The Computational Complexity of The Abelian Sandpile Model

69 Pages·2010·0.37 MB·English

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The Computational Complexity of The
Abelian Sandpile Model
Juan Andres Montoya, Carolina Mejia
June 2011
ii
ABSTRACT This is the (cid:133)rst volume of a series of studies related to the
complexity theoretical analysis of statisitical mechanics. In this (cid:133)rst work
we have considered some predicting tasks arising in statistical mechanics.
Predicting the evolution of dynamical systems is one of the foundational
tasks of natural sciences. Statistical mechanics is mainly concerned with
(cid:133)nite dynamical systems, which are more easy (from the conceptual point
of view) of analyzing. We have chosen to work with one speci(cid:133)c system of
statisitical mechanics: The Abelian Sandpile Model.
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Contents
0.1 Organization of the work. . . . . . . . . . . . . . . . . . . . vi
0.2 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . vi
1 Basics vii
1.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1.2 Complexity theory . . . . . . . . . . . . . . . . . . . . . . . viii
1.2.1 Parallel complexity . . . . . . . . . . . . . . . . . . . viii
1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
2 The Complexity of Predicting xi
2.1 Prediction problems and Boltzmann systems . . . . . . . . xi
2.2 Prediction problems and PSPACE . . . . . . . . . . . . . . xii
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
3 The Abelian Sandpile Model xvii
3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
4 Algorithmic problems xxvii
4.1 The algorithmic problems . . . . . . . . . . . . . . . . . . . xxvii
4.2 The relative hardness of sandpile prediction problems . . . xxix
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii
5 Statistics of critical avalanches xxxv
5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli
6 Dimension 1 xliii
iv Contents
6.1 GC[ ] belongs to logDCFL . . . . . . . . . . . . . . . . . xliii
1
L
6.2 SPA[ ] is TC0-hard . . . . . . . . . . . . . . . . . . . . . xlvi
1
L
6.3 A long remark: one-dimensional critical avalanches . . . . . xlviii
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlix
7 Dimension 2 li
7.1 Thehardnessoftwo-dimensionalsandpilepredictionproblems li
7.2 Two-dimensional critical avalanches . . . . . . . . . . . . . liii
7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv
8 Dimension 3 lv
8.1 RR[ ] is P-complete . . . . . . . . . . . . . . . . . . . . . lv
3
L
8.2 Strict P-completeness of SPP [ ] . . . . . . . . . . . . . . lvii
3
L
8.3 Three dimensional critical avalanches . . . . . . . . . . . . . lx
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxii
9 Open problems lxiii
9.1 Directed sandpiles . . . . . . . . . . . . . . . . . . . . . . . lxiii
9.2 The complexity of two-dimensional sandpiles . . . . . . . . lxv
References lxvii
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Preface
Predicting the evolution of dynamical systems is one of the main tasks of
natural science. Can we always predict? Chaos theory claims that there
are dynamical systems which are unpredictable. In this work we are in-
terested in analyzing the predictability of (cid:133)nite dynamical systems, which
are predictable. We are interested in a strong notion of predictability: a
dynamicalsystem isstrongly-predictable ifandonlyifwecanpredictthe
S
evolution of without parsimoniously simulating its dynamics. We focus
S
our attention on The Abelian Sandpile Model. Also, we consider the fol-
lowing technical question: Can we predict the evolution of a (cid:133)nite sandpile
inpolylogarithmicparalleltime?or,dopredictionproblemsrelatedtoThe
Abelian Sandpile Model belong to NC?
Our work is close related to the work of Moore and Machta (see for in-
stance [MMG], [Mo]), which is concerned with the algorithmic complexity
ofsimulating(cid:133)nitedynamicalsystems. Inthisworkwestudysomepredic-
tion problems related to The Abelian Sandpile Model, which can be con-
sidered as a class of (cid:133)nite cellular automata (Boltzmann Systems). There
are many works dealing with the algorithmic and physical complexity of
Sandpile Prediction Problems (see for instance [BTW], [D], [M], [Mo]).
In this work we prove that prediction problems belong to PSPACE;
we prove that the general prediction problem for Boltzmann Systems is
PSPACE complete, and we prove lower bounds for the restriction of The
SandpilePredictionProblemtolow-dimensionallattices. Furthermore,we
believe that we are de(cid:133)ning the basis of a research project concerned with
the analysis of the short term dynamics of (cid:133)nite cellular automata.
vi Contents
0.1 Organization of the work
0.2 Acknowledgements
Authors would like to thanks all the people that contribute with the writ-
ing of this booklet. We apologize if we forget to mention someone of our
co-workers, friends, students and supporters. We want to thanks the grad-
uate students that have taken part in our seminar (Computational Com-
plexity of Statistical Mechanics), special thanks go to Sergio Andres Mon-
toya, Francisco Javier Gutierrez and Sterling Castaæeda. Thanks go also
to our co-workers Carlos Arturo Rodriguez and Rafael Isaacs, and to our
sandpile-friend Anahi Gajardo. This work would not see the light of day
without the (cid:133)nancial support of VIE-UIS and Colciencias research project
111518925292.
Bucaramanga, Enero de 2011.
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1
Basics
In this chapter we introduce some of the notation that will be used along
the work, and some of the mathematical concepts that we will use in some
places of this work. Furthermore, we will introduce some basic facts, and
some basic concepts of complexity theory. We will focus our attention on
parallelcomplexity,whichprovideuswithaconceptualmachinerythatcan
be used to analyze the algorithmic hardness of most sandpile prediction
problems.
1.1 Lattices
Lattice graphs are discrete versions of the euclidean space, because of this
theyhaveplayedanimportantroleinstatisticalmechanics:latticesarethe
underlying graphs of most of the graphical models of statistical mechan-
ics. We will focus our attention on the restriction of the abelian sandpile
model to low-dimensional lattices, speci(cid:133)cally we will consider the restric-
tionoftheabeliansandpilemodeltolinearlattices,squarelatticesandcu-
biclattices(one-dimensional,two-dimensionalandthree-dimensionalcubic
lattices).
Given n 1; we use the symbol 1 to denote the linear lattice of order
(cid:21) Gn
n; whose vertex set is equal to [n]; where [n] = 1;:::;n and the edge
f g
relation is the nearest neighbor relation: We use the symbol 1 to denote
Ln
the linear sandpile lattice of order n; which is obtained from 1 by adding
Gn
a node s which is called the sink. Furthermore, given v; a node on the
viii 1. Basics
borderof 1;thereareoneedgein 1 connectingv ands:Notethatgiven
Gn Ln
v V 1 s ; we have that deg(v) = 2: We use the symbol to
2 Ln (cid:0)f g L1
denote the class 1 :n 1 :
(cid:0) (cid:1) Ln (cid:21)
We use the symbol 2 to denote the square lattice of order n; whose
(cid:8) Gn (cid:9)
vertex set is equal to [n] [n]: We use the symbol 2 to denote the square
(cid:2) Ln
sandpile lattice of order n; which is obtained from 2 by adding a node s
Gn
which is called the sink. Furthermore, given v a node on the border of 2;
Gn
there are 4 deg (v) edges in 2 connecting v and s: Note that given
v V 2 (cid:0) sG;n2 we have thatLdneg(v) = 4: We use the symbol to
2 Ln (cid:0)f g L2
denote the bounded class 2 :n 1 :
(cid:0) (cid:1) Ln (cid:21)
Finally, we use the symbol 3 to denote the cubic lattice of order n;
(cid:8) Gn (cid:9)
whosevertexsetisequalto[n] [n] [n]:Weusethesymbol 3 todenote
(cid:2) (cid:2) Ln
the cubic sandpile lattice of order n; which is obtained from 3 by adding
Gn
a node s called the sink. Furthermore, given v a node on the border of 3;
Gn
there are 6 deg (v) edges in 3 connecting v and s: Note that given
v V 3 (cid:0) sG;n3 we have thatLdneg(v) = 6: We use the symbol to
2 Ln (cid:0)f g L3
denote the bounded class 3 :n 1
(cid:0) (cid:1) Ln (cid:21)
(cid:8) (cid:9)
1.2 Complexity theory
Inthissectionweintroducesomebasicfacts(andconcepts)ofComplexity
Theory. Complexity Theory analyzes algorithmic problems with respect to
theirintrinsichardness.The(cid:133)nalgoalofComplexityTheoryistodetermine
the exact amount of computational resources required to solve a given
problem. A very good introduction to the (cid:133)eld is the reference [P].
1.2.1 Parallel complexity
Wewillanalyzeproblemsthatcanbesolvedinpolynomialtime.LetLbea
computational problem, knowing that L is polynomial time solvable is not
the last word, some additional questions can be stated: Can L be solved
using logarithmic space? Can L be solved in polylogarithmic time using a
polynomial number of processors? Actually, these are the questions that
weconsiderwhenanalyzingthepredictingtasksassociatedtoTheAbelian
Sandpile Model. Because of this, we want to use this preliminary chapter
to introduce the basic ideas of Parallel Complexity, which is a complexity
theorywellsuitedfordealingwiththekindofquestionsweareconsidering.
First at all we introduce the classes NCi; where i belongs to N. These
classescanbeintroducedviauniformfamiliesofcircuitswithsomespeci(cid:133)c
constraints related to size, depth, fan in and fan out. We will introduce
those classes from a pragmatic point of view as the classes of problems
that can be solved in polylogarithmic time using a polynomial number of
processors.
1.2 Complexity theory ix
De(cid:133)nition 1 Given i 1 we have that NCi is the class of problems that
can be solved in time O(cid:21)logi(n) using a polynomial number of processors.
Afundamentalnotion(cid:0)ofparal(cid:1)lelcomplexityisthenotionofNC-reduction,
intuitively NC reductions are Turing reductions that can be computed in
polylogarithmic time.
De(cid:133)nition 2 GivenLand(cid:5)twoproblems,wesaythatLisNCi reducible
to (cid:5) if and only if there exists a Turing reduction of L in (cid:5) which can be
computed in time O logi(n) using a polynomial number of processors.
There are some ot(cid:0)her poly(cid:1)logarithmic classes which will play an impor-
tant role in our work.
De(cid:133)nition 3 (further polylogarithmic classes)
1. L is the class of problems that can be solved using a logarithmic space
bounded deterministic Turing Machine.
2. NL is the class of problems that can be solved using a logarithmic
space bounded nondeterministic Turing Machine.
3. A logspace reduction is a Turing reduction which can be computed
using logarithmic space.
4. logDCFL is the class of problems which are logspace reducible to a
deterministic context free language.
5. TC0 is the class of problems which can be solved using a polynomial
size uniform family of circuits of bounded depth de(cid:133)ned on the logical
basis ; ; :
f^ _ (cid:24)g
6. We say that L is TC0 hard if and only if the majority function is
constant depth reducible to L; that is: L is TC0 hard if and only if
there exists a reduction of Maj, the problem consisting in computing
themajorityfunction,inLwhichcanbecomputedusingapolynomial
size uniform family of circuits of constant depth.
Remark 4 The de(cid:133)nition of TC0 hardness is based on Hastad-Sipser the-
orem, which says that Maj doesn(cid:146)t belong to TC0: Note that given L an
algorithmic problem, if L is TC0 hard we have that L = TC0; the problem
2
L requires unbounded depth.
It is known that
TC0 NC1 L NL logDCFL NC2 ::: NCi NCi+1 ::: P
(cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26)
Also, we have a hierarchy of complexity classes, and our goal is to (cid:133)nd
the right place, within this hierarchy, of each one of the problems that we
want to analyze.
x 1. Basics
A last concept is the important concept of P-completeness. Intuitively,
a P-complete problem is a ptime solvable problem which is very hard to
be solvable in polylogarithmic parallel time.
De(cid:133)nition 5 Given L P, we say that L is P-complete if and only if
2
given (cid:9) P there exists a NC reduction of (cid:9) in L:
2
Teorema 6 Let NC = NCi; and let L be a P-complete problem. We
i 1
have that NC =P if and[(cid:21)only if L NC:
2
Theproofoflasttheoremisstraightforward.LasttheoremsaysthatifL
isP-complete,thenitisveryunlikelythatL NC;sinceitisveryunlikely
2
that P =NC:
Let MCVP the problem de(cid:133)ned by
Problem 7 (MCVP; monotone circuit value problem)
Input: ( ;(cid:22)); where is a monotone, synchronous boolean circuit of
(cid:15) C C
fan in 2 and fan out 2:
Problem: Decide if ((cid:22))=1:
(cid:15) C
Teorema 8 (Cook(cid:146)s Reduction)
The problem MCVP is P-complete.
A proof of last theorem can be found in [P]. It is very easy to prove that
given L;(cid:5) P; if L is P-complete and L is NC reducible to (cid:5), then (cid:5)
2
is P-complete. Also, the problem MCVP can be used (and actually it is
used) as a pivot in P-completeness proofs.
1.3 Exercises
1. Prove theorem 6.
2. Prove that MCVP is P-complete.
3. Let L be a problem in P. Prove that if MCVP is NC reducible to
L; then L is P-complete.