1. Peter Banda banda@fmph.uniba.sk
Binary Cellular Automata Approach to Anonymous, Self-Stabilizing Leader Election on Rings
Peter Banda

2. Agenda Introduction Leader Election Problem Cellular Automaton
Computational Mechanics
Cellular Automata Evolution
Leader Election By Cellular Automata
Upper Bound Performance
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3. Introduction Reductionism Complexity Approach
Analytical decomposition – system reduced to its parts
Superstars of 20th century science – particle physics and molecular biology
Not sufficient
Complexity Approach
Holism – organization principles
Emergence - one H2O is not water, one neuron is not consciousness → system is > Σ constituting components
Circular causality, feedback loops
Unpredictability – small change in the cause implies dramatic effects
Structural simplicity and uniformity vs. complex global dynamics - Cellular automaton (CA)
Leader election problem – fundamental distributed problem
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4. Goals
Find solution for CA instance of leader election problem by employing genetic algorithms.
Analyze CA dynamics to understand basis of global, collective computation – Computational Mechanics.
Find barriers (upper bound performance) of binary CA approach.
Track complex phenomena and uncover information processing in nature.
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5. Leader Election – Distributed Algorithms
Definition of Leader Election
Distributed system is required to reach configuration in which the state of exactly one processor is, and then remains, within defined subset. The states of all other processors remain outside the subset.
Important prerequisite of other distributed algorithms
UID-Based Protocols
Processors are required to be distinguishable by (comparable) unique identifiers (UIDs); Le Lann, Chang & Roberts, Hirshberg & Sinclair etc.
Anonymous Protocols
(Angluin) No anonymous, deterministic algorithm (full symmetry of system).
Randomized model - random assignment of pseudo-IDs.
Self-stabilization
System eventually reaches legitimate state, no matter what initial configuration is.
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6. Leader Election – Our CA Model
Our distributed model of binary cellular automaton
Anonymous, deterministic, uniform, synchronous, self-stabilizing and shared-memory based operating on bidirectional rings.
Main aspects:
Deals with one of fully symmetric instances of problem that are principally insolvable.
Besides the inherent limitations on initial configurations, it is a one of the most fundamental system capable of leader election reaching very satisfactory performance of % on arbitrary initial configuration.
Minimal possible memory - binary state; O(N) time complexity
Compared to other distributed models does not require any additional prerequisites as centralized demon, oracle, randomization etc.
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7. Leader Election in Nature
Important role for global coordination, decision making and spatial orientation of variety of social and biological systems (sociobiology, development biology)
Shared consensus vs. despotic decision
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8. Cellular Automaton - Definition
ECA 110
Structurally simplest, discrete, dynamical system
Consists of the lattice of cells (size N)
Cell state sti  (i  N), conf. st = (st0,…,stN-1)
Neighborhood : N → n
Transition rule : n →  st+1i = (ti)
Global transition rule : N → N st+1 = (st)
Ensemble operator : 2 → 2  = N
CA – regular language processor
Neighborhood
r = 3
Local State sti
Configuration st t
Global transition
rule 
Transition table 
Configuration
st+1
t+1
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9. Cellular Automaton - Dynamics
Phenomenology
Wolfram’s classes, Edge of chaos concept
Global evolution
Ensemble operator update: L(Mt+1) = t+1 = (t)
FME algorithm: Mt+1 = [T o Mt]OUT
Limit sets -  = ∩t=0 t
Algebraic methods
Qualitative theory of dynamical systems
Computational Mechanics
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10. Computational Mechanics of CA
Analysis using concepts from the computation and dynamical system theories
Introduced by Crutchfield, Young and Hanson at SFI (1993)
The global, collective dynamics of CA can be understood and described in terms of space-time structures:
Regular domains – regular background of computation
Particles – carrier of information
Particle interactions – information processing
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11. Computational Mechanics – Regular Domains
Homogenous space-time region containing the same set of configurations appearing invariantly over and over again
Regular domain j is a process language consisting of a set of spatial configurations that fulfils the following two properties:
1. Temporal invariance - CA dynamics represented by  maps j to itself, i.e. p(j) = j, (time periodicity)
2. Spatial homogeneity - j is spatial translation invariant, the process language of a j is strongly connected, (spatial periodicity)
Can be identified either by eye inspection or by employing ε-machine reconstruction
Domains  = {0, 1,…} with associated DFAs can be filtered out from space-time diagrams by so called domain filter
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12. Computational Mechanics – Domain Filter
Domains identification
ε-machine reconstruction
Domain filter construction
Filtering space-time diagram
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13. Computational Mechanics - Particles
Particle (marked by letter of Greek alphabet) is a spatially localized and temporally periodic structure at the boundary of two domains
Displacement d - the number of cells, particle is shifted during one period
Velocity v is calculated as v = d / p
Particles code information about domains they separate
Geometric (“billiard”) model of computation
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14. Computational Mechanics - Interactions
Embedded Information processing in the computational mechanics
Interaction of particle α with particle β resulting to the production of particle  is denoted as α + β → 
Interaction types:
Decay - α → β + 
React - α + β →  + δ
Annihilate - α + β →  (i)
Different results according particle phases
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15. Computational Mechanics – Particle Catalog
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16. Evolutionary Cellular Automata
First performed by Packard in ‘88
Main research formed by EvCA research group at SFI
Chromosome
binary vector of length k2r+1 coding a transition table outputs (i)
Fitness defined by computational task T:
Performance
fraction of correctly computed ICs (in terms of computational task) to the total number of test ICs I (TMAX = 2N)
One-point cross-over and elitist selection
Example: Density classification, synchronization, prison dilemma etc.
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17. Evolutionary Cellular Automata (cont.)
Population
Individual
Genetic Algorithm Γ
Cellular Automaton 
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18. Evolution of Leader Election
Computational task T:
Binary CA with radius r = 3
Jump from the fitness of 0.4 to 0.8 – complexity transition
Various CA strategies found by evolutions were identified and examined
Localistic strategies
Strategy of mandatory function
Density reduction
Divide and eliminate strategy
Particle-based strategies
First particle-based strategy
Strategy of mirror particles
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19. Localistic Strategies
Strategy of mandatory function
includes 2r + 2 mandatory transitions securing the final leader configuration to be a stable point
Density reduction
reduces the number of active cells (density) to the minimum
contains zeros at the position of almost all bits (except ( ) and ( ))
Divide and eliminate strategy
division splits continuous regions of active or inactive cells into (10)+1, resp. (100)+1
elimination operates on these sequences and reduces them from one or both sides
Performance: ,
Performance decreases rapidly with regard to N
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20. First Particle-Based Strategy
Leader itself is considered as particle - 
The crucial particle  provides
Leader election function:  +   
Cleaning up functions:  +   ,
 +   ,  +   
Main issues
All particles except  have positive velocities
Particle  itself is very slow (just -1/3)
Particles  and  do not check the whole configuration before they create leader → more leaders possible
Performance decreases very slowly with regard to N
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21. First Particle-Based Strategy (Cont.)
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22. Strategy of Mirror Particles
Occurrence of "mirror" particles with opposite velocities
Leader election mechanism
Interaction  +    +  and  +   
Collision of  +  indicates that automaton is ready to enter the final phase
Particles  and  shift around the whole configuration and check remaining particles
Particles generations:
Interaction    +  and    +  + 
Particle phases (phase shift of )
High velocities of colliding particles, fosters overall performance.
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23. Strategy of Mirror Particles (Cont.)
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24. Strategy of Mirror Particles - N  5 mod 6 limitation
The product of crucial leader electing interactions  +  and  +  depends on the phases of colliding particles
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25. Upper Bound Performance – Homogenous ICs
Homogenous IC (0N, 1N)
Symmetry breaking issue (Angluin)
insolvable by any 1D, deterministic, uniform CA
All cells are always in the same state → no leader can be elected
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26. Upper Bound Performance – Stable Point Limit.
Leader Election definition → Mandatory transition function
ICs with loose-coupled active cells that keep a distance  2r + 1 must be stable points, hence they are insolvable
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27. Upper Bound Performance
Upper bound performance of binary CA with ICs generated as
Density-uniform
Uniform
where
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28. Upper Bound Performance
Comparison with performance of Improved strategy of mirror particles:
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29. Thank you for your attention :)
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30. Questions ???
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