two dimensional signed majority
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Two dimensional signed majority Universidad Adolfo Ibanez- Chile antonio.chacc@gmail.com Remains constant at 1 If 0 otherwise Decision problem PRE: given an initial configuration and a specific node at value 0. does there


  1. Two dimensional signed majority Universidad Adolfo Ibanez- Chile antonio.chacc@gmail.com

  2. Remains constant at 1 If 0 otherwise

  3. Decision problem PRE: given an initial configuration and a specific node at value 0. does there exist T>0 such that this node becomes 1?

  4. Theorem (P. Montealegre, I. Todinca, E:G (1911)) Given an undirected graph G if the maximum degree ≥ 5, PER is P-complete. Else PRE belongs to NC

  5. Clearly PRE belongs to P, because in almost O(n) steps the dynamics reaches the steady state. The proof of P-Completeness consist to simulate the monotone circuits behavior inside the strict majority dynamics.

  6. 1 1 DIODE

  7. Information only flows to the right

  8. 1 1 OR gate And Gate Diode arc

  9. For the case maximun degree ≤ 4 one may reduce the problem to compute connected and biconnected components in the graph, which one may do in a PRAM in See Jaja …………

  10. Max degree ≤ 4 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 Decision site 0 0 0 0 1 1 Alliances 0 Its vertices never change

  11. 0’s Connected component 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Decision site 0 0 0 0 0

  12. The Complexity of the majority vote rule for planar graphs

  13. Decision problem PRE: given an initial configuration and a specific node at value 0. Does there exist T>0 such that this node becomes 1?

  14. We consider the similar decision problem PER This problem has been studied by C. Moore for d-dimensional regular lattices with nearest interactions Von Neumann neighborhood Nearest neighborhood in 2D In 3D PER is P-Complete for d ≥ 3 open for d = 2 (C. Moore)

  15. For planar graphs PRE is P-Complete (P. Montealegre, E:G, 2012)

  16. PRE is in P Majority is a particular case of a threshold network: Since G is undirected W is a nxn symmetric matrix and the threshold: Odd neighborhood Even neighborhood The parallel dynamic is driven by Which is strictly decreasing and bounded So PRE is in P

  17. GADGETS FOR CIRCUITS wire Duplicate a signal = diode

  18. AND-gate OR-gate

  19. The cross-over gadget (traffic light ) diode

  20. Cross-over from a to e

  21. We will study 2D majority with signs

  22. Symmetric majority Antisymmetric majority Periodic configuration Asymmetric majority Initial Several attractor condition steps

  23. EG,1980 E.G. P. Montealegre, I Todinca, 2013 E.G., P. Montealegre, 2014

  24. F 1 simulation of AND OR gates (no cross over)

  25. F 5

  26. Gracias !!!

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