Two dimensional signed majority Universidad Adolfo Ibanez- Chile - PowerPoint PPT Presentation

Apr 23, 2023 •20 likes •380 views

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

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 exist T>0 such that this node becomes 1?
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
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.
1 1 DIODE
Information only flows to the right
1 1 OR gate And Gate Diode arc
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 …………
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
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
The Complexity of the majority vote rule for planar graphs
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?
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)
For planar graphs PRE is P-Complete (P. Montealegre, E:G, 2012)
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
GADGETS FOR CIRCUITS wire Duplicate a signal = diode
AND-gate OR-gate
The cross-over gadget (traffic light ) diode
Cross-over from a to e
We will study 2D majority with signs
Symmetric majority Antisymmetric majority Periodic configuration Asymmetric majority Initial Several attractor condition steps
EG,1980 E.G. P. Montealegre, I Todinca, 2013 E.G., P. Montealegre, 2014
F 1 simulation of AND OR gates (no cross over)
F 5
Gracias !!!

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