

The size of ~C is defined as

|~C| = max{|C0|, . . . , |Cn|}

Space bound for a P system

DefinitionLet Π be a (confluent or non-confluent) P systemwith active membranes such that every computationof Π halts. The size of Π is defined as

|Π| = max{|~C| : ~C is a computation of Π}

If |Π| ≤ n for some n ∈ N then Π is said to operatewithin space n.

Space bound for a family of P systems

DefinitionLet Π = {Πx : x ∈ Σ?} be a polynomiallysemi-uniform family of recogniser P systems, and letf : N→ N. Then Π is said to operate within spacebound f iff |Πx | ≤ f (|x |) for each x ∈ Σ?.

Space complexity classes

DefinitionLet D be a class of P systems (e.g. with activemembranes but without division rules), f : N→ Nand L ⊆ Σ?. Then L ∈ MCSPACE?

D(f ) iff L = L(Π)for some polynomially semi-uniform family Π ⊆ Dof confluent recogniser P systems operating withinspace bound f .

DefinitionThe corresponding class for non-confluentP systems is NMCSPACE?

D(f ).

Abbreviations

Definition

PMCSPACE?D = MCSPACE?

D(poly)

NPMCSPACE?D = NMCSPACE?

D(poly)

EXPMCSPACE?D = MCSPACE?

D(2poly)

NEXPMCSPACE?D = NMCSPACE?

D(2poly)

. . . and so on.

Preliminary results (1)

PropositionMCSPACE?

D(f ) ⊆ NMCSPACE?D(f ) for all

f : N→ N.

PMCSPACE?D ⊆ NPMCSPACE?

DEXPMCSPACE?

D ⊆ NEXPMCSPACE?D

PMCSPACE?D ⊆ EXPMCSPACE?

DNPMCSPACE?

D ⊆ NEXPMCSPACE?D

Preliminary results (2)

PropositionPMCSPACE?

D, NPMCSPACE?D, EXPMCSPACE?

D andNEXPMCSPACE?

D are all closed under polytimereductions.

PropositionMCSPACE?

D(f ) is closed under complementation foreach f : N→ N.

Preliminary results (3)

PropositionNPMC?

NAM ⊆ EXPMCSPACE?EAM.

Proof.NPMC?

NAM = NP and SAT ∈ EXPMCSPACE?EAM,

which is closed under polytime reductions.

Problems (1)

ProblemPMCSPACE?

NAM = PMCSPACE?EAM =

PMCSPACE?AM.

Ideas.Can we resolve all membrane divisions atconstruction time (by precomputing the finalmembrane structure)?

What about “conditional” divisions and “cyclic”behaviour (divide → dissolve → divide → · · · )?

Problems (2)

ProblemIs PMC?

NAM = PMCSPACE?NAM?

Is NPMC?NAM = NPMCSPACE?

NAM?

Idea.Maybe when the size of the P system is polynomialthere is no need to compute for a superpolynomialamount of time.

Problems (3)

ProblemDoes any of the following hold?

PMC?EAM 6= PMCSPACE?

EAMPMC?

AM 6= PMCSPACE?AM

Problems (4)

And, of course. . .

ProblemWhat are the relations between these newcomplexity classes and traditional ones like P, NP,PSPACE, EXP, EXPSPACE?