Simulating EXPSPACE Turing machines using P systems with active - - PowerPoint PPT Presentation

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)

a a a b b c a b c c c a b c

h1 h2 h0 h3 + −

3/16

Computation rules

a

h α

w a

h α

b

h β

a

h α

b

h β

a

h α

b

h β

c

h γ

Object evolution Communication (send-out) Communication (send-in) Elementary division 4/16

Parallelism and efficiency

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

in linear time

◮ So we can solve hard problems in polynomial time

by trading space for time

5/16

Parallelism and efficiency

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

in linear time

◮ So we can solve hard problems in polynomial time

by trading space for time

PSPACE PMCAM PPP PP NP

includes PH

5/16

Parallelism and efficiency

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

in linear time

◮ So we can solve hard problems in polynomial time

by trading space for time

PSPACE PMCAM PPP PP NP

includes PH ◮ Membrane division is provably needed

5/16

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 1 1 1

x ∈ Σn

M2 1

yes no

aab

aab

input multiset

n ∈ N

6/16

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

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

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.

◮ 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.

◮ 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

◮ 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

n

• p(n)

b a b a q

1 2 3 4 5

9/16

Simulating polynomial-space TMs

b a b a q

1 2 3 4 5 − 1 + 2 + 3 − 4 5

q2

h labels provide order and identify cells

9/16

The problem with superpolynomial space bounds

◮ We cannot use exponentially many labels (polytime uniformity) ◮ We must create the tape-membranes at runtime

via membrane division

◮ But the new membranes are indistinguishable from the outside

(they all have the same label)

◮ Solution: order and identify membranes according

to their contents

10/16

Encoding exponential space configurations n

• 2p(n)

b a b a q

010 011 100 101 110 111 11/16

Encoding exponential space configurations

b a b a q

010 011 100 101 110 111

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 02 a 12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔ q

+ +

10 11 02 11

11/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 02 a 12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔ q

+ +

10 11 02 11

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 02 a 12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ +

10 11 02 11 q1

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 02 a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ +

10 11 02 11 q2

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ +

1′′ 1′′

1

02 11 q3 0′

2

1′ 1′

1

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11 q4 0′

2

1′ 1′

1

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11 q5 0′

2

1′ 1′

1

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11 q6 0′

2

1′

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11 q7 e 1′

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11

+

q8 e e

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11

+

q9 e

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11

+

q10

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ + +

1′′ 1′′

1

02 11 q′

11

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

− + +

1′′ 1′′

1

02 11 q′

12

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a′

+

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

− + +

1′′ 1′′

1

02 11 q′

13

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b 0′′

2

a

−

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ +

1′′ 1′′

1

02 11 q′

14

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

+ +

02 11 q 02 10 11

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b

−

a

−

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

− + +

02 11 q 02 10 11

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b

−

a

−

12 01 00 a 12 01 10 b 12 11 00 ⊔ 12 11 10 ⊔

− + +

02 11 q1 02 10 11

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b

−

a

−

12 01 00 a 12 01 10 b 12 11 00 ⊔

+

12 11 10 ⊔

− + +

02 11 02 10 11 q2

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b

−

a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

− + +

02 11 02 10 11 q3 1′

2 1′ 1 0′

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q4 1′

2

1′

1 0′ − −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q5 1′

2

1′

1

0′

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q6 1′

1

0′

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q7 1′

1 − −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q8

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q9

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q10

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ + +

02 11 02 10 11 q11

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

⊔

+

12 11 10 ⊔

+ +

02 11 02 10 11 q12

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

b′

+

12 11 10 ⊔

+ +

02 11 02 10 11 q13

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

b′

+

12 11 10 ⊔

+ +

02 11 02 10 11 q14

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b 1′′

2 1′′ 1 0′′

b

−

12 11 10 ⊔

+ +

02 11 02 10 11 q⊔

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ +

02 11 02 10 11 q⊔ 12 11 00

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ + +

02 11 02 10 11 q⊔

1

12 11 00

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ + +

02 11 02 10 11 q⊔

1

12 11 00

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ +

02 11 02 10 11 q⊔

3

12 11 00

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a

−

12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ +

02 11 02 10 11 q⊔

4

12 11 00 t0 t1 c2

− −

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a 12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ +

02 11 02 10 11 q⊔

5

12 11 00 t0 t1 c1 c1

−

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a 12 01 00 a 12 01 10 b b

−

12 11 10 ⊔

+ +

02 11 02 10 11 q⊔

6

12 11 00 c0 c0 c0 c0

−

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a 12 01 00 a 12 01 10 b b 12 11 10 ⊔

+ +

02 11 02 10 11 q⊔

7

12 11 00 c0 c0 c0 c0

12/16

Simulating a computation step of the TM δ(q, ⊔) = (r, b, ⊳)

t0 t1 t t t t ⊔ b a 2 1 e s

00 b a 12 01 00 a 12 01 10 b b 12 11 10 ⊔

+ +

02 11 02 10 11 r 12 11 00

12/16

Initialising the system

b a qinit

010 011 100 101 110 111

t0 t1 t ⊔ b a 2 1 e s

00 b0 02 a1 12 d1 d0 ⊔ z2 10 11 02 11 +

13/16

Initialising the system

b a qinit

010 011 100 101 110 111

t0 t1 t ⊔ b a 2 1 e s

00 b 02 a 12 d1 00 ⊔ z1

+

10 11 02 11 +

t

12 d1 10 ⊔

13/16

Initialising the system

b a qinit

010 011 100 101 110 111

t0 t1 t ⊔ b a 2 1 e s

00 b 02 a 12 01 00 ⊔ qinit

+

10 11 02 11

t

12 01 10 ⊔

t

12 11 00 ⊔

t

12 11 10 ⊔

13/16

Main result

Theorem

Let M be a single-tape deterministic Turing machine working in time t(n) and space s(n), where s(n) ≤ n + 2p(n) for some polynomial p. Then there exists a uniform family of confluent P systems with restricted elementary active membranes Π

• perating in time O
• t(n)s(n) log s(n)
• and space O
• s(n) log s(n)
• such that L(Π) = L(M)

Corollary

Both exponential-space TMs and exponential-space P systems solve exactly the problems in EXPSPACE

14/16

Conclusions and open problems

◮ P systems and TMs have identical computing power

both in polynomial and in exponential space

◮ Preliminary results for sub-polynomial space

◮ A weaker uniformity condition is needed

◮ Is there a general equivalence in power wrt all space bounds?

◮ Probably, though the simulation must be improved ◮ Possible solution: unary encoding of tape cell numbers

15/16

Grazie per l’attenzione! Thanks for your attention!