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Set Domain Solvers

Set Domain Solvers. ic_sets - the set constraint solver. Load the ic_sets library by using :- lib(ic_sets). at the beginning of your code, or type lib(ic_sets). at the top-level ECLiPSe prompt. Sets must be finite and contain only integers We write sets as sorted lists

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Set Domain Solvers

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  1. Set Domain Solvers

  2. ic_sets - the set constraint solver • Load the ic_sets library by using • :- lib(ic_sets). • at the beginning of your code, or type • lib(ic_sets). • at the top-level ECLiPSe prompt. • Sets must be finite and contain only integers • We write sets as sorted lists • Note: there is another set solver lib(conjunto) which is being phased out, and lib(fd_sets) which is identical to ic_sets, but works with lib(fd)

  3. Functionality The ic_sets library implements • variables with finite integers set domains • basic constraints over set expressions • constraints over set cardinality and weights • a search primitive

  4. Set domain variables • An integer (ground) set • X = [1,7,12,19] • A set variable • Set :: [7]..[1,7,12,19] • attaches an initial domain to a variable or • intersect their old domain with the new one. • Set variables have a lower and upper bound • Both bounds are (ground) sets • lower bound: set of elements that are definitelyin the set • upper bound: set of elements that may be in the set • They also have a separate lower and upper bound on cardinality.

  5. The basic set of constraints • X in S, X notin S • Set membership. • S1 sameset S2, S1 subset S2, S1 disjoint S2 • Set equality, subset, disjointness. • all_union(Sets,Union), all_disjoint(Sets) • Generalised union and disjointness. • #(Set, Card), weight(Set, ElementWeights, Weight) • Set cardinality and set weight. • Set expressions are allowed: • Intersection S1 /\ S2, Union S1 \/ S2, Difference S1 \ S2

  6. Bound propagation • Constraints affect set bounds: • ?- S::[]..[1,2,3,4,5]. • S = S{([] .. [1, 2, 3, 4, 5]) : C{[0 .. 5]}} • yes. • ?- S::[]..[1,2,3,4,5], 3 in S, 5 notin S. • S = S{[3] \/ ([] .. [1, 2, 4]) : C{[1 .. 4]}} • yes. • ?- S::[]..[1,2,3,4,5], S subset [1,5,6]. • S = S{([] .. [1, 5]) : C{[0 .. 2]}} • yes. • ?- S::[2,4]..[1,2,3,4,5], #(S,2). • S = [2, 4] • yes.

  7. Basic search support • insetdomain(SetVar, Options…) • Try to instantiate SetVar to a ground set. • [eclipse 23]: A :: []..[1,2,3,4,5], insetdomain(A,_,_,_).A = [1, 2, 3, 4, 5] More? (;) A = [1, 2, 3, 4] More? (;) A = [1, 2, 3, 5] More? (;)

  8. Set example – Steiner triplets • Ternary Steiner system of order n: the set of triplets taking values between 1 and n such that all pairs included in any triplets are different. •  a triplet can share at most one number with any other triplet. • Set has n(n-1)/6 triplets. • Not easy to express generically (for any n) with fd only….

  9. Steiner Triplets (Program) :- lib(ic_sets).steiner(N, LS) :-NB is N * (N-1) // 6, % compute number of set variables needed intsets(LS, NB, 1, N), % NB sets can take values between 1 and N( foreach(S,LS) do #(S,3) ), % sets are triplets ( fromto(LS,[S1|Ss],Ss,[]) do ( foreach(S2,Ss), param(S1) do % no triplet can share more than one number with another #(S1 /\ S2,C), ic: (C #=< 1) ) ), label_sets(LS).label_sets([]).label_sets([S | LS]) :- insetdomain(S,_,_,_), label_sets(LS).

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