Rule Primality, Minimal Generating Sets, Turing-Universality and Causal Decomposition in Elementary Cellular Automata
RRule Primality, Minimal Generating Sets,Turing-Universality and Causal Decomposition inElementary Cellular Automata
J¨urgen Riedel ∗ Algorithmic Nature Group, LABORES, Paris, France; andAlgorithmic Dynamics Lab, Centre for MolecularMedicine, Karolinska Instituteand
Hector Zenil † Information Dynamics Lab, Unit of Computational Medicine,SciLifeLab, Centre for Molecular Medicine, Departmentof Medicine Solna, Karolinska Institute;Department of Computer Science, University of Oxford, UK; andAlgorithmic Nature Group, LABORES, Paris, France.
Abstract
We introduce several concepts such as prime and composite rule , toolsand methods for causal composition and decomposition. We discoverand prove new universality results in ECA, namely, that the Booleancomposition of ECA rules 51 and 118, and 170, 15 and 118 can emulateECA rule 110 and are thus Turing-universal coupled systems. We con-struct the 4-colour Turing-universal cellular automaton that carries theBoolean composition of the 2 and 3 ECA rules emulating ECA rule 110under multi-scale coarse-graining. We find that rules generating the ECArulespace by Boolean composition are of low complexity and comprise prime rules implementing basic operations that when composed enablecomplex behaviour. We also found a candidate minimal set with only 38ECA prime rules —and several other small sets—capable of generating allother (non-trivially symmetric) 88 ECA rules under Boolean composition.
Keywords:
Causal composition; multi-scale coarse-graining; renormal-ization; Boolean composition; Elementary Cellular Automata; Turing-universality; ECA algebraic and group-theoretic properties. ∗ [email protected] † [email protected] a r X i v : . [ n li n . C G ] F e b Notation and Preliminaries
The following definitions follow the notation in [5]. A cellular automaton (CA) isa tuple (cid:104) S, ( L , +) , T, f (cid:105) with a set S of states, a lattice L with a binary operation+, a neighbourhood template T , and a local rule f .The set of states S is a finite set with elements s taken from a finite alphabetΣ with at least two elements. It is common to take an alphabet composedentirely of integers modulo s : Σ = Z s = { , ..., s − } . An element of the lattice i ∈ L is called a cell. The lattice L can have D dimensions and can be eitherinfinite or finite with cyclic boundary conditions.The neighbourhood template T = (cid:104) η , ..., η m (cid:105) is a sequence of L . In particular,the neighbourhood of cell i is given by adding the cell i to each element of thetemplate T : T = (cid:104) i + η , ..., i + η m (cid:105) . Each cell i of the CA is in a particularstate c [ i ] ∈ S . A configuration of the CA is a function c : L → S . The set of allpossible configurations of the CA is defined as S L .The evolution of the CA occurs in discrete time steps t = 0 , , , ..., n . Thetransition from a configuration c t at time t to the configuration c ( t +1) at time t + 1 is induced by applying the local rule f . The local rule is to be taken asa function f : S | T | → S which maps the states of the neighbourhood cells oftime step t in the neighbourhood template T to cell states of the configurationat time step t + 1: c t +1 [ i ] = f ( c t [ i + η ] , ..., c t [ i + η m ]) (1)The general transition from configuration to configuration is called the globalmap and is defined as: F : S L → S L .In the following we will consider only 1-dimensional (1-D) CA as introducedby Wolfram [7, 8]. The lattice can be either finite, i.e. Z N , having the length N , or infinite, Z . In the 1-D case it is common to introduce the radius ofthe neighbourhood template which can be written as (cid:104)− r, − r + 1 , ..., r − , r (cid:105) and has length 2 r + 1 cells. With a given radius r the local rule is a function f : Z | S | (2 r +1) | S | → Z | S | with Z | S | (2 r +1) | S | rules. The so called Elementary Cellular Au-tomata (ECA) with radius r = 1 have the neighbourhood template (cid:104)− , , (cid:105) ,meaning that their neighbourhoods comprise a central cell, one cell to the leftof it and one to the right. The rulespace for ECA contains 2 = 256 rules.Here we consider non-equivalent rules subject to the operations complementa-tion, reflection, conjugation and joint transformation (combining reflection andconjugation) (see Supplementary Information). For example, the number ofreduced rules for ECA is 88 (see Supplementary Information).In order to keep the notation simple, we adopt the following definitions [3].A cellular automaton at time step t A = ( a ( t ) , { S A } , f A ) is composed of a lattice a ( t ) of cells that can each assume a value from a finite alphabet S A . A singlecell is referenced as a n ( t ). The update rule f A for each time step is defined as f A : { S r +1 } → { S A } with a n ( t + 1) = f A [ a n − ( t ) , a n ( t ) , a n +1 ( t )]. The entirelattice gets updated through the operation f A a ( t ).2 .1 CA Typical Behaviour and Wolfram’s Classes Wolfram also introduced [8] an heuristic for classifying computer programs byinspecting the behaviour of their space-time diagrams. Computer programs be-have differently for different inputs. It is possible, and not uncommon, however,to analyze the behaviour of a program asymptotically according to an initialcondition metric [9, 10].Wolfram’s classes can be characterized as follows: • Class 1. Symbolic systems which rapidly converge to a uniform state.Examples are rules 0, 32 and 160. • Class 2. Symbolic systems which rapidly converge to a repetitive or stablestate. Examples are rules 4, 108 and 218. • Class 3. Symbolic systems which appear to remain in a random state.Examples are rules 22, 30, 126 and 189. • Class 4. Symbolic systems which form areas of repetitive or stable states,but which also form structures that interact with each other in complicatedways. Examples are rules 54 and 110.We use the concept of A Wolfram class as a guiding index that popularlyassigns some typical behaviour to every ECA even though we have also shownthat such distinction is not of fundamental nature [6]. Here, however, it willbe useful to study this idea of typical behaviour of rules that are capable ofemulating others when included in minimal emulation sets.In [1, 8], it was shown that at least one ECA rule can perform Turing uni-versal computation. It is still an open question whether other rules are capableof Turing universality, but some evidence suggests that they may be, and thatcellular automata rules and random computer programs are, in general, highlyprogrammable and candidates of computation universality [6].Here we extend results reported in [4] regarding the Boolean compositionof ECA. We introduce minimal generating sets as candidates for being able togenerate all ECA rules, and we introduce new Turing-universality results in ECAby composition and an associated non-ECA Turing-universal CA implementingthe composition.
Rule composition for a pair of CA, i.e. rule C = rule A ◦ rule B, is defined as f C a (0) = f B ◦ ( f A a (0)). The lattice output of rule A is the input of rule B . Onecan say that the rule composition of rule A and rule B yields rule C . Rule C canbe composed out of rule A and rule B . The whole evolution of the compositerule A ◦ rule B is f B ◦ f A a (2 t ) = f C a ( t ) which is as long as the whole evolutionof rule C. More generally one can compose rule A out of n rules A n : f A a (0) = f A ◦ f A . . . ◦ f A n a (0). The whole evolution is f C a ( t ) = f A ◦ f A . . . ◦ f A n a ( nt ).3n order to find the CA in a higher rule space that implement the Booleancomposition of CA in lower rule spaces (e.g. ECA) we introduce the concept ofcausal separability. Definition 1.
A space time evolution of a function C : S → S (cid:48) (e.g. a CA) isminimally causally separable if, and only if, the rule icon network (see Fig. 2)of the rule R of C is the smallest rulespace in which R is separable into | S | disconnected networks. Fig. 2 illustrates the basics of (non-)separability under a rule compositionof ECA rules, an example demonstrating that the ECA rulespace is not closedunder Boolean composition. This will help us find the CA rule in a higher spacethat implements the emulation of rule 110 under composition of ECA rules (seeFig. 7 in the Supplementary Material).
The questions driving our experiments thus led us to the problem of finding theminimal rule set that generates the full ECA space.The implementation of the algorithm, and thus checking for rule N -primality,is equivalent to the formal equivalence checking process in electronic designautomation to formally prove that two representations of a circuit design exhibitexactly the same behaviour, which is reducible to the Boolean satisfiabilityproblem (SAT). Since the SAT problem is NP-complete, only algorithms withexponential worst-case complexity can carry this test.We therefore proceeded by sampling methods. One strategy is to start froma subset of ECA rules, composing and finding the emulations that fall back intothe ECA rulespace, because only a subset of all possible rule pairs of a givenrulespace lead to a rule which itself is a member of the same rulespace, thusclearly indicating that the ECA under composition is not an algebraic structure,as it is not closed under composition. If a rule composition remains in the samerulespace, the rule tuples, after application of the successive rules of the rulecomposition, map to a cell state which is one-to-one (see Fig. 2.4). However,this mapping is not one-to-one all the time and the resulting rule compositionleaves the rulespace of the constituent rules (again Fig. 2.4). In this paper wewill focus on both cases. We investigate the former case where the compositerule remains in the same rulespace, and set our focus on Turing-universality.Another strategy is to start from a subset of ECA rules and start finding thepair of rules that can emulate the rules in question, then move to anotherrule, knowing the compositions that were already found in the previous steps.However, all sampling approaches have their own potential problems becausethe result may be dependant on the initial subset of rules chosen to start theexploration. So we also employed a procedure akin to bootstrapping, wherewe re-sampled the ECA rulespace with different seeds (initial ECA rules) toinfer the compositions from a new sample of the same rulespace and sought outconvergence patterns. 4e also explore the mapping to a higher colour rulespace in order to analyzethe full richness of the rulespace induced by rule composition. The sampling algorithm 1 is based on the frequency of rule 51 in all rule pairs(i.e. 2-tuple) emulating another rule. On one hand, the number of distinct rulescomposed of rule 51 is the highest (see Fig. 2.3.1 (a)) (besides rule 204, whichis the identity rule). On the other hand, the number of distinct rule forming acomposition pair is the highest as well (see Fig. 2.3.1 (b)). Here rules 204 theidentity rule, and 0, the annihilator rule, have the same number of distinct rulesand are not of importance in this context.The algorithm searches alongside all rule pairs containing rule 51. If no rulepair containing rule 51 is found, the algorithm searches for other valid rule pairs.It tries to substitute (fold) each rule pair with other rule pairs found in orderto form a reduced n -tuples. The goal of the algorithm is to minimize set of ruletuples.Since rule 51 can be composed of rules 15 and 170, the algorithm substitutesalways rule 51 with rule pair (15,170).The pseudo-code for the first sampling algorithm is as follows:1. create empty sets prime.rules and ensemble.tuples as well as the set all.ECA.rulescontaining all 88 ECA rules.2. do(a) Draw a set containing rule tuples which each having composing rulesdifferent than the composed rules and the set as a whole containingall 88 ECA rules. This set is called tuple.pool(b) select all tuples from tuple.pool the composing rules of which are notin the set prime.rules(c) if selection (from step 2b) = empty then break(d) draw(one random sample) from tuple set(e) add the composing rule to the set prime.rules(f) add whole tuple to set ensemble.tuples3. for test.tuples in ensemble.tuples(a) extract all distinct composing rules from set and put them intoprime.rules doi. select at random a rule prime.rule from the set test.tuples not in { , , } .ii. set valid.tuples = (get all tuples which have prime.rule as thecomposite rule)iii. remove all tuples from set valid.tuples which have composingrules in set prime.rules test.tuples = (in all tuples of set test.tuplesreplace the composed rule with the composing rules of set valid.tuples)5a) Distinct count ( y axis) of ECA rules which can be emulatedby other ECA rules ( x axis).(b) Distinct count ( y axis) of ECA rules which form pairswith other ECA rules ( x axis).Figure 1: (a) Showing the non-uniform distribution count ( y axis) of ECArules which can be emulated by ECA rules ( x axis) pairing with another ECArule. For example, rule pairs containing rule 51 or rule 204 can emulated themost ECA rules. (b) Showing the distinct count ( y axis) of ECA rules whichform pair with ECA rules ( x axis). For example rules 0, 51, and 204 form themost distinct pairs with other ECA rules.iv. remove prime.rule from set prime.rulesv. remove all tuples from test.tuples which have repeating rules.vi. break if order of set prime.rules = 04. prime.rules = extract all distinct composing tuples from set test.tuples5. prime.rules = prime.rules + rules not contained in prime.tuples fromall.ECA.rules6. comp.tuples = test.tuples + 6for rule in primes.rulestuples = tuples + { rule, , } end for). This algorithm does not rely on special insight in the of rule composition pairsfor ECA rules as it is only relying on random sampling of the whole set of rulecomposition pairs.Pseudo-code for the second sampling algorithm is as follows:1. select all tuples from set all.tuples which do not have repeating rules2. set prime.pool = all 88 ECA rules3. initialize as empty sets all.tuples and new.tuples = { }
4. initialize as empty set selected.rules = { } (a) set rule = (pick a random rule from prime.pool excluding selected.rules)(b) set selected.rules = selected.rules + rule(c) initialize new.primes, new.rules and rules as empty sets(d) rules = { rule } (start with set containing one rule)i. tuples = (draw for each rule in rules a tuple from prime.poolwhich composes rule)ii. check if set tuples only contains composing rules which are notalready compositesiii. If length(tuples)=0, breakiv. set rules = (all composed rules in set tuples)v. set new.rules = newrules + tuplesvi. else break at 100 steps(e) new.tuple = fold (definition below) set new.rules to form one tupleadhering to causal order(f) primes = select composing rules of new.tuple(g) set new.tuples = new.tuples + new.tuple(h) if number of distinct rules in set new.tuple = 88 then break(i) else break at max steps5. all.tuples = all.tuples + new.tuples6. stop at maximal number of trials maxThe folding function in this algorithm refers to the substitution of compositerules previously found to be emulated by other prime rules in previous iterationsof the same the algorithm, i.e. the substitution of rules that can be decomposedin other prime rules. 7a) Rule mapping ECA rule 54.(b) Time evolution of ECA rule 54.(c) Non-causal rule mapping rule 50 ◦ ◦ .4 Primality and Rule (De)Composition Many rules can be composed from rule tuples not involving the composite ruleitself. As in the case of PCA all ECA rules can be composed from other ECArules. For example, for rule pairs there are 88 = 7744 of which 7744 −
736 =7008 are not in the ECA rulespace. One could investigate these rules anddetermine to which Wolfram class they belong. For example, the rule pair (50,37) (see Fig. 2.4) behaves like a Wolfram class 4 CA and seems potentially ofhigh complexity.
Definition 2.
A rule R ∈ S in rulespace S is N -prime if, and only if, it can onlybe simulated by itself or an equivalent rule under trivial symmetric transforma-tions (see Supplementary Material) in S , i.e. no composition exists to simulate R ∈ S up to the N -compositional iteration (see Algorithms in Subsection 2.3)other than (possibly) a composition of R itself. Definition 3.
A rule R (cid:48) ∈ S is N -composite if, and only if, it can be decomposedinto a composition of other rules in S non-equivalent to R (cid:48) under trivial sym-metry transformations (see Supplementary Material) up to an N -compositionaliteration (see Algorithms in Subsection 2.3). It follows that prime and composite rules are disjoint subsets in the rulespaceset of ECA.
We use two order parameters that do not play any fundamental role in themain results yet offer a guide to the type of accepted general knowledge aboutgenerating rules and space-time evolution dynamics in ECA. Wolfram [8] in-troduced a heuristic of behavioural class based on the typical behaviour of aCA for a random, equal density of non-zero (for binary) states/colours. Class1 is the most simple (exhibiting the most trivial behaviour), followed by class2 (converging to a uniform state); class 3 is random-looking and class 4 showspersistent structures and is thus considered the most complex.
We are interested in those rules which map back to ECA rules. We ask whichis the minimal ECA rule set which produces all necessary tuples to composeall other ECA rules. One way to find such a set of ‘prime’ rules is to create agraph having the rules as vertices and the edges created from the pairing of eachtuple element to the composite rule. By looking at the vertex-in and vertex-outdegrees one can eliminate the vertices which have a vertex-in degree > bc de f Figure 3: (a,b) Distribution of primes by Wolfram class [8] according to algo-rithms 1 (a) and 2 (b). (c,d) Distribution of vs composite rules by Wolframclass according to algorithms 1 (a) and 2 (b). Distributions of primes as usedto generate all other rules under Boolean composition for algorithms 1 (e) and2 (f). All bins are normalized by number of elements in each class for all 88non-trivially symmetrical ECA rules.
Table. 8 shows the minimal ECA generating set with the prime rules and rulecompositions for the composite rules.
Observation 1.
None of the prime rules is of Wolfram class 4, i.e. all class 4rules can be composed by prime rules of a lower class. In other words, all class4 rules are composite.
Fig. 3 shows the distribution of Wolfram classes for prime and compositerules building the ECA space. Among the ECA prime rules used to generate allothers, most belong to class 1 and 2, and these are the only 2 distributions forwhich sampling algorithms 1 and 2 produced the most different results. Yet in10oth cases rules 1 and 2 are the building blocks in the minimal ECA-generatingsets.Table. 8 in the Supplementary Information provides all the compositionsfound and the breakdown of ECA prime and composite rules. It is common tofind that rule permutations under compositions yield the same ECA rule. Forexample, rule 110 can be composed out of the prime tuple 170 , , > . Fig.4(c,d) shows the ECA prime rules with the highest frequency and their asso-ciated space-time evolutions, starting from a typical (i.e. random, 0.5 non-zerodensity) (Fig.4(e)) among the ‘building blocks’ able to generate the full ECAspace (88 non-trivially symmetric rules or 256 rules counting all). They canbe classified into two apparent main groups: (i) identity filters (e.g. rule 140,136 and 200) able to partially ‘silence’ or filter the communication of informa-tion from input to output and (ii) rules that transfer information diagonally or‘shifters’ (such as rules 170 and 14) at different speeds (e.g. slow, like rule 15,versus fast, like rules 14 and 184). The two most frequent ECA rules used tobuild all others are shifters that transfer information at different speeds with nocollisions and no loss of information (rules 15 and 170).The set of primes in the 38-rule minimal set able to produce all other ECArules (88 non-equivalent and 256 under trivial symmetries) is: 0, 1, 2, 3, 5, 7,11, 12, 13, 18, 19, 23, 24, 25, 27, 28, 29, 30, 34, 35, 38, 40, 42, 43, 46, 50, 51,57, 58, 72, 73, 74, 94, 104, 105, 108, 110, 128, 130, 132, 134, 136, 138, 154, 162,164, 178 and 204.The composite rules in the minimal set are: 4, 6, 8, 9, 10, 14, 15, 22, 26, 32,33, 36, 37, 41, 44, 45, 54, 56, 60, 62, 76, 77, 78, 90, 106, 122, 126, 140, 142, 146,150, 152, 156, 160, 168, 170, 172, 184, 200 and 232.The intersection between algorithm 1 and algorithm 2 is significant. Amongall the 38 and 40 prime rule sets produced by the 2 algorithms, 27 are the same:6, 9, 10, 14, 15, 22, 32, 37, 41, 60, 76, 77, 78, 90, 122, 126, 140, 142, 146, 152,156, 160, 168, 170, 184, 200 and 232.
Fig. 5 illustrates the way in which a set of simple ECA composed as buildingblocks in a Boolean circuit can produce rich behaviour, and how the differentelements can causally explain the behaviour of the final system in a top-downand bottom-up way through Boolean decomposition, fine- and coarse-graining.11 bcde
Figure 4: (a and b) Convergence to minimal sets, with the smallest of size 38ECA prime rules generating all other (256 or 88 non-equivalent) ECA rules fordifferent seeds and two different algorithms with algorithm 2 (b) producing thesmallest. (c) Top 40 most frequent prime rules able to produce all other ECAunder Boolean composition according to algorithm 1 (d) very similar ranking ofmost frequent rules but with algorithm 2 among not only the minimal 38-elementset but also all other 16 minimal sets of size at most 40. (e) ECA space-timeevolutions of rules that are the building blocks of all other rules in the ECAspace preceded by their rule icon running from random initial conditions forillustration purposes. 12 b c de f g h
Figure 5: Causal decomposition: A complex CA (not in the ECA rule space)built by Boolean composition of (e,f,g,h) 4 simple ECA rules: 15 ◦ ◦ ◦
45. (b,c,d) Pair compositions (15 ◦ ◦ ◦ An analytic proof that ECA rule 110 can be composed out of prime ECA (170,15, 118) is as follows:
Theorem 1. rule 110 = rule 51 ◦ rule 118 = ( rule 170 ◦ rule 15 ) ◦ rule 118 The ECA rulespace clearly does not form a group because it is not closedunder composition; no clear identity rule was found but identity candidatesformed of prime rules were found. In particular, rules 15 and 170 are wildcards as prime rules able to perform bit shifts, and are used in almost everycomposition as they have the ability to target and shift a rule’s bits.
Proof.
First, we show that rule 51 = rule 170 ◦ rule 15:13mulation of rule 110: Emulation of rule 54:Figure 6: Prime rule composition of ECA rule 110 (original rule on top) andemulation by composition of rules 15, 118 and 170 (middle) and of ECA rule54 (same arrangement, but composition of rules 15, 108 and 170) after coarse-graining the stereographic version of the emulated rule (bottom).(a): Given are the rules 170: ( p, q, r ) (cid:55)→ r and 15: ( p, q, r ) (cid:55)→ ¬ p
1. Applying 170: ( p, q, r ) (cid:55)→ r to the lattice, which shifts the lattice to theright: p → q, q → r , r → p .2. Applying now rule 15: ( p, q, r ) (cid:55)→ ¬ p we get: q = ¬ p → q = ¬ q (with 1.) which is rule 51 :( p, q, r ) (cid:55)→ ¬ q .Then, we show that rule 110 = rule 51 ◦ rule 118:(b): Given that the rule 51: ( p, q, r ) (cid:55)→ ¬ q and rule 118: ( p, q, r ) (cid:55)→ ( p ∨ q ∨ r ) (cid:89) ( q ∧ r ).1. Applying rule 51: ( p, q, r ) (cid:55)→ ¬ q (rule 51) to the lattice, which negates thelattice.2. Applying ( p, q, r ) (cid:55)→ ( p ∨ q ∨ r ) (cid:89) ( q ∧ r ) (rule 118) we get: ( ¬ p ∨ ¬ q ∨ ¬ r ) (cid:89) ( ¬ q ∧ ¬ r )3. By De Morgan’s Law: ¬ ( p ∧ q ∧ r ) (cid:89) ¬ ( q ∨ r )4. Expanding xor: ( ¬ ( p ∧ q ∧ r ) ∧ ( q ∨ r )) ∨ (( p ∧ q ∧ r ) ∧ ¬ ( q ∨ r ))14. =(( ¬ p ∨ ¬ q ∨ ¬ r ) ∧ ( q ∨ r )) ∨ (( p ∧ q ∧ r ) ∧ ( ¬ q ∧ ¬ r ))6. =(( ¬ p ∨ ¬ q ∨ ¬ r ) ∧ ( q ∨ r )) ∨ (( p ∧ q ∧ r ∧ ¬ q ∧ ¬ r ))7. =(( ¬ p ∨ ¬ q ∨ ¬ r ) ∧ ( q ∨ r )) ∨ (( p ∧ r ∧ ¬ r ))8. =(( ¬ p ∨ ¬ q ∨ ¬ r ) ∨ ( p ∧ r ∧ ¬ r )) ∧ (( q ∨ r ) ∨ ( p ∧ r ∧ ¬ r ))9. =(( ¬ p ∨ ¬ q ∨ ¬ r ∨ p ) ∧ ( ¬ p ∨ ¬ q ∨ ¬ r ∨ r ) ∧ ( ¬ p ∨ ¬ q ∨ ¬ r ∨ ¬ r )) ∧ (( q ∨ r ) ∨ ( p ∧ r ∧ ¬ r ))10. =(1 ∧ ∧ ( ¬ p ∨ ¬ q ∨ ¬ r ∨ ¬ r )) ∧ (( q ∨ r ) ∨ ( p ∧ r ∧ ¬ r ))11. =( ¬ p ∨ ¬ q ∨ ¬ r ) ∧ (( q ∨ r ) ∨ ( p ∧ r ∧ ¬ r ))12. (( q ∨ r ) ∨ ( p ∧ r ∧ ¬ r )) can be expanded as:13. ( q ∨ r ∨ p ) ∧ ( q ∨ r ∨ r ) ∧ ( q ∨ r ∨ ¬ r )14. =( q ∨ r ∨ p ) ∧ ( q ∨ r ) ∧ q ∨ r ) ∧ (1 ∧ ∨ p )16. =( q ∨ r )17. Substituting in (11) one gets: ( ¬ p ∨ ¬ q ∨ ¬ r ) ∧ ( q ∨ r )18. Starting now from rule 110: ( p, q, r ) (cid:55)→ ( q ∧ ¬ p ) ∨ ( q (cid:89) r ) by applying thedefinition of xor :19. =( q ∧ ¬ p ) ∨ ( q ∧ ¬ r ) ∨ ( ¬ q ∧ r )20. =( q ∧ ¬ p ∨ q ) ∧ ( q ∧ ¬ p ∨ ¬ r ) ∨ ( ¬ q ∧ r )21. =(( q ∧ ¬ p ∨ q ) ∧ ( q ∧ ¬ p ∨ ¬ r )) ∨ ¬ q ) ∧ (( q ∧ ¬ p ∨ q ) ∧ ( q ∧ ¬ p ∨ ¬ r )) ∨ r )22. The first part of (21) (( q ∧ ¬ p ∨ q ) ∧ ( q ∧ ¬ p ∨ ¬ r )) ∨ ¬ q ) can be expandedas:23. =(( q ∧ ¬ p ) ∨ q ∨ ¬ q ) ∧ (( q ∧ ¬ p ) ∨ ¬ r ∨ ¬ q )24. =(1 ∧ (( q ∧ ¬ p ) ∨ ¬ r ∨ ¬ q )25. =(( q ∧ ¬ p ) ∨ ¬ r ∨ ¬ q )26. =(( q ∨ ¬ r ) ∨ ( ¬ q ∨ ¬ r )) ∨ ¬ q
27. =( q ∨ ¬ r ∨ ¬ q ) ∨ ( ¬ p ∨ ¬ r ∨ ¬ q )28. =1 ∨ ( ¬ p ∨ ¬ r ∨ ¬ q )29. =( ¬ p ∨ ¬ r ∨ ¬ q )30. The second part of (21) (( q ∧ ¬ p ∨ q ) ∧ ( q ∧ ¬ p ∨ ¬ r )) ∨ r ) can be expandedas:31. =(( q ∧ ¬ p ∨ q ∨ r ) ∧ ( q ∧ ¬ p ∨ ¬ r ∨ r ))32. =(( q ∧ ¬ p ∨ q ∨ r ) ∧ q ∧ ¬ p ) ∨ q ∨ r )34. =( q ∨ q ∨ r ) ∧ ( ¬ p ∨ q ∨ r )35. =( q ∨ r ) ∧ (1 ∨ ¬ p )36. =( q ∨ r )37. Substituting in (21.) (29.) and (36.) one gets: ( ¬ p ∨ ¬ q ∨ ¬ r ) ∧ ( q ∨ r ).38. Since (37.) = (17.) this shows rule 110 = rule 51 ◦ rule 118In a similar fashion, one can prove that the other Wolfram class 4 ECA rules41, 54 and 106 are also composed of simpler prime rules.15a) Rule mapping of 4-colour rule (c) Space-time of 4-colour rule170 ◦ ◦
118 170 ◦ ◦
118 with gray scale.(b) Spacetime of 4-colour rule (d) Space-time of rule170 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ (cid:3) → (cid:3)(cid:3) , (cid:4) → (cid:4)(cid:4) , (cid:4) → (cid:3)(cid:4) and (cid:4) → (cid:4)(cid:3) . (c) Emulationof the 4-colour equivalent of rule 170 ◦ ◦
118 with colour re-mapping (cid:3) → (cid:3) , (cid:4) → (cid:4) , (cid:4) → (cid:4) and (cid:4) → (cid:4) . The non-ECA 4-colour Turing-universal cellularautomaton that simulates the rule space-time of the composition of ECA rules170 ◦ ◦ .5 Multicolour CA Emulating ECA 110 In order to find the CA in a higher rulespace that implements the Boolean com-position of ECA emulating ECA rule 110, let’s consider a block transformationof the form (cid:3)(cid:3) → (cid:3) , (cid:4)(cid:4) → (cid:4) , (cid:3)(cid:4) → (cid:4) and (cid:4)(cid:3) → (cid:4) which maps all combi-nations of the 2-colour pairs with the 4 colours of the larger rulespace. In orderto see if the rule icon of a CA generated by ECA rule composition is separable,one executes these steps:1. Choose the de Bruijn sequence for alphabet S A = { , , , } = { (cid:3) , (cid:4) , (cid:4) , (cid:4) } and sub-sequences of length n = 3 as the initial condition.2. Create 3-tuples representing the 4-colour rule tuples with range r = 13. Apply the transformation (cid:3) → (cid:3)(cid:3) , (cid:4) → (cid:4)(cid:4) , (cid:4) → (cid:3)(cid:4) and (cid:4) → (cid:4)(cid:3) toeach tuple.4. Let the CA evolve each tuple 1 step for the chosen rule composition.5. Apply back transformation (cid:3)(cid:3) → (cid:3) , (cid:4)(cid:4) → (cid:4) , (cid:3)(cid:4) → (cid:4) and (cid:4)(cid:3) → (cid:4) to each resulting output tuple.6. Identify middle cell for each input tuple and pair it with the correspondingoutput tuple7. Create a graph out off the resulting pairs.Fig. 7 illustrates the space-time evolution of this finer-grained CA capable ofemulating ECA rule 110 after coarse-graining. The rule icon network for 170 ◦ ◦
118 is separable in the 4-colour CA space and is thus a native CA belongingto the 2-colour rulespace with closest neighbour, the smallest rulespace in whichsuch an automaton can exist. Fig. A.2 in the Supplementary Material illustratesanother interesting example of causal decomposition.
We have introduced a notion of prime and composite rule that has allowed us toapproach ECA from a group-theoretic point of view whose set composition canemulate all other Elementary Cellular Automata, suggesting minimal generatingsets. While it was known that the set is not closed under Boolean compositionand the emulations are not commutative, an exhaustive exploration of the em-ulating minimal sets had not been undertaken. The exploration allowed us tofind some interesting emulations, including some Turing-universal compositionsby emulation of ECA rule 110.We found that two different sampling algorithms starting from different seedsreach and provide evidence that the smallest generating sets are close to 38elements if not exactly 38, suggesting that simple rules are the building blocksof more complex rules in minimal generating sets.17e have found features that appear to be essential in computation towardsuniversality making a cellular automaton capable of emulating another (univer-sal) cellular automata such as rule 110—and other complex rules—in the wayin which these rules need to be composed with rules capable of transfer infor-mation horizontally at different rates. The new universality result in ECA is acomposition of 2 and 3 rules but the actual Turing-universal cellular automatonis in a higher rulespace whose rule has been given in detail and is capable ofemulating ECA rule 110 under coarse-graining.We have introduced novel tools, concepts and methods to explore computa-tion by algebraic/boolean rule composition, and methods for causal compositionand decomposition. Our work suggests that novel model-based approaches tostudying computational behaviour of computer programs can shed light on fun-damental computational and causal processes underlying computing systemsand provide a set of powerful tools to study general systems from a computa-tional/informational perspective.
References [1] Cook, M. Universality in Elementary Cellular Automata,
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15: 1–40, 2004.[2] W.K Wootters, C.G. Langton, Is There a Sharp Phase Transition for De-terministic Cellular Automata?,
Physica D , 45, 1990.[3] Israel, N., Goldenfeld, N. Coarse-graining of cellular automata, emergence,and the predictability of complex systems,
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73, 026203, 2006.[4] Wolfram, S., Table of Cellular Automaton Properties,
Theory and Appli-cations of Cellular Automata , World Scientific, pp. 485–557, 1986.[5] Powley, E.J., and Stepney, S. Automorphisms of Transition Graphs forLinear Cellular Automata. Journal of Cellular Automata, 4(4):293–310,2009.[6] Riedel, J., Zenil, H. Cross-boundary Behavioural ReprogrammabilityReveals Evidence of Pervasive Turing Universality, arXiv:1510.01671(preprint).[7] Wolfram, S., Statistical Mechanics of Cellular Automata, Review of ModernPhysics, 55, 601–644, 1983.[8] Wolfram, S.
A New Kind of Science , Wolfram Science, Chicago, Il., 2002.[9] Zenil, H. Compression-based Investigation of the Dynamical Properties ofCellular Automata and Other Systems,
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Supplementary Material
A.1 Non-symmetric CA rules
Local rules define the dynamical behaviour of CA. However, not all rules showessentially different dynamical behaviour. To focus on the number of rules in arulespace which show essentially different dynamical properties, one can intro-duce the following symmetry transformations:
Reflection : f r ( x , x , ..., x n ) = f ( x n , ..., x , x ) (2) Conjugation : f c ( x , x , ..., x n ) = q − − f ( q − − x , q − − x , ..., q − − x n ) (3) Joint transformation , i.e., conjugation and reflection: f c ◦ f r ( x , x , ..., x n ) = q − − f ( q − − x n , ..., q − − x , q − − x ) (4)Under these transformations two CA rules are equivalent, and they induceequivalence classes in the rulespace. Taking from each equivalence class a singlerepresentative (by convention the one with the smallest rule number), one getsa set which contains essentially different rules, i.e. rules which exhibit differentglobal behaviour.Let G ( χ ( f r ) , χ ( f c ) , χ ( f c r )) be a group under the operation ◦ acting on a set X of all possible neighbourhood templates. Using the orbit counting theoremone can formulate the following theorem:1 | G | (cid:88) g ∈ G χ ( g ) = χ ( f I ) + χ ( f r ) + χ ( f c ) + χ ( f cr )4 (5)with χ ( g ) being the number of elements of X fixed by g . A.2 ECA Rulespace and Beyond
The ECA rulespace contains 88 essentially different rules and 9 linear (additive)rules (0, 15, 51, 60, 90, 105, 150, 170, 204). The Wolfram classification groupsthe ECA rules as follows: • •
65 Class 2 ECA rules (1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 23,24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 42, 43, 44, 46, 50, 51, 56, 57,58, 62, 72, 73, 74, 76, 77, 78, 94, 104, 108, 130, 132, 134, 138, 140, 142,152, 154, 156, 162, 164, 170, 172, 178, 184, 200, 204, 232) •
11 Class 3 ECA rules (18, 22, 30, 45, 60, 90, 105, 122, 126, 146, 150) • ◦
37 is not an ECA but a CA in a higher rulespace, we cancheck whether it also belongs to the 4-colour rulespace with range r = 1 byfinding the smallest separable icon network.20 ule:Composition Rule:Composition Rule:Composition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Figure 8: A minimal generating set of the ECA rulespace. Rules that are notcomposite are prime rules; all others are composites. Each list on the right handside is an ordered list of Boolean compositions for all 88 non-symmetric ECArules. Rules are not interchangeable except in a few cases, and therefore ECAis a space that is neither closed nor commutative under composition.21a) Rule mapping of 4-colour rule 50 ◦
37. (c) Space-time of 4-colour rule50 ◦
37 with gray scale.(b) Space-time of 4-colour rule 50 ◦
37. (d) Space-time of rule 50 ◦ ◦
37. Space-times start fromrandom initial conditions for illustration purposes (a) The rule icon is separableand therefore the resulting composition can be represented in the larger 4-colourCA rulespace. (b) Emulation of the 4-colour equivalent of rule 50 ◦
37 withcolour mapping (cid:3) → (cid:3)(cid:3) , (cid:4) → (cid:4)(cid:4) , (cid:4) → (cid:3)(cid:4) and (cid:4) → (cid:4)(cid:3) . (c) Emulation ofthe 4-colour equivalent of rule 50 ◦
37 with colour re-mapping (cid:3) → (cid:3) , (cid:4) → (cid:4) , (cid:4) → (cid:4) and (cid:4) → (cid:4) . This ‘simulates’ a black and white representation of therulespace-time displayed. (d) Emulation of 2-colour rule 50 ◦◦