Threshold Automata: dynamics and complexity Studium Institute-Orleans Universidad Adolfo Ibanez- Chile LIFO-Université d’Orléans Antonio.chacc@gmail.com
Topics: 1) Threshold Networks. 2) Updating schemes and dynamics over undirected graphs. 3) Characterization of the convergence to fixed points or cycles. 4) Related decision problems and computational complexity. .
Threshold networks x ∈ {0,1} n n for 1 ≤ i ≤ n ∑ x i = H ( w ij x j − b i ) " j = 1 W = ( w ij ) the weight integral matrix b = ( b i ) the threshold vector if H ( u ) = 1 u ≥ 0 0 otherwise
The dynamics Block- sequential updates : Consider a partition of the set {1, …, n} { I 1 ,..., I p } We update the blocks one by one: To update the k-th block we consider the new state of every sites belong to previous blocks. Parallel or synchronous update: only one block. Every site is updated at the same time. Sequential update: n-blocks of cardinality one: sites are updated one by one in a prescribed order .
_ 1 2 3 EXAMPLE {1,2,3} Some Block-Sequential {1} {2,3} {1,2} {3} partitions for three sites {1} {2} {3} F {1,2,3} ( x 1 , x 2 , x 3 ) = ( x 2 , x 1 + x 3 , ¬ x 2 ) F {1,2}{3} ( x 1 , x 2 , x 3 ) = ( x 2 , x 1 + x 3 ,( ¬ x 1 )( ¬ x 3 )) F {1}{2,3} ( x 1 , x 2 , x 3 ) = ( x 2 , x 2 + x 3 , ¬ x 2 ) F {1}{2,3} ( x 1 , x 2 , x 3 ) = ( x 2 , x 2 + x 3 ,( ¬ x 2 )( ¬ x 3 ))
Two cycle 101 111 001 010 101 F {1,2,3} 000 001 011 110 000 100 111 F 010 100 {1,2}{3} 011 110 101 111 F {1}{2,3} Block 000 001 011 110 sequential diagrams 100 010 000 001 010 100 011 110 101 111 F {1}{2}{3}
We consider only symmetric integral threshold networks. i.e. W being a symmetric matrix with integral entries. W=W(G) is the symmetric incidence matrix of a weighted graph G=(V,E) 2 # 0 2 1 0 & 1 2 % ( 2 0 5 1 % ( 1 1 W = 5 % 1 5 0 0 ( 3 4 % ( 0 1 0 − 1 $ ' -1
Example of dynamics for symmetric threshold networks We consider a 4x4 lattice with periodic conditions, nearest interactions, states 0 or 1, and the local majority function: If the number of ones is bigger or equal to the number of zeros then the site takes the value 1 x i − 1, j + x i + 1, j + x i , j − 1 + x i , j + 1 ≥ 2 x ' ij = 1 iff
Dynamics: two cycles and fixed points; different behavior for different updates
For arbitrary matrices W previous model may accept, Iterated in parallel or sequeneally, long period cycles and transients ….. But when W is symmetric the network converges to fixed point or two periodic cycles (parallel update), And, if diag(W)≥0 to fixed point (sequeneal update) . E.G, J. Olivos, Periodic behaviour of generalized threshold functions, Discrete mathematics, vol 30, pp 187-189, 1980. E.G., Fixed Point behavior of threshold functions on a finite set, SIAM Journal on Alg. And Discrete Methods, vol 3(4), pp 2554-2558, 1982.
Further for W symmetric the network admits an energy: Parallel update: n n n ∑ ∑ ∑ ( x i ( t ) + x i ( t − 1)) E ( x ( t )) = − x i ( t ) w ij x j ( t − 1) + b i i = 1 j = 1 i = 1 If diag (W) ≥ 0, Sequential update: n n n E ( x ) = − 1 ∑ ∑ ∑ x i w ij x i x j + b i 2 i = 1 j = 1 i = 1
Which implies that: 1) for the parallel updating the attractors are only Fixed points or two cycles. 2) For the sequential updating and diag(W)≥0 there are only fixed points. 3) In both situations transients are bounded by α ⎪⎪ W ⎪⎪ x ⎪⎪ b ⎪⎪ x ≠ x " If and only if x ( t ) ≠ x ( t − 2) Δ E = E ( x ( t )) − E ( x ( t − 1) < 0 And for the sequeneal iteraeon iff Δ E = E ( x ') − E ( x ) < 0 x ≠ x "
The most general dynamical result: Consider the block-sequential scheme s = { I 1 ,..., I p } The symmetrical threshold network T =(W, b, s ) Let the sub-matrix associated to the k-th block W ( I k ) If for every is non-negaeve-definite W ( I k ) k ∈ {1,..., p } The network converges to fixed points E. G., F. Fogelman-Soulie, D. Pellegrin, Decreasing energy functions as a tool For studying threshold networks, Discrete Applied Mathematics, vol 12, pp261-277, 1985 .
Sketch of the proof: x ' = ( x I 1 ,..., x I k − 1 ,. x ' I k , x I k + 1 ,..., x I p ) The update of the k-th block: n x j − b i ) − 1 ∑ ∑ ∑ ∑ Δ E = − ( x ' i − x i )( w ij ( x ' i − x i ) ( x j ' − x j ) 2 i ∈ I k j = 1 i ∈ I k i ∈ I k − 1 ∑ 2 y t W ( I k ) y Δ E = δ i i ∈ I k where y = ( x ' − x ) ∈ { − 1,0,1} n n ∑ δ i = − ( x ' i − x i )( w ij x j − b i ) j = 1 δ i ≤ − 1 x’ ≠x ⇒ there exists i ∈ {1,.., n } such that 2 (since W is an integral matrix) Then Δ E < 0
We will suppose now that every matrix is the incidence matrix of an undirected graph G=(V,E), so their entries belong to the set {0,1} W=W(G)= eventually with loops ( w ij ) ( w ii = 1) Consider the quantity: α ( G ) = − n − k + 2 m − 4 p n = |V|, m =|E|, (without loops) K = the number of loops, P = the minimum number of edges to remove such that the sub-graph becomes bipartite.
Example 2 1 k = 2 |V| = 4 |E| = 6 p = 2 3 4 2 1 Maximum bipartite sub-graph 3 4 α ( G ) = − 4 − 2 + 2 × 6 − 4 × 2 = − 2 < 0
Theorem-1 Consider an undirected graph G=(V,E), W=W(G), b being a threshold vector. and the network updated in parallel, N= (W, b, {1, …,n}) Fixed points for any For any G’ sub-graph of G (by deleting vertices) ⇒ α ( G ') < 0 threshold vector There exists a threshold vector ⇒ α ( G ') ≥ 0 such that two cycles appears
Parallel update f 1 ( x ) = H ( x 2 + x 3 + x 4 − 3 2) 2 1 f 2 ( x ) = H ( x 1 + x 3 − 1 Two-cycle 2) ⇒ ( x 1 , x 2 , x 3 , x 4 ) = (1,0,1,0) ↔ (0,1,0,1) f 3 ( x ) = H ( x 1 + x 2 + x 4 − 3 2) 3 4 f 4 ( x ) = H ( x 1 + x 3 + x 4 − 3 2) α ( G ) = − 5 + 2 × 5 − 4 = 1 ≥ 0 There exists a sub-graph with α ( G ) ≥ 0 α ( G ) = 0 α ( G ) = − 2 α ( G ) = − 2 2 1 2 2 1 1 f 1 ( x ) = H ( x 2 − 1 2) f 2 ( x ) = H ( x 1 − 1 2) 3 3 4 (1,0) ↔ (0,1) Two-cycle
Parallel updating on two families of graphs Biparete graphs (k=0) α ( G ) = − 2 n + 2 m n>m α ( K n ) < 0 ⇒ ⇒ Only fixed points (G is a forest) Complete graphs with n loops In this situation, the minimum number of edges to remove to obtain a bipartite graph for n=2q p = 2 q ( q − 1) p = 2 q 2 for n=2q+1 Complete graphs updated in ⇒ α ( K n ) < 0 Parallel converges to fixed points
Parallel Updating n=4 Fixed points Two-Cycles 0≤k≤2 3≤k≤4 k=number 1≤k≤4 k=0 of loops ∅ 0≤k≤4 3≤k≤4 0≤k≤2 1≤k≤4 k=0
Connected graphs for n=5 with 5 loops . α ( G ) = − n + m − 2 p 2 In red the edges to be removed for a maximum bipartite graphs
Theorem-II: attractors for every block-sequential update. Consider the block-sequential scheme s = { I 1 ,..., I p } The symmetrical threshold network T =(W, b, s ) Let the graph associated to the k-th block G ( I k ) fixed points ⇒ G ' ⊆ G ( I k ) k ∈ {1,..., p } ∀ ∀ α ( G ') < 0 k ∈ {1,..., p } and G ' ⊆ G ( I k ) such that cycles ∃ ⇒ α ( G ') ≥ 0
Corollary Consider an undirected graph G=(V,E) with every loop (diag(W)=n) and the block-sequential scheme s = { I 1 ,..., I p } ⇒ Fixed points ∀ | I k | ≤ 3 k ∈ {1,..., p } Otherwise, there exist graphs and threshold vectors such that cycles appear
Sketch of the proof: Partition size =1 directly from the fact that diag(W)≥0 Pareeon size = 2 α ( G ) = − 2 Partition size= 3 α ( G ) = − 2 α ( G ) = − 4
Cycles for block-sequential updates Every undirected graph with at least two connected vertices without loops admits cycles − 1 ∑ f 1 ( x ) = H ( x 2 + x j 2) j ∈ V 1 \{2} − 1 ∑ f 2 ( x ) = H ( x 1 + x j 2) j ∈ V 2 \{1} Every site {3, ..,n} 1 2 is constant at state 0 α ( G ({1,2},{(1,2)})) = − 2 + 2 × 1 = 0 ! ! ( x 1 , x 2 , ! x ) = (1,0, 0 ) ↔ (0,1, 0 ) Two cycle for any pareeon τ = {{1,2}, I 2 ,..., I p }
Non-Polynomial Cycles 2 3 4 5 8 1 6 7 staircase 3’ 6’ 2’ 4’ 5’ 7’ 8’ 1’ 2 3 4 Local majority at 3’ 2’ 4’ ’ each vertex f 3 ( x ) = H ( x 2 + x 3' + x 4 − 3 2) f 3' ( x ) = H ( x 2' + x 3 + x 4' − 3 2)
0 0 0 1 0 0 Local Majority 1 1 1 0 1 1 X X’ = Updated vertices 0 0 0 1 1 0 1 1 1 0 0 1 1 Travel to 0 The right 0 0 0 0 1 0 1 1 1 1 0 1
Block-Sequential updating τ = {{1,1'},{ n , n '},{ n − 1,( n − 1)'},...{3,3'},{2,2'}} 0 0 0 0 0 0 0 1 X(0) 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 X(1) 0 0 1 1 1 1 1 1 Cycle of period T=n-1
Union of the first l prime number’s staircases of size p 1 + 1 = 3; p 2 + 1 = 4; p 3 + 1 = 6, p 4 + 1 = 8,...., p l + 1 l ∪ So by considering the global partition τ = τ k k = 1 l ( ) | V ( G )|log| V ( G )| The period of the network is ∏ Ω T ≥ p k = e k = 1 Same arguments can be done for the transient time.
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