1. Introduction
2. Automata Networks
 $\Sigma =\{a,b,\cdots \}$ is the finite set of automata identifiers;
 For each $a\in \Sigma $, $\mathcal{S}\left(a\right)=\{{a}_{i},\cdots ,{a}_{j}\}$, is the finite set of local states of automaton a; $\mathcal{S}={\prod}_{a\in \Sigma}\mathcal{S}\left(a\right)$ is the finite set of global states and $\mathit{LS}$$={\cup}_{a\in \Sigma}\mathcal{S}\left(a\right)$ denotes the set of all of the local states.
 $\mathcal{T}=\{a\mapsto {\mathcal{T}}_{a}\mid a\in \Sigma \}$, where $\forall a\in \Sigma ,{\mathcal{T}}_{a}\subset \mathcal{S}\left(a\right)\times \wp (\mathit{LS}\setminus \mathcal{S}\left(a\right))\times \mathcal{S}\left(a\right)$ with $({a}_{i},\ell ,{a}_{j})\in {\mathcal{T}}_{a}\Rightarrow {a}_{i}\ne {a}_{j}$, is the mapping from automata to their finite set of local transitions.
3. Timed Automata Networks
 $\Sigma =\{a,b,\cdots \}$ is the finite set of automata identifiers;
 For each $a\in \Sigma $, $\mathcal{S}\left(a\right)=\{{a}_{i},\cdots ,{a}_{j}\}$, is the finite set of local states of automaton a; $\mathcal{S}={\prod}_{a\in \Sigma}\mathcal{S}\left(a\right)$ is the finite set of global states; $\mathit{LS}$$={\cup}_{a\in \Sigma}\mathcal{S}\left(a\right)$ denotes the set of all of the local states.
 $\mathcal{T}=\{a\mapsto {\mathcal{T}}_{a}\mid a\in \Sigma \}$, where $\forall a\in \Sigma ,{\mathcal{T}}_{a}\subset \mathcal{S}\left(a\right)\times \wp (\mathit{LS}\setminus \mathcal{S}\left(a\right))\times \mathbb{N}\times \mathcal{S}\left(a\right)$ with $({a}_{i},\ell ,\delta ,{a}_{j})\in {\mathcal{T}}_{a}\Rightarrow {a}_{i}\ne {a}_{j}$, is the mapping from automata to their finite set of timed local transitions.
4. Learning Timed Automata Networks
Algorithm
Algorithm 1 MoTAN: modeling timed automata networks. 

5. Refining
5.1. Synchronous Behavior
5.2. More Frequent Timed Automata Networks
5.3. Deterministic Influence
5.4. Several Delays
6. Case Study
 
 Detect biological components’ changes;
 
 Compute the candidate timed local transitions responsible for the network changes;
 
 Generate the minimal subset of candidate timed local transitions that can realize all changes;
 
 Filter the timed candidate actions.
7. ASP Encoding
7.1. ASP Syntax
7.2. All Models
1 % All network components with its levels after discretization 2 automatonLevel("a",0..1). automatonLevel("b",0..1). automatonLevel("c",0..1). 3 % Time series data or the observation of the components 4 obs("a",0,0). obs("a",0,1). obs("a",0,2). obs("a",0,3). obs("a",1,3). obs("a",1,4). obs("a",1,4). 5 obs("a",1,5). obs("a",0,5). obs("a",0,6). obs("a",0,7). obs("b",1,0). obs("b",1,1). obs("b",1,2). 6 obs("b",1,3). obs("b",1,4). obs("b",0,4). obs("b",0,5). obs("b",0,5). obs("b",0,6). obs("b",0,7). 7 obs("c",0,0). obs("c",0,1). obs("c",0,2). obs("c",1,2). obs("c",1,3). obs("c",1,4). obs("c",1,5). 8 obs("c",1,6). obs("c",0,6). obs("c",0,7).
9 % Changes identification 10 % initialization of all changes for each automaton at t=0 (assumption) 11 changeState(X,Val,Val,0) ← obs(X,Val,0). 12 changeState(X,0) ← obs(X,_,0). 13 % Compute all changes of each component according to the observations (chronogram) 14 changeState(X,Val1, Val2, T) ← obs(X, Val1, T), obs(X,Val2,T),obs(X, Val1, T1), obs(X, Val2, T+1), 15 Val1!=Val2. 16 changeState(X,T) ← changeState(X,_,_,T). 17 % Find all time points where changes occure (reduce complexity) 18 time(T) ← changeState(_,T). 19 delay(D) ← time(T1), time(T2), D=T2T1, T2>=T1. 20 % Observations processing 21 obs_normalized(X,Val,T1,T2) ← obs(X,Val,T1), obs(X,Val,T2), T1<T2, not existChange(X,Val,T1,T2), 22 time(T1), time(T2). 23 % Verify if X changes its level between two time points T1 and T2 24 existChange(X,Val,T1,T2) ← obs(X,Val,T1), obs(X,Val,T2), obs(X,Val1,T), T>T1, T<T2, Val!=Val1. 25 existChange(X,Val,T1,T2) ← obs(X,Val,T1), obs(X,Val,T2), changeState(X,T), T>T1, T<T2. 26 existChange(X,Val,T1,T2) ← obs(X,Val,T1), obs(X,Val,T2), T1<T2, changeState(X,Val,Val_,T1), Val!=Val_.
27 % Find the time step when the transition has started playing 28 % The last change of X such that W is influencing by X and X is influencing by Y 29 lastchange(X,Y,W,Max,T2) ← Max=#max{ T : changeState(Y,T;X,T;W,T), T<T2}, changeState(X, T2), 30 existInfluence(X,Y), existInfluence(W,X), Max>=0. 31 lastChangeAll(X,Y,Max,T2) ← lastchange(X,Y,W,Max,T2), transition(X,_,W,_,_,D, change(T3)), T3<T2, 32 T2U>D, lastConditionChange(X,Y,U,T2). 33 % Last change between X and its influencing component Y 34 lastConditionChange(X,Y,H,T2) ← H=#max{ T : changeState(Y,T;X,T) , T<T2}, changeState(X, T2), 35 existInfluence(X,Y), H>=0. 36 % Find the time point of the last change: it is the last change of X, or of the component influenced by X 37 % or of the components involved in the transition condition 38 lastChange(X,Y,Max,T2) ← lastChangeAll(X,Y,Max,T2). 39 lastChange(X,Y,H,T2) ← lastConditionChange(X,Y,H,T2), not lastChangeAll(X,Y,H,T2), H!=H, delay(H).
40 % Compute all models with all candidate timed local transitions 41 {transition(Y,Valy,X,Val1,Val2,D, change(T2))} ← obs_normalized(X,Val1,T1,T2), 42 obs_normalized(Y,Valy,T1,T2), changeState(X,Val1,Val2,T2), existInfluence(X,Y), 43 lastChange(X,Y,T1,T2), T2=T1+D, delay(D). 44 transition(Y,Valy,X,Val1,Val2,D) ← transition(Y,Valy,X,Val1,Val2,D, _). 45 % for each change keep only one transition (xOR) 46 getTransNumber(Tot,X,T)← Tot={transition(_,_,X,_,_,_, change(T))}, changeState(X,T), T!=0. 47 % Exactly one transition by change in a model 48 ← getTransNumber(Tot,X,T), changeState(X,T), Tot=0. 49 ← getTransNumber(Tot,X,T), changeState(X,T), Tot>1.
7.3. Refinement
7.4. Refinement According to the Semantics
50 % A component with the same level inhibits and activates the same component
51 ← transition(Y,Valy,X,Val1,Val2,_), transition(Y,Valy,X,Val3,Val4,_), Val1<Val2, Val3>Val4.
52 % A component with different levels influence another component with the same effect
53 ← transition(Y,Valy,X,Val1,Val2,_), transition(Y,Valy_,X,Val1,Val2,_), Valy_!=Valy.
54 % Last time points in the data 55 timeSeriesSize(Last) ← Last=#max{ T : obs(_,_,T) }. 56 step(0..Last)← timeSeriesSize(Last). 57 % Compute all the obs between all the time points in the data 58 obs_(X,Val,T1,T2) ← obs(X,Val,T1), obs(X,Val,T2), T1<T2, not existsChange(X,Val,T1,T2). 59 existsChange(X,Val,T1,T2) ← obs(X,Val1,T), T>T1, T<T2, Val1!=Val, obs(X,Val,T1), obs(X,Val,T2). 60 % There is a transition with different delays 61 existTransDiffDelays(Y,Valy,X,Val1,Val2,D1,D2) ← transition(Y,Valy,X,Val1,Val2,D1), 62 transition(Y,Valy,X,Val1,Val2,D2), D1!=D2. 63 % There is a transition in conflict with "transition(Y,Valy,X,Val1,Val2,D1)" 64 existTransInConflict(Y,Valy,X,Val1,Val2,D1,T1,T2) ← transition(Y,Valy,X,Val1,Val2,D1), 65 transition(X,Val1,_,_,_,D2,change(T3)), T3>=T2, D2>=D1, T3D2 <=T2, step(T2), 66 step(T1), T1<T2, D1=T2T1, obs_(X,Val1,T1,T2), obs_(Y,Valy,T1,T2). 67 % Eliminate all models that do not respect the semantics (Definitions 5–6) 68 ← transition(Y,Valy,X,Val1,Val2,D), obs_(X,Val1,T1,T2), obs_(Y,Valy,T1,T2), step(D), 69 not changeState(X,Val1,Val2,T2), not existTransInConflict(Y,Valy,X,Val1,Val2,D,T1,T2), 70 not existTransDiffDelays(Y,Valy,X,Val1,Val2,D,D2), changeState(X,Val1,Val2,T3), 71 T2!=Max, timeSeriesSize(Max), obs_(X,Val1,T1,T3), T3T1=D2, delay(D2), D=T2T1.
7.5. Refinement on the Delays
73 % No different delays for the same transition
74 ← transition(Y,Valy,X,Val1,Val2,D1), transition(Y,Valy,X,Val1,Val2,D2), D1!=D2.
75 % Transitions with the average of delays 76 % Number of the repetition of each transition 77 nbreTotTrans(Y,Valy,X,Val1,Val2,Tot) ← Tot={transition(Y,Valy,X,Val1,Val2,_,_)}, 78 transition(Y,Valy,X,Val1,Val2,_). 79 % The sum of the delays for each transition 80 sumDelays(Y,Valy,X,Val1,Val2,S) ← S=#sum{ D: transition(Y,Valy,X,Val1,Val2,D)}, 81 transition(Y,Valy,X,Val1,Val2,_), S!=0. 82 % Compute the average delay for each transition 83 transAvgDelay(Y,Valy,X,Val1,Val2,Davg) ← nbreTotTrans(Y,Valy,X,Val1,Val2,Tot), 84 sumDelays(Y,Valy,X,Val1,Val2,S), Davg=S/Tot.
85 % Compute the maximum value and the minimum value of the delays of a same transition
86 maxDelay(Y,Valy,X,Val1,Val2,Max) ← Max=#max{ D : transition(Y,Valy,X,Val1,Val2,D)},
87 transition(Y,Valy,X,Val1,Val2,_).
88 minDelay(Y,Valy,X,Val1,Val2,Min) ← Min=#min{ D : transition(Y,Valy,X,Val1,Val2,D)},
89 transition(Y,Valy,X,Val1,Val2,_).
90 transIntervalDelay(Y,Valy,X,Val1,Val2,interval(Min,Max)) ← minDelay(Y,Valy,X,Val1,Val2,Min),
91 maxDelay(Y,Valy,X,Val1,Val2,Max).
8. Evaluation
8.1. DREAM4
8.2. Results
8.3. DREAM8
8.4. Results
8.5. Discussion
9. Conclusions and Perspectives
Supplementary Materials
Acknowledgments
Author Contributions
Conflicts of Interest
Abbreviations
AN  Automata Network 
TAN  Timed Automata Network 
ASP  Answer Set Programming 
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t  $\mathit{B}(\mathcal{AN},\mathit{P},\mathit{t})$  $\mathit{F}(\mathcal{AN},\mathit{\zeta},\mathit{P},\mathit{t})$  SFS  $\mathit{FS}$  $\mathit{A}(\mathcal{AN},\mathit{t})$  $\mathbf{state}(\mathcal{AN},\mathit{t})$ 

0  ∅  $\{{\tau}_{1},{\tau}_{3}\}$  $\{\{\varnothing \},\left\{{\tau}_{1}\right\},\left\{{\tau}_{3}\right\},\{{\tau}_{1},{\tau}_{3}\}\}$  $\{{\tau}_{1},{\tau}_{3}\}$  $\{({\mathit{\tau}}_{\mathbf{1}},\mathbf{0}),({\mathit{\tau}}_{\mathbf{3}},\mathbf{0})\}$  $<{a}_{1},{b}_{0},{c}_{2},{d}_{2}>$ 
1  $\{{\tau}_{1},{\mathit{\tau}}_{\mathbf{2}},{\tau}_{3}\}$  ∅  $\{\varnothing \}$  ∅  $\{({\tau}_{1},0),({\tau}_{3},0)\}$  $<{a}_{1},{b}_{0},{c}_{2},{d}_{2}>$ 
2  $\left\{{\tau}_{3}\right\}$  $\{{\tau}_{2},{\tau}_{5}\}$  $\{\varnothing ,\left\{{\tau}_{2}\right\},\left\{{\tau}_{5}\right\}\}$  $\left\{{\tau}_{2}\right\}$  $\{({\tau}_{3},0),({\mathit{\tau}}_{\mathbf{2}},\mathbf{2})\}$  $<{a}_{1},{\mathit{b}}_{\mathbf{1}},{c}_{2},{d}_{2}>$ 
3  $\{{\tau}_{2},{\tau}_{3}\}$  $\left\{{\tau}_{5}\right\}$  $\left\{{\tau}_{5}\right\}$  $\left\{{\tau}_{5}\right\}$  $\{({\tau}_{3},0),({\tau}_{2},2),({\mathit{\tau}}_{\mathbf{5}},\mathbf{3})\}$  $<{a}_{1},{b}_{1},{c}_{2},{d}_{2}>$ 
4  $\{{\tau}_{2},{\tau}_{3},{\mathit{\tau}}_{\mathbf{5}}\}$  ∅  $\{\varnothing \}$  ∅  $\{({\tau}_{3},0),({\tau}_{2},2),({\tau}_{5},3)\}$  $<{a}_{1},{b}_{1},{c}_{2},{d}_{2}>$ 
5  ∅  $\left\{{\tau}_{4}\right\}$  $\{\varnothing ,{\tau}_{4}\}$  $\left\{{\tau}_{4}\right\}$  $\left\{({\mathit{\tau}}_{\mathbf{4}},\mathbf{5})\right\}$  $<{\mathit{a}}_{\mathbf{0}},{\mathit{b}}_{\mathbf{0}},{\mathit{c}}_{\mathbf{1}},{d}_{2}>$ 
6  $\left\{{\tau}_{4}\right\}$  ∅  $\{\varnothing \}$  ∅  $\left\{({\tau}_{4},5)\right\}$  $<{a}_{0},{b}_{0},{c}_{1},{d}_{2}>$ 
10  ∅  ∅  $\{\varnothing \}$  ∅  ∅  $<{a}_{0},{b}_{0},{c}_{1},{\mathit{d}}_{\mathbf{1}}>$ 
Benchmark  Number of Genes  MSE 

insilico_size10_1  10  0.086 
insilico_size10_2  10  0.080 
insilico_size10_3  10  0.076 
insilico_size10_4  10  0.039 
insilico_size10_5  10  0.076 
insilico_size100_1  100  0.052 
insilico_size100_2  100  0.042 
insilico_size100_3  100  0.033 
insilico_size100_4  100  0.033 
insilico_size100_5  100  0.052 
Discrete Levels  Run Time (s)  Mean RMSE 

2  9775 s  0.7054 
3  7078 s  0.6419 
4  15,941 s  0.5901 
5  14,102 s  0.5528 
6  19,356 s  0.5667 
7  20,963 s  0.5563 
Benchmarks  Mean RMSE  

2 Discrete Levels  5 Discrete Levels  
BT20  1.712  0.458 
BT549  1.507  0.449 
MCF7  0.713  0.310 
UACC812  12.6  3.391 
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