P systems simulating oracle computations

P systems simulating oracle computationsAntonio 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 on P M C AM(−d,−n) to P P P 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]αh Communication (send-in) a [ ]α → [b]β h h Communication (send-out) [a]α → [ ]β b h h Elementary division [a]α → [b]β [c]γ h h 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 n∈N 1 1 YES 1 aab M2 x∈Σ 0 1 aab 0 NO Input multiset 4/16

The complexity class P M C AM(−d,−n) It consists of the languages recognised in polytime by uniform families of P systems with restricted elementary membranes Contains N P problems [Zandron et al. 2000] (semi-uniform solution) Contains N P problems [Pérez-Jiménez et al. 2003] (ﬁrst uniform solution) Is contained in PSPACE [Sosík, Rodríguez-Patón 2007] Contains P P problems [Porreca et al. 2010, 2011] On the other hand, by using nonelementary division (class P M C AM ) we obtain exactly PSPACE 5/16

Solving 3 SAT Is ϕ(x1 , x2 , x3 ) satisﬁable? x1x2x3 6/16

Solving 3 SAT Is ϕ(x1 , x2 , x3 ) satisﬁable? x1 t2 x3 x1 f2 x3 6/16

Solving 3 SAT Is ϕ(x1 , x2 , x3 ) satisﬁable? t1 t2 x3 f1 t2 x3 x1 f2 t3 x1 f2 f3 6/16

Solving 3 SAT Is ϕ(x1 , x2 , x3 ) satisﬁable? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 6/16

Solving 3 SAT Is ϕ(x1 , x2 , x3 ) satisﬁable? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 t t t t 6/16

Solving 3 SAT Is ϕ(x1 , x2 , x3 ) satisﬁable? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 t t t YES 6/16

The complexity class P P Deﬁnition L ∈ PP if it is accepted in polytime by a nondeterministic TM such that more than half of its computations are accepting Solving P P 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 - 3 SAT is PP-complete 7/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 8/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 0 8/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 0 0 8/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 0 0 0 8/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 t t t t 0 0 0 8/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 t − − − t t t 8/16

Solving T H R E S H O L D - 3 SAT Does ϕ(x1 , x2 , x3 ) have more than 3 satisfying assignments? t1 t2 t3 f1 t2 t3 t1 f2 t3 t1 f2 f3 t1 t2 f3 f1 t2 f3 f1 f2 t3 f1 f2 f3 YES − − − t t t 8/16

Simulating Turing machines I q 0 1 0 0 − + − 0 0 0 1 2 3 4 H2,q S 9/16

Simulating Turing machines II − + H q1 ,i, [ ]i → [ H q2 ,i+1 ]i δ(q1 , 0) = (q2 , 1, ) [ H q2 ,i+1 ]+ → [ ]+ H q2 ,i+1 i i 0 − + − 0 0 0 1 2 3 4 H2,q1 S 10/16

Simulating Turing machines II − + H q1 ,i, [ ]i → [ H q2 ,i+1 ]i δ(q1 , 0) = (q2 , 1, ) [ H q2 ,i+1 ]+ → [ ]+ H q2 ,i+1 i i 0 − + + 0 0 H3,q2 0 1 2 3 4 S 10/16

Simulating Turing machines II − + H q1 ,i, [ ]i → [ H q2 ,i+1 ]i δ(q1 , 0) = (q2 , 1, ) [ H q2 ,i+1 ]+ → [ ]+ H q2 ,i+1 i i 0 − + + 0 0 0 1 2 3 4 H3,q2 S 10/16

Using P systems as subroutines 0 H2,q? − + − + 0 0 1 2 3 4 S 11/16

Using P systems as subroutines 0 0 0 0 0 0 Π Π Π Π Π 0 H2,q? A − + − + 0 0 1 2 3 4 S 11/16

Using P systems as subroutines 0 0 0 + 0 0 Π Π Π Π Π 0 H2,q? A − + − + 0 0 1 2 3 4 S 11/16

Using P systems as subroutines 0 0 0 − 0 0 Π Π Π Π Π 0 H2,q? AY E S A − + − + 0 0 1 2 3 4 S 11/16

Using P systems as subroutines 0 0 0 − 0 0 Π Π Π Π Π + H2,q? A − + − + 0 0 1 2 3 4 S 11/16

Main result Theorem P P P ⊆ PM C AM(−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) Q SAT instances have an arbitrary number of alternations of quantifiers By fixing the first quantifier (∀ or ∃) and the number of alternations, we get complete problems for all levels of the polynomial hierarchy Formulae with k alternations can be solved by P systems using nonelementary division and a membrane structure of depth Θ(k) Notice that k does not depend on the input size 13/16

Discussion II Let P H be the union of the levels of the polynomial hierarchy Toda’s theorem tells us that PH ⊆ P P P So we also have PH ⊆ PMC AM(−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 ⊆ PMC AM(−d,−n) and so P S PACE = PMC AM(−d,−n) ? Not immediately: PH is not known neither conjectured to be P S PAC E 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 PMC AM(−d,−n) = PSPACE even if P H = PSPACE But maybe it would be more interesting if it turns out that P M C AM(−d,−n) = P P P 15/16

Merci de votre attention!Thanks for your attention!

P systems simulating oracle computations

by Antonio E. Porreca on Aug 28, 2011

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P systems simulating oracle computations — Presentation Transcript