CABEAN

CABEAN: A Software Tool for the Control of Asynchronous Boolean Networks



Description

CABEAN is a software tool for the control of asynchronous Boolean networks, which are often used to model gene regulatory networks.

Features

The source-target control of asynchronous Boolean networks can be achieved either in one step, called one-step control, or in multiple steps, called sequential control. Taking difficulties in practical applications into consideration, we are particularly interested in the sequential control through other attractors, called attractor-based sequential control. Figure 1(a) and Figure 1(b) illustrate the one-step and attractor-based sequential control, respectively. Squares and rhombuses represent the source and target attractors. Circles represent the other attractors of the network. One-step control drives the network from the source directly to the target, while attractor-based sequential control employs other attractors as intermediate and identifies a sequence of perturbations. Based on the application time of control, we have instantaneous, temporary and permanent controls. Instantaneous control applies perturbations instantaneously; temporary control applies perturbations for sufficient time and then released; and permanent control applies the control for all the following time steps.


Figure 1. (a) One-step and (b) attractor-based sequential source-target control of Boolean networks.

Combining the two categories, there are six different control strategies. CABEAN provides the following methods to solve the six source-target control problems:

CABEAN provides the following target control methods:

Download

CABEAN is freely available. The newest version of CABEAN is 2.0.0, updated on October 26, 2020. You can download the files here, including executable binaries for Windows, Mac or Linux, a detailed user guide and examples. Starting from version 1.0.1, CABEAN supports model files under BoolNet format.

Requirements

1. Platform: x86 compatible 32 bit or 64 bit processor;

2. Operating system: Windows, Linux and Mac OSX (10.4 and above);

3. Required compilers: flex 2.6.4 or higher, GNU bison 3.0.4 or higher, GNU g++ 4.0.1 or higher, and Windows subsystem for Linux (Windows platform only).

Quickstart Guide

In this quick guide, we use two examples to illustrate the basic usages of CABEAN, including the syntax of the model file and how to compute different controls with and without practical constraints. Specifically, Example 1 focuses on the introduction of different control methods and Example 2 shows how to encode practical constraints. For detailed introduction of CABEAN, please download the user guide.

Example 1

How to define a Boolean network?

Nodes Functions Indices of attractors Attractors
x1     f1= x2 A1 000
x2    f2= x1A2 110
x3    f3= x2 ∧ x3 A3 111

Table 1. The Boolean functions and the attractors of Example 1.

The first two columns of Table 1 describe a Boolean network with three nodes, including x1, x2, x3. Each node is assigned with a Boolean function. CABEAN supports the BoolNet and ISPL (Interpreted Systems Programming Language) format. We recommend the BioLQM toolkits for the conversion of other formats, such as SBML-qual, Petri net, GINsim, to BoolNet format. BioLQM is available at http://colomoto.org/biolqm/.
BoolNet describes a Boolean network or probabilistic Boolean network in a standardized text file format. Here we only explain the syntax for Boolean networks. The model file of the example under BoolNet format is given below. The first line is a header: 'targets, factors'. The name and the Boolean function of each node are separated by a comma and given at each line. The header implies the format: 'targets' is the name of the node and 'factors' represents the Boolean function of the node. In the Boolean functions, the logical operators 'and', 'or' and 'not' are represented as '&', '|' and '!'. Users can initialise the value of an input node to either '1' or '0'.

  targets, factors
  x1, x2
  x2, x1
  x3, x2&x3
  
The model file of Example 1 under BoolNet format. (Download: toy.bnet)

The model file under ISPL format is shown below. It contains two sections: section 'Agent M' and section 'InitStates'. Section 'Agent M' defines the Boolean variables and the Boolean functions of the network; section 'InitStates' specifies the initial state. Section 'Agent M' includes four parts: 'Vars', 'Actions', 'Protocol', and 'Evolution'. The 'Vars' part defines the Boolean variables and the order of the variables is consistent with the order of the nodes in each state in the output. The 'Evolution' part specifies the Boolean functions for each variable. The functions are defined using parenthesis and three logical operations, including logical and '&', logical or '|', and logical not '~'. The 'Actions' and 'Protocol' parts are defined as '{none}'. The ISPL format is preferable than BoolNet format. In the following sections, we use 'toy.ispl' as the name of the model file.

  Agent M
    Vars:    
      x1: boolean;
      x2: boolean;    
      x3: boolean;
    end Vars
    Actions = {none};
    Protocol:
      Other: {none};
    end Protocol
    Evolution:
      x1=true  if  x2=true;
      x1=false if  x2=false;    
      x2=true  if  x1=true;
      x2=false if  x1=false;
      x3=true  if  x2&x3=true;
      x3=false if  x2&x3=false;
    end Evolution
  end Agent

  InitStates
      M.x1=true or M.x1=false;
  end InitStates
  
The model file of Example 1 under ISPL format. (Download: toy.ispl)


Figure 2. (a) The original ransition system (b) transition system under control C1={x2=1 x3=1} and (c) transition system under control C2={x1=1 x3=1} of Example 1. We omit self-loops for all the states except for state (101) in (a).

How to compute the attractors of the network?

CABEAN implements the decomposition-based attractor detection method [APBC18] that can identify all the exact attractors of the network. The attractors of the example can be computed using the following command:

./cabean -compositional 2 toy.ispl
The output of CABEAN is given below. This network has three attractors. In each state, the sequence of the nodes is consistent with the sequence of the nodes defined in the model file. The transition system of the network under asynchronous updating scheme is given in Figure 2(a). We can see that the attractors #1, #2 and #3 identified by our methods correspond to A1, A2 and A3 in Figure 2(a). From any initial state, the network will eventually settle down to one of the attractors.

Command line: ./cabean -compositional 2 toy.ispl
======================== find attractor #1 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
0-0-0-  1

======================== find attractor #2 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-0-  1

======================== find attractor #3 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-1-  1

number of attractors = 3
time for attractor detection=0.001 seconds
Attractors of Example 1. (Download: toy-attractors.txt)

How to compute the control for one pair of source and target attractors?

In the previous section, we showed how to use CABEAN for the identification of attractors. Below is a template command line for computing the control from the source attractor to the target attractor.

./cabean -compositional 2 -control <control method> -sin <indice of the source> -tin <indice of the target> <model file>

In the command line, '-compositional 2' implies that the strong basin of an attractors is computed using the decomposition-based computation methods [BCB18, TCBB19]; '-control <control method>' indicates the control method (OI, OT, OP, ASI, AST or ASP); '-sin <indice of the source> -tin <indice of the target>' indicates the indices of the source and target attractors; '<model file>' is the path of the model file and it is always placed at the end of the command line. In this way, the source and target correspond to one of the attractors of the network. The attractor can be either a single-state attractor or a cyclic attractor. If the source attractor is cyclic, CABEAN will identify the control to be applied to a state in the source attractor that requires the fewest number of perturbations. Currently, CABEAN does not support the control from/to a set of attractors.
CABEAN first identifies all the exact attractors of the network and then compute the control for the specified source and target attractors. For this example, let us assume A1 and A3 are the source and the target attractors, respectively. Next we show how to compute OI, OT, OP, ASI, AST and ASP from A1 to A3 with CABEAN.

Minimal OI control

We use the following command line to compute the minimal OI control from A1 to A3 for Example 1.

./cabean -compositional 2 -control OI -sin 1 -tin 3 toy.ispl

The results are shown below. CABEAN identifies one minimal OI control {x1=1 x2=1 x3=1}. To compute the minimal OI contol, our method computes the strong basin of the target attractor and then finds the intermediate state in the strong basin that has the shortest Hamming distance to the source state. The nodes that have different values in the source state and the intermediate state are the driver nodes. Figure 2(a) shows that the strong basin of A3 contains only one state (111). Thus, the minimal OI control drives the network from (000) to (111) and the associate control set is {x1=1 x2=1 x3=1}.

Command line: ./cabean -compositional 2 -control OI -sin 1 -tin 3 toy.ispl
======================== find attractor #1 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
0-0-0-  1

======================== find attractor #2 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-0-  1

======================== find attractor #3 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-1-  1

number of attractors = 3
time for attractor detection=0.001 seconds

====== ONE-STEP INSTANTANEOUS SOURCE-TARGET CONTROL (DECOMP) ======

source - 1 target - 3

PATH 1 - #perturbations: 3
  Control set: x1=1 x2=1 x3=1 
execution time of control  = 0 seconds
Results of the minimal OI control of Example 1 from A1 to A3. (Download: toy-OI.txt)

Minimal OT control

We use the following command line to compute the minimal OT control from A1 to A3 for Example 1.

./cabean -compositional 2 -control OT -sin 1 -tin 3 toy.ispl

The results are shown below. To compute the minimal OT contol, our method explores the weak basin of the target attractor starting from the state that has the shortest Hamming distance to the source state and verify if the corresponding control is a valid temporary control or not. If so, we are done and the control is the minimal OT control. Otherwise, we proceed to the next state that has the shortest Hamming distance to the source state and repeat the verification process. As shown in Figure 2(a), the weak basin of A3 consists of three states: (011), (101) and (111). States (011) and (101) have the shortest Hamming distance to A1. For state (011), the corresponding control set is C1={x2=1 x3=1}. The application control of C1 stirs the network from state (000) to (011) and reshapes the transition system from Figure 2(a) to Figure 2(b). During the control, the network will eventually and surely reach the target attractor A3. Therefore, C1 is a minimal OT control. Similarly, we can prove that the control C2={x1=1 x3=1} associated with the intermediate state (011) is also a minimal OT control. (Figure 2(c) shows the transition system under control C2. )

Command line: ./cabean -compositional 2 -control OT -sin 1 -tin 3 toy.ispl
======================== find attractor #1 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
0-0-0-  1

======================== find attractor #2 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-0-  1

======================== find attractor #3 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-1-  1

number of attractors = 3
time for attractor detection=0 seconds

====== ONE-STEP TEMPORARY SOURCE-TARGET CONTROL (DECOMP) ======

source - 1 target - 3

PATH 1 - #perturbations: 2
  Control set: x2=1 x3=1 

PATH 2 - #perturbations: 2
  Control set: x1=1 x3=1 
execution time for control = 0.001 seconds
Results of the minimal OT control of Example 1 from A1 to A3. (Download: toy-OT.txt)

Minimal OP control

We use the following command line to compute the minimal OP control from A1 to A3 for Example 1. The results of the minimal OP control are the same with the minimal OT control. To avoid duplication, we omit the output of OP control here (toy-OP.txt).

./cabean -compositional 2 -control OP -sin 1 -tin 3 toy.ispl

ASI control

We use the following command line to compute the minimal ASI control from A1 to A3 for Example 1.

./cabean -compositional 2 -control ASI -sin 1 -tin 3 toy.ispl
ASI control identifies a sequence of instantaneous perturbations to drive the dynamics from the source to the target passing through other attractors In the current implementation, the number of perturbations required by the minimal OI control is adopted as the threshold on the number of perturbations. We have shown that the minimal OI control from A1 to A3 requires at least three perturbatons. ASI identifies one sequential control path with three perturbations as shown below. It first flips the values of node x1 and x2 from 0 to 1, which drives the network from A1 to the intermediate attractor A2. Then it flips the value of x3, which leads the network from A2 to A3.
Command line: ./cabean -compositional 2 -control ASI -sin 1 -tin 3 toy.ispl
======================== find attractor #1 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
0-0-0-  1

======================== find attractor #2 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-0-  1

======================== find attractor #3 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-1-  1

number of attractors = 3
time for attractor detection=0.001 seconds

========= ATTRACTOR-BASED SEQUENTIAL INSTANTANEOUS SOURCE-TARGET CONTROL (DECOMP) =========

source - 1 target - 3

PATH 1 - #perturbations: 3
Sequence of the attractors: 1 -> 3
  STEP 1
    Control set 1: x1=1 x2=1 x3=1 

PATH 2 - #perturbations: 3
Sequence of the attractors: 1 -> 2 -> 3
  STEP 1
    Control set 1: x1=1 x2=1 
  STEP 2
    Control set 1: x3=1 
execution time of control=0 seconds
Results of ASI control of Example 1 from A1 to A3. (Download: toy-ASI.txt)

AST control

We use the following command to compute the minimal AST control from A1 to A3 for Example 1.

./cabean -compositional 2 -control AST -sin 1 -tin 3 toy.ispl
AST control identifies a sequence of temporary perturbations to drive the dynamics from the source to the target passing through other attractors of the network. In the current implementation, the number of perturbations required by the minimal OT control is adopted as the threshold for AST. For this example, AST identifies two sequential control paths with two perturbations as shown below. Similar to ASI control, AST control also employs A2 as the intermediate attractor. There exist two temporary control sets from A1 to A2: {x1=1} and {x2=1}. The control set from A2 to A3 is the same as that of ASI control.
Command line: ./cabean -compositional 2 -control AST -sin 1 -tin 3 toy.ispl
======================== find attractor #1 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
0-0-0-  1

======================== find attractor #2 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-0-  1

======================== find attractor #3 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-1-  1

number of attractors = 3
time for attractor detection=0 seconds

========= ATTRACTOR-BASED SEQUENTIAL TEMPORARY SOURCE-TARGET CONTROL (DECOMP) =========

source - 1 target - 3

PATH 1 - #perturbations: 2
Sequence of the attractors: 1 -> 3
  STEP 1
    Control set 1: x2=1 x3=1 
    Control set 2: x1=1 x3=1 

PATH 2 - #perturbations: 2
Sequence of the attractors: 1 -> 2 -> 3
  STEP 1
    Control set 1: x2=1 
    Control set 2: x1=1 
  STEP 2
    Control set 1: x3=1 
execution time of control=0.002 seconds
Results of AST control of Example 1 from A1 to A3. (Download: toy-AST.txt)

ASP control

We use the following command to compute the minimal ASP control from A1 to A3 for Example 1. The results of ASP control are the same with AST control. To avoid duplication, we omit the output of ASP control here (toy-ASP.txt).

./cabean -compositional 2 -control ASP -sin 1 -tin 3 toy.ispl

How to compute the target control?

TTC

We use the following command to compute TTC to A3 for Example 1.

./cabean -compositional 2 -control TTC -tin 3 toy.ispl
Command line: ./cabean -compositional 2 -control TTC -tin 3 toy.ispl
======================== find attractor #1 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
0-0-0-  1

======================== find attractor #2 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-0-  1

======================== find attractor #3 : 1 states ========================
: 4 nodes 1 leaves 1 minterms
1-1-1-  1

number of attractors = 3
time for attractor detection=0.002 seconds

=================== TEMPORARY TARGET COTNROL (DECOMP) ===================

************************************************************
TARGET ATTRACTOR #3
************************************************************

Control set 1: x1=1 x3=1 
Control set 2: x2=1 x3=1 

Time for temporary target control = 0.001 seconds
Results of TTC of Example 1 to A3. (Download: toy-TTC.txt)

Example 2

Model files

Nodes Functions Indices of attractors Attractors
x1     f1= x2 A1 001100
x2    f2= x1A2 001111
x3    f3= (~x2) ∨ x4 A3 110000
x4    f4= x3A4 110010
x5     f5= (x1 ∨ x6) ∧ x5 A5 111100
x6    f6=x4 ∧ x5 A6 111111

Table 2. The Boolean functions and the attractors of Example 2.

The first two columns of Table 2 describe a Boolean network with six nodes, including x1, x2, x3, x4, x5, and x6. The model file under BoolNet format is given below.

  targets, factors
  x1, x2
  x2, x1
  x3, (!x2)|x4
  x4, x3
  x5, (x1|x6)&x5
  x6, x4&x5
  
The model file of Example 2 under BoolNet format. (Download: example.bnet)

The model file under ISPL format is shown below.

  Agent M
    Vars:    
      x1: boolean;
      x2: boolean;    
      x3: boolean;
      x4: boolean;
      x5: boolean;
      x6: boolean;
    end Vars
    Actions = {none};
    Protocol:
      Other: {none};
    end Protocol
    Evolution:
      x1=true  if  x2=true;
      x1=false if  x2=false;    
      x2=true  if  x1=true;
      x2=false if  x1=false;
      x3=true  if  (~x2)|x4=true;
      x3=false if  (~x2)|x4=false;
      x4=true  if  x3=true;
      x4=false if  x3=false;
      x5=true  if  (x1|x6)&x5=true;
      x5=false if  (x1|x6)&x5=false;
      x6=true  if x4&x5=true;
      x6=false if x4&x5=false;
    end Evolution
  end Agent

  InitStates
      M.x1=true or M.x1=false;
  end InitStates
  
The model file of Example 2 under ISPL format. (Download: example.ispl)

Attractors

The attractors of Example 2 can be computed using the following command:

./cabean -compositional 2 example.ispl
The output of CABEAN is given below. This network has six attractors. Once the attractors are computed, users can use the provided functions to compute the minimal control for one pair of source and target attractors or for all combinations of source and target attractors.

Command line: ./cabean -compositional 2 example.ispl
======================== find attractor #1 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-0-0-  1

======================== find attractor #2 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-1-1-  1

======================== find attractor #3 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-0-0-  1

======================== find attractor #4 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-1-0-  1

======================== find attractor #5 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-0-0-  1

======================== find attractor #6 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-1-1-  1

number of attractors = 6
time for attractor detection=0.003 seconds
Attractors of the Example 2. (Download: example-attractors.txt)

How to compute the control for one pair of source and target attractors with constraints on perturbations?

We use the minimal OT control from A6 to A1 of Example 2 as an example to show how to compute the control for one pair of source and target attractors with constraints on perturbations. We first give the results of the minimal OT control without constraints below. We can see that CABEAN identifies eight minimal OT control sets and each control requires two perturbations.

Command line: ./cabean -compositional 2 -control OT -sin 6 -tin 1 example.ispl
======================== find attractor #1 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-0-0-  1

======================== find attractor #2 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-1-1-  1

======================== find attractor #3 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-0-0-  1

======================== find attractor #4 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-1-0-  1

======================== find attractor #5 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-0-0-  1

======================== find attractor #6 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-1-1-  1

number of attractors = 6
time for attractor detection=0.001 seconds

====== ONE-STEP TEMPORARY SOURCE-TARGET CONTROL (DECOMP) ======

source - 6 target - 1

PATH 1 - #perturbations: 2
  Control set: x2=0 x6=0 

PATH 2 - #perturbations: 2
  Control set: x2=0 x5=0 

PATH 3 - #perturbations: 2
  Control set: x2=0 x4=0 

PATH 4 - #perturbations: 2
  Control set: x2=0 x3=0 

PATH 5 - #perturbations: 2
  Control set: x1=0 x6=0 

PATH 6 - #perturbations: 2
  Control set: x1=0 x5=0 

PATH 7 - #perturbations: 2
  Control set: x1=0 x4=0 

PATH 8 - #perturbations: 2
  Control set: x1=0 x3=0 
execution time for control = 0.003 seconds
Results of the minimal OT control of Example 2 from A6 to A1. (Download: example-OT-6to1.txt)

CABEAN allows users to specify undesired perturbations and compute the minimal control excluding those perturbations. In the current version, user can define three types of perturbaions:

Suppose node x5 can not be flipped from 1 to 0 and nodes x4 and x6 can not be perturbed in any direction. The specification file is given below.

R0: 
R1: x5
R: x4, x6
File for the specification of undesired perturbations. (Download: rmPert.txt)

To compute the minimal OT control from A6 to A1 without undesired perturbations, we add the option '-rmPert <file name>' to the command line as follows:

./cabean -compositional 2 -control OT -sin 6 -tin 1 -rmPert rmPert.txt example.ispl

The output of CABEAN is given below. We can see that the minimal OT control still requires at least two perturbations and there are two solutions without undesired perturbations.

Command line: ./cabean -compositional 2 -control OT -sin 6 -tin 1 -rmPert rmPert.txt example.ispl
======================== find attractor #1 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-0-0-  1

======================== find attractor #2 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-1-1-  1

======================== find attractor #3 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-0-0-  1

======================== find attractor #4 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-1-0-  1

======================== find attractor #5 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-0-0-  1

======================== find attractor #6 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-1-1-  1

number of attractors = 6
time for attractor detection=0.002 seconds

====== ONE-STEP TEMPORARY SOURCE-TARGET CONTROL (DECOMP) ======

source - 6 target - 1

PATH 1 - #perturbations: 2
  Control set: x2=0 x3=0 

PATH 2 - #perturbations: 2
  Control set: x1=0 x3=0 
execution time for control = 0.001 seconds
Results of the minimal OT control of Example 2 from A6 to A1 with constraints on perturbations. (Download: example-OT-6to1-rmPert.txt)

How to compute the control for all pairs of source and target attractors?

CABEAN allows users to compute the control for all combinations of source and target attractors by replacing '-sin <indice of the source> -tin <indice of the target>' with '-allpairs' in the command line. The minimal OT control for all pairs of source and target attractors of the example can be computed using the following command line:

./cabean -compositional 2 -control OT -allpairs example.ispl
The results can be downloaded here (example-OT-allpairs.txt).

How to compute attractor-based sequential control with constraints on the intermediate attractors?

For attractor-based sequential control (ASI, AST and ASP), undesired attractors, such as attractors that correspond to diseased cells, can be avoided from being adopted as intermediate attractors. Suppose attractors A2 and A5 are the undesired ones, we can define them in the file 'rmID.txt' as follows (the indices are separated by comma):

2,5
File for the speicification of undesired attractors. (Download: rmID.txt)

Taking ASI control as an example, we first show the results without any predefined constraints below. Note that ASI control use the minimal number of perturations required by the minimal OI control as the threshold and computes all the sequential paths within the threshold. We can see that besides the one-step control paths, there are two sequential control paths from A6 to A1, where A2 and A5 act as the intermediate attractors, respectively.

Command line: ./cabean -compositional 2 -control ASI -sin 6 -tin 1 example.ispl
======================== find attractor #1 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-0-0-  1

======================== find attractor #2 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-1-1-  1

======================== find attractor #3 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-0-0-  1

======================== find attractor #4 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-1-0-  1

======================== find attractor #5 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-0-0-  1

======================== find attractor #6 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-1-1-  1

number of attractors = 6
time for attractor detection=0.002 seconds

========= ATTRACTOR-BASED SEQUENTIAL INSTANTANEOUS SOURCE-TARGET CONTROL (DECOMP) =========

source - 6 target - 1

PATH 1 - #perturbations: 3
Sequence of the attractors: 6 -> 1
  STEP 1
    Control set 1: x1=0 x2=0 x5=0 

PATH 2 - #perturbations: 3
Sequence of the attractors: 6 -> 2 -> 1
  STEP 1
    Control set 1: x1=0 x2=0 
  STEP 2
    Control set 1: x5=0 

PATH 3 - #perturbations: 3
Sequence of the attractors: 6 -> 5 -> 1
  STEP 1
    Control set 1: x5=0 
  STEP 2
    Control set 1: x1=0 x2=0 
execution time of control=0.004 seconds

Results of ASI control of Example 2 from A6 to A1. (Download: example-ASI-6to1.txt)

Now, we add constraints on the intermediate attractors by inserting '-rmID rmID.txt' to the command line and the output is given below. The two sequential paths passing though the undesired attractors are eliminated from the results.


Command line: ./cabean -compositional 2 -control ASI 
-sin 6 -tin 1 -rmID rmID.txt example.ispl
======================== find attractor #1 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-0-0-  1

======================== find attractor #2 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-1-1-  1

======================== find attractor #3 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-0-0-  1

======================== find attractor #4 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-1-0-  1

======================== find attractor #5 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-0-0-  1

======================== find attractor #6 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-1-1-  1

number of attractors = 6
time for attractor detection=0.001 seconds

========= ATTRACTOR-BASED SEQUENTIAL INSTANTANEOUS SOURCE-TARGET CONTROL (DECOMP) =========

IDs of undesired attractors
2 5 
source - 6 target - 1

PATH 1 - #perturbations: 3
Sequence of the attractors: 6 -> 1
  STEP 1
    Control set 1: x1=0 x2=0 x5=0 
execution time of control=0.002 seconds

Results of ASI control of Example 2 from A6 to A1 with constraints on the intermediate attractors. (Download: example-ASI-6to1-rmID.txt)

How to compute target control?

We take TTC as the representative to show how to compute target control. We compute TTC to A1 of Example 2 with the folloing command line:

./cabean -compositional 2 -control TTC -tin 1 example.ispl

The results are shown below.

Command line: ./cabean -compositional 2 -control TTC -tin 1 example.ispl
======================== find attractor #1 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-0-0-  1

======================== find attractor #2 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
0-0-1-1-1-1-  1

======================== find attractor #3 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-0-0-  1

======================== find attractor #4 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-0-0-1-0-  1

======================== find attractor #5 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-0-0-  1

======================== find attractor #6 : 1 states ========================
: 7 nodes 1 leaves 1 minterms
1-1-1-1-1-1-  1

number of attractors = 6
time for attractor detection=0.001 seconds

=================== TEMPORARY TARGET COTNROL (DECOMP) ===================

************************************************************
TARGET ATTRACTOR #1
************************************************************
Control set 1: x1=0 x3=0 
Control set 2: x2=0 x3=0 
Control set 3: x1=0 x4=0 
Control set 4: x2=0 x4=0 
Control set 5: x1=0 x5=0 
Control set 6: x2=0 x5=0 
Control set 7: x1=0 x6=0 
Control set 8: x2=0 x6=0 

Time for temporary target control = 0.01 seconds
Results of TTC of Example 2 to A1. (Download: example-TTC-A1.txt)

References