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Conjunctive networks Complexity of limit cycle problems with - - PowerPoint PPT Presentation

Jan 30, 2023 38 likes •491 views

Conjunctive networks Complexity of limit cycle problems with different schedules Julio Aracena, Florian Bridoux, Luis G omez, Lilian Salinas Florian BRIDOUX Conjunctive networks 2020 1/18 Boolean networks and interaction digraph

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Conjunctive networks

Complexity of limit cycle problems with different schedules Julio Aracena, Florian Bridoux, Luis G´

  • mez,

Lilian Salinas

Florian BRIDOUX Conjunctive networks 2020 1/18

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Boolean networks and interaction digraph

Boolean networks: Global function: f : {0, 1}n → {0, 1}n. Local functions: f1, . . . , fn : {0, 1}n → {0, 1}. f (x) = (f1(x), f2(x), . . . , fn(x)) Local functions: f1 : x → 1. f2 : x → x2. f3 : x → x1∨x2. Interaction digraph Df : 1 3 2 Nin(1) = ∅, Nin(2) = {x2} and Nin(3) = {x1, x2}.

Florian BRIDOUX Conjunctive networks 2020 2/18

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Conjunctive networks

A conjunctive networks f : {0, 1}n → {0, 1}n: ∀j ∈ [n], fj : x →

  • i∈Nin(j)

xi (if Nin(j) = ∅, fj(x) = 0). Local functions: f1 : x → 0. f2 : x → x2. f3 : x → x1∨x2. Interaction digraph Df : 1 3 2

Florian BRIDOUX Conjunctive networks 2020 3/18

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Limit cycles

x ∈ {0, 1}n is in a limit cycle of f of length k if ∀1 ≤ q < k, f q(x) = x, and ∀f k(x) = x. Notations: φk(f ): number of limit cycles of length k of f : . Φk(f ): configurations in a limit cycle of length k of f : We have φk(f ) = |Φk(f )|/k.

Florian BRIDOUX Conjunctive networks 2020 4/18

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Decision problems

For any constant k, we define the following problems. Definition:k-Parallel Limit Cycle problem (k-PLC) Given a conjunctive network f , does φk(f ) ≥ 1? Definition:k-Block-sequential Limit Cycle problem (k-BLC) Given a conjunctive network f , does there exist a block-sequential schedule w such that φk(f w) ≥ 1? Definition:k-Sequential Limit Cycle problem (k-SLC) Given a conjunctive network f , does there exist a sequential schedule w such that φk(f w) ≥ 1? Remark All this problems are trivial for k = 1.

Florian BRIDOUX Conjunctive networks 2020 5/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. See: Disjunctive networks and update schedules, Eric Goles and Mathilde Noual, 2011.

Florian BRIDOUX Conjunctive networks 2020 6/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 2 4 3

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 2 4 3

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 1 2 4 3

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 1 2 4 3

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 1 1 2 4 3

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 1 1 2 4 3

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 1 1 2 4 3 1 1

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Theorem For all k ≥ 2, The k-PLC problem can be resolved in polynomial time. When Df is strongly connected, it is equivalent to know if there exists a function c : [n] → [0, k − 1] such that for all i, j ∈ [n], i ∈ Nin(j) = ⇒ c(j) = c(i) + 1 mod k. Example: 2-PLC problem for the two following interaction digraphs. 1 2 4 3 1 1 1 2 4 3 1 1

Florian BRIDOUX Conjunctive networks 2020 7/18

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k-PLC

Proof of: c exists ⇒ φk(f ) ≥ 1. 1 2 3 4 5 1 1 2 3 x(t) : ∀i ∈ [n], c(i) = t ⇐ ⇒ x(t)

i

= 1. x(0)

i f

− →x(1)

i f

− →x(2)

i f

− →x(3)

i f

− →x(0)

i

10000

f

− →01100

f

− →00010

f

− →00001

f

− →10000

Florian BRIDOUX Conjunctive networks 2020 8/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 ?11?

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 0110

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 0110 0011

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 0110 0011 1001

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 0110 0011 1001 1100

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 0110 0011 1001 1100 1 2 3

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-PLC

Proof of: φk(f ) ≥ 1 ⇒ c exists . Let x ∈ Φk(f ). Periodic trace: for all i ∈ [n], pi(x) = (xi, fi(x), f 2

i (x)), . . . , (f k−1 i

(x))) In this example, consider that i ∈ [2] with the maximum 1 in its periodic trace is 2. 1 2 3 4 5 1100 0110 0011 1001 1100 1 2 3 Is to possible to have 0 − → 2 for example? = ⇒ No, because otherwise the period would not be minimum.

Florian BRIDOUX Conjunctive networks 2020 9/18

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k-BLC et k-SLC

Lemma [ Eric Goles and Mathilde Noual, 2011] For any disjunctive network f , there exists a block-sequential update schedule w such that f w only has fixed points. Theorem The k-BLC et k-SLC problems are NP-complete. To resolve these two problems when Df is strongly connected, it is sufficient to execute the following non-deterministic polynomial time algorithm. Chose a (block)-sequential update schedule w. Chose a configuration x ∈ {0, 1}n. Verify that (f w)k(x) = x and that for all q ∈ [1, k − 1], (f w)q(x) = x. This problems are thus in NP.

Florian BRIDOUX Conjunctive networks 2020 10/18

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2-BLC and 2-SLC

When Df is strongly connected, it is equivalent to find a update digraph and a function c : [n] → [0, k − 1] such that i

⊕

− → j = ⇒ c(j) = c(i) + 1 mod k. i

⊖

− → j = ⇒ c(j) = c(i). Lemma An upgrade digraph corresponds to a sequential update schedule if when we reverse every negative arcs, the digraph becomes acyclic. Lemma An upgrade digraph corresponds to a block-sequential update schedule if when we reverse every negative arcs, the only remaining cycles are only composed of positive arcs.

Florian BRIDOUX Conjunctive networks 2020 11/18

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2-BLC and 2-SLC

1 2 7 3 5 6 8 4

⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕

V1 V2 V3 V4

Florian BRIDOUX Conjunctive networks 2020 12/18

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2-BLC and 2-SLC

1 2 7 3 5 6 8 4

⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕

V1 V2 V3

Florian BRIDOUX Conjunctive networks 2020 12/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

Florian BRIDOUX Conjunctive networks 2020 13/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3

Florian BRIDOUX Conjunctive networks 2020 13/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4

Florian BRIDOUX Conjunctive networks 2020 13/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4

Florian BRIDOUX Conjunctive networks 2020 13/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a x1 x2 ¯ x1 ¯ x2 ¯ x3 x3 c1,1 c1,2 c1,3 c1,4 c2,1 c2,2 c2,3 c2,4

Florian BRIDOUX Conjunctive networks 2020 13/18

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SLIDE 34

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 14/18

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SLIDE 35

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 14/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 14/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 f1 t1 ¯ x1 x2 f2 t2 f2 t2 ¯ x2 x3 f3 t3 f3 t3 ¯ x3 c1,1 ℓ1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 ℓ1,2 ℓ1,3 c2,1 ℓ2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 ℓ2,2 ℓ2,3 b a

Florian BRIDOUX Conjunctive networks 2020 14/18

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Reduction of 2-BLC and 2-SLC from 3-SAT

Lemma Let f be a conjunctive network and k an integer. Then, φk(f (w1...wn−1wn)) = φk(f (wnw1...wn−1)).

Florian BRIDOUX Conjunctive networks 2020 15/18

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SLIDE 39

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 16/18

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SLIDE 40

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 16/18

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SLIDE 41

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 17/18

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SLIDE 42

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 17/18

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SLIDE 43

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 f1 t1 ¯ x1 x2 f2 t2 ¯ x2 x3 f3 t3 ¯ x3 c1,1 ℓ1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 c2,1 ℓ2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 b a

Florian BRIDOUX Conjunctive networks 2020 17/18

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SLIDE 44

Reduction of 2-BLC and 2-SLC from 3-SAT

3-SAT problem: (λ1 ∨ λ2 ∨ λ3) ∧ (¬λ1 ∨ ¬λ2 ∨ λ3)

x1 f1 t1 f1 t1 ¯ x1 x2 f2 t2 f2 t2 ¯ x2 x3 f3 t3 f3 t3 ¯ x3 c1,1 ℓ1,1 ℓ1,1 c1,2 ℓ1,2 c1,3 ℓ1,3 c1,4 ℓ1,2 ℓ1,3 c2,1 ℓ2,1 ℓ2,1 c2,2 ℓ2,2 c2,3 ℓ2,3 c2,4 ℓ2,2 ℓ2,3 b a

Florian BRIDOUX Conjunctive networks 2020 17/18

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Conclusion

Results: The k-PLC problem can be resolved in polynomial time. The k-BLC and k-SLC problems are NP-complete for any k ≥ 2. Ongoing: Not strongly connected. Future works: Does the complexity change when k is not a constant but a problem parameter? The problem of computing φk(f w) is it difficult?

Florian BRIDOUX Conjunctive networks 2020 18/18