global constraints and constraint programming michael maher loyola university chicago
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Constraint Personalities
Type A
aggressive, time-sensitive
Type B
relaxed, deliberate
There are two kinds of constraints:
Type A Constraints
Composition of constraints is totally understood by a constraint solver Intervals in Solver, CHIP, … Domains in CSP Linear inequalities in CLP(R) Term equations in Prolog Linear inequalities in Math Prog
These constraints can be solved algorithmically (at low cost).
Constraint Solver
Constraint store holding all active Type A constraints.
On-line algorithms for deciding consistency, implication, …
Constraint store changes by addition/undo/deletion of constraints.
Type B Constraints
Composition of constraints is at best partly understood Global constraints Arithmetic constraints in Solver,
CHIP, … Constraints in CSP Integrality in Math Prog
These are the reason we do search.
In FD solvers, arithmetic inequalities are Type B constraints.
From x + y ≤ z, z ≤ y + 2 the system does not understand x ≤ 2.
Intervals are Type A constraints.
From x :: 1..5 and x :: 3..7 the system understands x :: 3..5.
Type B constraints are implemented as propagators of Type A constraints
x :: 1..3, y :: 2..6 x + y ≤ z
z ≤ y + 2 z :: 3..z :: 3..8
Global Constraints
Collections of many small constraints that are treated as a unit alldifferent cumulative cycle etc
Not global !
Global Constraints
Implementations of global constraints are encapsulations of on-line algorithms alldifferent cumulative cycle etc
Often (?) the implementations are based on CO techniques.
Constraint Programming is a software architecture that supports the combination of CO algorithms to solve problems.
Arc Consistency
Ideally, an implementation will propagate all information expressible with Type A constraints that is a consequence of the Type B constraint.
If B & c → c’ then c → c’
This is a general form of arc consistency.
Arc Consistency
The definition covers many existing definitions:
Arc consistency in CSP Interval consistency in FD Echidna consistency Hull consistency Rule consistency
Arc Consistency
Existing forms of arc consistency only admit unary Type A constraints. They are all weakenings of arc consistency in CSP.
We can also use other Type A constraints (like ordering x ≤ y) to get different, not weaker, consistencies.
Minimum
min(x, y, z)
For AC we need the following propagations, among others:
min(x, y, z) → z ≤ x, z ≤ y x ≤ y, min(x, y, z) → z = x u ≤ x, u ≤ y, min(x, y, z) → u ≤ z
Implementing non-unary AC is hard.
Improve on AC
Stronger consistency conditionsconsider more than one global constraint at a time
Weaker consistency conditions
Improve on AC
Stronger consistency conditionsconsider more than one global constraint at a time
Weaker consistency conditionsavoid or delay complete propagation
Exercises for COs
Define and implement new global constraints
Define and implement constraint solver for new class of constraints
Introduce us to the Dark Arts
Exercises for BRs
Formulate and implement BR in a conjunctive constraint context
Extend BR to situations where tests for consistency are not complete.