Constraint-satisfaction search

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1 Lecture 6 onstraint-satisfaction search Milos Hauskrecht milos@cs.pitt.edu 539 Sennott Square Search problem search problem: Search space (or state space): a set of objects among which we conduct the search; Initial state: an object we start to search from; Operators (actions): transform one state in the search space to the other; Goal condition: describes the object we search for Possible metric on a search space: measures the quality of the object with regard to the goal Search problems occur in planning optimizations learning

onstraint satisfaction problem (SP) onstraint satisfaction problem (SP) is a configuration search problem where: state is defined by a set of variables Goal condition is represented by a set

2 onstraint satisfaction problem (SP) onstraint satisfaction problem (SP) is a configuration search problem where: state is defined by a set of variables Goal condition is represented by a set constraints on possible variable values Special properties of the SP allow more specific procedures to be designed and applied for solving them ample of a SP: N-queens Goal: n queens placed in non-attacking positions on the board Variables: Represent queens one for each column: Q1 Q Q3 Q Values: Row placement of each queen on the board {1 3 } Q1 = Q = onstraints: Qi Q j Q Q i j i j Two queens not in the same row Two queens not on the same diagonal

3 Satisfiability (ST) problem etermine whether a sentence in the conjunctive normal form (N) is satisfiable (can evaluate to true) Used in the propositional logic (covered later) ( P Q R) ( P R S ) ( P Q T )K Variables: Propositional symbols (P R T S) Values: True alse onstraints: very conjunct must evaluate to true at least one of the literals must evaluate to true ( P Q R) True ( P R S ) True K Other real world SP problems Scheduling problems:.g. telescope scheduling High-school class schedule esign problems: Hardware configurations VLSI design More comple problems may involve: real-valued variables additional preferences on variable assignments the optimal configuration is sought

4 Map coloring olor a map using k different colors such that no adjacent countries have the same color Variables:? Variable values:? onstraints:? Map coloring olor a map using k different colors such that no adjacent countries have the same color Variables: Represent countries Values: K -different colors {Red lue Green..} onstraints:?

5 Map coloring olor a map using k different colors such that no adjacent countries have the same color Variables: Represent countries Values: K -different colors {Red lue Green..} onstraints: etc n eample of a problem with binary constraints onstraint satisfaction as a search problem ormulation of a SP as a search problem: States. ssignments(partial complete) of values to variables. Initial state. No variable is assigned a value. Operators. ssign a value to one of the unassigned variables. Goal condition. ll variables are assigned no constraints are violated. onstraints can be represented: plicitly by a set of allowable values Implicitly by a function that tests for the satisfaction of constraints

6 Solving SP as a standard search Unassigned: Q Q Q Q ssigned: 1 3 Unassigned: Q Q Q3 ssigned: Q 1 = 1 Unassigned: Q Q Q3 ssigned: Q 1 = Unassigned: ssigned: Q Q Q 3 = Q 1 = Solving a SP through standard search Maimum depth of the tree (m):? epth of the solution (d) :? ranching factor (b) :? Unassigned: Q Q Q Q ssigned: 1 3 Unassigned: Q Q3 Q ssigned: Q = 1 1 Unassigned: Q Q Q3 ssigned: Q = 1 Unassigned: Q 3 Q ssigned: Q = Q 1 =

7 Solving a SP through standard search Maimum depth of the tree: Number of variables of the SP epth of the solution: Number of variables of the SP ranching factor: if we fi the order of variable assignments the branch factor depends on the number of their values Unassigned: Q Q Q Q ssigned: 1 3 Unassigned: Q Q3 Q ssigned: Q = 1 1 Unassigned: Q Q Q3 ssigned: Q = 1 Unassigned: ssigned: Q Q Q 3 = Q 1 = Solving a SP through standard search What search algorithm to use:? epth of the tree = epth of the solution=number of vars Unassigned: Q Q Q Q ssigned: 1 3 Unassigned: Q Q3 Q ssigned: Q = 1 1 Unassigned: Q Q Q3 ssigned: Q = 1 Unassigned: Q 3 Q ssigned: Q = Q 1 =

8 Solving a SP through standard search What search algorithm to use: epth first search Since we know the depth of the solution S in contet of SP is also referred to as backtracking Unassigned: Q Q Q Q ssigned: 1 3 Unassigned: Q Q3 Q ssigned: Q = 1 1 Unassigned: Q Q Q3 ssigned: Q = 1 Unassigned: ssigned: Q Q Q 3 = Q 1 = hecking constraint consistency The violation of constraints needs to be checked for each node either during its generation or before its epansion onsistency of constraints: urrent variable assignments together with constraints restrict remaining legal values of unassigned variables; The remaining legal and illegal values of variables may be inferred (effect of constraints propagates) To prevent blind search space eploration it is necessary to keep track of the remaining legal values so we know when the constraints are violated and when to terminate the search

9 onstraint propagation state (more broadly) is defined by a set of variables and their legal and illegal assignments Legal and illegal assignments can be represented through variable equations and variable disequations ample: map coloring quation = Red isequation Red onstraints + assignments can entail new equations and disequations = Red Red onstraint propagation: the process of inferring of new equations and disequations from eisting equations and disequations onstraint propagation ssign - equations - disequations

10 onstraint propagation ssign = - equations - disequations onstraint propagation ssign =lue =lue

11 onstraint propagation ssign =lue =? =lue onstraint propagation ssign =Green =Green =lue

12 onstraint propagation ssign =Green =Green =? =lue onstraint propagation ssign =Green =Green =? =lue onflict!!! No legal assignments available for and

13 onstraint propagation We can derive remaining legal values through propagation =Green =Green =? =? =lue onstraint propagation We can derive remaining legal values through propagation =? =? =lue =? =Green =Green =Red

14 onstraint propagation We can derive remaining legal values through propagation =lue =Green =Green =Red =Green =Green =Red onstraint propagation Three known techniques for propagating the effects of past assignments and constraints: Value propagation rc consistency orward checking ifference: ompleteness of inferences Time compleity of inferences.

15 onstraint propagation 1. Value propagation. Infers: equations from the set of equations defining the partial assignment and a constraint. rc consistency. Infers: disequations from the set of equations and disequations defining the partial assignment and a constraint equations through the ehaustion of alternatives 3. orward checking. Infers: disequations from a set of equations defining the partial assignment and a constraint quations through the ehaustion of alternatives Restricted forward checking: uses only active constraints (active constraint only one variable unassigned in the constraint) Value propagation: analysis Value propagation. Infers: equations from the set of equations defining the partial assignment and a constraint Procedure: t every step after a new variable gets assigned a value we check if a constrain does not imply a new equation on yet to be assigned variable. set of equations may be inferred

16 orward checking: analysis orward checking. Infers: disequations from a set of equations defining the partial assignment and a constraint quations through the ehaustion of alternatives Procedure: t every step after a new variable gets assigned a value (a new equation is created) we check a constraint if it does not imply a new disequation on yet to be assigned variable. fter all possible disequations are derived we check if a new equation is not implied by a set of disequations nd repeat till no new derivations can be made set of disequations and equations may be inferred rc consistency: analysis rc consistency. Infers: disequations from the set of equations and disequations defining the partial assignment and a constraint equations through the ehaustion of alternatives Procedure: t every step after a new equation or a disequation is generated we need to check constraints if a new disequation is not implied on yet to be assigned variable. fter all possible disequations are derived we check if a new equation is not implied by a set of disequations. nd repeat till no new derivations are possible set of disequations and equations may be inferred

17 Heuristics for SPs acktracking searches the space in the depth-first manner. ut we still can choose: Which variable to assign net? Which value to choose first? Heuristics Most constrained variable Which variable is likely to become a bottleneck? Least constraining value Which value gives us more fleibility later? amples: map coloring Heuristics for SP Heuristics Most constrained variable ountry is the most constrained one (cannot use Red Green) Least constraining value ssume we have chosen variable Red is the least constraining valid color for the future

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