Preference¶

This page documents how to declare and use preferences in the constraint handler.

In Optimization, we have shown how to declare optimization objectives such that the solver tries to find solutions that minimize or maximize certain values. Ultimately, this leads to a single optimal solution.

However, sometimes we may not want only the best solution but also the second or even third best solution. More precisely, we may want the solutions to have an ordering based on our preferences.

Declaring Preferences¶

The constraint handler provides multiple ways of specifying preferences. Here, we will cover the different ways of declaring preferences using preference values and how to combine them to express more complex preferences.

Variable Value¶

Declaration Label Support

One way to provide a preference is to assign a preference value to a variable having a specific value. For this, we can use the preference_variableValue/3 predicate.

preference_variableValue(Name, Expression, Value)

Name	Description
`Name`	The name of the variable for which we are specifying a preference.
`Expression`	The expression representing the value of the variable we are assigning a preference to.
`Value`	The preference value assigned to the variable having the specified value.

The preference value is a numeric value that indicates how much we prefer that variable to take that specific value. Higher preference values indicate stronger preferences.

Example

Consider a program that defines a variable color with possible values red, green, blue, and yellow.

variable_declare(color, fromFacts).
variable_domain(color, val(symbol, (red;green;blue;yellow))).

If we want to express that we prefer red the most, followed by blue and green, and finally yellow, we can assign preference values as follows:

preference_variableValue(color,val(symbol,red),5).
preference_variableValue(color,val(symbol,(green;blue)),3).
preference_variableValue(color,val(symbol,yellow),1).

In this example, the variable color has the highest preference value of 5 for the value red, a preference value of 3 for both green and blue, and the lowest preference value of 1 for yellow. When the solver evaluates solutions, it will prioritize those where color is red, followed by those where it is green or blue, and lastly those where it is yellow.

This leads to an ordering of solutions based on our specified preferences for the variable color with the highest values appearing first.

Holds¶

Declaration Label Support

In addition to assigning preference values to variable-value pairs, we can also express preferences based on certain conditions using the preference_holds/2 predicate.

preference_holds(Condition, Value)

Name	Description
`Condition`	An Expression representing the condition we are assigning a preference to.
`Value`	The preference value assigned when the condition holds true.

Example

Taking the previous example further, suppose we want to express a preference for a combination of colors color(a) and color(b).

variable_declare(color(a;b), fromFacts).
variable_domain(color(a;b), val(symbol, (red;green;blue;yellow))).

If we wanted to express that we prefer combinations where a and b are equal, we could do so as follows:

preference_holds(operation(eq, (variable(color(a)),(variable(color(b)),()))), 2).
preference_holds(operation(neq, (variable(color(a)),(variable(color(b)),()))), 1).

In this example, we express a preference value of 2 for the condition where color(a) is equal to color(b) and a preference value of 1 for the condition where they are not equal. Thus, when the solver evaluates solutions, it will prioritize those where color(a) and color(b) are the same, followed by those where they differ.

Thus, first models will appear where color(a) and color(b) are the same, followed by models where they differ.

More Complex Preferences¶

Preferences can also be combined to create more complex preference structures. For instance, we can assign different preference values to multiple conditions or variable-value pairs.

Example

Continuing from the previous examples, suppose we want to combine the preferences of the variable color with the preference for color(a) and color(b) being equal.

variable_declare(color(a;b), fromFacts).
variable_domain(color(a;b), val(symbol, (red;green;blue;yellow))).

preference_holds(operation(eq, (variable(color(a)),(variable(color(b)),()))), 10).
preference_variableValue(color(a;b),val(symbol,red),3).
preference_variableValue(color(a;b),val(symbol,(green;blue)),2).
preference_variableValue(color(a;b),val(symbol,yellow),1).

This describes both, a preference where color(a) and color(b) are equal, as well as individual preferences for the values of color(a) and color(b).

Preference Score¶

Declaring preference values is not enough. These values have to be aggregated into a total preference score for each model. This is done using the preference_score/1 result predicate.

preference_score(TotalScore)

Name	Description
`TotalScore`	The total aggregated preference score for the current model.

Example

Using the values of the previous examples, the total preference score for a model where color(a) and color(b) are both red would be calculated as follows:

Preference for color(a) = red: 3
Preference for color(b) = red: 3
Preference for color(a) = color(b): 10
Total Score: 3 + 3 + 10 = 16

Negative Values¶

So far, we have only used positive preference values. However, it is also possible to use negative values to express that certain conditions or variable-value pairs are undesirable.

However, instead of decreasing the overall score, negative preference values will increase the score of models that do not satisfy the corresponding condition or variable-value pair.

Example

Extending the previous example, suppose we want to express that we strongly dislike the color yellow appearing in any model.

variable_declare(color(a;b), fromFacts).
variable_domain(color(a;b), val(symbol, (red;green;blue;yellow))).

preference_holds(operation(eq, (variable(color(a)),(variable(color(b)),()))), 10).
preference_variableValue(color(a;b),val(symbol,red),3).
preference_variableValue(color(a;b),val(symbol,(green;blue)),2).
preference_variableValue(color(a;b),val(symbol,yellow),-5).

One might be inclined to believe that models containing yellow have 5 subtracted from their total score for each occurrence. This would lead the model containing the combination color(a) = yellow and color(b) = yellow to have a total score of 0.

However, this is not the case. Instead, models that do not contain yellow will have their total score increased by 5 for each occurrence of yellow that is avoided. Thus, the model with color(a) = yellow and color(b) = yellow will still have a total score of 10, because it satisfies the preference of color(a) = color(b). On the other hand, a model with color(a) = red and color(b) = red will have a total score of 26, which is calculated as follows:

Preference for color(a) = red: 3
Preference for color(a) != yellow: 5
Preference for color(b) = red: 3
Preference for color(b) != yellow: 5
Preference for color(a) = color(b): 10
Total Score: 3 + 5 + 3 + 5 + 10 = 26

Maximize Score¶

To actually utilize the declared preferences and their scores, we also need to instruct the solver to consider these preference values when searching for solutions. This can be done using the preference_maximizeScore/0 predicate.

Warning

When using preferences by running Clingo on the command line, it is currently necessary to also provide the --heuristic=domain option to ensure that the solver properly considers the preferences when generating models.

preference_maximizeScore.

This prints the found models in descending order of their preference scores.