P systems simulating oracle computations Antonio E. Porreca - - PowerPoint PPT Presentation

P systems simulating

• racle computations

Antonio E. Porreca Alberto Leporati Giancarlo Mauri Claudio Zandron

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

12th Conference on Membrane Computing Fontainebleau, France, 24 August 2011

Summary

◮ We show how to reuse existing recogniser P systems

as “subroutines”

◮ This allows us to simulate oracles ◮ The procedure is quite general

(though technical details may vary)

◮ As an application, we improve the lower bound

• n PMCAM(−d,−n) to PPP

2/16

P systems with active membranes

◮ Known for their ability to solve

computationally hard problems

◮ Here we focus on restricted elementary membranes

(no nonelementary division, no dissolution) Object evolution [a → w]α

h

Communication (send-in) a [ ]α

h → [b]β h

Communication (send-out) [a]α

h → [ ]β h b

Elementary division [a]α

h → [b]β h [c]γ h

3/16

Uniform families of recogniser P systems

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

receiving as input a multiset encoding x

◮ Both are constructed by fixed polytime Turing machines ◮ The resulting P system decides if x ∈ L M1 1 1 1

x ∈ Σ⋆

M2 1

Y E S N O

aab

aab

Input multiset

n ∈ N

4/16

The complexity class PMCAM(−d,−n)

It consists of the languages recognised in polytime by uniform families of P systems with restricted elementary membranes

◮ Contains NP problems [Zandron et al. 2000]

(semi-uniform solution)

◮ Contains NP problems [Pérez-Jiménez et al. 2003]

(first uniform solution)

◮ Is contained in PSPACE [Sosík, Rodríguez-Patón 2007] ◮ Contains PP problems [Porreca et al. 2010, 2011]

On the other hand, by using nonelementary division (class PMCAM) we obtain exactly PSPACE

5/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

x2x3 x1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

x2x3 x1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2x3 x1 f2x3 x1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2x3 x1 f2x3 x1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2x3 f1 f2 t3 x1 t2x3 t1 f2 f3 x1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2x3 f1 f2 t3 x1 t2x3 t1 f2 f3 x1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t t

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t t

6/16

Solving 3SAT

Is ϕ(x1, x2, x3) satisfiable?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t

Y E S

6/16

The complexity class PP

Definition

L ∈ PP if it is accepted in polytime by a nondeterministic TM such that more than half of its computations are accepting Solving PP is “essentially the same as” counting the number of solutions

Problem (T H R E S H O L D-3SAT)

Given a Boolean formula ϕ over m variables and an integer k < 2m, do more than k assignments out of 2m satisfy ϕ?

Theorem

T H R E S H O L D-3SAT is PP-complete

7/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t t

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t t

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t t

− − −

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t t

− − −

8/16

Solving T H R E S H O L D-3SAT

Does ϕ(x1, x2, x3) have more than 3 satisfying assignments?

t2 t3 f1 f2 t3 t1 t2 t3 t1 f2 f3 t1 t2 f3 f1 f2 t3 f1 t2 f3 t1 f2 f3 f1 t t t

Y E S

− − −

8/16

Simulating Turing machines I

q 1

S − + − 1 2 3 4

H2,q 9/16

Simulating Turing machines II

δ(q1, 0) = (q2, 1, ⊲)

• Hq1,i, [ ]−

i → [Hq2,i+1]+ i

[Hq2,i+1]+

i → [ ]+ i

Hq2,i+1 S − + − 1 2 3 4

H2,q1

10/16

Simulating Turing machines II

δ(q1, 0) = (q2, 1, ⊲)

• Hq1,i, [ ]−

i → [Hq2,i+1]+ i

[Hq2,i+1]+

i → [ ]+ i

Hq2,i+1 S − + − 1 2 3 4

H2,q1

10/16

Simulating Turing machines II

δ(q1, 0) = (q2, 1, ⊲)

• Hq1,i, [ ]−

i → [Hq2,i+1]+ i

[Hq2,i+1]+

i → [ ]+ i

Hq2,i+1 S − + + 1 2 3 4

H3,q2

10/16

Simulating Turing machines II

δ(q1, 0) = (q2, 1, ⊲)

• Hq1,i, [ ]−

i → [Hq2,i+1]+ i

[Hq2,i+1]+

i → [ ]+ i

Hq2,i+1 S − + + 1 2 3 4

H3,q2

10/16

Simulating Turing machines II

δ(q1, 0) = (q2, 1, ⊲)

• Hq1,i, [ ]−

i → [Hq2,i+1]+ i

[Hq2,i+1]+

i → [ ]+ i

Hq2,i+1 S − + + 1 2 3 4

H3,q2

10/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A −

AY E S

11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A −

AY E S

11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + − 11/16

Using P systems as subroutines

S − + − + 1 2 3 4

H2,q?

Π Π Π Π Π A + − 11/16

Main result

Theorem

PPP ⊆ PMCAM(−d,−n)

Proof.

◮ Any polytime TM M with a PP oracle can be simulated

by a polytime TM M′ with an oracle for T H R E S H O L D-3SAT and only one tape

◮ Just apply a reduction (which always exists,

since T H R E S H O L D-3SAT is PP-complete) before querying the oracle

◮ And we know how to simulate the TM M′

with a polytime AM(−d, −n) uniform family.

12/16

Discussion I

◮ We can solve QSAT (PSPACE-complete)

by using nonelementary division and a membrane structure of depth Θ(n)

◮ QSAT instances have an arbitrary number

• f alternations of quantifiers

◮ By fixing the first quantifier (∀ or ∃) and the number

• f alternations, we get complete problems for all levels
• f the polynomial hierarchy

◮ Formulae with k alternations can be solved by P systems

using nonelementary division and a membrane structure

• f depth Θ(k)

◮ Notice that k does not depend on the input size

13/16

Discussion II

◮ Let PH be the union of the levels

• f the polynomial hierarchy

◮ Toda’s theorem tells us that PH ⊆ PPP ◮ So we also have PH ⊆ PMCAM(−d,−n) ◮ This means that all levels of the polynomial hierarchy

can be solved by using P systems with only elementary division and membrane structure of depth 3

◮ Does this mean PSPACE ⊆ PMCAM(−d,−n)

and so PSPACE = PMCAM(−d,−n)?

◮ Not immediately: PH is not known neither conjectured

to be PSPACE

14/16

Open problems

◮ Prove that we can always do the oracle simulation ◮ If we can reset the “oracle P systems”

then we only need a single copy of it

◮ It might still be possible that PMCAM(−d,−n) = PSPACE

even if PH = PSPACE

◮ But maybe it would be more interesting if it turns out

that PMCAM(−d,−n) = PPP

15/16

Merci de votre attention! Thanks for your attention!