Limit cycles and update schedules in Boolean networks: Inverse - - PowerPoint PPT Presentation

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit cycles and update schedules in Boolean networks: Inverse Problem.

(Results of Luis G´

• mez’s Ph.D. Thesis)

Advisors: L. Salinas(UdeC), J. Demongeot (UG) and J. Aracena (UdeC).

Novembre 2014

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Summary

1 Definition and Notation 2 Limit Cycle Existence problem 3 Limit Cycle Non Existence problem 4 Feasible Limit Cycle problem

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Boolean Networks

A Boolean network is N = (F, s), where F = (fv)v∈V :: {0, 1}n → {0, 1}n, global transition function, V a set of n elements. fv(x) := F(x)v, ∀v ∈ V , local activation functions. s : V → {1, . . . , n} a deterministic update schedule (parallel,sequential, block-sequential).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Interaction Digraph

G F = (V , A) interaction digraph associated to a Boolean Network. (u, v) ∈ A if an only if fv depends on xu.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Interaction Digraph

G F = (V , A) interaction digraph associated to a Boolean Network. (u, v) ∈ A if an only if fv depends on xu.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Example 1

F : {0, 1}4 → {0, 1}4 f1(x) := x3 ∧ x4 f2(x) := x1 ∧ x3 f3(x) := (x1 ∧ x2) ∨ x4 f4(x) := x2 1 2 3 4

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Example 1

F : {0, 1}4 → {0, 1}4 f1(x) := x3 ∧ x4 f2(x) := x1 ∧ x3 f3(x) := (x1 ∧ x2) ∨ x4 f4(x) := x2 1 2 3 4

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Example 1

F : {0, 1}4 → {0, 1}4 f1(x) := x3 ∧ x4 f2(x) := x1 ∧ x3 f3(x) := (x1 ∧ x2) ∨ x4 f4(x) := x2 1 2 3 4

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Example 1

F : {0, 1}4 → {0, 1}4 f1(x) := x3 ∧ x4 f2(x) := x1 ∧ x3 f3(x) := (x1 ∧ x2) ∨ x4 f4(x) := x2 1 2 3 4

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Example 1

F : {0, 1}4 → {0, 1}4 f1(x) := x3 ∧ x4 f2(x) := x1 ∧ x3 f3(x) := (x1 ∧ x2) ∨ x4 f4(x) := x2 1 2 3 4

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Schedule

s : V → {1, . . . , n}, function.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Schedule

s : V → {1, . . . , n}, function. s(V ) = {1}, parallel. 1 2 3 4 5

s(1) = 1 s(2) = 1 s(3) = 1 s(4) = 1 s(5) = 1

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Schedule

s : V → {1, . . . , n}, function. s(V ) = {1}, parallel. s(V ) = {1, . . . , n}, sequential. 1 2 3 4 5

s(1) = 1 s(2) = 2 s(3) = 3 s(4) = 4 s(5) = 5

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Schedule

s : V → {1, . . . , n}, function. s(V ) = {1}, parallel. s(V ) = {1, . . . , n}, sequential. s(V ) = {1, . . . , m}, 1 < m < n, block-sequential. 1 2 3 4 5

s(1) = 1 s(2) = 1 s(3) = 2 s(4) = 3 s(5) = 3

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Iteration

Given x = (xv)v∈V ∈ {0, 1}n, the (k + 1)-iteration of x by F according to s is given by: xk+1

v

= fv(xlu

u : u ∈ V )

Where: lu =

• k

if s(v) ≤ s(u) k + 1 if s(v) > s(u)

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Iteration

Given x = (xv)v∈V ∈ {0, 1}n, the (k + 1)-iteration of x by F according to s is given by: xk+1

v

= fv(xlu

u : u ∈ V )

Where: lu =

• k

if s(v) ≤ s(u) k + 1 if s(v) > s(u)

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Dynamical Behavior

We can define f s

v (x) = fv(gs v,u(x): u ∈ V )

Where: gs

v,u(x) =

• xu

if s(v) ≤ s(u) f s

u (x)

if s(v) > s(u) F s is the dynamical behavior of N = (F, s).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Dynamical Behavior

We can define f s

v (x) = fv(gs v,u(x): u ∈ V )

Where: gs

v,u(x) =

• xu

if s(v) ≤ s(u) f s

u (x)

if s(v) > s(u) F s is the dynamical behavior of N = (F, s).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Dynamical Behavior

We can define f s

v (x) = fv(gs v,u(x): u ∈ V )

Where: gs

v,u(x) =

• xu

if s(v) ≤ s(u) f s

u (x)

if s(v) > s(u) F s is the dynamical behavior of N = (F, s).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Behavior

Fixed point: x ∈ {0, 1}n : F s(x) = x Limit Cycles: C =

• xkp

k=0 , xk ∈ {0, 1}n, p > 1:

xk+1 = F s(xk) ∧ xp ≡ x0 LC(N): set of limit cycles of N.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Behavior

Fixed point: x ∈ {0, 1}n : F s(x) = x Limit Cycles: C =

• xkp

k=0 , xk ∈ {0, 1}n, p > 1:

xk+1 = F s(xk) ∧ xp ≡ x0 LC(N): set of limit cycles of N.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Behavior

Fixed point: x ∈ {0, 1}n : F s(x) = x Limit Cycles: C =

• xkp

k=0 , xk ∈ {0, 1}n, p > 1:

xk+1 = F s(xk) ∧ xp ≡ x0 LC(N): set of limit cycles of N.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Digraph

Given a Boolean network (F, s), we define the associated labeled digraph G F

s = (G F, labs), called update digraph, where labs : A(G F) → {⊖, ⊕} is

defined as: labs(u, v) =

• ⊕

if s(u) ≥ s(v) ⊖ if s(u) < s(v) Example:

1 2 3 4

⊕ ⊖ ⊖ ⊖ ⊕ ⊕

s(i) = i, ∀i ∈ {1, . . . , n}.

It was proven in (Aracena, J., Goles, E., Moreira, A., Salinas, L., 2009. Biosystems 97, 1-8) that if two different updates schedules have the same update digraph, then they also have the same dynamical behavior.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Digraph

Given a Boolean network (F, s), we define the associated labeled digraph G F

s = (G F, labs), called update digraph, where labs : A(G F) → {⊖, ⊕} is

defined as: labs(u, v) =

• ⊕

if s(u) ≥ s(v) ⊖ if s(u) < s(v) Example:

1 2 3 4

⊕ ⊖ ⊖ ⊖ ⊕ ⊕

s(i) = i, ∀i ∈ {1, . . . , n}.

It was proven in (Aracena, J., Goles, E., Moreira, A., Salinas, L., 2009. Biosystems 97, 1-8) that if two different updates schedules have the same update digraph, then they also have the same dynamical behavior.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Update Digraph

Given a Boolean network (F, s), we define the associated labeled digraph G F

s = (G F, labs), called update digraph, where labs : A(G F) → {⊖, ⊕} is

defined as: labs(u, v) =

• ⊕

if s(u) ≥ s(v) ⊖ if s(u) < s(v) Example:

1 2 3 4

⊕ ⊖ ⊖ ⊖ ⊕ ⊕

s(i) = i, ∀i ∈ {1, . . . , n}.

It was proven in (Aracena, J., Goles, E., Moreira, A., Salinas, L., 2009. Biosystems 97, 1-8) that if two different updates schedules have the same update digraph, then they also have the same dynamical behavior.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Summary

1 Definition and Notation 2 Limit Cycle Existence problem 3 Limit Cycle Non Existence problem 4 Feasible Limit Cycle problem

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Inverse problems of update schedules

General problem Given F = (fv)v∈V : {0, 1}n → {0, 1}n. Does there exists an update schedule s such that (F, s) has a given dynamical property? Particular cases: Limit Cycle Existence problem (LCE) Limit Cycle Non Existence problem (LCNE) Feasible Limit Cycle (FLC)

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Inverse problems of update schedules

General problem Given F = (fv)v∈V : {0, 1}n → {0, 1}n. Does there exists an update schedule s such that (F, s) has a given dynamical property? Particular cases: Limit Cycle Existence problem (LCE) Limit Cycle Non Existence problem (LCNE) Feasible Limit Cycle (FLC)

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Limit Cycle Existence problem (LCE) Given F = (fv)v∈V : {0, 1}n → {0, 1}n. Does there exists an update schedule s such that LC(F, s) = ∅?

Previous works: The specific problem of determining the existence of limit cycles of a Boolean network with parallel update is known to be NP-Hard (Just, W., 2006. The steady state system problem is NP-Hard even for monotone quadratic Boolean dynamical systems. pre-print).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Limit Cycle Existence problem (LCE) Given F = (fv)v∈V : {0, 1}n → {0, 1}n. Does there exists an update schedule s such that LC(F, s) = ∅?

Previous works: The specific problem of determining the existence of limit cycles of a Boolean network with parallel update is known to be NP-Hard (Just, W., 2006. The steady state system problem is NP-Hard even for monotone quadratic Boolean dynamical systems. pre-print).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Theorem AND-OR LCE is NP-hard. Proof (idea): SAT ≤p AND-OR LCE.

vφ vC1 vC2 . . . vCm v1 ¯ v1 vn ¯ vn a1

• 1

. . . an

• n

A O z1 z2 z3

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Theorem AND-OR LCE is NP-hard. Proof (idea): SAT ≤p AND-OR LCE.

vφ vC1 vC2 . . . vCm v1 ¯ v1 vn ¯ vn a1

• 1

. . . an

• n

A O z1 z2 z3

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Remark: AND-OR LCE is NP-Hard even in the following cases: i.- Restricted to the parallel update schedule. ( In this case we remove the vertex z3 and we add an arc from z1 to z2.) ii.- Restricted to sequential update schedules. iii.- Restricted to limit cycles of length 2. iv.- Restricted to maximum in-degree equal to 2.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Theorem SYMMETRIC LCE is NP-Hard. Proof: The proof is similar with the following local activation functions: ∀i ∈ {1, . . . , n},

fvi(x) = xvi ∧ xvφ fvφ(x) = φ (xvi) ∧ (xz1 ∨ xz2) fz1(x) = xvφ ∧ xz2 fz2(x) = xvφ ∧ xz1

vφ v1 v2 vn . . . z1 z2

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Existence problem

Theorem SYMMETRIC LCE is NP-Hard. Proof: The proof is similar with the following local activation functions: ∀i ∈ {1, . . . , n},

fvi(x) = xvi ∧ xvφ fvφ(x) = φ (xvi) ∧ (xz1 ∨ xz2) fz1(x) = xvφ ∧ xz2 fz2(x) = xvφ ∧ xz1

vφ v1 v2 vn . . . z1 z2

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit cycles in symmetric AND-OR networks

Defn: Given F an AND-OR function with symmetric G F. Let G OR

1

, . . . , G OR

kOR be the non trivial connected components of

G [VOR(F)]. Let G AND

1

, . . . , G AND

kAND be the non trivial connected components of

G [VAND(F)]. We define the alternated nodes as VAO = V \ kOR

• i=1

V

• G OR

i

• ∪

kAND

• i=1

V

• G AND

i

• and we denote by G AO

1

, . . . , G AO

kAO, to the connected component of

G [VAO].

• G OR

1

, . . . , G OR

kOR, G AND 1

, . . . , G AND

kAND , G AO 1

, . . . , G AO

kAO

• is an AOA

(AND-OR ALTERNATED) decomposition of G F.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycles in AND-OR symmetric networks

Theorem Given F an AND-OR function with symmetric G F. There exists an update schedule s such that, LC(F, s) = ∅ if and only if there exists a bipartite element (without loops) in the AOA decomposition of G F. Corollary SYMMETRIC AND-OR LCE is polynomial. Remark: In AND-OR networks with symmetric G F with block-sequential update there are limit cycles of length super-polynomial.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycles in AND-OR symmetric networks

Theorem Given F an AND-OR function with symmetric G F. There exists an update schedule s such that, LC(F, s) = ∅ if and only if there exists a bipartite element (without loops) in the AOA decomposition of G F. Corollary SYMMETRIC AND-OR LCE is polynomial. Remark: In AND-OR networks with symmetric G F with block-sequential update there are limit cycles of length super-polynomial.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Summary

1 Definition and Notation 2 Limit Cycle Existence problem 3 Limit Cycle Non Existence problem 4 Feasible Limit Cycle problem

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Non Existence problem

Limit Cycle Non Existence problem (LCNE) Given F = (fv)v∈V : {0, 1}n → {0, 1}n. Does there exists an update schedule s such that: LC (F, s) = ∅?

Previous works: Symmetric threshold networks with sequential update schedule: Fogelman, F., Goles, E., , Pellegrin, D., 1985. Decreasing energy functions as a tool for studying threshold networks. Discrete Applied Mathematics 12, 261-277. OR networks: Goles, E., Noual, M., 2012. Disjunctive networks and update schedules. Advances in Applied Mathematics 48, 646-662.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Non Existence problem

Limit Cycle Non Existence problem (LCNE) Given F = (fv)v∈V : {0, 1}n → {0, 1}n. Does there exists an update schedule s such that: LC (F, s) = ∅?

Previous works: Symmetric threshold networks with sequential update schedule: Fogelman, F., Goles, E., , Pellegrin, D., 1985. Decreasing energy functions as a tool for studying threshold networks. Discrete Applied Mathematics 12, 261-277. OR networks: Goles, E., Noual, M., 2012. Disjunctive networks and update schedules. Advances in Applied Mathematics 48, 646-662.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Non Existence problem

Theorem LCNE is NP-Hard. Proof (idea):

1 2 3 . . . vn

− 1

vn vφ

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Non Existence problem

Theorem LCNE is NP-Hard. Proof (idea):

1 2 3 . . . vn

− 1

vn vφ

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Non Existence problem

Theorem Let F be an OR (AND) function. Then, there exists a sequential update schedule s such that N = (F, s) has only have fixed points as attractors. Proof: Let V ′ be a minimum FVS of the cycles of G F = (V , A) and let consider N = (F, s), with s = sV ′ defined by: sV ′(u, v) = ⊕ if (u, v) ∈ AV ′ ⊖

• therwise

where AV ′ is a minimal FAS from N+(V ′).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Limit Cycle Non Existence problem

Remark: The previous Theorem is not valid int the case of AND-OR networks as shown in the following example:

G F f1(x) = x6 ∨ x9 ∨ x11 f2(x) = x6 ∨ x7 ∨ x12 f3(x) = x6 ∨ x8 ∨ x10 f4(x) = x4 f5(x) = x5 f6(x) = x4 ∧ x5 f7(x) = x1 ∧ x4 f8(x) = x2 ∧ x4 f9(x) = x3 ∧ x4 f10(x) = x1 ∧ x5 f11(x) = x2 ∧ x5 f12(x) = x3 ∧ x5

1 2 3 4 5 6 7 8 9 10 11 12

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Summary

1 Definition and Notation 2 Limit Cycle Existence problem 3 Limit Cycle Non Existence problem 4 Feasible Limit Cycle problem

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Feasible Limit Cycle problem

Feasible Limit Cycle problem (FLC) Given a set V of n elements and F = (fv)v∈V : {0, 1}n → {0, 1}n and a sequence C =

• xkp

k=0 such that xk ∈ {0, 1}n, xk are

pairwise distinct and xp ≡ x0. Does there exist an update schedule s such that C ∈ LC (F, s)?

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Feasible Limit Cycle problem

Theorem OR FLC is NP-Complete. Proof (idea):

φ z0 z1 z2 v1 v2 v3 v4 C 0 C 0

1

C 0

2

C 1 C 2 C 1

1

C 2

1

C 1

2

C 2

2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Feasible Limit Cycle problem

C0

0 C1 0 C2 0 C0 1 C1 1 C2 1 C0 2 C1 2 C2 2

z0 z1 z2 v1 v2 v3 v4 vφ x0,0 1 1 1 1 1 1 1 x1,0 1 1 1 1 1 1 x2,0 1 1 1 1 x0,1 1 1 1 1 1 1 1 x1,1 1 1 1 1 x2,1 1 1 1 1 1 x0,2 1 1 1 1 1 1 1 x1,2 1 1 1 x2,2 1 1 1 1 1 x0,3 1 1 1 1 1 1 1 φ (w) = (w1 ∨ w2 ∨ ¬w3 ∨ w4) ∧ (¬w2 ∨ w3 ∨ ¬w4) ∧ (¬w1 ∨ ¬w3).

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

Feasible Limit Cycle problem

Theorem FLC is polynomial in the following cases: OR Symmetric OR and cycle of length two. AND-OR and cycle of length two.

Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem

The End Merci

Nice 2014 Limit cycles and update schedules in Boolean networks