Simulating EXPSPACE Turing Machines Using P Systems with Active Membranes

Simulating EXPSPACE Turing machines using P systems with active membranes

Artiom Alhazov1,2 Alberto Leporati1 Giancarlo Mauri1 Antonio E. Porreca1 Claudio Zandron1

1Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi di Milano-Bicocca

2Institute of Mathematics and Computer Science Academy of Sciences of Moldova

13th Italian Conference on Theoretical Computer Science 19–21 September 2012, Varese, Italy Outline

1. P systems 2. Space complexity and the time-space trade-oﬀ 3. Simulating Turing machines 4. Conclusions and open problems

2/16 P systems (with restricted elementary active membranes)

0 + − a a a 0 c b b c c a b

c h3

h1 a b c

3/16 Computation rules

α Object evolution a w

α β Communication (send-out) a b

h h

α β Communication (send-in) a b

h h

α β γ Elementary division a b c

h h h

4/16 PSPACE PMC AM PPP includes PH PP NP

I Membrane division is provably needed

Parallelism and eﬃciency

I The rules are applied in a maximally parallel way I There may be nondeterminism, but we require conﬂuence I Division may create exponentially many processing units in linear time

I So we can solve hard problems in polynomial time by trading space for time

5/16 I Membrane division is provably needed

Parallelism and eﬃciency

I The rules are applied in a maximally parallel way I There may be nondeterminism, but we require conﬂuence I Division may create exponentially many processing units in linear time

I So we can solve hard problems in polynomial time by trading space for time

PSPACE PMC AM PPP includes PH PP NP

5/16 Parallelism and eﬃciency

I The rules are applied in a maximally parallel way I There may be nondeterminism, but we require conﬂuence I Division may create exponentially many processing units in linear time

I So we can solve hard problems in polynomial time by trading space for time

PSPACE PMC AM PPP includes PH PP NP

I Membrane division is provably needed 5/16 M1 n N ∈ 1 1 1 yes

aab

M2 n x Σ 0 1 aab ∈ 0 no

input multiset

Uniformity

We decide membership in some language L by using a uniform family of P systems

6/16 Uniformity

We decide membership in some language L by using a uniform family of P systems

M1 n N ∈ 1 1 1 yes

aab

M2 n x Σ 0 1 aab ∈ 0 no

input multiset

6/16 space = #membranes + #objects

Formalising the time-space trade-oﬀ

What’s the exact meaning of trading space for time?

time = #computation steps

7/16 Formalising the time-space trade-oﬀ

What’s the exact meaning of trading space for time?

time = #computation steps

space = #membranes + #objects

7/16 Proof.

I A family of P systems working in space f (n) can be simulated by a TM working in O(f (n) log f (n)) space

I A TM working in space f (n) can be simulated by a family of P systems in space O(f (n)) as long as f is a polynomial

Known results on space complexity

Theorem Polynomial-space P systems and polynomial-space Turing machines solve the same class of decision problems, namely PSPACE

8/16 I A TM working in space f (n) can be simulated by a family of P systems in space O(f (n)) as long as f is a polynomial

Known results on space complexity

Theorem Polynomial-space P systems and polynomial-space Turing machines solve the same class of decision problems, namely PSPACE Proof.

I A family of P systems working in space f (n) can be simulated by a TM working in O(f (n) log f (n)) space

8/16 Known results on space complexity

Theorem Polynomial-space P systems and polynomial-space Turing machines solve the same class of decision problems, namely PSPACE Proof.

I A family of P systems working in space f (n) can be simulated by a TM working in O(f (n) log f (n)) space

I A TM working in space f (n) can be simulated by a family of P systems in space O(f (n)) as long as f is a polynomial

8/16 Simulating polynomial-space TMs

a a b b 2 1 3 5 0 4

| {z } p(n) | {z }

9/16 Simulating polynomial-space TMs

a a b b 2 1 3 5 0 4

0 + + 0 0 − − 0 1 2 3 4 5 q2 h labels provide order and identify cells

9/16 The problem with superpolynomial space bounds

I We cannot use exponentially many labels (polytime uniformity)

I We must create the tape-membranes at runtime via membrane division

I But the new membranes are indistinguishable from the outside (they all have the same label)

I Solution: order and identify membranes according to their contents

10/16 Encoding exponential space conﬁgurations

q a a b b 100 011 101 111 010 110

| {z } 2p(n) | {z } 11/16 Encoding exponential space conﬁgurations

q a a b b 100 011 101 111 010 110

0 0 0 0 0 0 0 02 11 00 02 11 10 12 01 00 12 01 10 12 11 00 12 11 10 b a a b t0 t1 t t t t t t

+ + 0 0 q 0 0 0 2 1 0 e a b t s

11/16 Simulating a computation step of the TM

δ(q, ) = (r, b,/) t

0 0 0 0 0 0 0 02 11 00 02 11 10 12 01 00 12 01 10 12 11 00 12 11 10 b a a b t0 t1 t t t t t t

+ + 0 0 q 0 0 0 2 1 0 e a b t s