P systems with elementary active membranes:Beyond NP and coNP

Antonio E. Porreca Alberto Leporati Giancarlo MauriClaudio Zandron

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

11th Conference on Membrane ComputingJena, Germany, 25 August 2010

Summary

I P systems with active membranes are thoroughly investigatedfrom a complexity-theoretic standpoint

I They have been known to solve NP and coNP problemsin polytime, using elementary division

I We improve this result by solving a PP-complete problem

PP ⊆ PMCAM(−d,−n)

2/18

Outline

P systems with elementary active membranes

Recogniser P systems and uniformity

The complexity class PP

Solving a PP-complete problem

Conclusions and open problems

3/18

Membrane structure and its contents

I Membranes have a fixed label and a changeable chargeI The charges regulate which set of rules can be appliedI In each membrane we have the usual multiset of objects

h0

h1

h2

+ −0

aaabbc

bcc

abc

4/18

Rules for restricted elementary active membranes

Object evolution [a→ w]αh

Send out [a]αh → [ ]βh b

Send in a [ ]αh → [b]βh

Elementary division [a]αh → [b]βh [c]γh

No dissolution or nonelementary divisionMaximally parallel application of rules

5/18

Uniform families of recogniser P systems

I For each input length n = |x| we construct a P system Πn

receiving as input a multiset encoding xI Both are constructed by fixed polytime Turing machinesI The resulting P system decides if x ∈ L

M1

1 11

x ∈ Σ?M2

0 10

Y E S

N O

aab

aab

Input multiset

1|x| ∈ {1}?

6/18

Timeline of P systems with active membranes

I Attacking (and solving) NP-complete problems [Paun 1999],uses dissolution and nonelementary division

I Solving NP-complete problems [Zandron et al. 2000],no dissolution nor nonelementary division

I Solving NP-complete problems [Pérez-Jiménez et al. 2003],uniform, no dissolution nor nonelementary division

I PSPACE upper bound [Sosík, Rodríguez-Patón 2007]I Solving PP-complete problems [Alhazov et al. 2009],

no nonelementary division, uses either cooperationor postprocessing

7/18

The PP complexity class

DefinitionPP is the class of languages decided by polytime probabilisticTuring machines with error probability strictly less that 1/2

Definition (equivalent)PP is the class of languages decided by polytimenondeterministic Turing machines such thatmore than half of the computations accept

8/18

How large is PP?

P

NP coNP

PSPACE

PP

9/18

The S Q R T -3SAT problem

Problem (S Q R T -3SAT)Given a Boolean formula of m variables in 3CNF,do more that

√2m assignments satisfy it?

FactS Q R T -3SAT is PP-complete

10/18

Encoding S Q R T -3SAT instances

I There are(m

3

)sets of 3 variables out of m

I Each variable can be positive or negated (23 ways)I Hence there are n = 8

(m3

)possible clauses

I We can represent a 3CNF formula by an n-bit stringI Checking well-formedness and recovering m from n

are easy (polytime)

11/18

An example

I If we have 3 variables, the number of clauses is 8(3

3

)= 8

x1 ∨ x2 ∨ x3 x1 ∨ x2 ∨ ¬x3 x1 ∨ ¬x2 ∨ x3

x1 ∨ ¬x2 ∨ ¬x3 ¬x1 ∨ x2 ∨ x3 ¬x1 ∨ x2 ∨ ¬x3

¬x1 ∨ ¬x2 ∨ x3 ¬x1 ∨ ¬x2 ∨ ¬x3

I Then the formula

ϕ = (x1 ∨ ¬x2 ∨ x3)︸ ︷︷ ︸3rd

∧ (¬x1 ∨ x2 ∨ ¬x3)︸ ︷︷ ︸6th

∧ (¬x1 ∨ ¬x2 ∨ x3)︸ ︷︷ ︸7th

is encoded as〈ϕ〉 = 0010 0110

12/18

A membrane computing algorithm for S Q R T -3SAT

AlgorithmLet ϕ be a 3CNF formula of m variables

1. Generate 2m membranes, one for each assignment

2. Evaluate ϕ in parallel in each of these membranes,send out object t from them if it is satisfied

3. Erase d√

2me − 1 instances of t

4. Output Y E S if an instance of t remains and N O otherwise

Phase 3 was first proposed by Alhazov et al. 2009using cooperative rewriting rules

13/18

Overview of the computation

0

1

00

x1x2 · · ·

14/18

Overview of the computation

0

0

1

0t1x2 · · ·

1

0f1x2 · · ·

14/18

Overview of the computation

0

0

1

0t1t2 · · ·

1

0f1t2 · · ·

1

0t1f2 · · ·

1

0f1f2 · · ·

14/18

Overview of the computation

0

0

2m assignments

1

0t1t2 · · ·

1

0f1t2 · · ·

1

0t1f2 · · ·

1

0f1f2 · · ·

14/18

Overview of the computation

0

0

2

0

1

0t1t2 · · ·

1

0f1t2 · · ·

1

0t1f2 · · ·

1

0f1f2 · · ·

14/18

Overview of the computation

0

0

1

0t1t2 · · ·

1

0f1t2 · · ·

1

0t1f2 · · ·

1

0f1f2 · · ·

2

0

2

0

d√

2me − 1 copies

14/18

Overview of the computation

0

0

2

0

2

0

1

0

1

0

1

0

1

0

t t t

14/18

Overview of the computation

0

0

2

0

2

0

1

0

1

0

1

0

1

0

t t t

14/18

Overview of the computation

0

0

2

−2

−

1

0

1

0

1

0

1

0

t t

t

14/18

Overview of the computation

0

0

2

−2

−

1

0

1

0

1

0

1

0

t t

t

14/18

Overview of the computation

0

0

2

−2

−

1

0

1

0

1

0

1

0

t t

YES

14/18

Our main result

PropositionThere is a uniform construction of the family of P systemssolving S Q R T -3SAT

PropositionS Q R T -3SAT ∈ PMCAM(−d,−n)

TheoremPP ⊆ PMCAM(−d,−n)

15/18

In other words. . .

P

NP coNP

PSPACE

PMCAM(−d,−n)

PP

16/18

Conclusions and open problems

I We solved a PP-complete problem in polytime usingP systems with restricted active membranes

I As a consequence PP ⊆ PMCAM(−d,−n) ⊆ PSPACE holdsI However, neither inclusion is known to be strict,

and a full characterisation is still missingI This class is possibly larger than PPI Maybe even PMCAM(−d,−n) = PSPACE holds?

17/18

Thanks for your attention!