Shift-Symmetric Configurations in Two-Dimensional Cellular Automata: Irreversibility, Insolvability, and Enumeration
aa r X i v : . [ n li n . C G ] M a r Shift-Symmetric Configurations in Two-Dimensional Cellular Automata:Irreversibility, Insolvability, and Enumeration
Peter Banda ∗ Luxembourg Centre For Systems BiomedicineUniversity of Luxembourg
John Caughman † Department of Mathematics and StatisticsPortland State University
Martin Cenek ‡ Department of Computer Science and EngineeringUniversity of Alaska Anchorage
Christof Teuscher § Department of Electrical and Computer EngineeringPortland State University (Dated: August 28, 2018)The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin ofregular, repeating configuration patterns have been for a long time a central focus of complexityscience, and physics.Here, we introduce group-theoretic concepts to identify and enumerate the symmetric inputs,which result in irreversible system behaviors with undesired effects on many computational tasks.The concept of so-called configuration shift-symmetry is applied on two-dimensional cellular au-tomata as an ideal model of computation. The results show the universal insolvability of “non-symmetric” tasks regardless of the transition function. By using a compact enumeration formulaand bounding the number of shift-symmetric configurations for a given lattice size, we efficiently cal-culate how likely a configuration randomly generated from a uniform or density-uniform distributionturns shift-symmetric. Further, we devise an algorithm detecting the presence of shift-symmetry ina configuration.The enumeration and probability formulas can directly help to lower the minimal expected errorfor many crucial (non-symmetric) distributed problems, such as leader election, edge detection,pattern recognition, convex hull/minimum bounding rectangle, and encryption. Besides cellularautomata, the shift-symmetry analysis can be used to study the non-linear behavior in varioussynchronous rule-based systems that include inference engines, Boolean networks, neural networks,and systolic arrays.
Keywords: configuration shift-symmetry, two-dimensional cellular automata, insolvability, irreversibility,enumeration, symmetry detection, prime factorization, prime orbit, mutually-independent generators, leaderelection
I. INTRODUCTION
Symmetry is a synonym for beauty and rarity, andgenerally perceived as something desired. In this paperwe investigate an opposing side of symmetry and showhow it can irreversibly corrupt a computation and restrictthe system’s dynamics.The structure of the computational rules that result inregular, repeating system configurations has been stud-ied by many, yet the question of how the natural andengineered system organize into symmetric structures is ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] not completely known. To understand the role of sym-metry of the starting configurations (the inputs), howthey are processed (the machine), and produce the fi-nal configurations with desired properties (the outputs)we use a cellular automata (CA) as a simple distributedmodel of computation. First introduced by John vonNeumman, CAs allowed to explore logical requirementsfor machine self-replication and information processing innature [38]. Despite having no central control and limitedcommunication among the components, CAs are capableof universal computation and can exhibit various dynam-ical regimes [8, 38, 46, 52]. As one of the structurallysimplest distributed systems, CAs have become a funda-mental model for studying complexity in its purist form[16, 53]. Subsequently, CAs have been successfully em-ployed in numerous research fields and applications, suchas modeling artificial life [29], physical equations [19, 47],and social and biological simulations [18, 25, 42, 43].The CA input configurations define a language thatis processed by the machine. Exploring the structuralsymmetries of the input language not only translates toan efficient machine implementation, but allows us topresent a theoretical argument of a problem insolvabilityand the irreversibility of computation.In this paper, we introduce the concept of shift-symmetry and show that any standard CA maintains aconfiguration shift-symmetry due to uniformity and syn-chronicity of cells. We prove that once a system reaches asymmetric, i.e., spatially regular configuration, the com-putation will never revert from this attractor and willfail to solve all problems that require the asymmetric so-lutions. As a result, the number of symmetries of thesystem is non-decreasing.Using group theory, we prove that the CA’s configura-tion space irreversibly folds causing a permanent regime“shift” when a configuration slips to a symmetric, re-peating pattern. Consequently, a non-symmetric solu-tion cannot be reached from a shift-symmetric configura-tion. A more general implication is that a configurationis unreachable (even if symmetric) if a source configu-ration has a symmetry not contained in a target one.Non-symmetric tasks, such as leader election or patternrecognition, i.e., tasks expecting a final configuration tobe non-symmetric, are therefore principally insolvable,since for any lattice size there always exist input config-urations that are symmetric.We develop three progressively more efficient enumer-ation techniques based on mutually independent gen-erators to answer the question of how many potentialshift-symmetric configurations there are in any given two-dimensional CA lattice. As a side product, we demon-strate that the shift-symmetry is closely linked to primefactorization. We introduce and prove lower and up-per bounds for the number of shift-symmetric configu-rations, where the lower bound (local minima) is tightand reached only for prime lattice sizes. We enumerateshift-symmetric configurations for a given lattice size andnumber of active cells.Finally, we derive a formula and bounds for the proba-bility of selecting shift-symmetric configuration randomlygenerated from a uniform or density-uniform distribu-tion. We develop a shift-symmetry detection algorithmand derive its worst and average-case time complexities. A. Applications
Since all the formulas and proofs presented in this pa-per assume two-dimensional CAs with any number ofstates, and arbitrary uniform transition and neighbor-hood functions, our results are widely applicable.Knowing the number of shift-symmetric configura-tions, we can directly determine the probability of ran-domly selecting a shift-symmetric configuration. Thisprobability then equals an error lower bound or expectedinsolvability for any non-symmetric task. As we show the shift-symmetry-caused insolvability is rapidly asymptot-ically decreasing with the lattice size for a uniform distri-bution. For instance, the probability is 0 . × . × − for a 10 ×
10 lattice.Since the number of shift-symmetric configurations heav-ily depends on the prime factorization of a lattice size,the probability function is non-monotonously decreasing.To minimize the occurrence of shift-symmetries for uni-form distribution, our general recommendation is to useprime lattices, or at least avoid even ones. On the otherside, the probability for a density-uniform distribution isquite high, regardless of primes: it is around 10 − evenfor a 45 ×
45 lattice.The distribution error-size constraints have importantconsequences for designing robust and efficient compu-tational procedures for non-symmetric tasks. A classof non-symmetric tasks covers many crucial distributedproblems, such as leader election [5, 46], pattern recog-nition [41], edge detection [45], convex hull/minimumbounding rectangle [11], and encryption [49]. For thesetasks an expected final configuration, e.g., reproductionof a certain two-dimensional image, is in a general casenon-shift-symmetric, and therefore unreachable from asymmetric configuration.Practical implications of these properties includesperformance degradation of systolic CPU arrays andnanoscale multicore systems [56]. Our results span tothe hardware implementations of synchronous CAs withFPGAs, used, e.g., for traffic signals control [26], randomnumber generation [44], and reaction-diffusion model[24]; and spintronics, where computation is achieved bycoupled oscillators [9, 48]. Also, current efforts of im-plementing two or three-dimensional cellular automatausing DNA tiles [20, 50] and/or gel-separated compart-ments in so-called gellular automata [21, 28] might faceproblems related to configuration shift-symmetry if asynchronous update is considered.
B. Related Work
In his seminal work, Packard [39] identified the impor-tance of symmetry and showed that CA’s global proper-ties emerge as a function of transition function’s reflectiveand rotational symmetries. The fundamental algebraicproperties of additive and non-additive CAs were studiedby Martin et al. [34], who demonstrated on simple casesthat there is a connection between the global behaviorand the structure of the configuration transitions. Wolzand deOliveira [54] exploited the structure and symme-try in the transition table to design an efficient evolution-ary algorithm that found the best results for the densityclassification and parity problems. Marquez-Pita et al. [32, 33] used a brute-force approach to find similar in-put configurations that produce the same outputs. Theirresults are a compact transition function re-descriptionschema that used wild-cards to represent the many-to-one computation rules on a majority problem. Bagnoli et al. [4] explored different methods of master-slave syn-chronization and control of totalistic cellular automata.CA computation-theoretic results were summarized byCulik II et al. [17], who investigated CAs through theeyes of set theory and topology. The concept of sym-metry in number theory has been applied to so-calledtapestry design and periodic forests [3, 35], which relatesto CA configurations. However, the triangular topologyand geometric branching differs from a discrete toroidalCartesian topology typically used for CAs.Despite a substantial focus on the symmetry of transi-tion functions , and the design of transition functions re-sulting in regular or synchronized patterns, the theoreti-cal CA research did not address the general structure andimplications of the shift-symmetric configurations with-out assuming anything specific about a transition func-tion, as we do here.One of our main motivations is the pioneering work ofAngluin [2], who noticed that a ring containing anony-mous components (processors), which are all in the samestate, will never break its homogeneous configuration andelect a leader. This intuitive observation is, in fact, a spe-cial case of our concept of configuration shift-symmetryfor CAs. We will show that Angluin’s homogeneous state,which corresponds to a configuration of all zeros or allones in a binary CA, is the most symmetric configura-tion for a given lattice size.The concept of shift-symmetry is related to the notionof regular domains in computational mechanics [15, 22].A shift-symmetric configuration is essentially a (global)regular domain spread to a full lattice. Although we can-not apply the results directly to regular domains at thelevel of sub-configurations because we pay no attentionto local symmetries and non-cyclic and non-regular bor-ders, the number of possible shift-symmetric configura-tions gives at least an upper bound on the number ofpossible regular domains.In our previous work [7] we proved that configura-tion shift-symmetry along with loose-coupling of activecells prevents a leader to be elected in a one-dimensionalCA [5]. The leader election problem, first introduced bySmith [46], requires processors to reach a final configu-ration where exactly one processor is in a leader state(one) and all others are followers (zero). Leader electionis a representative of a problem class where a solution isan asymmetric, non-homogeneous, transitionally and ro-tationally invariant system configuration. A final fixed-point configuration is asymmetric, since it contains onlyone processor in a leader state. Clearly, leader electionand symmetry are enemies , and, in fact, leader electionis often called symmetry-breaking.To enumerate shift-symmetric configurations for a one-dimensional case [7] we employed only basic combina-torics. Here, in order to span to two dimensions, we ex-tended our enumeration machinery to group theory andindependent generators. We show that the insolvabilitycaused by configuration symmetry extends beyond leaderelection to a whole class of non-symmetric problems.
FIG. 1. Schematic of the configuration update for a binarytwo-dimensional cellular automaton.
C. Model
By definition, a CA [13] consists of a lattice of N com-ponents, called cells , and a state set Σ. A state of the cellwith index i is denoted s i ∈ Σ. A configuration is then asequence of cell states: s = ( s , s , . . . , s N − ) . Given a topology for the lattice and the number ofneighbors b , a neighborhood function η : N × Σ N → Σ b maps any pair ( i, s ) to the b -tuple η i ( s ) of cells’ statesthat are accessible (visible) to cell i in configuration s .Note that each cell is usually its own neighbor.The transition rule φ : Σ b → Σ is applied in parallelto each cell’s neighborhood, resulting in the synchronousupdate of all of the cells’ states s t +1 i = φ ( η i ( s ) t ). Thetransition rule is represented either by a transition table,also called a look-up table, or a finite state transducer[23]. Here we focus exclusively on uniform CAs, whereall cells share the same transition function. The globaltransition rule
Φ : Σ N → Σ N is defined as the transitionrule with the scope over all configurations s t +1 = Φ( s t ) . In this paper we analyze two-dimensional CAs, wherecells are topologically organized on a two-dimensionalgrid with cyclic boundaries, i.e., we treat them as tori.The true power of our analysis is that it applies totwo-dimensional CAs with arbitrary neighborhood andtransition functions. We rely only on their uniformity: t t t t t t t t FIG. 2. Example space-time diagrams of a leader-electingCA on lattice size N = 40 [6]. Figures show a CA compu-tation starting with a random initial configuration (time t ),followed by 7 configuration snapshots. The CA reaches a finalconfiguration with a single active cell (leader) at time t . each cell has the same neighborhood and transition func-tion; and synchronous update, the attributes typicallyassumed for a standard CA.Figure 1 shows the update mechanism for a two-dimensional binary CA with a Moore neighborhood, asquare neighborhood with radius r = 1 containing 9 cells.The dynamics of two-dimensional CAs are illustrated asa series of configuration snapshots, where an active cellis black and an inactive cell white (Figure 2). II. SHIFT-SYMMETRIC CONFIGURATIONS
As stated by Angluin [2], homogeneous configurationsare insolvable by any anonymous deterministic algorithm(including CAs). The CA uniformity can be embeddedin its transition function, the deterministic update, syn-chronicity, topology, configuration, and cells’ anonymity.Intuitively, a fully uniform system in terms of its struc-ture, configuration, and computational mechanisms can-not produce any reasonable or complex dynamics.We show that Angluin’s homogeneous configurations of0 N and 1 N belong to a much larger class of so-called shift-symmetric configurations. In this section we formalizethe concept of configuration shift-symmetry by employ-ing vector translations and group theory. Figure 3 showsa CA computation on a two-dimensional shift-symmetricconfiguration. Compared to the one-dimensional case [7],two dimensions are more symmetry potent.It is important to mention that we deal with square configurations only. Nevertheless, we suggest most ofthe lemmas and theorems could be extended to incor-porate arbitrary rectangular shapes. Also, the formulasand methodology to enumerate two-dimensional shift-symmetric configurations could be generalized to arbi-trarily many dimensions. For consistency, however, weleave the rectangular as well as n -dimensional extensionsfor future consideration. First, we formally define a shift-symmetric (square)configuration by a given vector as shown in Figure 4. Definition II.1
For a non-zero vector (pattern shift) v ∈ Z n × Z n we denote by S n × n ( v ) = { s ∈ Σ n × n | ∀ u ∈ Z n × Z n : s u = s u ⊕ v } the set of all shift-symmetric square configurations of size N = n relative to v over the alphabet Σ , where ⊕ denotes coordinate-wise addition on Z n × Z n . (cid:3) Note that as opposed to our previous work [6], we re-named symmetric configurations to shift-symmetric con-figurations to avoid confusion with reflective or rotationalsymmetries. These two symmetry types, unlike shift-symmetry, are not generally preserved by a transitionfunction unless we impose certain “symmetric” proper-ties on the transitions.Since any translation by a non-zero vector v definesa configuration symmetry, we can study shift-symmetricconfigurations with the techniques of group theory. Fromnow on, we will call such a vector v that we use for statetranslation a generator . Lemma II.1
For any non-zero vector (generator) v ∈ Z n × Z n , S n × n ( v ) = { s ∈ Σ n × n | ∀ u ∈ Z n × Z n ∀ w ∈ h v i : s u = s u ⊕ w } , where h v i is the cyclic subgroup of Z n × Z n generated by v . (cid:3) Lemma II.2
For any non-zero v = ( l , l ) ∈ Z n × Z n ,the following hold:(i). | S n × n ( v ) | = | Σ | n |h v i| . (ii). |h v i| = n gcd( l ,l ,n ) ,(iii). | S n × n ( v ) | = | Σ | n gcd( l ,l ,n ) . Proof (i). When v = ( l , l ) is repeatedly applied toany cell in the lattice, an orbit is generated, consistingof |h v i| cells that must share a common state for anyconfiguration in S n × n ( v ). The number of distinct orbits t t t FIG. 3. Space-time diagrams of CA computation on a two-dimensional binary shift-symmetric configuration showing alattice at three consecutive time steps. Once reached, a shift-symmetry cannot be broken. of cells in the lattice is simply n |h v i| . Any configuration in S n × n ( v ) is thus uniquely determined by choosing a statefrom Σ for each orbit of cells, so (i) follows.(ii). For l ∈ Z n it is easily shown that |h l i| = n gcd( l,n ) ,so |h v i| = lcm (cid:18) n gcd( l , n ) , n gcd( l , n ) (cid:19) = n gcd( l , l , n ) , where lcm denotes the least common multiple.(iii). By (ii), the exponent in (i) becomes n |h v i| = n n gcd( l ,l ,n ) = n gcd( l , l , n )as desired. (cid:3) Lemma II.3
Fix any non-zero vector v ∈ Z n × Z n andany shift-symmetric square configuration s ∈ S n × n ( v ) .Then for any w ∈ Z n × Z n , the neighborhoods satisfy η w ( s ) = η w ⊕ v ( s ) . Proof
Suppose the neighborhood function, which is uni-formly shared by all cells, is defined by (relative) vec-tors u , . . . , u m , i.e., η w ( s ) = ( s w ⊕ u , . . . , s w ⊕ u m ) andassume the lemma does not hold, i.e., there exists w forwhich η w ( s ) = η w ⊕ v ( s ). Then( s w ⊕ u , . . . , s w ⊕ u m ) = ( s ( w ⊕ v ) ⊕ u , . . . , s ( w ⊕ v ) ⊕ u m )and so there exists some u j such that s w ⊕ u j = s ( w ⊕ v ) ⊕ u j , i.e., s w ⊕ u j = s ( w ⊕ u j ) ⊕ v , which contradictsthe assumption that s ∈ S n × n ( v ). (cid:3) Theorem II.4 If s ∈ S n × n ( v ) then Φ( s ) ∈ S n × n ( v ) forany uniform global transition rule Φ . Proof
Suppose q = Φ( s ) is not symmetric by v . Then,there exists u ∈ Z n × Z n , such that q u = q u ⊕ v . ByLemma II.3, η u ( s ) = η u ⊕ v ( s ), and so q u = φ ( η u ( s )) = φ ( η u ⊕ v ( s )) = q u ⊕ v , which is a contradiction. (cid:3) Corollary II.5 If s ∈ S n × n ( v ) and q / ∈ S n × n ( v ) thena non-symmetric configuration q is unreachable from ashift-symmetric configuration s for any uniform globaltransition rule Φ , i.e., for ∀ i ∈ N , Φ i ( s ) = q , where Φ i ( s ) denotes i applications of Φ on s . Proof
By induction Φ i ( s ) ∈ S n × n ( v ) q . (cid:3) Corollary II.6
Leader election from a symmetric squareconfiguration s is impossible for n > . Proof
A target configuration q for leader election con-tains exactly one cell in the leader state a ∈ Σ. Thisconfiguration is asymmetric for n >
1, and therefore un-reachable from a shift-symmetric configuration s definedby any vector v . (cid:3) FIG. 4. Schematic of a shift-symmetric two-dimensional con-figuration generated by the vector v = (3 ,
4) on Z × . III. ENUMERATING SHIFT-SYMMETRICCONFIGURATIONS
In this section we will further investigate shift-symmetric two-dimensional configurations and ask howmany there are in a square lattice of size N = n . First,to generalize shift-symmetry and lay a solid ground forgroup-centric analysis we define the symmetric configu-rations over several generators. Then, we construct effec-tive generators using prime factors of n and prove theyare mutually independent. Finally, we enumerate shift-symmetric configurations using inclusion-exclusion prin-ciple as stated in Lemma III.10 and III.11, and concludein the final formula in Theorem III.12. Definition III.1
Let L ⊆ Z n × Z n . We define the set of L -symmetric configurations to be the set S n × n ( L ) = { s ∈ Σ n × n | ∀ u ∈ Z n × Z n , ∀ v ∈ h L i : s u = s u ⊕ v } , where h L i = { c v ⊕ . . . ⊕ c | L | v | L | | c i ∈ Z n } . In otherwords, S n × n ( L ) denotes the set of all shift-symmetricsquare configurations of size N = n over the alphabet Σ with generator set L . (cid:3) Corollary III.1
For any subset L ⊆ Z n × Z n , | S n × n ( L ) | = | Σ | n |h L i| . Proof
Similar to Lemma II.2(i). (cid:3)
Corollary III.2
For any u , v ∈ Z n × Z n S n × n ( u ) ∩ S n × n ( v ) = S n × n ( { u , v } ) . Proof
Immediate from Definition III.1. (cid:3)
Lemma III.3
For any u , v ∈ Z n × Z n , the followinghold:(i). |h u , v i| = |h u i||h v i||h u i∩h v i| . (ii). | S n × n ( u ) ∩ S n × n ( v ) | = | Σ | n |h u i∩h v i||h u i||h v i| . (iii). | S n × n ( u ) ∪ S n × n ( v ) | = | Σ | n ( |h u i| + |h v i| − |h u i∩h v i||h u i||h v i| ) . Proof (i). By definition, h u , v i = { c u + c v | c , c ∈ Z n } , hence there are |h u i||h v i| selections of vectors from h u i and h v i . However, each vector from h u , v i is included |h u i ∩ h v i| times.(ii). Immediate from (i) and Corollary III.2.(iii). Inclusion-exclusion using (ii). (cid:3) Lemma III.4
For any u , v ∈ Z n × Z n S n × n ( u ) ⊆ S n × n ( v ) ⇐⇒ h v i ≤ h u i . Proof ( ⇒ ). Suppose S n × n ( u ) ⊆ S n × n ( v ). Then S n × n ( u ) = S n × n ( u ) ∩ S n × n ( v ) = S n × n ( { u , v } )by Corollary III.2. But then |h u i| = |h u , v i| by Corol-lary III.1, which forces h u i = h u , v i , so that v ∈ h u i and h v i ≤ h u i as desired.( ⇐ ). By way of contradiction, suppose that S n × n ( u ) S n × n ( v ) and h v i ≤ h u i . Let s ∈ S n × n ( u ) such that s S n × n ( v ). Then s is symmetric under u but not un-der v . Consequently, there exists w ∈ Z n × Z n such that s w = s w ⊕ v . But s ∈ S n × n ( u ) and v ∈ h u i by assump-tion, so Lemma II.1 implies that s w = s w ⊕ v , which is acontradiction. (cid:3) Definition III.2
We denote by S n × n the set of allsquare shift-symmetric configurations of length N = n over the alphabet Σ , so that S n × n = [ = v ∈ Z n × Z n S n × n ( v ) . (cid:3) Lemma III.5
For any prime p that divides n and any i (0 ≤ i < n ) , the cyclic group h ( np , i np ) i is simple, i.e., ithas no nontrivial proper subgroups. Proof
By Lemma II.2(ii), we see that h ( np , i np ) i has order p , and by Lagrange’s Theorem, any group with primeorder is simple. (cid:3) Remark:
By swapping the coordinates, the proof appliesalso to each subgroup of the form h ( i np , np ) i . Definition III.3
Fix any natural number n and let n = Q ω ( n ) j =1 p α j j be the prime factorization of n , where ω ( n ) denotes the number of distinct prime factors. For eachprime divisor p j we define G n ( p j ) = (cid:26)(cid:18) , np j (cid:19)(cid:27) ∪ (cid:26)(cid:18) np j , i np j (cid:19) : 0 ≤ i ≤ p j − (cid:27) , and we define G n = S ω ( n ) j =1 G n ( p j ) . (cid:3) Corollary III.6
For any natural number n , | G n | = ω ( n ) + ω ( n ) X i =1 p i . Proof
Immediate from Definition III.3. (cid:3)
Lemma III.7
Fix any natural number n and let n = Q ω ( n ) j =1 p α j j be the prime factorization of n , where ω ( n ) denotes the number of distinct prime factors. Then S n × n = [ w ∈ G n S n × n ( w ) , where G n is defined as in Definition III.3. Proof
See Appendix A. (cid:3)
Lemma III.8
Fix any n ∈ N . For any distinct u , v ∈ G n , |h u i ∩ h v i| = 1 . Proof
See Appendix A. (cid:3)
Lemma III.9
Fix any n ∈ N and any prime divisor p of n . Let ˆ n = n/p . Then for any distinct u , v ∈ G n ( p ) , h u , v i = h (ˆ n, , (0 , ˆ n ) i . In particular, |h u , v i| = p . Proof
See Appendix A. (cid:3)
Definition III.4
Given any v , w ∈ Z k , we write v E w whenever the coordinates satisfy v i ≤ w i for every i (1 ≤ i ≤ k ) . We write v ⊳ w if v E w and v = w . We denotethe sum of the coordinates by | v | = P ki =1 v i , and for any m ∈ Z , we write m for the k -tuple whose coordinates allequal m . (cid:3) Lemma III.10
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then | S n × n | = X ⊳ v E p + ( − | v | k Y i =1 (cid:18) p i + 1 v i (cid:19) | Σ | f ( v ) , where p = ( p , . . . , p k ) and f ( v ) = n Q ki =1 p − min( v i , i . Proof
See Appendix A. (cid:3)
Lemma III.11
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then an alternative counting of | S n × n | is | S n × n | = X ⊳ v E | Σ | g ( v ) X v E u E top( v ) ( − | u | k Y i =1 (cid:18) p i + 1 u i (cid:19) where g ( v ) = n Q ki =1 p − v i i and top( v ) ∈ Z k has i thcoordinate top( i ) = ( v i if v i < p i + 1 if v i = 2 . Proof
See Appendix A. (cid:3)
Theorem III.12
Let n = Q ki =1 p α i i be the prime fac-torization of n , where k = ω ( n ) , the number of distinctprime factors of n . Then | S n × n | = X ⊳ v E ( − | v | | Σ | g ( v ) k Y i =1 r ( i ) where g ( v ) = n Q ki =1 p − v i i and r ( i ) = if v i = 0 p i + 1 if v i = 1 p i if v i = 2 . Proof
See Appendix A. (cid:3)
Corollary III.13
Let n = p , where p is a prime. Then | S n × n | = | Σ | n ( n + 1) − | Σ | n Proof
For v = (1), g ( v ) = n and b ( v ) = ( n + 1), andfor v = (2), g ( v ) = 1 and b ( v ) = − n . (cid:3) The alternative counting method presented in LemmaIII.11 is more efficient than the method from LemmaIII.10 because of the grouping of the exponential ele-ments, which are costly to calculate. The final formulapresented in Theorem III.12 is the most efficient because,besides having the exponential elements grouped, it alsoreduces the inner binomial sum to a simple expression r ( i ). This is illustrated for n = 2 α α in Appendix B.Also, note that a one-by-one lattice offers no symme-tries since there exists no non-zero shift in Z × Z .
10 20 30 40 50 60 70 80 90 10010 , , , Size (Square) C o un t FIG. 5. Number of shift-symmetric two-dimensional binary( | Σ | = 2) configurations for the lattice sizes 2 to 100 withan inset focused on the area 2 to 10 . Note the local minimafor prime and local maxima for even sizes. FIG. 6. All 26 binary shift-symmetric configurations forthe lattice size 3 grouped into 5 classes based on thegenerating vector(s). The vectors are from left to right:(0 , , (1 , , (1 , , (1 ,
2) and the one at the bottom contain-ing all of them. The arrows show allowed transitions. Notethat for prime-size binary lattices | S n × n | = 2 n ( n + 1) − n . A. Bounding the Number of Shift-SymmetricConfigurations
In the previous section we derived a closed and efficientformula for counting the number of shift-symmetric con-figurations in a square lattice N = n . To get a deeperand more qualitative insight we now bound this numberfrom the top and the bottom. We prove that the lowerbound is tight and reached only on prime lattices. Lemma III.14
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n , and for each m (1 ≤ m < k ) , let q mn × n = X ⊳ v E ( − | v | | Σ | g ( v ) m Y i =1 r ( i ) , where v ∈ Z m and g ( v ) and r ( i ) are defined as before.Then q mn × n ≤ q m +1 n × n . Note that | S n × n | = q kn × n . Proof
See Appendix A. (cid:3)
Lemma III.15 | Σ | n ( n + 1) − | Σ | n ≤ | S n × n | , where equality holds if and only if n is a prime. Proof If k = 1, i.e., n is a prime, the equality holds asshown in Corollary III.13. If k > p < n, p ≤ n | S n × n | = q kn × n ≥ q k − n × n ≥ . . . ≥ q n × n = | Σ | n p − ( p + 1) − | Σ | n p − p > | Σ | n n − ( n + 1) − | Σ | n n − n (cid:3) Lemma III.16
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then | S n × n | ≤ k X i =1 | Σ | n p − i ( p i + 1) . Proof
See Appendix A. (cid:3)
Lemma III.17
Let p be a prime divisor of n . Then | Σ | n p − ( p + 1) ≤ | Σ | n ( p − − p. Proof
Let B = | Σ | n p − . Then B pp − p − B ( p + 1) = B ( B p − p − ( p + 1)) ≥ B ( | Σ | np − p − ( p + 1)) ≥ B (2 np − p − ( p + 1)) | Σ | ≥ ≥ B (2 p − ( p + 1)) np − ≥ ≥ (cid:3) Corollary III.18
Let p be a prime of n . Then | Σ | n p − ( p + 1) ≤ | Σ | n . (cid:3) Lemma III.19
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then | S n × n | ≤ ( n ) | Σ | n . Proof
By Lemma III.16 and Corollary III.18 | S n × n | ≤ k X i =1 | Σ | n p − i ( p i +1) ≤ k | Σ | n ≤ ( n ) | Σ | n (cid:3) Corollary III.20
The number of shift-symmetric con-figurations | S n × n | satisfies | Σ | n ( n + 1) − | Σ | n ≤ | S n × n | ≤ ( n ) | Σ | n . Proof
By Lemma III.15 and Lemma III.19. (cid:3)
10 20 30 40 50 60 70 80 90 10010 − , − , − , − , − , − − Size (Square) P r o b a b ili t y − − − − − FIG. 7. Probability of selecting a shift-symmetric two-dimensional binary ( | Σ | = 2) configuration using uniform dis-tribution for the lattice sizes 2 to 100 with an inset focusedon the area 2 to 10 . Note the local minima for prime andlocal maxima for even sizes ( n > B. Probability of Selecting Shift-SymmetricConfiguration over Uniform Distribution
To calculate the probability that a randomly drawnconfiguration is shift-symmetric, we use a uniform dis-tribution, where the probability of selecting each symbolfrom Σ for s i in configuration s is the same. For non-symmetric tasks, this probability directly equals a leastexpected insolvability (or error lower bound). Lemma III.21
The probability of selecting a shift-symmetric configuration in a square lattice of size N = n over uniform distribution is P unif n × n = | S n × n || Σ | n . Proof
Overall, there exist | Σ | n configurations and eachconfiguration is equally likely. (cid:3) Lemma III.22
The probability of selecting a shift-symmetric configuration in a square lattice of size N = n over uniform distribution satisfies n | Σ | − n + n ≤ P unif n × n ≤ ( n ) | Σ | − n . Proof
By Corollary III.20 and n | Σ | n ≤ | Σ | n ( n + 1) −| Σ | n . (cid:3) As shown in Figure 7 and proved by Lemma III.22, theprobability P unif n × n decreases rapidly: square-exponentiallyby n or exponentially by the lattice size N = n . Since | S n × n | depends on the prime factorization of n the prob-ability is non-monotonous. Similarly to | S n × n | the prob-ability P unif n × n reaches local minima for prime and localmaxima for even lattices ( n > IV. ENUMERATING SHIFT-SYMMETRICCONFIGURATIONS FOR k ACTIVE CELLS
Having enumerated all shift-symmetric configurationswe now tackle a subproblem of enumerating configura-tions with a specific number of cells in a given state,such as the state active . The motivation behind thisendeavour is to calculate the probability of selecting ashift-symmetric configuration for a density-uniform dis-tribution. Similarly to Section III we present three pro-gressively more efficient counting techniques based onmutually-independent generators of prime factors of n in Lemma IV.3, Lemma IV.5, and Theorem IV.6. Definition IV.1
For any state a ∈ Σ and n, k ∈ N ,define D an × n,k to be the set of all square configurationswith exactly k sites in state a : D an × n,k = { s ∈ Σ n × n | a s = k } . Accordingly, let S an × n,k be the set of such configurationsthat are symmetric: S an × n,k = S n × n ∩ D an × n,k . And for any v ∈ Z n × Z n , let S an × n,k ( v ) denote the set ofconfigurations in S an × n,k that are generated by v , so that S an × n,k ( v ) = S n × n ( v ) ∩ D an × n,k . (cid:3) Corollary IV.1
For any a ∈ Σ , any n, k ∈ N , and v =( l , l ) ∈ Z n × Z n , S an × n,k ( v ) = ∅ iff |h v i| = n gcd( l ,l ,n ) is an integer that divides k . (cid:3) Lemma IV.2
For any a ∈ Σ , any k ∈ N , and v ∈ Z n × Z n such that |h v i| divides k (cid:12)(cid:12) S an × n,k ( v ) (cid:12)(cid:12) = (cid:18) n |h v i| k |h v i| (cid:19) ( | Σ | − n − k |h v i| . Proof
Let s ∈ S an × n,k ( v ). Then the number of selec-tions of state in s , i.e., the pattern size, is n / |h v i| . Toenumerate the number of such configurations, we firsthave to choose k/ |h v i| out of n / |h v i| sites to be in state a , and then fill the remaining n / |h v i|− k/ |h v i| sites withstates from Σ \ { a } . (cid:3) Lemma IV.3
Pick n, k ∈ N with k ≤ n and let d =gcd( k, n ) . Let n = Q ω ( n ) i =1 p α i i , k = Q ω ( k ) i =1 q β i i , and d = Q ω ( d ) i =1 r γ i i be the prime factorizations of n , k , d , re-spectively. Then for any a ∈ Σ , | S an × n,k | = X ⊳ u E r + ( − | u | ω ( d ) Y i =1 (cid:18) r i + 1 u i (cid:19)(cid:18) n h ( u ) kh ( u ) (cid:19) ( | Σ | − n − kh ( u ) , where r = ( r , . . . , r ω ( d ) ) and h ( u ) = Q ω ( d ) i =1 r min( u i , i . Proof
See Appendix A. (cid:3)
Corollary IV.4
For any state set Σ and state a ∈ Σ , theset S an × n, equals the set S n × n for the state set Σ \ { a } . (cid:3) Lemma IV.5
Pick n, k ∈ N with k ≤ n and let d =gcd( k, n ) . Let n = Q ω ( n ) i =1 p α i i , k = Q ω ( k ) i =1 q β i i , and d = Q ω ( d ) i =1 r γ i i be the prime factorizations of n , k , d , re-spectively. Then for any a ∈ Σ , | S an × n,k | = X ⊳ v E E u E top( v ) ( − | u | (cid:18) n h ( v ) kh ( v ) (cid:19) ( | Σ | − n − kh ( v ) ω ( d ) Y i =1 (cid:18) r i + 1 u i (cid:19) , where h ( v ) = Q ω ( d ) i =1 r min( v i , i and top( v ) ∈ Z ω ( d ) has i th coordinate top( i ) = ( v i if v i < r i + 1 if v i = 2 . Proof
Similar to the proof of Lemma III.11. (cid:3)
Theorem IV.6
Pick n, k ∈ N with k ≤ n and let d = gcd( k, n ) . Let n = Q ω ( n ) i =1 p α i i , k = Q ω ( k ) i =1 q β i i , and d = Q ω ( d ) i =1 r γ i i be the prime factorizations of n , k , d , re-spectively. Then for any a ∈ Σ , | S an × n,k | = X ⊳ v E ( − | v | (cid:18) n h ( v ) kh ( v ) (cid:19) ( | Σ | − n − kh ( v ) ω ( d ) Y i =1 r ( i ) , where h ( v ) = Q ω ( d ) i =1 r min( v i , i and r ( i ) = if v i = 0 p i + 1 if v i = 1 p i if v i = 2 . Proof
Similar to the proof of Theorem III.12. (cid:3)
Corollary IV.7
The number of binary symmetric con-figurations ( | Σ | = 2 ) with k sites in state a is given by | S an × n,k | = X ⊳ v E ( − | v | (cid:18) n h ( v ) kh ( v ) (cid:19) ω ( d ) Y i =1 r ( i ) . (cid:3) As in Section III, the alternative counting method pre-sented in Lemma IV.5 is more efficient than the methodfrom Lemma IV.3 due to the grouping of the exponen-tial elements. The final counting Theorem IV.6 is themost efficient, since it further simplifies the formula bycollapsing the inner binomial sum to a simple expression r ( i ). This evolution is illustrated for n = 2 α α and k = 2 β β , β ≤ α , β ≤ α in Appendix B.0
10 20 30 40 50 60 70 80 90 10010 − − − − Size (Square) P r o b a b ili t y − FIG. 8. Probability of selecting a shift-symmetric two-dimensional binary ( | Σ | = 2) configuration using density-uniform distribution for the lattice sizes 2 to 100 with aninset focused on the area 2 to 10 . A. Probability of Selecting Shift-SymmetricConfiguration over Density-Uniform Distribution
Besides a uniform distribution, a CA’s performanceis commonly evaluated using a so-called density-uniformdistribution. In a density-uniform distribution the prob-ability of selecting k active cells ( a s = k ) is uniformlydistributed, therefore, each density is equally likely. Lemma IV.8
The probability of selecting a shift-symmetric configuration in a square lattice of size N = n over a density-uniform distribution is P dens n × n = 1 n + 1 n X k =0 | S an × n,k | (cid:0) n k (cid:1) ( | Σ | − n − k . Proof
For a density k ∈ { , . . . , n } there exist (cid:0) n k (cid:1) ( | Σ | − n − k configurations and each density (thenumber of active cells) is equally likely. (cid:3) As presented in Figure 8, the probability for density-uniform distribution decreases a magnitude slower thanfor the uniform one and reaches 0 .
001 even for N = 45 .That is due to the fact that density-uniform distributionselects configurations with a few or many active cells,which are combinatorially more symmetric, more often. V. DETECTING A SHIFT-SYMMETRICCONFIGURATION
For practical reasons, e.g., to find whether a currentsystem’s configuration is shift-symmetric and take an ac-tion (restart), we provide an algorithm to effectively de-tect an occurrence of shift-symmetry.First, to find out whether a configuration is shift-symmetric by a shift v we start at a cell w = (0 ,
0) and check if all the cells at the orbit w ⊕ i v are in thesame state. If yes, we repeat this process for the next or-bit and so on, moving in arbitrary but fixed order (e.g.,left-right up-down), until we check all the cells. If a cellhas been visited before we skip it and move on until wefind an unvisited cell, which marks a start of the next or-bit. Also, if the test fails at any point, a configuration isnon-shift-symmetric (by v ), and the process can be ter-minated. Otherwise, the property holds for all the cellsand a configuration is shift-symmetric.To determine whether a configuration is shift-symmetric globally, a naive way would be to try all possi-ble non-zero vectors v and check if any of them passes theaforementioned procedure. Luckily, as we discovered inSection II each configuration shift-symmetry “overlaps”with mutually-independent generators from G n . Recallthat these generators are defined by prime factors G n = ω ( n ) [ j =1 G n ( p j ) . The number of these generators | G n | = ω ( n )+ P ω ( n ) i =1 p i is significantly smaller than n . Now, we calculate theworst and average-case time complexity of the shift-symmetry test using the generators from G n . Theorem V.1
The worst-case time complexity of theshift-symmetry detection algorithm for a square config-uration of size N = n is O ( n ) . Proof
In a worst-case scenario, when a configuration isnon-shift-symmetric and there is only one cell breakingsymmetry, each test requires to visit potentially all n cells. The overall worst-case time complexity is there-fore O ( | G n | n ). We know that the sum of distinct primefactors sopf( n ) = P ω ( n ) i =1 p i also known as the integer log-arithm is at most n (if n is prime), which gives us O ( | G n | n ) = O ( ω ( n ) + ω ( n ) X i =1 p i ) n = O ((log ( n ) + sopf( n )) n )= O ( n ) . (cid:3) Theorem V.2
The average-case time complexity of theshift-symmetry detection algorithm for a square configu-ration of size N = n generated from a uniform distribu-tion is O ( n ) . Proof
See Appendix A. (cid:3) O ( n ) and O ( n ) respectively, which translate to O ( √ N N ) and linear O ( N ) when interpreted by the op-tics of the number of cells N = n . The function sopf( n ),which plays a crucial role in both O formulas, is of a log-arithmic nature in “most of the cases,” but n for primes.Since the number of primes is infinite we could not useany tighter asymptote than n . However, for randomlychosen n we expect the time complexities to be just O ( log ( n ) n ) and O ( log ( n )) respectively.It is worth mentioning that the presented algorithm de-tects if a configuration is shift-symmetric but does notcount the number of shift-symmetries in a configuration.The validity of the detection holds because we know that any shift-symmetric configuration must obey at least oneof the prime generators from G n . Nevertheless, to de-termine the number of shift-symmetries, i.e., the num-ber of vectors with distinct vector spaces in Z n × Z n forwhich the cells at a same orbit share the same state, wewould need to consider also sub-vectors, whose satisfia-bility cannot be generally inferred from the prime genera-tors. Construction of a counting algorithm is addressablebut goes beyond the scope of this paper. VI. DISCUSSION AND CONCLUSION
We showed that shift-symmetry decreases the system’scomputational capabilities and expressivity, and is gen-erally good to be avoided. For each shift-symmetry, asystem falls into, a configuration folds by the order ofsymmetry and “independent” computation shrinks to asmaller, prime fraction of the system. The rest of thesystem is mirrored and lacks any intrinsic computationalvalue or novelty. The number of reachable configurationsshrinks proportionally as well.One of the key aspects of shift-symmetry is that it ismaintained (irreversible) for any number of states, andany uniform transition and neighborhood functions. Itmeans that the occurrence of shift-symmetry is rootedin the CA model itself, specifically, in the cells’ unifor-mity, synchronous update, and toroidal topology. Shift-symmetry is preserved as along as a transition func-tion is uniform (shared among the cells), even if non-deterministic. In other words, during each step a tran-sition function can be discarded and regenerated at ran-dom. However, within the same synchronous update itmust be consistent, i.e., two cells whose neighborhood’ssub-configurations are the same must be transitioned tothe same state.We showed that a non-symmetric solution is unreach-able from a shift-symmetric configuration, which ren-ders the non-symmetric tasks, such as leader election[5, 46], several image processing routines including pat-tern recognition [41], and encryption [49], insolvable in ageneral sense. These non-symmetric procedures are fun-damental parts of many distributed protocols and algo-rithms. Additionally, leader election contributes to deci- sion making of biological societies [14, 31], and is a keydriver of cell differentiation [30, 37] responsible for struc-tural heterogeneity and the specialization of cells.To determine how likely a configuration randomly gen-erated from a uniform distribution is shift-symmetric,hence insolvable, we efficiently enumerated and boundedthe number of shift-symmetric configurations using mu-tually independent generators. We also introduced alower, tight prime-size bound, and an upper bound, andshowed that even-size lattices are locally most likely shift-symmetric. Overall, shift-symmetry is not as rare as onewould think, especially for small or non-prime lattices, orwhen a configuration is generated using density-uniformdistribution. Asymptotically, the probability for uniformdistribution drops exponentially with the lattice size buta magnitude slower for a density-uniform distribution.For instance the probability for a 100 square lattice isaround 10 − using uniform and 2 × − using density-uniform distribution.To detect whether a configuration is shift-symmetricwe constructed an algorithm, which, by using the baseprime generators, can effectively determine a presence ofshift-symmetry in linear O ( N ) time for prime and just O (( log( N )) ) for randomly chosen N on average.Further, we need to emphasize that shift-symmetrydoes not necessarily have to be harmful for all the tasks.For instance, the density classification [12, 16, 36, 40],which is widely used as a CA benchmark problem, re-quires a final configuration to be either 1 N if the ma-jority of cells are initially in the state 1, and 0 N oth-erwise. Since the expected homogeneous configurationsare fully shift-symmetric, they can be reached poten-tially from any configuration. Naturally, that depends onthe structure of a transition function but shift-symmetrydoes not impose any strong restrictions here. The abil-ity of reaching a valid answer does not mean reachinga correct answer. However, for the density classification,shift-symmetry tolerates the latter as well. It is because ashift-symmetric configuration consists purely of repeatedsub-configurations, and so the density (ratio of ones) ina sub-configuration is the same as in the whole.By moving from one to two dimensions we general-ized our machinery to vector translations, which can beextended to the n -dimensional case [10]. It is expectedthat the number of shift-symmetric configurations willgrow with the dimensionality of lattice. It will be inter-esting to investigate this relation from the perspective ofprime-exponent divisors.An important implication of shift-symmetry is thatcyclic behavior must occur only within the same sym-metry class defined by a set of prime shifts (vectors) asillustrated in Figure 6. Note that we count no-symmetryas a class as well. This leads to the realization that oncea CA gains a symmetry, i.e., a configuration crosses sym-metry classes, it cannot be injective and reversible, andthere must exist a configuration without a predecessor,a so-called “Garden of Eden” configuration [1, 27]. Itmeans that the only way for the CA to stay injective2is to decompose all the configurations into cycles, eachfully residing in a certain shift-symmetry class. Againone large class would contain all the non-shift-symmetricconfigurations. Open question is for which lattices, i.e.,for how many shift-symmetric configurations, CAs arenon-injective, thus irreversible, on average. As opposedto our shift-symmetric endeavour, which applies to anytransition function, investigating injectivity would re-quire to assume something about the transition function,e.g., that is generated randomly. Trivially, for any lat-tice there always exists an injective transition function.An example is an identity function.As shown, the number of symmetries in any syn-chronous toroidal CA is non-decreasing but could it beincreasing in the “average” case for a random transitionfunction? We know that the expected behavior of ran-domly generated CA is most likely chaotic and the attrac-tor length is exponential to the lattice size N , as opposedto ordered or complex CAs with linear or quadratic at-tractors [55]. Would the length of attractor be sufficientto discover a shift-symmetry if we keep a random CA run-ning long enough, potentially | Σ | N time steps? As seen inFigure 7, the ratio of shift-symmetric configurations as-suming a uniform distribution is exponentially decreas-ing with the lattice size, and prime lattices could pro-duce “only” around n | Σ | n symmetric configurations. Fora randomly chosen lattice size, dimensions, and cell con-nectivity, we expect the number of reachable symmetriesto be significantly smaller than the total number of sym-metries available. However, for symmetry-rich lattices,we speculate that toroidal synchronous uniform systems,such as CAs, could undergo spontaneous symmetrization contracting an initial configuration to a fully homoge-neous state (reverse Big Bang). If proven, it would di-rectly imply the system’s non-injectivity and irreversibil-ity, and would bind symmetrization with Gibbs entropy.This hypothesis will be addressed in our future work.We suggest that several phenomena observed in CA dy-namics, such as irreversibility, emergence of structured“patterns”, and self-organization could be explained orcontributed to shift-symmetry. As demonstrated by Wol-fram [51] on 256 elementary one-dimensional CAs, whenrun long enough, most of these CAs condensate to or-dered structures: homogeneous configurations and self-similar patterns, which are in fact shift-symmetric.A straightforward way to fight symmetry would beto introduce noise, i.e., to break the uniformity of cellsand/or to use an asynchronous update. Based on theamount of noise, this could, however, disrupt the consis-tency of local, particle-based, interactions, which give riseto a global computation. Clearly, asynchronicity makesa system more robust but sacrifices the information pro-cessing by algebraic structures, which could exist onlydue to synchronous update. Using our enumeration for-mulas and probability calculations we could in principleminimize a desired shift-symmetry insolvability and thenumber of resources needed based on distributed appli-cation requirements. ACKNOWLEDGMENTS
This material is based upon work supported by theNational Science Foundation under grants
Appendix A: Proofs
Lemma III.7
Fix any natural number n and let n = Q ω ( n ) j =1 p α j j be the prime factorization of n , where ω ( n ) denotes the number of distinct prime factors. Then S n × n = [ w ∈ G n S n × n ( w ) , where G n is defined as in Definition III.3. Proof ( ⊆ ). Let s ∈ S n × n , so that s ∈ S n × n ( v ) forsome nonzero v = ( a, b ) ∈ Z n × Z n . It suffices to showthat h w i ≤ h v i for some w ∈ G n , since this fact, byLemma III.4, implies S n × n ( v ) ⊆ S n × n ( w ) and therefore s ∈ S n × n ( w ).Without loss of generality, we may assumegcd( a, b, n ) = 1. Otherwise, we simply divide ev-erything by d = gcd( a, b, n ) to obtain ˆ v = (ˆ a, ˆ b ), andˆ n , respectively. Once we show that h ˆ w i ≤ h ˆ v i for someˆ w ∈ G ˆ n , we multiply throughout by d to obtain thedesired result. Case 1.
Suppose gcd( a, n ) = 1. Then ai ≡ n b and aj ≡ n i , j ∈ Z . Also, n v ≡ n (0 , |h v i| divides n . Let p be any prime divisor of |h v i| and write n = pm for some m ∈ Z . Let w = ( m, im ) and note that w ∈ G n ( p ). Also observe v = a (1 , i ) and w = m (1 , i ), sothat mj v = mja (1 , i ) = m (1 , i ) = w . Therefore w ∈ h v i and thus h w i ≤ h v i as desired. Case 2.
Suppose gcd( a, n ) = 1. Let p be any primedivisor of both a and n , so that a = pa ′ and n = pn ′ for some a ′ , n ′ ∈ Z . Let w = (0 , n ′ ) and note that w ∈ G n ( p ). Observe that an ′ = a ′ pn ′ = a ′ n ≡ n
0, so n ′ v = n ′ ( a, b ) = ( an ′ , bn ′ ) = (0 , bn ′ ) = b w . Therefore n ′ v ∈ h w i . But by Lemma II.2(ii), |h w i| = p ,a prime. So if n ′ v is nonzero, then it generates h w i . But n ′ v is indeed nonzero, since its second coordinate is bn ′ ,and if bn ′ ≡ n
0, then n | bn ′ . Dividing by n ′ , we see p | b .But recall that p divides a and n , and we assumed at thebeginning (without loss of generality) that gcd( a, b, n ) =1. So p cannot divide b . This contradiction shows n ′ v isnonzero and so n ′ v generates h w i . Thus w ∈ h w i = h n ′ v i ≤ h v i . So h w i ≤ h v i as desired.( ⊇ ). Immediate by Definition III.2. (cid:3) Lemma III.8
Fix any n ∈ N . For any distinct u , v ∈ G n , |h u i ∩ h v i| = 1 . (A1) Proof
First, suppose that u ∈ G n ( p ) and v ∈ G n ( q ),where p = q . By Lemma II.2(i), |h u i| = p and |h v i| = q . Since |h u i ∩ h v i| must divide both of these primes, line(A1) must hold as claimed.Next, suppose u , v ∈ G n ( p ) and write n = ˆ np for someˆ n ∈ Z . Suppose u = (ˆ n, i ˆ n ) and v = (ˆ n, j ˆ n ) for some0 ≤ i < j < p . If x ∈ h u i ∩ h v i then ∃ k, l (0 ≤ k, l < p )such that x = k u = l v . But then ( k ˆ n, ki ˆ n ) = ( l ˆ n, lj ˆ n ) , so k ˆ n ≡ n l ˆ n and thus k ≡ p l . But also, ki ˆ n ≡ n lj ˆ n , sothat ki ≡ p lj . Since i p j , this forces k ≡ p
0, so that x = 0 and (A1) must hold as claimed.Finally, suppose u , v ∈ G n ( p ) and suppose u = (0 , ˆ n )and v = (ˆ n, i ˆ n ) for some 0 ≤ i < p . If x ∈ h u i ∩ h v i then ∃ k, l (0 ≤ k, l < p ) such that x = k u = l v . But then(0 , k ˆ n ) = ( l ˆ n, li ˆ n ) , so 0 ≡ n l ˆ n and thus 0 ≡ p l . But also, k ˆ n ≡ n li ˆ n , so that k ≡ p li and therefore k ≡ p
0. Now x = 0 and (A1) must hold as claimed. (cid:3) Lemma III.9
Fix any n ∈ N and any prime divisor p of n . Let ˆ n = n/p . Then for any distinct u , v ∈ G n ( p ) , h u , v i = h (ˆ n, , (0 , ˆ n ) i . In particular, |h u , v i| = p . Proof ( ⊆ ). First suppose u = (ˆ n, i ˆ n ) and v = (ˆ n, j ˆ n )for some 0 ≤ i < j < p . Then u = (ˆ n,
0) + i (0 , ˆ n )and v = (ˆ n,
0) + j (0 , ˆ n ). So h u , v i ⊆ h (ˆ n, , (0 , ˆ n ) i asdesired. A similar argument holds when u = (ˆ n, i ˆ n ) and v = (0 , ˆ n ).( ⊇ ). Again suppose u = (ˆ n, i ˆ n ) and v = (ˆ n, j ˆ n ) forsome 0 ≤ i < j < p . Then u − v ∈ h (0 , ˆ n ) i . But u − v = 0 and |h (0 , ˆ n ) i| = p , so u − v generates h (0 , ˆ n ) i .Thus (0 , ˆ n ) ∈ h u − v i ⊆ h u , v i . Likewise, (ˆ n, ∈ h j u − i v i ⊆ h u , v i , so the desired containment holds. A similarargument can be made when u = (ˆ n, i ˆ n ) and v = (0 , ˆ n ),showing that (ˆ n, ∈ h u − i v i , which implies the desiredresult. (cid:3) Lemma III.10
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then | S n × n | = X ⊳ v E p + ( − | v | k Y i =1 (cid:18) p i + 1 v i (cid:19) | Σ | f ( v ) , where p = ( p , . . . , p k ) and f ( v ) = n Q ki =1 p − min( v i , i . Proof
By Lemma III.7, inclusion-exclusion, and Corol-lary III.2, | S n × n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ w ∈ G n S n × n ( w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X ∅6 = J ⊆ G n ( − | J | +1 | S n × n ( J ) | . Since G n = S kj =1 G n ( p j ), we have k = ω ( n ) sets fromwhich to choose the elements of J , so | S n × n | = X J ⊆ G n ( p ) ...J k ⊆ G n ( p k ) ( − P ki =1 | J i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S n × n k [ i =1 J i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , J i = ∅ for all i . It follows from Corollary III.2 that S n × n ( S J i ) = T S n × n ( J i ) and so Corollary III.1 gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S n × n k [ i =1 J i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | Σ | n |h S ki =1 Ji i| . But by Lemma III.8 we know |h J i i∩h J j i| = 1 when i = j ,so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* k [ i =1 J i +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k Y i =1 |h J i i| . Since J i ⊆ G n ( p i ), recall that h J i i = h ( np i , , (0 , np i ) i when | J i | ≥ |h J i i| = 1, p i , and p i when | J i | = 0, 1, and ≥
2, respectively. Therefore k Y i =1 |h J i i| = k Y i =1 p min( | J i | , i Substituting all this into the expression for | S n × n | , weobtain | S n × n | = X J ⊆ G n ( p ) ...J k ⊆ G n ( p k ) ( − P ki =1 | J i | | Σ | n Q ki =1 p min( | Ji | , i ! Now, because the content of J i is irrelevant and wecare only about the cardinality | J i | , for each size v i = | J i | we have (cid:0) | G n ( p i ) | v i (cid:1) = (cid:0) p i +1 v i (cid:1) ways of choosing v i elementsfrom G n ( p i ), which produces the final formula as re-quired. (cid:3) Lemma III.11
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then an alternative counting of | S n × n | is | S n × n | = X ⊳ v E | Σ | g ( v ) X v E u E top( v ) ( − | u | k Y i =1 (cid:18) p i + 1 u i (cid:19) where g ( v ) = n Q ki =1 p − v i i and top( v ) ∈ Z k has i thcoordinate top( i ) = ( v i if v i < p i + 1 if v i = 2 . Proof
We know that the exponent of each p i in S n × n from Lemma III.10 is at most 2. Therefore for given v , . . . , v k ∈ { , , } we can combine all binomial ex-pressions associated with | Σ | n Q ki =1 pvii . If v i ≤ (cid:0) p i +1 v i (cid:1) selections from G n ( p i ), and S p i +1 u i =2 (cid:0) p i +1 u i (cid:1) for v i = 2. These two expressions could be generalizedas S top( i ) u i = v i (cid:0) p i +1 u i (cid:1) using the top function defined above.Therefore the total coefficient of | Σ | n Q ki =1 pvii is X v ≤ u ≤ top(1) ...v k ≤ u k ≤ top( k ) ( − P ki =1 u i k Y i =1 (cid:18) p i + 1 u i (cid:19) as required. (cid:3) Theorem III.12
Let n = Q ki =1 p α i i be the prime fac-torization of n , where k = ω ( n ) , the number of distinctprime factors of n . Then | S n × n | = X ⊳ v E ( − | v | | Σ | g ( v ) k Y i =1 r ( i ) where g ( v ) = n Q ki =1 p − v i i and r ( i ) = if v i = 0 p i + 1 if v i = 1 p i if v i = 2 . Proof
For a given vector v with v , . . . , v k ∈ { , , } we define b ( v ) = X v E u E top( v ) ( − | u | k Y i =1 (cid:18) p i + 1 u i (cid:19) , where top( i ) = v i if v i < p i + 1 if v i = 2.Using Lemma III.11, we are left to show that b ( v ) = ( − | v | k Y i =1 r ( i ) . We prove it by induction on k . As the induction basiswe choose k = 1, and so n = p , where p is a prime. Since b ( v ) = r (0) we need to confirm it equals − , p + 1 , or − p for three different cases of v defined by the function r .If v = (0), top( v ) = (0) and the only u is u = (0),which gives b ( v ) = − (cid:0) p +10 (cid:1) = −
1. If v = (1), top( v ) =(1) and the only u is u = (1), and so b ( v ) = (cid:0) p +11 (cid:1) = p +1.If v = (2), top( v ) = ( p + 1) and u ranges from (2) to( p + 1). Therefore b ( v ) = p +1 X u =2 ( − u (cid:18) p + 1 u (cid:19) = p +1 X u =0 ( − u (cid:18) p + 1 u (cid:19)| {z } + (cid:18) p + 10 (cid:19) − (cid:18) p + 11 (cid:19) = − p For the induction step we prove: b ( v ) = ( − | v | k Y i =1 r ( i ) ⇒ b ( w ) = ( − | w | k +1 Y i =1 r ( i ) , w = ( v , . . . , v k , v k +1 ). Similarly to the inductionbasis we need to consider three cases for v k +1 :If v k +1 = 0 b ( w ) = (cid:18) p k +1 + 10 (cid:19) X v E u E top( v ) ( − | u | k Y i =1 (cid:18) p i + 1 u i (cid:19)| {z } b ( v ) = ( − | v | k Y i =1 r ( i ) by induction step= ( − | w | k +1 Y i =1 r ( i )If v k +1 = 1 b ( w ) = − (cid:18) p k +1 + 11 (cid:19) X v E u E top( v ) ( − | u | k Y i =1 (cid:18) p i + 1 u i (cid:19)| {z } b ( v ) = − ( p k +1 + 1)( − | v | k Y i =1 r ( i ) by induction step= ( − | w | k +1 Y i =1 r ( i )If v k +1 = 2 b ( w ) = p k +1 +1 X u k +1 =2 ( − u k +1 (cid:18) p k +1 + 1 u k +1 (cid:19)| {z } p k +1 b ( v )= p k +1 ( − | v | k Y i =1 r ( i ) by induction step= ( − | w | k +1 Y i =1 r ( i ) (cid:3) Lemma III.14
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n , and for each m (1 ≤ m < k ) , let q mn × n = X ⊳ v E ( − | v | | Σ | g ( v ) m Y i =1 r ( i ) , where v ∈ Z m and g ( v ) and r ( i ) are defined as before.Then q mn × n ≤ q m +1 n × n . Note that | S n × n | = q kn × n . Proof
Let v = ( v , . . . , v m ) and w = ( v , . . . , v m , v m +1 ).Then q m +1 n × n = X ⊳ w E ( − | w | | Σ | g ( w ) m +1 Y i =1 r ( i )= X v m +1 =0 (cid:18) ( − v m +1 r ( m + 1) X ⊳ v E ( − | v | | Σ | g ( v ) p − vm +1 m +1 m Y i =1 r ( i ) (cid:19) + X v m +1 =1 ( − v m +1 | Σ | n p − vm +1 m +1 r ( m + 1)We split the expression into five parts: q m +1 n × n = x + x + x + y + y and define, for any c ∈ R q mn × n ( c ) = X ⊳ v E ( − | v | | Σ | g ( v ) c m Y i =1 r ( i ) , i.e., q mn × n = q mn × n (1). Then x = q mn × n x = − ( p m +1 + 1) q mn × n ( p − m +1 ) x = p m +1 q mn × n ( p − m +1 ) y = | Σ | n p − m +1 ( p m +1 + 1) y = −| Σ | n p − m +1 p m +1 Now we show that y + x + y ≥ A = n p − m +1 . Then ( y + x + y )( p m +1 + 1) − ≥ | Σ | A − q mn × n ( p − m +1 ) − | Σ | Apm +1 ≥ | Σ | A − X ⊳ v E | Σ | g ( v ) p − m +1 m Y i =1 r ( i ) − | Σ | Apm +1 ≥ | Σ | A − X ⊳ v E | Σ | Apl − | Σ | Apm +1 p l = min { p , . . . , p m } ≥ | Σ | A − m | Σ | Apl − | Σ | Apm +1 | v | = m ≥ | Σ | A − | Σ | m | Σ | Apl − | Σ | Apm +1 | Σ | ≥ ≥ | Σ | A − | Σ | m + Apl + Apm +1 ≥ | Σ | A − | Σ | ( n )+ Apl + Apm +1 m ≤ k − < log ( n ) ≥ | Σ | A − | Σ | ( n )+ A + A p l , p m +1 ≥ , p l = p m +1 ≥ x is non-negative we can conclude that q m +1 n × n = x |{z} q mn × n + x + y + y | {z } ≥ + x |{z} ≥ ≥ q mn × n (cid:3) Lemma III.16
Let n = Q ki =1 p α i i be the prime factoriza-tion of n , where k = ω ( n ) , the number of distinct primefactors of n . Then | S n × n | ≤ k X i =1 | Σ | n p − i ( p i + 1) . Proof
As in the proof of Lemma III.14 we employ thefunction q n × n , which can be decomposed into five partsas defined earlier q m +1 n × n = x + x + x + y + y Now we show that y ≥ x + x + y Let A = n p − m +1 . Then ( y − x − x − y )( p m +1 + 1) − ≥ | Σ | A + q mn × n ( p − m +1 ) | {z } ≥ − q mn × n ( p − m +1 ) + | Σ | Apm +1 | {z } ≥ ≥ | Σ | A − q mn × n ( p − m +1 ) ≥ | Σ | A − X ⊳ v E | Σ | g ( v ) p − m +1 m Y i =1 r ( i ) ≥ | Σ | A − X ⊳ v E | Σ | Aplpm +1 p l = min { p , . . . , p m } ≥ | Σ | A − m | Σ | Aplpm +1 | v | = m ≥ | Σ | A − | Σ | m | Σ | Aplpm +1 | Σ | ≥ ≥ | Σ | A − | Σ | m + Aplpm +1 ≥ | Σ | A − | Σ | ( n )+ Aplpm +1 m ≤ k − < log ( n ) ≥ | Σ | A − | Σ | ( n )+ A p l , p m +1 ≥ , p l = p m +1 ≥ y ≥ x + x + y q m +1 n × n = x + x + x + y + y ≤ x + 2 y . By substituting x and y we obtain a recursive in-equality q m +1 n × n ≤ q mn × n + 2 | Σ | n p − m +1 ( p m +1 + 1) ≤ q m − n × n + 2 | Σ | n p − m ( p m + 1) + 2 | Σ | n p − m +1 ( p m +1 + 1) . . . ≤ m +1 X i =1 | Σ | n p − i ( p i + 1)To finalize the proof we use | S n × n | = q kn × n . (cid:3) Lemma IV.3
Pick n, k ∈ N with k ≤ n and let d =gcd( k, n ) . Let n = Q ω ( n ) i =1 p α i i , k = Q ω ( k ) i =1 q β i i , and d = Q ω ( d ) i =1 r γ i i be the prime factorizations of n , k , d , re-spectively. Then for any a ∈ Σ , | S an × n,k | = X ⊳ u E r + ( − | u | ω ( d ) Y i =1 (cid:18) r i + 1 u i (cid:19)(cid:18) n h ( u ) kh ( u ) (cid:19) ( | Σ | − n − kh ( u ) , where r = ( r , . . . , r ω ( d ) ) and h ( u ) = Q ω ( d ) i =1 r min( u i , i . Proof
Using Definition IV.1, Lemma III.7, and Corol-lary IV.1, S an × n,k = [ w ∈ G n S n × n ( w ) ! \ D an × n,k = ω ( n ) [ i =1 [ w ∈ G n ( p i ) S an × n,k ( w )= ω ( d ) [ i =1 [ w ∈ G n ( r i ) S an × n,k ( w ) . By the inclusion-exclusion principle | S an × n,k | = X J ⊆ G n ( r ) ...J ω ( d ) ⊆ G n ( r ω ( d ) ) ( − P ω ( d ) i =1 | J i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) \ w ∈∪ i J i S an × n ( w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now, by Corollary III.2 \ w ∈∪ ω ( d ) i =1 J i S an × n,k ( w ) = \ w ∈∪ ω ( d ) i =1 J i S n × n ( w ) ∩ D an × n,k = S n × n ( ∪ ω ( d ) i =1 J i ) ∩ D an × n,k = S an × n,k ( ∪ ω ( d ) i =1 J i )Finally let m = |h∪ ω ( d ) i =1 J i i| , then using Lemma IV.2, | S an × n,k ( ∪ ω ( d ) i =1 J i ) | = (cid:18) n mkm (cid:19) ( | Σ | − n − km , where m = |h∪ ω ( d ) i =1 J i i| = Q ω ( d ) i =1 r min( | J i | , i . (cid:3) Theorem V.2
The average-case time complexity of theshift-symmetry detection algorithm for a square configu-ration of size N = n generated from a uniform distribu-tion is O ( n ) . Proof
Let m = n p − be the number of orbits for aprime p . Assuming a uniform distribution the probabilityof passing an orbit is Q = | Σ | − p . If successful we move toa next orbit, otherwise we terminate with the probability1 − Q . The probability of terminating at i th orbit can betherefore generalized as P i = ( (1 − Q ) Q i − if i < mQ m − if i = m. It is easy to show that these probabilities sum to 1, i.e.,we must terminate at one of m orbits. Further, the prob-ability of successfully passing the test for all the orbits—the probability that a configuration generated from a uni-form distribution is shift-symmetric by a vector with anorder p —equals | Σ | n ( p − − .By using the formula for a geometric sum we can provethat n − X i =0 ( i + 1) r i = 1 − r n (1 + n (1 − r ))(1 − r ) . We apply this to calculate the expected number of vis-ited orbits as E p [ orbits ] = m X i =1 iP i = (1 − Q ) m − X i =0 ( i + 1) Q i + mQ m − = 1 − Q m − ( m − Qm + Q ) + (1 − Q ) mQ m − − Q = 1 − Q m − Q .
Owing to
Q < p as E p [ orbits ] ≤ (1 − Q ) − = (1 − | Σ | − p ) − . Each p -orbit contains p cells and so the expected num-ber of visited cells is simply E p [ cells ] ≤ p (1 − | Σ | − p ) − . Note that while moving from one orbit to a next onewe can potentially revisit some cells, however, becausethe order is fixed we can visit each cell at most twice.The overall expected number of visited cells, i.e., theaverage-case time complexity in O-notation is ω ( n ) X i =1 ( p i + 1) p i (1 − | Σ | − p i ) − . Since the expression (1 − | Σ | − p i ) − is at most 2 ( p i ≥
2) and the integer logarithm sopf( n ) is at most n , theaverage-case time complexity of the shift-symmetry testis O ( ω ( n ) X i =1 p i + ω ( n ) X i =1 p i ) = O (sopf ( n ) + sopf( n ))= O ( n ) . (cid:3) Appendix B: Examples
Example:
Let n = 2 α α , then using counting fromLemma III.10, | S n × n | = (cid:18) (cid:19) | Σ | n + (cid:18) (cid:19) | Σ | n − (cid:18) (cid:19) | Σ | n − (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
22 3 − (cid:18) (cid:19) | Σ | n + (cid:18) (cid:19) | Σ | n + (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 3 + (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
22 32 + (cid:18) (cid:19) | Σ | n − (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 3 − (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 32 − (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
22 32 − (cid:18) (cid:19) | Σ | n + (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 32 + (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 32 + (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
22 32 − (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 32 − (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 32 + (cid:18) (cid:19)(cid:18) (cid:19) | Σ | n
222 32 by Lemma III.11, | S n × n | = | Σ | n (cid:20) + (cid:18) (cid:19)(cid:21) + | Σ | n (cid:20) + (cid:18) (cid:19)(cid:21) + | Σ | n
22 3 (cid:20) − (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) + | Σ | n (cid:20) − (cid:18) (cid:19) + (cid:18) (cid:19)(cid:21) + | Σ | n (cid:20) − (cid:18) (cid:19) + (cid:18) (cid:19) − (cid:18) (cid:19)(cid:21) + | Σ | n
222 3 (cid:20) + (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) + | Σ | n
22 32 (cid:20) + (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) + | Σ | n
222 32 (cid:20) − (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) and finally by Theorem III.12, | S n × n | = | Σ | n | Σ | n − | Σ | n
22 3 · − | Σ | n − | Σ | n | Σ | n
222 3 · | Σ | n
22 32 · − | Σ | n
222 32 · Example:
Let n = 2 α α , a ∈ Σ, and k = 2 β β ,where β ≤ α , β ≤ α , and σ = | Σ | −
1. Then usingcounting from Lemma IV.3 | S an × n,k | = ! n k ! σ n − k + ! n k ! σ n − k − ! n k ! σ n − k − ! ! n k ! σ n − k − ! n k ! σ n − k + ! n k ! σ n − k + ! ! n k ! σ n − k
22 3 + ! ! n k ! σ n − k + ! n k ! σ n − k − ! ! n k ! σ n − k
22 3 − ! ! n k ! σ n − k
22 32 − ! ! n k ! σ n − k − ! n k ! σ n − k + ! ! n k ! σ n − k
22 32 + ! ! n k ! σ n − k
22 32 + ! ! n k ! σ n − k − ! ! n k ! σ n − k
22 32 − ! ! n k ! σ n − k
22 32 + ! ! n k ! σ n − k
22 32 by Lemma IV.5 | S an × n,k | = (cid:18) n k (cid:19) σ n − k (cid:20) + (cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k (cid:20) + (cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k (cid:20) − (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k (cid:20) − (cid:18) (cid:19) + (cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k (cid:20) − (cid:18) (cid:19) + (cid:18) (cid:19) − (cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k
22 3 (cid:20) + (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k (cid:20) + (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) + (cid:18) n k (cid:19) σ n − k
22 32 (cid:20) − (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) − (cid:18) (cid:19)(cid:18) (cid:19) + (cid:18) (cid:19)(cid:18) (cid:19)(cid:21) | S an × n,k | = (cid:18) n k (cid:19) σ n − k (cid:18) n k (cid:19) σ n − k − (cid:18) n k (cid:19) σ n − k · − (cid:18) n k (cid:19) σ n − k − (cid:18) n k (cid:19) σ n − k (cid:18) n k (cid:19) σ n − k
22 3 · (cid:18) n k (cid:19) σ n − k · − (cid:18) n k (cid:19) σ n − k
22 32 · [1] Amoroso, S. and Cooper, G., Proceedings of the Ameri-can Mathematical Society , 158 (1970).[2] Angluin, D., in Proceedings of the Twelfth Annual ACMSymposium on Theory of Computing , STOC ’80 (ACM,New York, NY, USA, 1980) pp. 82–93.[3] ApSimon, H., Philosophical Transactions of the RoyalSociety of London A: Mathematical, Physical and Engi-neering Sciences , 113 (1970).[4] Bagnoli, F., Rechtman, R., and El Yacoubi, S.,Physical Review E , 066201 (2012).[5] Banda, P., in Advances in Artificial Life. Darwin Meetsvon Neumann , Lecture Notes in Computer Science, Vol.5778, edited by G. Kampis, I. Karsai, and E. Szathm´ary(Springer Berlin / Heidelberg, 2011) pp. 310–317.[6] Banda, P.,
Anonymous Leader Election in One- and Two-Dimensional Cellular Automata , Ph.D. thesis, ComeniusUniversity (2014).[7] Banda, P., Caugmann IV, J., and Pospichal, J., Journalof Cellular Automata , 1 (2015).[8] Berlekamp, E. R., Conway, J. H., and Guy, R. K., “Whatis Life?” in Winning Ways for Your Mathematical Plays,Vol. 2: Games in Particular (Academic Press, London,1982) Chap. 25.[9] Boehme, C. and Lupton, J. M., Nature Nanotechnology , 612 (2013).[10] Brunnet, L. G. and Chat´e, H.,Physical Review E , 057201 (2004).[11] Cenek, M., Information Processing in Two-DimensionalCellular Automata , Ph.D. thesis, Portland State Univer-sity (2011).[12] Cenek, M. and Mitchell, M., in
Encyclopedia of Complex-ity and Systems Science (2009) pp. 3233–3242.[13] Codd, E. F.,
Cellular automata (Academic Press, 1968).[14] Conradt, L. and List, C., Philosophical Transactions ofthe Royal Society B: Biological Sciences , 719 (2008).[15] Crutchfield, J. P. and Hanson, J. E.,Physica D , 279 (1993).[16] Crutchfield, J. P., Mitchell, M., and Das, R., in Evolu-tionary dynamics: Exploring the interplay of selection,accident, neutrality, and function (Oxford, 2003) pp.361–411.[17] Culik II, K., Hurd, L., and Yu, S.,Physica D: Nonlinear Phenomena , 357 (1990).[18] Ermentrout, G. B. and Edelstein-Keshet, L., Journal oftheoretical Biology , 97 (1993).[19] Fredkin, E. and Toffoli, T., International Journal of the-oretical physics , 219 (1982).[20] Gu, H., Chao, J., Xiao, S.-J., and Seeman, N. C., Naturenanotechnology , 245 (2009).[21] Hagiya, M., Wang, S., Kawamata, I., Murata, S.,Isokawa, T., Peper, F., and Imai, K., in InternationalConference on Unconventional Computation and Natu-ral Computation (Springer, 2014) pp. 177–189.[22] Hanson, J. E. and Crutchfield, J. P.,Journal of statistical physics , 1415 (1992).[23] Hordijk, W., Dynamics, emergent computation, and evo-lution in cellular automata , Ph.D. thesis, University ofNew Mexico, Albuquerque, NM (2000).[24] Ishimura, K., Komuro, K., Schmid, A., Asai, T., andMotomura, M., Nonlinear Theory and Its Applications,IEICE , 252 (2015). [25] Kalogeiton, V., Papadopoulos, D., Georgi-las, I., Sirakoulis, G., and Adamatzky, A.,International Journal of General Systems , 354 (2015),http://dx.doi.org/10.1080/03081079.2014.997527.[26] Kalogeropoulos, G., Sirakoulis, G. C., and Karafyllidis,I., The Journal of Supercomputing , 664 (2013).[27] Kari, J., Physica D: Nonlinear Phenomena , 379(1990).[28] Kawamata, I., Yoshizawa, S., Takabatake, F., Sugawara,K., and Murata, S., in International Conference onUnconventional Computation and Natural Computation (Springer, 2016) pp. 168–181.[29] Langton, C. G., Physica D: Nonlinear Phenomena , 12(1990).[30] Lawrence, P. A., The Making of a Fly: The Genetics ofAnimal Design (Wiley-Blackwell, 1992).[31] Lusseau, D. and Conradt, L., Behavioral Ecology andSociobiology (2009).[32] Marques-Pita, M., Mitchell, M., and Rocha, L. M.,
The role of conceptual structure in designing cellularautomata to perform collective computation (Springer,2008).[33] Marques-Pita, M. and Rocha, L. M., in
Artificial Life(ALIFE), 2011 IEEE Symposium on (IEEE, 2011) pp.233–240.[34] Martin, O., Odlyzko, A. M., and Wolfram, S., Commu-nications in mathematical physics , 219 (1984).[35] Miller, J. C., Philosophical Transactions of the Royal So-ciety of London A: Mathematical, Physical and Engineer-ing Sciences , 63 (1970).[36] Mitchell, M., Crutchfield, J. P., and Das, R., in Hand-Book of Evolutionary Computation , edited by T. Baeck,D. Fogel, and Z. Michalewicz (Oxford University Press,1997).[37] Nagpal, R., in
Engineering Self-Organising Systems , Lec-ture Notes in Computer Science, Vol. 2977 (Springer,2003) pp. 53–62.[38] Neumann, J. V.,
Theory of self-reproducing automata -edited and completed by Burks (Illinois Press, 1966).[39] Packard, N. H. and Wolfram, S., Journal of StatisticalPhysics , 901 (1985).[40] Reynaga, R. and Amthauer, E., Pattern recognition let-ters , 2849 (2003).[41] Rosin, P. L., IEEE Transactions on Image Processing ,2076 (2006).[42] de Sales, J. A., Martins, M. L., and Stariolo, D. A.,Physical Review E , 3262 (1997).[43] dos Santos, R. M. Z. and Coutinho, S., Physical ReviewLetters , 168102 (2001).[44] Shackleford, B., Tanaka, M., Carter, R. J., and Snider,G., in Proceedings of the 2002 ACM/SIGDA tenth inter-national symposium on Field-programmable gate arrays (ACM, 2002) pp. 106–112.[45] Slatnia, S., Batouche, M., and Melkemi, K. E., “Appli-cations of fuzzy sets theory: 7th international workshopon fuzzy logic and applications, wilf 2007, camogli, italy,july 7-10, 2007. proceedings,” (Springer Berlin Heidel-berg, Berlin, Heidelberg, 2007) Chap. Evolutionary Cel-lular Automata Based-Approach for Edge Detection, pp.404–411.[46] Smith, A., Association of Computing Machinery Journal , 339 (1971).[47] Vichniac, G. Y., Physica D: Nonlinear Phenomena ,96 (1984).[48] Vodenicarevic, D., Locatelli, N., Grollier, J., and Quer-lioz, D., in Neural Networks (IJCNN), 2016 InternationalJoint Conference on (IEEE, 2016) pp. 2015–2022.[49] Wang, X. and Luan, D., Communications in NonlinearScience and Numerical Simulation , 3075 (2013).[50] Wei, B., Dai, M., and Yin, P., Nature , 623 (2012).[51] Wolfram, S., Reviews of modern physics , 601 (1983). [52] Wolfram, S., Physica D: Nonlinear Phenomena , 1(1984).[53] Wolfram, S., Theory and Application of Cellular Au-tomata (Singapore: World Scientific, 1986).[54] Wolz, D. and De Oliveira, P., Journal of Cellular Au-tomata (2008).[55] Wuensche, A., Complexity , 47 (1999).[56] Zhirnov, V., Cavin, R., Leeming, G., and Galatsis, K.,IEEE computer41