Simply modified GKL density classifiers that reach consensus faster
SSimply modified GKL density classifiers that reach consensus faster
J. Ricardo G. Mendonça a,b, ∗ a LPTMS, UMR 8626, CNRS, Université Paris-Sud, Université Paris Saclay, 91405 Orsay CEDEX, France b Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, Rua Arlindo Bettio 1000, 03828-000 São Paulo, SP, Brazil
Abstract
The two-state Gacs-Kurdyumov-Levin (GKL) cellular automaton has been a staple model in the study of complex systems due toits ability to classify binary arrays of symbols according to their initial density. We show that a class of modified GKL models overextended neighborhoods, but still involving only three cells at a time, achieves comparable density classification performance butin some cases reach consensus more than twice as fast. Our results suggest the time to consensus (relative to the length of the CA)as a complementary measure of density classification performance.
Keywords:
Cellular automata, density classification problem, spatially distributed computing, emergenceIn 1978, Gacs, Kurdyumov, and Levin (GKL) introducedthe density classification problem for cellular automata (CA)in the literature [1–3]. The problem consists in classifying ar-rays of symbols according to their initial density using localrules, and is completed successfully if all the cells of the CAconverge to the initial majority state in linear time in the sizeof the input array. Density classification is a nontrivial task forCA composed of autonomous and memoryless cells becausethe cells have to achieve a global consensus cooperating lo-cally; emergence of collective behavior is required. The GKLtwo-state model, or GKL-II for short, became a staple modelin the theory of complex systems related with the concepts ofcommunication, efficiency, and emergence [4–6]. It has beendemonstrated that the density classification problem cannot besolved correctly of the times by uniform two-state CA,although no upper bound on the maximum possible efficiencyhas been set [7, 8]. Solutions involving nonuniform CA andless strict criteria for what a solution to the problem means ex-ist [9, 10]. Recent reviews on the density classification prob-lem for CA are given in [11, 12].The GKL-II CA is a finite one-dimensional array of 𝑛 ≥ cells under periodic boundary conditions evolving by the ac-tion of a transition function Φ II ∶ {0 , 𝑛 → {0 , 𝑛 that giventhe state 𝒙 𝑡 = ( 𝑥 𝑡 , … , 𝑥 𝑡𝑛 ) of the CA at instant 𝑡 determines itsstate 𝒙 𝑡 +1 = Φ II ( 𝒙 𝑡 ) at instant 𝑡 + 1 by the majority rule 𝑥 𝑡 +1 𝑖 = { maj( 𝑥 𝑡𝑖 −3 , 𝑥 𝑡𝑖 −1 , 𝑥 𝑡𝑖 ) , if 𝑥 𝑡𝑖 = 0 , maj( 𝑥 𝑡𝑖 , 𝑥 𝑡𝑖 +1 , 𝑥 𝑡𝑖 +3 ) , if 𝑥 𝑡𝑖 = 1 , (1)where maj( 𝑝, 𝑞, 𝑟 ) = ⌊ ( 𝑝 + 𝑞 + 𝑟 ) ⌋ for - variables 𝑝 , 𝑞 , 𝑟 . The CA classifies density if 𝒙 𝑡 → = (0 , … , or =(1 , … , depending whether, respectively, the initial density ∗ Permanent address: Escola de Artes, Ciências e Humanidades,Universidade de São Paulo, SP, Brazil. Email: [email protected] . 𝜌 = 𝑛 −1 ∑ 𝑖 𝑥 𝑖 < or 𝜌 > . We do not require adefinite behavior when 𝜌 = 1∕2 . The CA is supposed to reachconsensus in 𝑂 ( 𝑛 ) time steps. In [1, 3], the authors prove thatthe GKL-II CA on the infinite lattice ℤ displays the eroderproperty, washing out finite islands of the minority phase infinite time and eventually leading the CA to one of the twoinvariant states or . In an array of 𝑛 = 149 cells (odd lengthto avoid ties), GKL-II scores an average density classificationperformance of . over random initial conditions with eachcell initialized in the state or equally at random (Bernoulliproduct measure), taking on average
86 ∼ 0 . 𝑛 time stepsto reach consensus. Details on the GKL-II performance aregiven in [4–6, 11–14].We now modify the neighborhood in the GKL-II. Instead ofevaluating the majority vote of cell 𝑖 with its nearest 𝑖 ± 1 andthird 𝑖 ± 3 neighbours, we pick neighbors 𝑖 ± 𝑗 and 𝑖 ± 𝑘 , with 𝑘 > 𝑗 ≥ . The rules for the modified CA read 𝑥 𝑡 +1 𝑖 = { maj( 𝑥 𝑡𝑖 − 𝑘 , 𝑥 𝑡𝑖 − 𝑗 , 𝑥 𝑡𝑖 ) if 𝑥 𝑡𝑖 = 0 , maj( 𝑥 𝑡𝑖 , 𝑥 𝑡𝑖 + 𝑗 , 𝑥 𝑡𝑖 + 𝑘 ) if 𝑥 𝑡𝑖 = 1 . (2)We refer to this CA as GKL( 𝑗, 𝑘 ) ; GKL(1 , recovers the orig-inal GKL-II model. To the best of our knowledge these mod-els have never been considered in the literature before. Wemeasured the average density classification performance ⟨ 𝑓 ⟩ of GKL( 𝑗, 𝑘 ) over random initial states close to the crit-ical density ( 𝑥 𝑖 = 0 or equally at random) in an array of 𝑛 = 299 cells to minimize finite-size effects that show up inthe rules with larger ( 𝑗, 𝑘 ) . Our results appear in Table 1. Wesee that the GKL( 𝑗, 𝑘 ) with 𝑘 = 3 𝑗 , i. e., the GKL( 𝑗, 𝑗 ) mod-els, all display virtually the same density classification and rel-ative time to consensus ( ⟨ 𝑡 ∗ ⟩ ∕ 𝑛 ) performances. Otherwise, the GKL(1 , and GKL(1 , models display almost the samedensity classification performance as GKL-II but achieve con-sensus in about half the time. Explicitly, GKL(1 , is justabout . less efficient than GKL-II but is ∼2 . times faster. Submitted to Elsevier on January 15, 2019 Published in Physics Letters A 383 (19) (2019) 2264–2266 a r X i v : . [ n li n . C G ] M a y able 1: Best density classification performances of GKL( 𝑗, 𝑘 ) in the range ≤ 𝑗 ≤ , 𝑗 < 𝑘 ≤ in an array of 𝑛 = 299 cells averaged over random initial configurations near the critical density 𝜌 = 1∕2 . The uncertainty in the performance ⟨ 𝑓 ⟩ is ±0 . . GKL-II figures are displayedin bold for comparison. ( 𝑗, 𝑘 ) (4 ,
12) (3 ,
9) (2 ,
6) (5 ,
15) ( , ) (1 ,
9) (1 ,
11) (2 ,
14) (2 ,
10) (3 ,
15) (1 ,
7) (1 , ⟨ 𝑓 ⟩ . . . . . . . . . . . . ⟨ 𝑡 ∗ ⟩ ∕ 𝑛 . . . . . . . . . . . . l l l l l l l l l l l l l l l
0% 1% 2% 3% 4% 5% % % % % % % d n Æ f æ l (1,3)(1,5)(1,7)(1,9)(1,11) l l l l l l l l l l l l l l l
0% 1% 2% 3% 4% 5% . . . . . . . . d n Æ t * æ n l (1,3)(1,5)(1,7)(1,9)(1,11) Figure 1: Density classification performance ⟨ 𝑓 ⟩ and time to consen-sus ⟨ 𝑡 ∗ ⟩ ∕ 𝑛 averaged over random initial configurations of some GKL(1 , 𝑘 ) CA of length 𝑛 = 299 as a function of the relative imbal-ance 𝛿 ∕ 𝑛 = ( 𝑛 − 𝑛 )∕ 𝑛 in the initial configurations. Error bars aremuch smaller than the symbols shown. From Table 1 we conclude that if quality is critical, then
GKL(4 , is the best CA in its class, while if one needs speed,then GKL(1 , or GKL(1 , becomes the CA of choice.Figure 1 displays the average classification performance ofthe GKL(1 , 𝑘 ) CA as a function of the imbalance 𝛿 = ( 𝑛 − 𝑛 ) between the number of cells in states and in the initial con-figuration. Here the initial density 𝜌 = 1∕2 + 𝛿 ∕ 𝑛 is fixed butthe configurations are random. By symmetry, the performanceof the CA depends only on the magnitude of 𝛿 , not on its sign.The data show that the density classification performance ofall these CA are close over a range of initial densities, differingsignificantly, however, on the time to consensus. Space-timediagrams of some GKL( 𝑗, 𝑘 ) CA are displayed in Figure 2.We do not currently have a sound explanation for the ef-ficient combinations of 𝑗 , 𝑘 found. The efficiency of the GKL( 𝑗, 𝑗 ) can be related with that of GKL(1 , in one ormore sublattices, although the fast convergence of GKL(1 , and GKL(1 , cannot be immediately related with any sub- lattice dynamics. Intuitively, in the GKL( 𝑗, 𝑘 ) CA informationabout the dynamics of the interfaces between islands of s and s can jump over longer distances (i. e., move faster) with in-creased 𝑘 − 𝑗 . Data from Table 1 for the time to consensus for GKL(1 , 𝑘 ) with 𝑘 = 3 , , , , and corroborates this idea.Note that the metric ⟨ 𝑡 ∗ ⟩ ∕ 𝑛 is not unique—one could considerthe alternative timings given by ⟨ 𝑡 ∗ ⟩ ∕ 𝑛𝑘 , with 𝑘 the radius ofthe CA, as well as ⟨ 𝑡 ∗ ⟩ ∕ 𝑛𝑧 , with 𝑧 the number of cells thatenter the local rule ( 𝑧 = 3 for all GKL( 𝑗, 𝑘 ) ). A characteriza-tion of the “computational mechanics” of the GKL( 𝑗, 𝑘 ) CA[4–6, 15] may help to understand their eroder mechanism andtheir efficiency better. It would also be of interest to assessthe robustness of the
GKL( 𝑗, 𝑘 ) against noise and whether theensuing probabilistic CA display an ergodic-nonergodic transi-tion, a long-standing unsettled issue for one-dimensional den-sity classifiers [1–3, 12–18]. These and related questions (e. g.,how ⟨ 𝑓 ( 𝑛, 𝛿 ) ⟩ → for any 𝛿 ≠ as 𝑛 ↗ ∞ , see [8]) will bethe subject of forthcoming publications. Acknowledgments
The author thanks Nazim Fatès (LORIA, Nancy) for usefulconversations and an anonymous reviewer for valuable sugges-tions improving the manuscript. The author also acknowledgesthe LPTMS for kind hospitality during a sabbatical leave inFrance and FAPESP (Brazil) for partial support through grantno. 2017/22166-9.
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GKL( 𝑗, 𝑘 ) with 𝑛 = 299 , ≤ 𝑡 ≤ (time flows downwards), and random initial conditions with 𝜌 = 150∕299 .From left to right, top to bottom, ( 𝑗, 𝑘 ) = (1 , (the usual GKL-II) and (2 , , (3 , and (4 , , (1 , and (1 , . The diagrams displayed are thosethat reached or would eventually reach the majority state of all- cells (in black).[7] M. Land, R. K. Belew, No perfect two-state cellular automatafor density classification exists, Phys. Rev. Lett. 74 (25) (1995)5148–5150.[8] A. Busić, N. Fatès, I. Marcovici, J. Mairesse, Density classifica-tion on infinite lattices and trees, Electron. J. Probab. 18 (2013)51.[9] H. Fukś, Solution of the density classification problem with twocellular automata rules, Phys. Rev. E 55 (3) (1997) R2081–R2084.[10] M. Sipper, M. S. Capcarrere, E. Ronald, A simple cellular au-tomaton that solves the density and ordering problems, Int. J.Mod. Phys. C 9 (7) (1998) 899–902.[11] P. P. B. de Oliveira, On density determination with cellular au-tomata: Results, constructions and directions, J. Cell. Autom.9 (5–6) (2014) 357–385.[12] N. Fatès, Stochastic cellular automata solutions to the densityclassification problem – When randomness helps computing, Theory Comput. Syst. 53 (2) (2013) 223–242.[13] J. R. G. Mendonça, Sensitivity to noise and ergodicity of an as-sembly line of cellular automata that classifies density, Phys.Rev. E 83 (3) (2011) 031112.[14] J. R. G. Mendonça, R. E. O. Simões, Density classification per-formance and ergodicity of the Gacs-Kurdyumov-Levin cellularautomaton model IV, Phys. Rev. E 98 (1) (2018) 012135.[15] P. Gács, I. Törmä, Stable multi-level monotonic eroders,arXiv:1809.09503 [math.PR].[16] J. Mairesse, I. Marcovici, Around probabilistic cellular au-tomata, Theor. Comput. Sci. 559 (2014) 42–72.[17] R. Fernández, P.-Y. Louis, F. R. Nardi, Overview: PCA mod-els and issues, in: P.-Y. Louis, F. R. Nardi (Eds.), ProbabilisticCellular Automata, Springer, Cham, 2018, pp. 1–30.[18] I. Marcovici, M. Sablik, S. Taati, Ergodicity of some classes ofcellular automata subject to noise, Electron. J. Probab. 24 (2019)41.cells (in black).[7] M. Land, R. K. Belew, No perfect two-state cellular automatafor density classification exists, Phys. Rev. Lett. 74 (25) (1995)5148–5150.[8] A. Busić, N. Fatès, I. Marcovici, J. Mairesse, Density classifica-tion on infinite lattices and trees, Electron. J. Probab. 18 (2013)51.[9] H. Fukś, Solution of the density classification problem with twocellular automata rules, Phys. Rev. E 55 (3) (1997) R2081–R2084.[10] M. Sipper, M. S. Capcarrere, E. Ronald, A simple cellular au-tomaton that solves the density and ordering problems, Int. J.Mod. Phys. C 9 (7) (1998) 899–902.[11] P. P. B. de Oliveira, On density determination with cellular au-tomata: Results, constructions and directions, J. Cell. Autom.9 (5–6) (2014) 357–385.[12] N. Fatès, Stochastic cellular automata solutions to the densityclassification problem – When randomness helps computing, Theory Comput. Syst. 53 (2) (2013) 223–242.[13] J. R. G. Mendonça, Sensitivity to noise and ergodicity of an as-sembly line of cellular automata that classifies density, Phys.Rev. E 83 (3) (2011) 031112.[14] J. R. G. Mendonça, R. E. O. Simões, Density classification per-formance and ergodicity of the Gacs-Kurdyumov-Levin cellularautomaton model IV, Phys. Rev. E 98 (1) (2018) 012135.[15] P. Gács, I. Törmä, Stable multi-level monotonic eroders,arXiv:1809.09503 [math.PR].[16] J. Mairesse, I. Marcovici, Around probabilistic cellular au-tomata, Theor. Comput. Sci. 559 (2014) 42–72.[17] R. Fernández, P.-Y. Louis, F. R. Nardi, Overview: PCA mod-els and issues, in: P.-Y. Louis, F. R. Nardi (Eds.), ProbabilisticCellular Automata, Springer, Cham, 2018, pp. 1–30.[18] I. Marcovici, M. Sablik, S. Taati, Ergodicity of some classes ofcellular automata subject to noise, Electron. J. Probab. 24 (2019)41.