PVS-Types

MiniBrass revolves around the central concept of partial valuation structures (also called preference valuation structures, PVS). A PVS instance represents a coherent set of soft constraints (or preferences) that all map a solution to the element type of the underlying PVS-type. The element type must also offer a (partial) ordering and a combination operation to aggregate several individual valuations.

But let’s first consider an example implemented in the MiniBrass standard library.

Weighted constraints assign a weight w[c] to every soft constraint c that acts as a penalty if violated. The overall violation is found by summing up all penalties.

To see it in action, consider the following example

include "defs.mbr";

PVS: user1Prefs = new WeightedCsp("user1Prefs") {
soft-constraint c1: 'x > y' :: weights('3');
soft-constraint c2: 'x = y + 2' :: weights('1');
};

output '["x = \(x), y = \(y)"]';

solve user1Prefs;

based on a simplistic constraint model

include "soft_constraints/minibrass.mzn";
include "pvsTypes_o.mzn";

var 0..3: x;
var 0..3: y;

solve search miniBrass();

It can be interpreted as “it’s pivotal that x is greater than y” and it would be even better if the difference were exactly 2.

Our output shows the results to this model

Intermediate solution:x = 0, y = 0
Valuations: mbr_overall_user1Prefs = 4
----------
Intermediate solution:x = 2, y = 0
Valuations: mbr_overall_user1Prefs = 0
----------
==========

The first found solution (x = 0, y = 0) violates both x > y with penalty 3 and x = y + 2 with penalty 1. Hence the score 4.

The second solution (`x = 2, y = 0’) satisfies both soft constraints, thus resulting an overall score of 0.

But how is the type WeightedCsp actually defined?

Inspecting an existing type

To see this, we take a look at its definition in defs.mbr, found in the mbr_std folder in the mbr2mzn.jar directory (see here).

type WeightedCsp = PVSType<bool, int> =
params {
int: k :: default('1000');
array[1..nScs] of int: weights :: default('1');
} in
instantiates with "soft_constraints/mbr_types/weighted_type.mzn" {
times -> weighted_sum;
is_worse -> is_worse_weighted;
top -> 0;
}
offers {
heuristics -> getSearchHeuristicWeighted;
};

WeightedCsp shows all important characteristics :

• An element type (the scale, weight, etc. by which we want to order solutions) which here corresponds to int
• A specification type which refers to the data type of the soft constraint expressions. Most often, this is bool but sometimes we want something like a cost function that directly maps to int
• A combination operation weighted_sum, which means that we build a some of all violated soft constraints, weighted by their importance
• An ordering relation is_worse_weighted which orders the elements of the element type and a top element (here 0)

The actual implementations of weighted_sum and is_worse_weighted are a MiniZinc function and a MiniZinc predicate, respectively, which are defined in mbr_types/weighted_type.mzn (see here).

function var int: weighted_sum(array[int] of var bool: b,
par int: nScs, par int: k, array[int] of par int: weights) =
sum(i in index_set(b)) ( (1 - bool2int(b[i])) * weights[i]);

predicate is_worse_weighted(var int: x, var int: y,
par int: nScs, par int: k, array[int] of par int: weights) =
x > y;

These ingredients are all encoded in the PVS type and thus reusable for, e.g., many WeightedCsp instances but also compatible with other PVS types.

Common PVS types

The following table highlights the commonalities of PVS types. By convention, we always read a predicate a is_worse_than b from left to right. Hence, for weighted CSP, a is_worse_than b if and only if a > b (since there is a higher violation).

PVS-Type Spec. / Element type times is_worse top
Weighted CSP bool / int weighted_sum is_greater_than k (max violation)
Cost Function Networks int sum is_greater_than k
Set-Based Max CSP bool / set of int set_union superset {}
Fuzzy CSP [0.0 1.0] min is_less_than 1.0

All standard types can be found here.