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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!