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 h2 h1 a b c h0 3/16 Computation rules α Object evolution a w h 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 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 PSPACE PMC AM PPP includes PH PP NP 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 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 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 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 q a a b b 2 1 3 5 0 4 n | {z } p(n) | {z } 9/16 Simulating polynomial-space TMs q 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 n | {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 12/16 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 q1 a a b t0 t1 t t t t t t + + 0 0 0 0 0 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 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 0 0 0 q2 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b 00 10 10 a b t0 2 1 0 t1 t t t t t t a + + 0 0 0 0 0 q3 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b 00 10 a b t0 2 0 t1 t t t t t t 110 + + 0 0 + 0 0 q4 a 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b 10 a b t0 0 t1 t t t t t t 020 + + 0 0 q + 0 0 10 5 a 2 1 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t 10 + + 0 0 q + 0 0 00 6 a 2 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t e + + 0 0 q + 0 0 10 7 a 2 1 0 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t e + + 0 + + 0 0 e q8 a 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t + + 0 + + 0 0 e q9 a 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t + + 0 + + 0 0 q10 a 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t q110 + + + 0 0 a 0 0 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t + + 0 0 0 0 a q0 − 2 1 0 e 12 a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 + 0 0 0 0 02 11 00 0200 1100 1000 12 01 00 12 01 10 12 11 00 12 11 10 b a b t0 t1 t t t t t t a0 + + 0 0 0 0 q0 − 2 1 0 e 13 a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 0 0 0 02 11 00 0200 1100 1000 − 12 01 00 12 01 10 12 11 00 12 11 10 b a a b t0 t1 t t t t t t q140 + + 0 0 0 0 0 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 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 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 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 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 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 q1 t0 t1 t t t t t t + + 0 0 0 0 0 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 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 0 0 0 q2 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 + 02 11 00 − 02 11 10 − 12 01 00 12 01 10 1200 1100 0000 12 11 10 − b a a b 10 10 00 t0 t1 t t 2 1 0 t t t + + 0 0 0 0 t 0 q3 2 1 0 e a b t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 + 02 11 00 − 02 11 10 − 12 01 00 12 01 10 1200 1100 0000 12 11 10 − b a a b 10 00 t0 t1 t t 1 0 t t t 120 + + 0 0 0 0 + q4 2 1 0 e a b t t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 + 02 11 00 − 02 11 10 − 12 01 00 12 01 10 1200 1100 0000 12 11 10 − b a a b 10 t0 t1 t t 1 t t t 00 + + 0 0 q 0 0 + 10 5 2 2 1 0 e a b t t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 + 02 11 00 − 02 11 10 − 12 01 00 12 01 10 1200 1100 0000 12 11 10 − b a a b t0 t1 t t t t t 110 + + 0 0 q 0 0 + 00 6 2 1 0 0 e a b t t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 + 02 11 00 − 02 11 10 − 12 01 00 12 01 10 1200 1100 0000 12 11 10 − b a a b t0 t1 t t t t t + + 0 0 q 0 0 + 10 7 2 1 1 0 e a b t t s 12/16 Simulating a computation step of the TM δ(q; ) = (r; b;/) t 0 0 0 + 02 11 00 − 02 11 10 − 12 01 00 12 01 10 1200 1100