How hard is it to predict sandpiles on lattices? A survey
HHow hard is it to predict sandpiles on lattices? A survey.
Enrico Formenti and K´evin Perrot Universit Cte dAzur, CNRS, I3S, France. Aix-Marseille Univ., Toulon Univ., CNRS, LIS, Marseille, France.
Abstract
Since their introduction in the 80s, sandpile models have raised interest for theirsimple definition and their surprising dynamical properties. In this survey we focuson the computational complexity of the prediction problem, namely, the complexityof knowing, given a finite configuration c and a cell x in c , if cell x will eventuallybecome unstable. This is an attempt to formalize the intuitive notion of “behavioralcomplexity” that one easily observes in simulations. However, despite many effortsand nice results, the original question remains open: how hard is it to predict thetwo-dimensional sandpile model of Bak, Tang and Wiesenfeld? Langton proposed to describe complex dynamical systems as being at the “edge of chaos”[44]. Complexity arises in a context that is neither too ordered, i.e. not exhibiting a rigidstructure allowing to efficiently understand and predict the future state of the system,nor completely chaotic, i.e. avoiding pseudo-random behaviour and uncomputable long-term effects. From a computer science point of view, complex dynamical systems are anobject of great interest because they precisely model physical systems that are able toperform non-trivial computation.In 1990, Moore et. al. started to formalize the intuitive notion of “complexity”of a system, through the computational complexity of predicting the behaviour of thesystem [41, 48, 49, 50, 55, 56] (for other kinds of complexity in dynamical systems,see for instance [17, 18, 20, 1, 16, 11, 10, 19]). Computational complexity theory isactually a perfect fit to capture the complexity of systems able to compute. In turnedout that the hierarchy of complexity classes offers a very precise way to characterize thebehavioural complexity of discrete dynamical systems. For example if a system has a P -hard prediction problem, it means that it is able to efficiently simulate a general purposesequential computer (such as a Turing machine), whereas if the prediction problem ismuch below in the hierarchy, let say in L , then the system can only compute under severespace restriction, and therefore cannot perform efficiently any computation if we assume P (cid:54) = L . In this precise sense the former would be more complex than the latter.1 a r X i v : . [ c s . D M ] S e p t the same time, the sandpile model of Bak, Tang and Wiesenfeld [2, 3] gainedinterest. It exhibits both a “complex” behaviour and a very elegant algebraic struc-ture [22]. Unsurprisingly, sandpile models are capable of universal computation [34]. In1999, Moore and Nilsson began to apply the computational complexity vocabulary tocapture the intuitive “complexity” of sandpile models [54]. Moreover, they observed adimension sensitivity that received great attention. It is the purpose of this survey toreview such very interesting results and to generalise some of them.Sandpile models are a subclass of number-conserving cellular automata where we aregiven a d -dimensional lattice ( Z d ) with a finite amount of sand grains at each cell. Alocal rule applied in parallel at every cell let grains topple: if the sand content at a cellis greater or equal to 2 d , then the cell gives one grain to each of the 2 d cells it touches(two cells in each dimension). This is the very first sandpile model of Bak, Tang andWiesenfeld, which they defined for d = 2. This is also the sandpile model studied byMoore and Nilsson, for which they proved the following foundations. • In dimension one it is possible to predict efficiently the dynamics with a parallelalgorithm (complexity class NC ). • In dimension three or more it is not possible to predict efficiently the dynamics witha parallel algorithm, unless some classical complexity conjecture is wrong (unless P = NC , since prediction is proven to be a P -hard problem). In other terms, inthis case the dynamics is inherently sequential.This survey concentrates on lattice Z d , because of this interest in the dimensionsensitivity. In a more general setting than lattices, sandpiles can very easily embedarbitrary computation and become almost always hard to predict from a computationalcomplexity point of view [35].Sandpile models have close relatives, the family of majority cellular automata. In-deed, though the latter model is not number-conserving, open questions on its two-dimensional prediction are remarkably similar [53]. Goles et. al. made progress in var-ious directions to capture the essence of P -completeness in majority cellular automata[33, 36, 37, 38, 39], with notable applications of NC algorithms from [43]. Cellularautomata with finite support are very close to sandpiles when considered under the se-quential update policy. However, in this context we witness a general increase of thecomplexity of the (decidable) questions about the dynamics which seems not to happenin sandpiles [18].The paper is structured as follows. In Section 2 we define sandpile models on latticeswith uniform neighborhood, formulate three versions of the prediction problem, intro-duce some classical considerations, and briefly review the complexity classes at stake.Subsequent sections survey known results, and generalise some of them or propose con-jectures. All prediction problems are in P (Section 3). The dimension sensitivity forarbitrary sandpile models generalizes as follows: in dimension one prediction is in NC (Section 4), in dimension three or above it is P -complete (Section 5), and in two dimen-sions the precise complexity classification remains open for the original sandpile modelwith von Neumann neighborhood, though insightful results have been obtained aroundthis question (Section 6). Finally, we briefly mention how undecidability may arise when2he finiteness condition on the initial configuration is relaxed (Section 8). Since their introduction by Bak, Tang and Wiesenfeld in [2], sandpiles underwent manygeneralizations. In this survey we propose a general framework which tries to cover allsuch models. However, we will focus only over lattices of arbitrary dimension and uniform(in space and time) number-conserving local rules. This section includes formalizationof folklore terminology and considerations extended to this general setting.
Let N + (resp. N − ) denote the set of strictly positive (resp. negative) integers. Forany dimension d ∈ N + , a cell is a point in Z d . A configuration is an assignment ofa finite number of sand grains to each cell i.e. it is an element of N Z d . A sandpilemodel is a structure (cid:104)N , D , θ (cid:105) where N is a finite subset of Z d \ { d } called neighborhood and D ∈ N + N is distribution of sand grains w.r.t. the neighborhood N (it is requiredthat dom ( D ) = N ) and θ = (cid:80) x ∈N D ( x ) is the stability threshold . To avoid irrelevanttechnicalities, we will consider only complete neighborhoods N , that is to say such that span + ( N ) = Z d , (1)where span + ( N ) is the set of positive integer linear combinations of cell coordinates from N . Moreover, remark that, for simplicity sake, we assumed that 0 d / ∈ N i.e. cells do notbelong to their own neighborhoods. Indeed, allowing 0 d ∈ N would only correspond tohaving irremovable grains in each cell.The dynamics associated with a sandpile model is the parallel application of thefollowing local rule: if a cell has at least θ grains, then it redistributes θ of its grainsto its neighborhood x + N , according to the distribution D . More formally, denote F : N Z d → N Z d the global rule which associates any configuration c ∈ N Z d with aconfiguration F ( c ) ∈ N Z d defined as follows ∀ x ∈ Z d : (cid:0) F ( c ) (cid:1) ( x ) = c ( x ) − θ H ( c ( x ) − θ ) + (cid:88) y ∈N D ( y ) H ( c ( x + y ) − θ ) (2)where H ( n ) equals 1 if n ≥
0, and equals 0 otherwise (the classical Heaviside function).From the Equation (2), it is clear that the knowledge of the distribution D suffices tocompletely specify the dynamics since from the domain of D one can deduce N andthe dimension d , and from D one finds θ . However, we shall prefer to provide explicitly (cid:104)N , D , θ (cid:105) , at least in this introductory material.The system characterized by Equation (2) is number-conserving in the sense the totalnumber of sand grains is conserved along its evolution as stated in the following. Proposition 1 (Number conservation) . For any configuration c ∈ N Z d + , it holds F ( c )) = c )3 ,
0) 1 2 4 0 3 10 1 6 1 4 22 5 0 3 2 01 3 5 5 3 1 F (cid:55)→ F (cid:55)→ Figure 1: An example of sandpile model neighborhood and distribution (left, θ = 6),and two steps of the global rule from a finite configuration (right, outside the picturedregion the configuration is considered as initially empty). Unstable cells are marked.Only the interesting portion of the configuration is drawn. where c ) = (cid:80) x ∈ Z d c ( x ) . Note that c ) could be positive infinity. Figure 1 provides an illustration of a simple sandpile model and its dynamics.
Remark 1.
In this general framework, the Moore sandpile model of dimension d andradius r corresponds to (cid:10) N M , D , (2 r + 1) d − (cid:11) where N M = {− r, . . . , r } d \ (cid:8) d (cid:9) and D is the constant function equal to for any element of its domain. The von Neumannsandpile model of dimension d and radius r corresponds to (cid:104)N VN , D , rd (cid:105) where N VN = (cid:110) ( x , . . . , x d ) ∈ {− r, . . . , r } d | ∃ i : ( x i (cid:54) = 0 and ∀ j (cid:54) = i : x j = 0) (cid:111) (this is the original model of Bak, Tang and Wiesenfeld [2] for d = 2 and r = 1 ). Denote c → c (cid:48) whenever c (cid:48) = F ( c ), and let ∗ → be the reflexive and transitive closureof → . Given a cell x , remark that its neighbors might play different roles. Indeed, onecan distinguish the out-neighbors of x as the set of cells x + N = { y | ( y − x ) ∈ N } fromthe in-neighbors which are x − N = { y | ( x − y ) ∈ N } . Remark 2.
Another even more general way to define sandpile models is on an arbitrarymulti-digraph G = ( V, A ) (where A is a multiset), where each vertex has finite in-degreeand finite out-degree. In this case, configurations c are taken in N V and the local rulewould be: if v ∈ V contains at least d + ( v ) grains ( d + ( v ) is the out-degree of node v )then it gives one grain along each of its out-going arcs. When the graph supportingthe dynamics is not a lattice, sandpile models are also called chip firing games in theliterature [6, 7]. A cell x is stable if c ( x ) < θ , and unstable otherwise. A configuration is stable when all cells are stable, and is unstable if at least one cell is unstable. Remark thatstable configurations are fixed points of the global rule F . From the Equation (2), itis clear that the system is deterministic and therefore given a configuration c and astable configuration c (cid:48) either there is a unique sequence of configurations c = c → c → . . . → c n = c (cid:48) or c (cid:54) ∗ → c (cid:48) . However, one can consider also other types updating policies.The sequential policy consists in choosing non-deterministically a cell from the unstableones and in updating only this chosen cell. Then, repeat the same update policy onthe newly obtained configuration and so on. It is clear that the new dynamics might4e very different from the one obtained from Equation (2). Sandpiles models in whichthe sequential update and the parallel update policies produce the same set of stableconfigurations with the same number of topplings are called Abelian . Recall that theterms firing and toppling are employed to describe the action of moving sand grainsfrom unstable cells to other cells. The stabilization of a configuration c is the process ofreaching a stable configuration. A finite configuration contains a finite number of grains,or equivalently its number of non-empty cells is finite.The topplings counter, usually called shot vector or odometer function in the liter-ature, started at an initial configuration configuration c is a very useful formal tool forthe analysis of sandpiles and it is defined as follows. For all configurations c, c (cid:48) , c (cid:48)(cid:48) if c → c (cid:48) then, ∀ x ∈ Z d : odo ( c, c (cid:48) )( x ) = H ( c ( x ) − θ ) , and if c ∗ → c (cid:48) → c (cid:48)(cid:48) then ∀ x ∈ Z d : odo ( c, c (cid:48)(cid:48) )( x ) = odo ( c, c (cid:48) )( x ) + odo ( c (cid:48) , c (cid:48)(cid:48) )( x )with the convention odo ( c, c )( x ) = 0 for all x ∈ Z d .Let c x (cid:42) c (cid:48) denote application of the sequential update policy at cell x i.e. , one has c x (cid:42) c (cid:48) if and only if c (cid:48) ( y ) = c ( y ) − θ H ( c ( y ) − θ ) if y = xc ( y ) + D ( y − x ) H ( c ( x ) − θ ) if y ∈ x + N c ( y ) otherwise.We also simply denote c (cid:42) c (cid:48) when there exists x such that c x (cid:42) c (cid:48) . Moreover, let ∗ (cid:42) denote the reflexive transitive closure of (cid:42) , and odo (cid:42) ( c, c (cid:48) ) denote the odometer functionunder the sequential update policy (it counts the number of topplings occurring at eachcell to reach c (cid:48) from c ). The Abelian property can be formally stated as follows. Proposition 2.
For any sandpile model, given a configuration c , if c → c (cid:48) then c ∗ (cid:42) c (cid:48) .Moreover, if c ∗ → c (cid:48) and c (cid:48) is a stable configuration, then1. c ∗ (cid:42) c (cid:48) ,2. c (cid:54) ∗ (cid:42) c (cid:48)(cid:48) for any other stable configuration c (cid:48)(cid:48) ,3. odo ( c, c (cid:48) ) = odo (cid:42) ( c, c (cid:48) ) . We stress that Proposition 2 is an important feature in our context. Indeed, itstates that the (non-deterministic) sequential policy always leads to the same stableconfiguration as the parallel policy, with exactly the same number of topplings at eachcell. Relaxing this requirement deeply changes the dynamics and the structure of thephase space (it has no more a lattice structure for example). For non-abelian models seefor example [30, 27].Endowing the set of configurations with binary addition + (given two configurations c, c (cid:48) , ( c + c (cid:48) )( x ) = c ( x ) + c (cid:48) ( x ) for all x i.e. grain content is added cell-wise), N Z d is acommutative monoid. It is also the case of the set of stable configurations, where the5ddition is defined as addition followed by stabilization. The famous Abelian sandpilegroup of recurrent configurations appears when a global sink is added to the multi-digraph supporting the dynamics. This subject goes beyond the scope of the presentsurvey, for more see [24, 23]. Given a sandpile model (cid:104)N , D , θ (cid:105) , the basic prediction problem asks if a certain cell x will become unstable when the system is started from a given finite initial configuration c . More formally, Prediction problem (
PRED ). Input: a finite configuration c ∈ N Z d and a cell x ∈ Z d . Question: will cell x eventually become unstable during the evolution from c ?One of the most intriguing features of sandpiles is that they are a paradigmaticexample of self-organized criticality . Indeed, starting from an initial configuration andadding grains at random positions, the system reaches a stable configuration c fromwhich a small perturbation (an addition of a single grain at some cell) may trigger anarbitrarily large chain of reactions commonly called an avalanche . The distribution ofsizes of avalanches (when grains are added at random) follow a power law []. We areinterested in the computational complexity of deciding if a given cell x will topple duringthis process. More formally, one can ask the following. Stable prediction problem ( S - PRED ). Input: a stable finite configuration c ∈ N Z d , two cells x, y ∈ Z d . Question: does adding one grain on cell y from c trigger a chain of reactionsthat will eventually make x become unstable?A variant of S - PRED is obtained when the cell y is fixed from the very beginning.When y is the lexicographically minimal cell ( i.e. the cell of the finite configuration withthe lexicographically minimal coordinates), one has the following. First column stable prediction problem ( st col - S - PRED ). Input: a stable finite configuration c ∈ N Z d and a cell x ∈ Z d . Question: does adding one grain on the (lexicographically) minimal cell of c trigger achain of reactions that will eventually make x become unstable?Adding the grain at an extremity of the configuration (the lexicographically minimal cell) implies a strong monotonicity of the dynamics, which has been especially useful inone-dimensional proofs of NC ness (see Subection 2.3 and Proposition 4)). st col - S - PRED has also been called the
Avalanche problem .Finally, one can simply ask how hard it is to compute the stable configuration reached6hen starting from a given finite initial configuration. In other words,
Computational prediction problem (
Compute - PRED ). Input: a finite configuration c ∈ N Z d . Output: what is the stable configuration reached when starting at c ?Before stepping to the detailed study of the complexity of the prediction problemsseen above, one shall discuss about the input coding and the input size. Indeed, it isconvenient (at the cost of a polynomial increase in size) to consider that input configu-rations c are given on finite d dimensional hypercubes of side n placed at the origin, letus call them elementary hypercubes (they have a volume of n d cells). We also assumethat the number of sand grains stored at each cell of a configuration is strictly smallerthan 2 θ , which is a constant, as this is an invariant (see Proposition 3) that • allows to consider constant time basic operations, • preserves the “dynamical complexity”, within P and with P -hard problems. Un-bounded values bring considerations of another kind (namely, of computing fixedpoints from a single column of sand grains, as in [45, 46, 47, 58]) not necessary tocapture the intrinsic complexity of the problem.Cell positions are also admitted to be given using O (log( n d )) bits which is o ( n d ) (seeLemma 3 for a polynomial bound on the most distant cell that can receive a grain). Thetotal size of any input is therefore O ( n d ), the total number of cells. Proposition 3.
For all c ∈ N Z d and x ∈ Z d , if c ( x ) < θ then F ( c )( x ) < θ .Proof. In one time step a cell x gains at most θ grains (if all its in-neighbor topple), andit looses θ grains if c ( x ) ≥ θ because it is unstable. For convenience sake, denote by { y } the indicator function of cell y ∈ Z d , so that addingone grain to cell y of a configuration c ∈ N Z d translates into considering the configuration c + { y } .The notion of avalanche naturally arises in the dynamics of sandpiles. It representsthe chain of reactions which originates from some grain addition to a stable configuration. Definition 1.
Given a stable configuration c ∈ N Z d and an index y ∈ Z d , the avalanchegenerated by adding one grain at cell y of c is given by odo ( c + { y } , c (cid:48) ) where c ∗ → c (cid:48) and c (cid:48) is a stable configuration. Avalanches are especially related to S - PRED and st col - S - PRED where only one grainis added to a stable configuration. In order to study the dynamics of avalanches, it isuseful to consider sequential iterations and to introduce a canonical sequence of cell top-plings (recall that under the sequential update policy, the system is non-deterministic).
Definition 2.
The avalanche process associated with an avalanche odo ( c + { y } , c (cid:48) ) isthe lexicographically minimal sequence ( z , . . . , z t ) such that: c + { y } z (cid:42) c z (cid:42) . . . z t (cid:42) c (cid:48) . emark that t = (cid:80) x ∈ Z d odo ( c + { y } , c (cid:48) )( x ) . The avalanche corresponding to an instance of st col - S - PRED verifies the followingstrong monotonicity property, in the sense that the dynamics is very contrained and theodometer function is incremented at most once at every cell, until a stable configurationis reached.
Proposition 4.
Given a finite configuration c ∈ N Z d within the elementary hypercube,the dynamics starting from c + { d } to a stable configuration topples any cell at mostonce. Formally, for all c (cid:48) such that c + { d } ∗ → c (cid:48) it holds ∀ x ∈ Z d : odo ( c + { d } , c (cid:48) )( x ) ∈ { , } . Proof.
Let z ∈ Z d be one of the chronologically first cells to topple twice, and t and t be the times of the toppling event. Cell z needs all its in-neighbors to topple betweentimes t (included) and t (excluded). If z (cid:54) = 0 d , then at least one in-neighbor z (cid:48) of cell z is fired before it. This is a contradiction since z (cid:48) must topple for a second time before t and z was supposed to be the first cell with that property. If z = 0 d , then we use thefact that c is in an elementary hypercube and the grain addition is done at the origin:since the neighborhood spans the whole lattice (Equation 1) there is an in-neighbor z (cid:48) of z with c ( z (cid:48) ) = 0, which cannot topple unless all its in-neighbors topple before it,and z (cid:48) has an in-neighbor z (cid:48)(cid:48) with c ( z (cid:48)(cid:48) ) = 0 which in its turn cannot topple unless allits in-neighbors topple before it, etc. This leads to an infinite chain of consequencescontradicting the fact that the dynamics converges to a stable configuration in a finitetime (even in polynomial time, see Theorem 1).From the proof of Proposition 4, one can also notice that, for S - PRED , multiple top-plings always originate from cells starting in an unstable state (the sequential statementis stronger).
Proposition 5.
Given an instance ( c, x, y ) of S - PRED , cell y which receives a grain isalways the most toppled cell throughout the evolution from c + { y } . Formally, for all c (cid:48) such that c + { y } ∗ (cid:42) c (cid:48) we have ∀ z ∈ Z d : odo (cid:42) ( c + { y } , c (cid:48) )( y ) ≥ odo (cid:42) ( c + { y } , c (cid:48) )( z ) . This section quickly recalls the main definitions and results in complexity theory thatwill be used in the sequel. For more details, the reader is referred to [40, 43, 60]. P is the class of decision problems solvable in polynomial time by a deterministicTuring machine, or equivalently in polynomial time by a random-access stored-programmachine (RASP, a kind of RAM, i.e. a sequential machine with constant time memoryaccess). For i ∈ N , NC i is the class of decision problems solvable by a uniform family ofBoolean circuits, with polynomial size, depth O (log i ( n )), and fan-in 2, or equivalently intime O (log i ( n )) on a parallel random-access machine (PRAM) using O ( n i ) processors (it8s not important to consider how the PRAM handles simultaneous access to its sharedmemory). For i ∈ N , AC i is the class of decision problems solvable by a non-uniformfamily of Boolean circuits, with polynomial size, depth O (log i ( n )), and unbounded fan-in. NC = ∪ i ∈ N NC i and AC = ∪ i ∈ N AC i . Some hardness results will also employ TC ,the class of decision problems solvable by polynomial-size, constant-depth circuits withunbounded fan-in, which can use and , or , and not gates (as in AC ) as well as thresholdgates (a threshold gate returns 1 if at least half of its inputs are 1, and 0 otherwise). Tocomplete the picture, let us define the space complexity class L which consists in decisionproblems solvable in logarithmic space on a deterministic Turing machine, and NL itsnon-deterministic version. Classical relations among the above classes can be resumedas follows (see [40] for details): NC (cid:40) AC (cid:40) TC ⊆ NC ⊆ L ⊆ NL ⊆ AC ⊆ . . . NC i ⊆ AC i ⊆ NC i +1 · · · ⊆ P . Intuitively, problems in NC are thought as efficiently computable in parallel , whereas P -complete problems (under NC reductions or below) are inherently sequential . Thisdistinction is interesting also in the context of sandpiles: when can we efficiently paral-lelize the prediction?In the PRAM model, processors can write the output of the computation on theirshared memory, and in circuit models, for each input size there is a fixed number ofnodes to encode the output of the computation. We denote A ≤ C B when there is amany-one reduction from A to B computable in C . Remark that reductions in NC (constant depth and constant fan-in) are the most restrictive we may consider: eachbit of output may depend only on a constant number of bits of input (for example itcannot depend on the input size). From a decision problem point of view, the answer toa problem in NC depends only on a constant part of its input. Computing the parityand majority of n bits is not in NC , nor in AC as proved in [31].To give a lower bound on the complexity of solving a problem in parallel, a TC -hardness (under AC reduction) result means that the dynamics is sufficiently complexto perform non-trivial computation, such as the parity or majority of n bits (indeed, TC is the closure of Majority under constant depth reductions). TC -hard problemsare not in AC − (cid:15) for any constant (cid:15) > Open question 1. NC (cid:54) = P ? (It is not even known whether NC (cid:54) = NP or NC = NP .) Open question 2.
Are NC i and AC i proper hierarchies of classes or do they collapseat some level i ∈ N ? The
Circuit value problem ( CVP ) is the canonical P -complete problem (under AC reductions): predict the output of the computation of a given a circuit with identifiedinput gates and one output gate. It remains P -complete when restricted to monotone Majority is the problem of deciding, given a word of n bits, if it contains a majority of ones or not. MCVP ), when restricted to planar circuits (
PCVP ), but not both:
MPCVP is in
LOGCFL ⊆ AC [25, 62].Since the early studies of Banks in [4], the reduction from MCVP is the mostwidespread method (if not the only one) to prove the P -completeness of prediction prob-lems in discrete dynamical systems (in particular for sandpiles). This reduction techniqueis often referred to as Banks’ approach (see Section 5 for applications of Banks’ approachto sandpiles).There are obvious reductions among the decision versions of the prediction problems,giving a hierarchy of difficulties.
Proposition 6. st col - S - PRED ≤ AC S - PRED ≤ AC PRED . The functional version of the prediction problem,
Compute - PRED , seems harder thanjust answering a yes/no question about one cell. It relates a function problem to adecision problem. We propose a clear statement.
Conjecture 1.
PRED ∈ NC Compute - PRED . In other words, the decision problem
PRED can be solved in NC with a Compute - PRED oracle (which is a function problem).
In the opposite direction (relating the complexity of a harder problem to an easierone), Proposition 5 hints at a decomposition of the prediction of an arbitrary sandpileinto a succession of avalanches. However, it is not clear if the hierarchy of problems isproper or if some decision problem are equivalent in terms of computational difficulty.To our knowledge there is no example of a sandpile model for which the computationalcomplexity of any two of these problems would be different.
Open question 3.
Is the hierarchy given in Proposition 6 and Conjecture 1 proper? Inother terms, are there sandpile models such that
Compute - PRED (resp.
PRED , S - PRED )is strictly harder than
PRED (resp. S - PRED , st col - S - PRED )? P The starting point of complexity studies in sandpiles is a paper of Moore and Nils-son in 1999. It gives the global picture [54] (for von Neumann sandpile model and
Compute - PRED ): sandpile prediction is in NC in dimension one, and it is P -completefrom dimension three and above. The two-dimensional case is somewhat surprisinglyopen. Many results appeared since [54] made the picture more precise. In this sectionwe prove that prediction problems for all sandpile models are in P regardless of thedimension, because the sandpile dynamics runs for a polynomial number of steps beforeit stabilizes.One easily gets the intuition that from any finite configuration, the dynamics con-verges to a stable configuration because sand grains spread all over the lattice (Equation1). In [61], Tardos proved a polynomial bound on the convergence time when the graphsupporting the dynamics is finite and undirected (the original bound is 2 vek for a graphconsisting of v vertices, e edges, and having diameter k , when the dynamics converges10o a stable configuration). The proof idea generalizes, not so trivially, using a series oflemma as follows. Lemma 1.
For any finite non-empty set X ⊂ Z d of cells, there exists a bijection λ : X in → X out with X in = { ( y, x ) | y / ∈ X, x ∈ X and y ∈ x − N } (arcs from c X to X ), X out = { ( x, y ) | x ∈ X, y / ∈ X and y ∈ x + N } (arcs from X to c X ),such that λ (( y, x )) = ( x (cid:48) , y (cid:48) ) implies x + x (cid:48) = y + y (cid:48) .Proof. For all ( y, x ) ∈ X in , the bijection λ is defined as λ (( y, x )) = ( x + k ∗ ( x − y ) , x + ( k ∗ + 1)( x − y ))with k ∗ = min { k ∈ N | ( x + k ( x − y ) , x + ( k + 1)( x − y )) ∈ X × c X } . First of all, re-mark that λ is well defined. Indeed, for any edge ( y, x ) in X in , since we are consideringa lattice and since X is finite, there must exist a path containing ( y, x ) and passingthrough some edge ( x (cid:48) , y (cid:48) ) belonging to X out . Then, k is the number of edges between( y, x ) and ( x (cid:48) , y (cid:48) ). Finally, λ is bijective since R = { (( y, x ) , ( x + k ( x − y ) , x + ( k + 1)( x − y ))) | k ∈ Z } is an equivalence relation, and the equivalence classes verify | [( y, x )] R ∩ X in | equals | [( y, x )] R ∩ X out | and is finite, for any y, x (this is the number of times the associatedline enters and exits X ).The following lemma is stronger when expressed in the sequential context. Lemma 2.
For any pair of finite configurations c, c (cid:48) ∈ N Z d such that c ∗ (cid:42) c (cid:48) , and pairof cells x, y ∈ Z d such that y ∈ x + N , it holds | odo (cid:42) ( c, c (cid:48) )( x ) − odo (cid:42) ( c, c (cid:48) )( y ) | ≤ (cid:88) z ∈ Z d c ( z ) . Proof.
Suppose odo (cid:42) ( c, c (cid:48) )( x ) < odo (cid:42) ( c, c (cid:48) )( y ) (the other case is symmetric). Let X = { z ∈ Z d | odo (cid:42) ( c, c (cid:48) )( z ) ≤ odo (cid:42) ( c, c (cid:48) )( x ) } be the set of cells that toppled at most as much as x and define X in , X out and the bijection λ as in the proof of Lemma 1. We have x ∈ X and y / ∈ X therefore ( x, y ) ∈ X out . Then,using λ , one can count the number of grains that moved in and out of X , (cid:88) z ∈ X c (cid:48) ( z ) ≥ (cid:88) ( y (cid:48) ,x (cid:48) ) ∈ X in odo (cid:42) ( c, c (cid:48) )( y (cid:48) ) D ( x (cid:48) − y (cid:48) ) − (cid:88) ( x (cid:48) ,y (cid:48) ) ∈ X out odo (cid:42) ( c, c (cid:48) )( x (cid:48) ) D ( y (cid:48) − x (cid:48) )= (cid:88) ( y (cid:48) ,x (cid:48) ) ∈ X in λ (( y (cid:48) ,x (cid:48) ))=( x (cid:48)(cid:48) ,y (cid:48)(cid:48) ) (cid:104) odo (cid:42) ( c, c (cid:48) )( y (cid:48) ) D ( x (cid:48) − y (cid:48) ) − odo (cid:42) ( c, c (cid:48) )( x (cid:48)(cid:48) ) D ( y (cid:48)(cid:48) − x (cid:48)(cid:48) ) (cid:105) . λ (( y (cid:48) , x (cid:48) )) = ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) implies x (cid:48) + x (cid:48)(cid:48) = y (cid:48) + y (cid:48)(cid:48) , for each term of the last sum wehave D ( x (cid:48) − y (cid:48) ) = D ( y (cid:48)(cid:48) − x (cid:48)(cid:48) ), and by definition of X , one finds 0 ≤ odo (cid:42) ( c, c (cid:48) )( y (cid:48) ) ≥ odo (cid:42) ( c, c (cid:48) )( x (cid:48)(cid:48) ), therefore each term is positive. The result follows because in c (cid:48) therecannot be more grains at cells in X than total number of sand grains in c ( ∗ (cid:42) is number-conserving).The last argument in the proof of Tardos [61] is concerned with the finiteness of theunderlying graph and, of course, it does not apply here. However, it is possible to boundthe region of the lattice that may eventually receive at least one sand grain. Lemma 3.
For any finite configuration c ∈ N Z d belonging to the elementary hypercubeof size n d , if c ∗ → c (cid:48) for some c (cid:48) ∈ N Z d , then c (cid:48) ( x ) = 0 for any | x | ∞ > θrn d , with r = max {| v | ∞ | v ∈ N } .Proof. The configuration c may contain at most (2 θ − n d sand grains since it belongsto the elementary hypercube of size n d . Let us prove that:1. sand grains cannot all leave one place,2. there is no isolated sand grain.Consider an encompassing hypercube R = {− r, . . . , r } d around the neighborhood N .Let us first show that,if for some x ∈ Z d , c ( x ) > ∃ y ∈ x + R such that c (cid:48) ( y ) > . By contradiction, consider any of the cells that where the last to topple inside x + R ,from Equation 1 it must have send at least one grain inside x + R which thus cannot beempty. An analogous argument shows that,if c (cid:48) ( x ) > ∃ y ∈ x + 2 R such that c (cid:48) ( y ) > . We can therefore conclude that in c (cid:48) there is a grain within cells {− r, . . . , n + r } d , andany other grain cannot be at distance greater than (2 θ − n d r in the max norm.Now the previous lemmas can be exploited to fit the argumentation of Tardos [61]in this general framework. Theorem 1.
Given any finite configuration c ∈ N Z d of size n d , a stable configuration isreached within at most (2 dθrn ) O ( d ) time steps, a polynomial in the size of c .Proof. According to Lemma 3, two cells which have toppled cannot be at distance(through a chain of neighbors in the (cid:96) -norm) greater than 8 dθrn d . By repeated ap-plications of Lemma 2, one finds that no cell can topple more than 8 dθrn d (cid:80) z ∈ Z d c ( z )times. Since no more than (8 dθrn d ) d different cells can topple (Lemma 3 again), and thenumber of grains is upper bounded by 2 θ n d , we get precisely 2 d +4 d d +1 θ d +2 r d +1 n d +2 d which is upper-bounded for any d by (2 dθrn ) O ( d ) .Theorem 1 provides an upper bound on all the prediction problems for any sandpilemodel. Corollary 1.
PRED , S - PRED , st col - S - PRED , Compute - PRED ∈ P . NC in dimension one In one dimension, prediction problems on sandpile dynamics have been proven to beefficiently computable in parallel, i.e. they lie in NC . As mentioned in the introductionof Section 3, the whole story began in 1999 with a study of the computation variantof the prediction problems, and it has been successively extended to more restrictivevariants. Theorem 2 ([54], improved in [52]) . For von Neumann sandpile model of radius onein dimension one (see Remark 1),
Compute - PRED is in NC , and it is not in AC − (cid:15) forany constant (cid:15) > . The last part comes from a simple constant depth reduction of the
Majority of n bits problem (given x ∈ B n , decide if there is a majority of one)to sandpile dynamics, proving that the problem is TC -hard. These results rely ona clever technical study of predicting the dynamics of a stripe of 1s containing a single2 (because θ = 2 in this model): a 0 appears within the stripe of 1s which is enlarged,such that the center of mass is unchanged. Generalizing these results seems to be atechnically challenging task, but we conjecture that they do. Conjecture 2.
For any one-dimensional sandpile model, prediction is efficiently com-putable in parallel, i.e.
Compute - PRED , PRED , S - PRED , st col - S - PRED ∈ NC . Three results from the literature support this conjecture, they are expressed on st col - S - PRED and exploit the strong monotonicity of avalanches in this case (Propo-sition 4) to prove that the problem is in NC for a large class of models: • Kadanoff sandpile models ([26] completed in [28]), • extended to any decreasing sandpile model (in [29]). Remark 3.
In dimension one, the radius r Kadanoff sandpile model corresponds to (cid:104)N K , D K , r + 1 (cid:105) with N K = {− , r } and D K ( −
1) = r, D K ( r ) = 1 (see [28] for anillustration). A decreasing sandpile model simply has a neighborhood N such that N ∩ N − = {− } . The name decreasing sandpile model comes from an interpretation of thesand content at each cell as the slope between consecutive columns of sand grains (it addsone artificial dimension to the picture), such that it preserves a monotonous form. Theorem 3 ([26, 28, 29]) . For any one-dimensional decreasing sandpile model, st col - S - PRED ∈ NC . This result can be generalized to any one-dimensional sandpile model (Theorem 4),which supports Conjecture 2. However the generalization of the idea presented in [28, 29]is not straightforward. Decreasing sandpile models have the feature that
N ∩ N − = {− } which, together with the strong monotonicity Proposition 4, implies a pseudo-linear dynamics of the avalanche process (lexicographically minimal sequence of topplingsunder the sequential update policy): using a sliding window of width r = max {| v | ∞ | v ∈N } , one can compute the whole avalanche from cell 0 to the maximal index of a toppled13 Figure 2: Example of avalanche process for a one-dimensional sandpile model (left, θ = 7), which is not pseudo-linear: the avalanche process may topple cells arbitrarilyfar on the right before going backwards to topple cells on the left end (right, instance of st col - S - PRED where arrows depict the avalanche process). x yr r r r Figure 3: When every cell topples at most once, knowing which cells among [ x − r, x − y +1 , y + r ] topple allows to compute which cells among [ x, x +2 r −
1] and [ y − r +1 , y ]topple, where r = max {| v | ∞ | v ∈ N } (Lemma 4).cell. Then, it is possible to precompute the topplings locally (via functions of constantsize telling what happens around cell i + 1, called status at i + 1 in [28, 29], according towhat happens around cell i , i.e. from the status at i ), and compose these informationsaccording to a binary tree of logarithmic height, each level of the composition beingcomputed in constant time, hence resulting in an NC algorithm.This pseudo-linear dynamics does not hold any more for general sandpile models indimension one, as shown in the example in Figure 2. Nevertheless, it is possible to geta similar result with a more involved construction presented below.In dimension one, for any c, c (cid:48) ∈ N Z , let odo ( c, c (cid:48) ) | [ x,y ] denote the restriction of odo ( c, c (cid:48) ) to the interval [ x, y ], with x < y two cells in Z . Lemma 4.
For any one-dimensional sandpile model (cid:104)D , N , θ (cid:105) and any configurations c, c (cid:48) such that c is within the elementary hypercube, c ∗ → c (cid:48) and c (cid:48) is stable, with r =max {| v | ∞ | v ∈ N } , knowing • odo ( c, c (cid:48) ) | [ x − r,x − and odo ( c, c (cid:48) ) | [ y +1 ,y + r ] for cells x, y ∈ Z such that x + r ≤ y , allows to compute • odo ( c, c (cid:48) ) | [ x,x +2 r − and odo ( c, c (cid:48) ) | [ y − r +1 ,y ] in time O ( y − x ) on one processor. Figure 3 provides a graphical illustration of the statement of Lemma 4.
Proof.
If one knows all topplings that may influence the topplings within [ x, y ] (givenby the assumptions odo ( c, c (cid:48) ) | [ x − r,x − and odo ( c, c (cid:48) ) | [ y +1 ,y + r ] , where r is the radius ofthe sandpile model), it is possible to compute all topplings occurring within [ x, y ]. Fur-thermore, from Proposition 4, any cell topples at most once and hence the number oftopplings is upper bounded by y − x , which in its turn is an upper bound on the numberof computation steps. The condition x + r < y ensures that the output is made of odo ( c, c (cid:48) ) values within the interval [ x − r, y + r ].14 heorem 4. For any one-dimensional sandpile model, st col - S - PRED ∈ NC .Sketch. Let us describe how to derive an NC algorithm for st col - S - PRED from Lemma 4.Let c, x be the instance, and c (cid:48) be the stable configuration such that c ∗ → c (cid:48) . For simplicitysake, assume that c is c + { } , and let r = max {| v | ∞ | v ∈ N } .1. Compute, in parallel for every y ∈ N multiple of r , the function taking as input odo ( c, c (cid:48) ) | [ y − r,y − and odo ( c, c (cid:48) ) | [ y +4 r,y +5 r − , and outputting odo ( c, c (cid:48) ) | [ y,y +2 r − and odo ( c, c (cid:48) ) | [ y +2 r,y +4 r − (see picture below for a partial example with some y multiple of r and z = y + 4 r , on two copies of the configuration for clarity). FromLemma 4, each computation needs a constant time on one processor, and there arelinearly many. y y + r y +2 r y +3 r y +4 rr r r r z z + r z +2 r z +3 r z +4 rr r r ri i i i i i i i i i
2. Compose them according to some binary tree: each function is of constant size(from Proposition 4, input is 2 r bits and output is 4 r bits), and we compose func-tions having some carefully chosen overlap. Let us denote i , i , i , i , i , i , i , i , i , i the respective portions of odo ( c, c (cid:48) ) of size r under consideration (see pictureabove), then the two functions are respectively: i , i (cid:55)→ i , i , i , i and i , i (cid:55)→ i , i , i , i . With these two we can compute i , i , i , i (cid:55)→ i , i , i , i , i , i , i , i ,and fortunately the fixed point regarding i , i is uniquely determined by i , i ,according to a constant time application of Lemma 4 (remark that there may bean arbitrarily large gap between i and i , and also between i and i ). Each com-position deals with a constant number of functions of constant size, hence it takesa constant time. As there is a logarithmic number of levels in the composition, theoverall process takes a logarithmic parallel time.Now remark that no cell within [ − r, −
1] nor [ n + 1 , n + r ] topple, with n the size ofconfiguration c , hence odo ( c, c (cid:48) ) | [ − r, − and odo ( c, c (cid:48) ) | [ n +1 ,n + r ] are fixed. In order to getthe answer of whether cell x = kr + k (cid:48) topples (for some unique k, k (cid:48) ∈ N with 0 ≤ k (cid:48) < r ),we consider two binary trees of compositions: one such that the root gives the functionwith input [ − r, − , [ kr, ( k + 1) r − k − r, kr − , [ n + 1 , n + r ]. The fixed point of their conjunction, resolvedwith the function with input [( k − r, ( k − r − , [( k + 1) r, ( k + 2) r −
1] (again aconstant time application of Lemma 4), tells whether odo ( c, c (cid:48) )( x ) equals 0 or 1 (seepicture below). 15 ( k − r ( k − r kr ( k +1) rr r x ( k − r kr ( k +1) r ( k +2) rr r As mentioned above, the generalization of the Theorem 4 to other prediction prob-lems requires non-trivial extensions, because of multiple topplings at a cell which some-how should be handled in constant time computation. The proof of TC -hardness from[52] also makes heavy use of multiple topplings, and as a consequence it does not general-ize for free to an arbitrary one-dimensional sandpile model. We nevertheless conjecturethat it also does. Conjecture 3.
For any one-dimensional sandpile model,
PRED is TC -hard for AC reductions, and therefore not in AC − (cid:15) for any constant (cid:15) > . P -completeness in dimension three and above During his PhD thesis in the 1970s, Banks started to implement circuit computation us-ing discrete dynamical systems working on grids (namely cellular automata) [4]. The in-tuition behind such implementations is quite straightforward: a sequence of cells changestate in a chain of reactions to transport information; two flows of information caninteract to create logic gates.For simplicity, circuits are restricted to have the following characteristics: • gates have fan in and fan out 2, • layered (information flows from one end to the other, layer by layer from inputs tooutput, without going backward), • arranged on a grid layout.Even with the previous constraints the prediction problems ( SAM2CVP in [40]) are still P -complete. Moore and Nilsson ported Banks’ technique to sandpile models in 1999[54]. They used an adaptation to the grid of the computation encoding idea developedby Bitar, Goles and Margenstern in [5, 34]. Theorem 5 ([54]) . For von Neumann neighborhood N VN of radius one and dimension d ≥ , the problem st col - S - PRED is P -complete.Sketch. The reduction is from
MCVP (see Section 2.4), in the case of fan in and fan out2, layered, on a grid. We will present the proof for dimension 3 in details, and explainat the end how it generalizes.There are different types of gadgets to implement within sandpiles: wires, turns, and gates, or gates, plus diodes (to prevent unintended backward propagation of information)and multipliers (to get fan out 2). All these elements are presented as macrocells in16igure 4 (two top rows), and are embedded in a two-dimensional plane of the three-dimensional configuration. The non-planarity of the circuit requires (recall that MPCVP is in NC , hence it is important that the circuit may not be planar) that wires cross eachother, which is achieved using the third dimension (Figure 4 bottom row).Now the idea is that by replacing circuit elements with macrocells we can have asingle grain addition triggering a wire that can be multiplied to implement the inputconstants on the first layer, which can be connected to gate on the second layer, etc ,until the output gate which is simply a wire with the questionned cell in the center. Asnoted in [53], diodes should be added between layers to prevent backward propagationof information (specificaly to prevent false 1).To finish the reduction, we would like to insist on the fact that crossing of wires inthe third dimension is only necessary to overcome the non-planarity of MCVP . In fact,it is equivalent to have a crossing or a negation gate: • In [40] on
PCVP : “
A planar xor circuit can be built from two each of and , or , and not gates; a planar cross-over circuit can be built from three planar xor circuits ”. • In [54] Moore and Nilsson wrote: “
Using a double-wire logic where each variablecorresponds to a pair of wires carrying x and ¯ x , [Bitar, Goles and Margenstern]implement negation as well by crossing the two wires ” .We prefer to imagine a single-wire logic, with cross-over of wires in the third dimension.Cell topplings correspond to the circuit computation (truth value 1 transmitted fromlayer to layer, going through gates), and the circuit outputs 1 if and only if the questionedcell topples.This reduction is performed in constant parallel time (in AC ): each constant sizepart of the circuit is converted to a sandpile sub-configuration (macrocell) of constantsize placed at a fixed position, hence in a PRAM model each processor can handle onesuch part (there are polynomially many) in constant time.In order to generalize the proof to any dimension d ≥
3, simply remark that the sameconstruction can be embedded in only three dimensions among many, with coordinatezero in all other dimensions, provided one adapts the sand content of non-empty cellsaccording to θ (plus two grains for any dimension above three).From the proof above we can notice that to simulate a circuit (instance of MCVP ),von Neumann sandpile model of radius one and dimension d ≥ • only three dimensions are used (mainly two), • only cell contents 0, θ − θ − • the strong monotonicity of Proposition 4 holds (any cell topples at most once), • we have wires and gates organized in successive layers from inputs to output, • all directions of information are known in the reduction, • there are diodes everywhere so that information flows in exactly one direction. interestingly, the next sentence in this quote is: “ Since this violates planarity, negation does notappear to be possible in two dimensions ”. → → → → ↑ wire (0 or idle) wire (transmits 1) turn → ↓ → → ↓ → → → → ↑ → and gate or gate diode multiplier → a → a ↓ b ↓ b
555 5 0 5 555 555 cross-over cross-over cross-over( − rd dimension) (0 in 3 rd dimension) (1 in 3 rd dimension)Figure 4: Implementation of wires and gates with von Neumann neighborhood of radiusone, in three dimensions ( θ = 6). Only the cross-over gate uses the third dimension (itshould be thought as three stacked layers), all other gates can be considered to live atcoordinate 0 in the third dimension. A wire is implemented as a simple chain of reactionsof cell topplings: when it is triggered it transmits a 1, if not ( i.e. it remains stable) ittransmits a 0. Arrows indicate fan in and fan out (input and output pairs are identifiedin the cross-over gate: a signal coming from input a (resp. b ) results in a signal goingto output a (resp. b ), independently of each other), empty cells contain no sand grain.Remark that appart from central cells of cross-overs, initialy empty cells will not topple.18s a consequence we can consider that the evolution of the obtained st col - S - PRED instance ( c, x ) verifies very restrictive conditions. Let us consider that there is a diodebetween every pair of gates presented on Figure 4. Then the flow of information betweengates and layers is totally fixed, and we can almost claim that for any pair of neighbouringcells it is known which one would topple first (in the case both topple). However thisis not completely accurate, as in or gates for example: if only one of the two inputstransmits a 1 signal, then some cells going to the other input topple “backward”. This isclearly not an issue nor an important feature. We formalize how another sandpile modelwith d ≥ Lemma 5.
If a sandpile model of neighborhood N has three linearly independent x, y, z ∈N such that ax + by + cz / ∈ N for any a, b, c ∈ Z , except when: a = ± , b = 0 , c = 0 or a = 0 , b = ± , c = 0 or a = 0 , b = 0 , c = ± , then it has a P -complete st col - S - PRED problem.Proof.
Let us add two modifications to the circuit simulation of von Neumann radiusone in three dimensions presented in the proof of Theorem 5, so that it fits the presentcontext. First, up to a straightforward layout shift, we can embed all planar gates fromFigure 4 with only two directions of information transmission: x and y . Second, thethird direction z is used for the cross-over, which shifts in this direction the rest of thesandpile implementation of the circuit (see Figure 5). We can simply assume that thereis one cross-over per layer to fix the z -shift everywhere. These coordinate changes donot change the computational complexity of the problem.A sandpile model with such x, y, z can simulate at a local level von Neumann of radiusone in three dimensions (neighbours are given by x, y, z ), itself simulating a circuit as inFigure 5. Indeed, information flows in at most one direction ( x, y, z ∈ N implies thatinformation flows in the expected direction, and the reverse direction may or may notbe in N , as it is nor important nor an issue), and in other cases transmissions do notinterfere one with another from the condition that ax + by + cz / ∈ N .Hence taking cells from the three-dimensional grid generated by { x, y, z } , and placing: • θ − D ( u ) grain with u ∈ { x, y, z } for cells with 5 grains in von Neumann ( θ − u , or two in-neighbors andthe minimum number of sand grains received from one of them, • θ − D ( u ) − D ( v ) grains with u, v ∈ { x, y, z } for cells with 4 grains in von Neumann( θ − u, v , • no grain for empy cells in von Neumann,the sandpile model can simulate von Neumann radius one in three dimensions, itselfsimulating a circuit. The transformation is performed in constant parallel time, AC .It follows that all d -dimensional sandpile models with d ≥ P -complete to predict. 19 nputs diodes gates diodes gates diodes gates101 ∨∧∧ ∧∧ ∨ ? x yz Figure 5: von Neumann radius one in three dimensions simulating a layered circuit, withinformation flowing in three directions which will correspond to x, y, z . Light coloredcubes contain 5 grains ( θ − θ − { } d and the questioned cell (they both contain 5 = θ − ×
5, and a z -shift of 2 units. Arrows indicate thedirection of information flow. 20 orollary 2. st col - S - PRED , S - PRED and
PRED are P -complete for any sandpile modelin dimension d ≥ .Proof. Let M = (cid:104)N , D , θ (cid:105) be a sandpile model in dimension d ≥
3, and let (cid:107) u (cid:107) denotethe Euclidean norm ( (cid:96) -norm) of cell u ∈ Z d . We define x, y, z as follows. • x is a cell inside N of maximal norm, i.e. x ∈ arg max u ∈N {(cid:107) u (cid:107)} . • y is a cell inside N of maximal norm when projected onto the ( d − { } d , x , i.e. y ∈ arg max u ∈N {(cid:107) p x ⊥ ( u ) (cid:107)} with p x ⊥ the projection onto { v ∈ Z d | v · x = 0 } . • z is a cell inside N of maximal norm when projected onto the ( d − { } d , x, y , i.e. z ∈ arg max u ∈N {(cid:107) p xy ⊥ ( u ) (cid:107)} with p xy ⊥ the projection onto { v ∈ Z d | ∀ a, b ∈ Z : v · ( ax + by ) = 0 } .From Equation (1) such x, y, z exist and are non-colinear.Furthermore, it is always possible to choose x, y, z such that ax + by + xz / ∈ N for a, b, c ∈ Z different than in the statement of Lemma 5. Indeed, a linear combinationwith two non-null components such that ax + by + xz ∈ N either gives another vector ofmaximal norm that may replace one of y, z (if the projection onto x ⊥ or xy ⊥ is negativeon all components, then it becomes positive on one component), or contradicts themaximality of x , y or z (if the projection onto x ⊥ or xy ⊥ is positive on one component).From Lemma 5 we can conclude that st col - S - PRED is P -complete, and Proposition 6implies that S - PRED and
PRED are also P -complete.Writting formal proofs of P -completeness via reduction from some CVP problemrequires a substantial amount of precisions, and some obvious details are often notmentionned, though they may be key in other contexts (such as the symmetry of vonNeumann neighborhood and the dynamics of layers equipped with diodes in the proofof Theorem 5). In [38] the authors present a general framework to prove P -completenessusing Banks’ technic in cellular automata. A neat formalism is introduced, defining whatis meant by “ simulating a gate set ” . Then results are presented, of the form: if a cellularautomaton can simulate a set of gates A (for example A = { and , or , cross-over } ), then itsprediction problem is hard for some class C (in this example C = P ). A novel distinctionappears to be fundamental for the dynamical complexity of discrete dynamical systems: in a nutshell: a macrocell must be in some valid state, and depending on the state of its neighboringmacrocells, change to another valid state, within some common delay so that the simulation remainssynchronised. Valid states ensure that nothing unexpected happens. re-usable simulation of wires and gates, or not ( weaksimulation ). Indeed, re-usable simulation brings P -completeness from and and or gatesonly, because it is possible to build a planar cross-over gadget with re-usable simulation ofmonotones gates. This result is surprising compared to the characterization of Booleangates allowing planar cross-over presented in [51], which is not the case of any set ofmonotone gates. It comes from the dynamical nature of circuit simulation with discretedynamical systems, and the possibility to re-use wires that is not present in the originalcircuit model.In the context of sandpile models, circuit simulation in two dimensions is harder toachieve precisely because of the difficulty to create cross-over (see Section 6). The planarcross-over gadget in re-usable simulation seems however difficult to apply to sandpilemodels, since it exploits delays in signal transmissions, which is in contradiction withthe Abelian Proposition 2 telling that order of topplings do not matter. No two-dimensional sandpile model is known to be efficiently predictable in parallel, i.e. such that its prediction problem is in NC . As we will see, some slight extensionsof von Neumann sandpile model of radius one turn out to have P -complete predictionproblems, but the computational complexity of predicting the original model of Bak,Tang and Wiesenfeld [2] remains open. This is considered as the major open problemregarding the complexity of prediction in sandpile models. It is also open for Mooreneighborhood of radius one. Open question 4.
Consider the von Neumann and Moore sandpile models in dimensiontwo. Are
PRED , S - PRED and st col - S - PRED in NC , P -complete, or neither? The third possibility comes from the fact that under the assumption NC (cid:54) = P , thereexist problems in P that are neither in NC nor P -complete [59]. The difficulty in applyingBanks’ approach here is to overcome planarity imposed by the two-dimensional grid withvon Neumann or Moore neighborhood, since the monotone planar circuit value problem( MPCVP ) is in NC . In fact, it has been proven to be impossible to perform elementaryforms of signal cross-over in this model [32]. The precise statement is a bit technical tostate , but corresponds neatly to the intuition of having two potential sequences of celltopplings representing two wires that may convey a bit of information, and cross eachother without interacting ( i.e. they independently transport information). We shall callthese elementary forms of signaling since other ways to encode the transportation ofinformation in sandpile may be found, but as emphasized by Delorme and Mazoyer in[15] to quantify over such possible encodings and give fully general impossibility results,is an issue. Theorem 6 ([32]) . It is impossible to perform elementary forms of signal cross-over invon Neumann and Moore neighborhoods of radius one. because an impossibility result requires to define in full generality what is considered as a cross-over. Theorem 7 ([26, 32]) . In two dimensions, von Neumann neighborhood of radius r ≥ has P -complete prediction problems st col - S - PRED , S - PRED and
PRED . It is also thecase for the non-deterministic Kadanoff sandpile model of radius r ≥ . In dimension two, the radius r Kadanoff sandpile model is defined in [26] as the non-deterministic application of the one-dimensional Kadanoff sandpile model (see Remark3) in the two directions of the plane, plus a monotonicity property that needs to bepreserved, and the prediction problem asks for the existence of an avalanche reachingthe questioned cell (in the circuit implementation any cell topples at most once, henceit corresponds to st col - S - PRED ).Augmenting a little bit the neighborhood allows to perform cross-over and simulate
MCVP instances. More has been said in [57] on cross-over impossibility, and as animmediate corollary we have that elementary forms of signal cross-over are impossiblewhen the graph supporting the dynamics is planar (which is the case for von Neumannof radius one, but not of greater radii).
Lemma 6 ([57]) . If a sandpile model can implement an elementary form of cross-over ,then it can implement one such that the two elementary signals do not have any cell incommon. This is true on any Eulerian digraph.
Also, the fact that a neighborhood can or cannot perform cross-over (given thedistribution D sending 1 grain to each out-neighbor) is intrinsically discrete. Indeed,if one thinks about the shape of some neighborhood as a continuous two-dimensionalregion that we can scale and place on a grid to get a (discrete) neighborhood, thenimpossibility to perform cross-over cannot be characterized in terms of shape. Theorem 8 ([57]) . Any shape can perform cross-over starting from some scaling ratio.
Even a circle, with a big enough scaling ratio, gives a neighborhood that can performcross-over and have a P -complete prediction problem.The precise conditions for P -completeness of the prediction problems in two-dimensionsare still to be found. Having a precise characterization would be of great interest to shedlight on the universality of Banks’ approach: if a sandpile model has a P -complete pre-diction problem, then for sure it can simulate circuits. But are there ways of doing sowith non-elementary forms of signaling ?In [54], after recalling that MPCVP (monotone planar
CVP ) is in NC , Moore and Nils-son gave insights on the possibilities to reduce circuit simulation to sandpile dynamicsfor von Neumann radius one in two dimensions: “ two-dimensional sandpiles differ fromplanar Boolean circuits in two ways. First, connections between sites are bidirectional.Secondly, a given site can change a polynomial number of times in the course of the sand-pile’s evolution. Thus we really have a three-dimensional Boolean circuit of polynomialdepth, with layers corresponding to successive steps in the sandpile’s space-time .”23et us conclude this section with a lower bound on the complexity of two-dimensionalsandpile models. As noticed by Miltersen in [52], a corollary of Theorem 5 is that thetwo-dimensional von Neumann sandpile model of radius one is NC -hard since it can sim-ulate monotone planar circuits (an MPCVP instance can be reduced to a st col - S - PRED instance with an AC algorithm, without implementing cross-over), and MPCVP is NC -hard since evaluating a Boolean formula (for which the circuit is a tree) is NC -complete[8]. From Lemma 5 and Corollary 2 this observation extends to any two-dimensionalsandpile model, since the third dimension is used exclusively for cross-over gates. Theorem 9.
For any two-dimensional sandpile model, st col - S - PRED , S - PRED and
PRED are NC -hard. As seen in the previous sections the prediction problems for sandpile models are di-mension sensitive but once the dimension is fixed they turn out to be closely related toone another. This section provides a notion of simulation between sandpile models andshow that simulating a run of a sandpile model by another with different parameters(neighborhood or distribution) has a polynomial cost. Only sequential update policy isconsidered in this section. Moreover, all finite configurations mentioned are intended tobelong to the elementary hypercube. For this reason those hypothesis is omitted fromthe statements.Given a configuration c ∈ N Z d , a firing sequence for c is a sequence ( x , x , . . . , x n ) ∈ Z d of cells such that there exist configurations c = c, c , . . . , c n such that c i x i (cid:42) c i +1 forall i ∈ { , . . . , n − } and1. c ( x ) = θ − x i +1 − x i ∈ N for all i ∈ { , , . . . , n − } ;3. c i ( x i ) = θ for i ∈ { , . . . , n } .Remark that when a firing sequence has been triggered i.e. when a sand grain has beenadded at x in c , grains may be distributed on set of cells which are in the neighborhoodof some element of the sequence. The hitting set collects precisely this information. Moreformally, the hitting set induced by a firing sequence ( x , x , . . . , x n ) ∈ Z d for a finiteconfiguration c is the set of pairs ( x, u ) where x ∈ Z d is a cell and u ∈ N \ { } is the totalnumber of grains that x has received when all the cells of the firing sequence have beenfired. An N -path from the cell x ∈ Z d to y ∈ Z d is an sequence ( z , z , . . . , z n ) of cellssuch that z = x , z n = y and z i +1 − z i ∈ N for all i ∈ { , . . . , n − } . Remark that if N is complete, then there always exists an N -path between any pair of cells. Given a finiteconfiguration c , let V ( c ) ⊂ Z d be the convex hull of points x ∈ Z d such that c x (cid:54) = 0.A detector cell y for a finite configuration c with hitting set H is a cell such that thereexists an integer u > y, u ) ∈ H and y / ∈ V ( c ). In other words, a detectorcell for a finite configuration c is a cell which is “outside” c and which may receive asand grain if some cell z “inside” c is triggered. Of course, if given a finite configuration24 , for all x ∈ Z d we have that c + x is stable, then c has no detector cells. Figure 6illustrates all these recent notions.
11 21 1cell at(0 , F (cid:55)→ F (cid:55)→ F (cid:55)→ F (cid:55) → F (cid:55)→ Figure 6: Sandpile model with θ = 6, neighborhood and distribution function as depictedon bottom left. The evolution starts from the finite configuration on top left (cells notdrawn are supposed to contain 0). Dark-grayed cells (taken in the order indicated by (cid:55)→ ) are the firing sequence. In the final stable configuration, cells in the hitting set havetheir content in bold face, while detector cells have a light-gray background.Given two sandpile models M ≡ (cid:104)N , D , θ (cid:105) and M ≡ (cid:104)N , D , θ (cid:105) , we say that M simulates M if there exist a computable transformation h : N Z d → N Z d such thatfor all finite configurations c, c (cid:48) ∈ N Z d and all pair of cells x, y ∈ Z d such that1. c, c (cid:48) are stable for M ;2. c + x ∗ (cid:42) c (cid:48) according to M ;3. y is a detector cell for c ;and there exists a finite configuration c (cid:48)(cid:48) such that1. h ( c ) , c (cid:48)(cid:48) are stable for M ;2. h ( c ) + x ∗ (cid:42) c (cid:48)(cid:48) according to M ;3. if c (cid:48) ( y ) > c (cid:48)(cid:48) ( y ) > M simulates M if starting on a computable encoding of an initialconfiguration c of M the stable configuration which is reached afterwards contains thesame bit of information in the detector cell y as M . Lemma 7.
Consider two complete neighborhoods N , N ⊂ Z d such that N = N ∪ { u } for u ∈ Z d . For any sandpile model M ≡ (cid:104)N , D , θ (cid:105) , there exists a sandpile model M ≡ (cid:104)N , D , θ (cid:105) which simulates M .Proof. Let c ∈ N Z d be a stable configuration for M and choose x ∈ Z d . Let c (cid:48) be suchthat c + x ∗ (cid:42) c (cid:48) according to M and let y be a detector cell for c (cid:48) . Let F be the firing25equence F ≡ ( x , . . . , x n ) associated with c + x and let H be the induced hittingset. The idea is to take the same firing sequence also for the model M that we aregoing to define but to craft the configuration c (cid:48) realizing the circuit for M in such a wayto prevent unnecessary supplementary firings or before-time firings because of grainsdropped forward by the u component of the neighborhood. Hence, let F = F be thefiring sequence for the model M . Define θ = θ + 1 and ∀ v ∈ N , D ( v ) = (cid:40) D ( v ) if v ∈ N c (cid:48)(cid:48) as follows ∀ z ∈ Z d , c (cid:48)(cid:48) ( z ) = θ − z = xθ − v if z ∈ F and ( z, v ) ∈ H c (cid:48)(cid:48)(cid:48) be the stable configuration such that c (cid:48)(cid:48) ∗ (cid:42) c (cid:48)(cid:48)(cid:48) according to M . It is clear that c (cid:48) ( y ) > c (cid:48)(cid:48)(cid:48) ( y ) >
0. Indeed, if c (cid:48) ( y ) >
0, then there exists x i ∈ F = F suchthat y − x i ∈ N ⊂ N . Hence, c (cid:48)(cid:48)(cid:48) ( y ) > Lemma 8.
Consider a neighborhood
N ⊆ Z d (not necessarily complete) and a sandpilemodel M ≡ (cid:104)N , D , θ (cid:105) . Define the model M ≡ (cid:104)N , D , θ + k (cid:105) for some k ∈ N + andsuch that1. there exists a unique u ∈ N for which D ( u ) = D ( u ) + k ;2. ∀ v ∈ N , ( u (cid:54) = v ) ⇒ D ( v ) = D ( v ) .Then, M simulates M . Lemma 9.
Consider a neighborhood
N ⊆ Z d (not necessarily complete) and a sandpilemodel M ≡ (cid:104)N , D , θ (cid:105) . Define the model M ≡ (cid:104)N , D , θ (cid:105) such that1. there exists a unique u ∈ N for which D ( u ) = D ( u ) − k > for some k ∈ N + ;2. ∀ v ∈ N , ( u (cid:54) = v ) ⇒ D ( v ) = D ( v ) ;3. θ = θ − k .Then, M simulates M . Remark that from a computational complexity point of view, the constructions of theprevious lemma come at no cost. However, each single simulation has a cost which es-sentially consists in computing the firing sequence for the original system (the hitting setcan be computed at a constant multiplicative cost while computing the firing sequence).By Theorem 1, this can be done in polynomial time (using a stack for example).The notion of simulation between sandpile models induces a preorder structure onthe set of sandpile models in the same dimension. It is an interesting research directionto explore the properties of such an order and see if and under which form there existsa notion of universality. 26
Undecidability on infinite configurations
A further generalisation of sandpile models consists in relaxing the finiteness of theinitial configuration. As noted in [9], allowing any configuration initially written on thetape of the Turing machine would make the prediction problem trivially undecidablein any dimension, and restricting to periodic configurations comes down to studying afinite region on a torus and is therefore decidable in any dimension (since given a fixednumber of sand grains there are finitely many configurations). On the other hand, whenthe initial configuration given as input to the prediction problem is ultimately periodic ( i.e. periodic except on a finite region), then the following result holds. Theorem 10 ([9]) . PRED extended to ultimately periodic configurations (the input con-sists in the finite non-periodic region, plus the finite periodic pattern repeated all around)in dimension three is undecidable.
Finally, if we further relax the number-conservation property and allows the distri-bution functions to “eat” grains, then one obtains sand automata [12]. Without goinginto the details of their precise definition (which is a bit involved), we can just recallthat they are a special type of cellular automata particularly adapted to the sandpile“playground” [21, 20]. The following result is interesting in our context.
Theorem 11 ([13, 14]) . Ultimate (temporal) periodicity is undecidable for sand au-tomata on finite configurations.
This last result tells that (somewhat unsurprisingly) sand automata are highly un-predictable.
Acknowledgments
The authors thank the
Young Researcher project ANR-18-CE40-0002-01 “FANs”, theproject ECOS-CONICYT C16E01, the project STIC AmSud CoDANet 19-STIC-03(Campus France 43478PD).
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