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Academic Citation

Please cite the following paper when using this tool.

Ibrahim, A., Rehwald, S., Pretschner, A.: Efficiently checking actual causality with sat solving. In: Dependable Software Systems Engineering (2019),

@incollection{ibrahim2019, author={Ibrahim, Amjad and Rehwald, Simon and Pretschner, Alexander}, title = {Efficiently Checking Actual Causality with SAT Solving}, booktitle = {Dependable Software Systems Engineering}, year = {2019}, url ={}, }


Build Status


This library allows to determine actual causality according to the modified Halpern-Pearl definition of causality [1] . The used examples in the unit test cases (specifically in CausalitySolverInstanceTest) are described here.


Currently, this library is not published in a Maven repository. Please build it manually from source:

$ mvn install

Then, you can import it using Maven:


Alternatively, a pre-built .jar is offered in the release section of this repository.



Creation of a causal model

// instantiate a new FormulaFactory
FormulaFactory f = new FormulaFactory();

// create exogenous variables; using _exo is not required, but used to distinguish them
Variable BTExo = f.variable("BT_exo");
Variable STExo = f.variable("ST_exo");

// create endogenous variables; technically, there is no difference to exogenous ones
Variable BT = f.variable("BT");
Variable ST = f.variable("ST");
Variable BH = f.variable("BH");
Variable SH = f.variable("SH");
Variable BS = f.variable("BS");

// create the formula/function for each endogenous variable
Formula BTFormula = BTExo;
Formula STFormula = STExo;
Formula SHFormula = ST;
Formula BHFormula = f.and(BT, f.not(SH));
Formula BSFormula = f.or(SH, BH);

// create the equations of the causal model: each endogenous variable and its formula form an equation
Equation BTEquation = new Equation(BT, BTFormula);
Equation STEquation = new Equation(ST, STFormula);
Equation SHEquation = new Equation(SH, SHFormula);
Equation BHEquation = new Equation(BH, BHFormula);
Equation BSEquation = new Equation(BS, BSFormula);

Set<Equation> equations = new HashSet<>(Arrays.asList(BTEquation, STEquation, SHEquation,
    BHEquation, BSEquation));
Set<Variable> exogenousVariables = new HashSet<>(Arrays.asList(BTExo, STExo));

// instantiate the CausalModel
CausalModel causalModel = new CausalModel("RockThrowing", equations, exogenousVariables, f);

Check whether ST = 1 is a cause of BS = 1 in the previously created causal model given ST_exo, BT_exo = 1 as context

// IMPORTANT: Use the same FormulaFactory instance as in the above!

 * Create positive literals for ST_exo and BT_exo. If ST_exo, BT_exo = 0, we would create negative ones,
 * e.g. f.literal("ST_exo", false). Using f.variable("ST_exo") would be a shortcut for f.literal("ST_exo", true)
Set<Literal> context = new HashSet<>(Arrays.asList(f.literal("BT_exo", true),
    f.literal("ST_exo", true)));

 * Similar as for the context, we specify f.literal("ST", true) as cause and f.variable("BS") as phi, as we 
 * want to express ST = 1 and BS = 1, respectively.
Set<Literal> cause = new HashSet<>(Collections.singletonList(f.literal("ST", true)));
Formula phi = f.variable("BS");

// finally, call isCause on the causal model using the SAT-based algorithm
CausalitySolverResult causalitySolverResult =
    CauscausalModel.isCause(context, phi, cause, SolvingStrategy.SAT);

Use other algorithms

The SolvingStrategy enum contains all currently supported algorithms/strategies:

public enum SolvingStrategy {

Just call the isCause-method with the respective SolvingStrategy:

// Brute-Force
CausalitySolverResult causalitySolverResult =
    CauscausalModel.isCause(context, phi, cause, SolvingStrategy.BRUTE_FORCE);

// SAT-based
CausalitySolverResult causalitySolverResult =
    CauscausalModel.isCause(context, phi, cause, SolvingStrategy.SAT);

// SAT-based returning a minimal W for AC2
CausalitySolverResult causalitySolverResult =
    CauscausalModel.isCause(context, phi, cause, SolvingStrategy.SAT_MINIMAL);

// SAT-based where checking AC2 and AC3 is combined
CausalitySolverResult causalitySolverResult =
    CauscausalModel.isCause(context, phi, cause, SolvingStrategy.SAT_COMBINED);

// SAT-based where AC3 check does not require ALL-SAT
CausalitySolverResult causalitySolverResult =
    CauscausalModel.isCause(context, phi, cause, SolvingStrategy.SAT_OPTIMIZED_AC3);

Important Notes

  • When working with a causal model, always use the same FormulaFactory instance. If not, an exception might occur.
  • When creating a CausalModel, it is checked whether the latter is valid. It needs to fulfill the following characteristics; otherwise an exception is thrown:
    • Each variable needs to be either exogenous or defined by exactly one equation.
    • The causal model must be acyclic. That is, no variables are allowed to mutually depend on each other (directly and indirectly)
    • Variables must not be named with "_dummy".


[1] J. Y. Halpern. "A Modification of the Halpern-Pearl Definition of Causality." In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, July 25-31, 2015. 2015, pp. 3022–3033.


A java library for actual causality computation.





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