From a technical point of view, the main difference between tree search and local (or move-based) search is that tree search adds assignments while local search changes them. During tree search constraints get tightened when going down the tree, and this is undone in reverse order when backing up the tree to a parent node. This fits well with the idea of constraint propagation.
It is characteristic of local search that a move produces a small change, but it is not clear what effect this will have on the constraints. They may become more or less satisfied. We therefore need implementations of the constraints that monitor changes rather than propagate instantiations.
Local search can be implemented quite naturally in ECLiPSe using the
In essence, the difference between implementing tree search techniques
search in ECLiPSe is that, instead of instantiating variables during
search, local search progresses by changing tentative values of
For the satisfiability example of the last section, we can change
local_search by simply replacing the
guess predicate by the
local_search(Vars) :- conflict_constraints(cs,List), ( List =  -> set_to_tent(Vars) ; List = [Constraint|_] -> term_variables(Constraint,[Var|_]), move(Var), local_search(Vars) ). move(Var) :- Var tent_get Value, NewValue is (1-Value), Var tent_set NewValue.
There is no guarantee that this move will reach a better assignment, since NewValue may violate more constraints than the original Value.
To find a neighbour which overall increases the number of satisfied
constraints we could replace
local_search with the predicate
hill_climb(Vars) :- conflict_constraints(cs,List), length(List,Count), ( Count = 0 -> set_to_tent(Vars) ; try_move(List,NewCount), NewCount < Count -> hill_climb(Vars) ; write(’local optimum: ’), writeln(Count) ). try_move(List,NewCount) :- select_var(List,Var), move(Var), conflict_constraints(cs,NewList), length(NewList,NewCount). select_var(List,Var) :- member(Constraint,List), term_variables(Constraint,Vars), member(Var,Vars).
Some points are worth noticing:
The code for
try_move is very inefficient, because it
repeatedly goes through the whole list of conflict constraints to
count the number of constraints in conflict.
The facility to propagate tentative values supports more efficient
maintenance of the number constraints in conflict.
This technique is known as maintenance of invariants (see
For the propositional satisfiability example we can maintain the
number of satisfied clauses to make the hill climbing implementation
The following program not only monitors each clause for conflict, but
it also records in a boolean variable whether the clause is satisfied.
Each tentative assignment to the variables is propagated to the
tentative value of the boolean.
The sum of the boolean
BSum records for any tentative
assignment of the propositional variables, the number of satisfied
This speeds up hill climbing because, after each move, its effect on
the number of satisfied clauses is automatically computed by the
propagation of tentative values.
prop_sat_2(Vars) :- Vars = [X1,X2,X3], tent_init(Vars), clause_cons(X1 or neg X2 or X3,B1), clause_cons(neg X1 or neg X2,B2), clause_cons(X2 or neg X3,B3), BSum tent_is B1+B2+B3, hill_climb_2(Vars,BSum). clause_cons(Clause,B) :- Clause $= 1 r_conflict cs, B tent_is Clause. hill_climb_2(Vars,BSum) :- conflict_constraints(cs,List), BSum tent_get Satisfied, ( List= -> set_to_tent(Vars) ; select_var(List,Var), move(Var), tent_get(BSum) > Satisfied -> hill_climb_2(Vars,BSum) ; write(’local optimum: ’), writeln(Count) ).
To check whether the move is uphill, we retrieve the tentative
BSum before and after the move is done.
Remember that, since the move operator changes the tentative values of
some variable, the
tent_is primitive will automatically
This code can be made more efficent by recording more invariants, as described in .
Local search can be implemented in ECLiPSe with the repair library. Invariants can be implemented by tentative value propagation using tent_is/2.