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problems concrete problems P α γ α γ( (P)) α α (P) = P Fig. 3. The on rete and the abstra t problem. In Figure 3, P is the starting SCSP problem. Then with the mapping we get ~ P = (P ), whi h is an abstra tion of P . By applying the mapping to ~ P , we get the problem ( (P )). Let us rst noti e that these two problems (P and ( (P ))) are related by a pre ise property, as stated by the following theorem. Theorem 3. Given an SCSP problem P over S, we have that P vS ( (P )). Proof. Immediately follows from the properties of a Galois insertion, in parti ular from the fa t that x S ( (x)) for any x in the on rete latti e. In fa t, P vS ( (P )) means that, for ea h tuple in ea h onstraint of P , the semiring value asso iated to su h a tuple in P is smaller (w.r.t. S) than the orresponding value asso iated to the same tuple in ( (P )). 2 Noti e that this implies that, if a tuple in ( (P )) has semiring value 0, then it must have value 0 also in P . This holds also for the solutions, whose semiring value is obtained by ombining the semiring values of several tuples. Corollary 1. Given an SCSP problem P over S, we have that Sol(P ) vS Sol( ( (P )). Proof. We re all that Sol(P ) is just a onstraint, whi h is obtained asN(C) + on. Thus the statement of this orollary omes from the monotoni ity of and +. 2 Therefore, by passing from P to ( (P )), no new in onsisten ies are introdu ed: if a solution of ( (P )) has value 0, then this was true also in P . However, it is possible that some in onsisten ies are forgotten (that is, they appear to be onsistent after the abstra tion pro ess). If the abstra tion preserves the semiring ordering (that is, applying the abstra tion fun tion and then ombining gives elements whi h are in the same ordering as the elements obtained by ombining only), then there is also an interesting relationship between the set of optimal solutions of P and that of (P ). In fa t, if a ertain tuple is optimal in P , then this same tuple is also optimal in (P ). Let us rst investigate the meaning of the order-preserving property. De nition 12 (order-preserving abstra tion). Consider two sets I1 and I2 of on rete elements. Then an abstra tion is said to be order-preserving if ~ Y x2I1 (x) ~ S ~ Y x2I2 (x) ) Y x2I1 x S Y x2I2 x where the produ ts refer to the multipli ative operations of the on rete (Q) and the abstra t ( ~ Q) semirings. In words, this notion of order-preservation means that if we rst abstra t and then ombine, or we ombine only, we get the same ordering (but on di erent semirings) among the resulting elements. Example 3. An abstra tion whi h is not order-preserving an be seen in Figure 4. Here, the on rete and the abstra t sets, as well as the additive operations of the two semirings, an be seen from the pi ture. For the multipli ative operations, we assume they oin ide with the glb of the two semirings. In this ase, the on rete ordering is partial, while the abstra t ordering is total. Fun tions and are depi ted in the gure by arrows going from the on rete semiring to the abstra t one and vi e versa. Assume that the on rete problem has no solution with value 1. Then all solutions with value a or b are optimal. Suppose a solution with value b is obtained by omputing b = 1 b, while we have a = 1 a. Then the abstra t ounterparts will have values (1) 0 (b) = 1 0 0 = 0 and (1) 0 (a) = 1 0 1 = 1. Therefore the solution with value a, whi h is optimal in the on rete problem, is not optimal anymore in the abstra t problem. 1 a b 0 1 0 γ γ α α α α Fig. 4. An abstra tion whi h is not order-preserving. Example 4. The abstra tion in Figure 2 is order-preserving. In fa t, onsider two abstra t values whi h are ordered, that is 0 0 1. Then 1 = 1 01 = (x) 0 (y), where both x and y must be greater than 0:5. Thus their on rete ombination (whi h is the min), say v, is always greater than 0:5. On the other hand, 0 an be obtained by ombining either two 0's (therefore the images of two elements smaller than or equal to 0:5, whose minimum is smaller than 0:5 and thus smaller than v), or by ombining a 0 and a 1, whi h are images of a value greater than 0:5 and one smaller than 0:5. Also in this ase, their ombination (the min) is smaller than 0:5 and thus smaller than v. Thus the order-preserving property holds. Example 5. Another abstra tion whi h is not order-preserving is the one that passes from the semiring hN [f+1g;min; sum; 0;+1i, where we minimize the sum of values over the naturals, to the semiring hN [ f+1g;min;max; 0;+1i, where we minimize the maximum of values over the naturals. In words, this abstra tion maintains the same domain of the semiring, and the same additive operation (min), but it hanges the multipli ative operation (whi h passes from sum to max). Noti e that the semiring orderings are the opposite as those usually used over the naturals: if i is smaller than j then j S i, thus the best element is 0 and the worst is +1. The abstra tion fun tion is just the identity, and also the on retization fun tion . In this ase, onsider in the abstra t semiring two values and the way they are obtained by ombining other two values of the abstra t semiring: for example, 8 = max(7; 8) and 9 = max(1; 9). In the abstra t ordering, 8 is higher than 9. Then, let us see how the images of the ombined values (the same values, sin e is the identity) relate to ea h other: we have sum(7; 8) = 15 and sum(1; 9) = 10, and 15 is lower than 10 in the on rete ordering. Thus the order-preserving property does not hold. Noti e that, if we redu e the sets I1 and I2 to singletons, say x and y, then the order-preserving property says that (x) ~ S (y) implies that x S y. This means that if two abstra t obje ts are ordered, then their on rete ounterparts have to be ordered as well, and in the same way. Of ourse they ould never be ordered in the opposite sense, otherwise would not be monotoni ; but they ould be in omparable. Therefore, if we hoose an abstra tion where in omparable obje ts are mapped by onto ordered obje ts, then we don't have the order-preserving property. A onsequen e of this is that if the abstra t semiring is totally ordered, and we want an order-preserving abstra tion, then the on rete semiring must be totally ordered as well. On the other hand, if two abstra t obje ts are not ordered, then the orresponding on rete obje ts an be ordered in any sense, or they an also be not omparable. Noti e that this restri tion of the order-preserving property to singleton sets always holds when the on rete ordering is total. In fa t, in this ase, if two abstra t elements are ordered in a ertain way, then it is impossible that the orresponding on rete elements are ordered in the opposite way, be ause, as we said above, of the monotoni ity of the fun tion. Theorem 4. Consider an abstra tion whi h is order-preserving. Given an SCSP problem P over S, we have that Opt(P ) Opt( (P )). Proof. Let us take a tuple t whi h is optimal in the on rete semiring S, with value v. Then v has been obtained by multiplying the values of some subtuples. Suppose, without loss of generality, that the number of su h subtuples is two (that is, we have two onstraints): v = v1 v2. Let us then take the value of this tuple in the abstra t problem, that is, the abstra t ombination of the abstra tions of v1 and v2: this is v0 = (v1) 0 (v2). We have to show that if v is optimal, then also v0 is optimal. Suppose then that v0 is not optimal, that is, there exists another tuple t00 with value v00 su h that v0 S0 v00. Assume v00 = v00 1 0 v00 2 . Now let us see the value of tuple t00 in P . If we set v00 i = ( vi), then we have that this value is v = v1 v2. Let us now ompare v with v. Sin e v0 ~ S v00, by order-preservation we get that v S v. But this means that v is not optimal, whi h was our initial assumption. Therefore v0 has to be optimal. 2 Therefore, in ase of order-preservation, the set of optimal solutions of the abstra t problem ontains all the optimal solutions of the on rete problem. In other words, it is not allowed that an optimal solution in the on rete domain beomes non-optimal in the abstra t domain. However, some non-optimal solutions ould be ome optimal by be oming in omparable with the optimal solutions. Thus, if we want to nd an optimal solution of the on rete problem, we ould nd all the optimal solutions of the abstra t problem, and then use them on the on rete side to nd an optimal solution for the on rete problem. Assuming that working on the abstra t side is easier than on the on rete side, this method ould help us nd an optimal solution of the on rete problem by looking at just a subset of tuples in the on rete problem. Another important property, whi h holds for any abstra tion, on erns omputing bounds that approximate an optimal solution of a on rete problem. In fa t, any optimal solution, say t, of the abstra t problem, say with value ~ v, an be used to obtain both an upper and a lower bound of an optimum in P . In fa t, we an prove that there is an optimal solution in P with value between (~ v) and the value of t in P . Thus, if we think that approximating the optimal value with a value within these two bounds is satisfa tory, we an take t as an approximation of an optimal solution of P . Theorem 5. Given an SCSP problem P over S, onsider an optimal solution of (P ), say t, with semiring value ~ v in (P ) and v in P . Then there exists an optimal solution t of P , say with value v, su h that v v (~ v). Proof. By lo al orre tness of the multipli ative operation of the abstra t semiring, we have that v S (~ v). Sin e v is the value of t in P , either t itself is optimal in P , or there is another tuple whi h has value better than v, say v. We will now show that v annot be greater than (~ v). In fa t, assume by absurd that (~ v) S v. Then, by lo al orre tness of the multipli ative operation of the abstra t semiring, we have that ( v) is smaller than the value of t in (P ). Also, by monotoni ity of , by (~ v) S v we get that ~ v ~ S ( v). Therefore by transitivity we obtain that ~ v is smaller than the value of t in (P ), whi h is not possible be ause we assumed that ~ v was optimal. Therefore there must be an optimal value between v and (~ v). 2 Thus, given a tuple t with optimal value ~ v in the abstra t problem, instead of spending time to ompute an exa t optimum of P , we an do the following: { ompute (~ v), thus obtaining an upper bound of an optimum of P ; { ompute the value of t in P , whi h is a lower bound of the same optimum of P ; { If we think that su h bounds are lose enough, we an take t as a reasonable approximation (to be pre ise, a lower bound) of an optimum of P . Noti e that this theorem does not need the order-preserving property in the abstra tion, thus any abstra tion an exploit its result. 4.2 Working on the abstra t problem Consider now what we an do on the abstra t problem, (P ). One possibility is to apply an abstra t fun tion ~ f , whi h an be, for example, a lo al onsisten y algorithm or also a solution algorithm. In the following, we will onsider fun tions ~ f whi h are always intensive, that is, whi h bring the given problem loser to the bottom of the latti e. In fa t, our goal is to solve an SCSP, thus going higher in the latti e does not help in this task, sin e solving means ombining onstraints and thus getting lower in the latti e. Also, fun tions ~ f will always be lo ally orre t with respe t to any fun tion fsol whi h solves the on rete problem. We will all su h a property solutionorre tness. More pre isely, given a problem P with onstraint set C, fsol(P ) is a new problem P 0 with the same topology as P whose tuples have semiring values possibly lower. Let us all C 0 the set of onstraints of P 0. Then, for any onstraint 0 2 C 0, 0 = (NC) +var( 0). In other words, fsol ombines all onstraints of P and then proje ts the resulting global onstraint over ea h of the original onstraints. De nition 13. Given an SCSP problem P over S, onsider a fun tion ~ f on (P ). Then ~ f is solutionorre t if, given any fsol whi h solves P , ~ f is lo ally orre t w.r.t. fsol. We will also need the notion of safeness of a fun tion, whi h just means that it maintains all the solutions. De nition 14. Given an SCSP problem P and a fun tion f : PL ! PL, f is safe if Sol(P ) = Sol(f(P )). It is easy to see that any lo al onsisten y algorithm, as de ned in [4℄, an be seen as a safe, intensive, and solutionorre t fun tion. From ~ f( (P )), applying the on retization fun tion , we get ( ~ f( (P ))), whi h therefore is again over the on rete semiring (the same as P ). The following property says that, under ertain onditions, P and P ( ~ f( (P ))) are equivalent. Figure 5 des ribes su h a situation. In this gure, we an see that several partial order lines have been drawn: { on the abstra t side, fun tion ~ f takes any element loser to the bottom, be ause of its intensiveness; { on the on rete side, we have that P ( ~ f( (P ))) is smaller than both P and ( ~ f( (P ))) be ause of the properties of ; ( ~ f( (P ))) is smaller than ( (P )) be ause of the monotoni ity of ; ( ~ f( (P ))) is higher than fsol(P ) be ause of the solutionorre tness of ~ f ; fsol(P ) is smaller than P be ause of the way fsol(P ) is onstru ted; if is idempotent, then it oin ides with the glb, thus we have that P ( ~ f( (P ))) is higher than fsol(P ), be ause by de nition the glb is the higher among all the lower bounds of P and ( ~ f( (P ))). (P) α f( ~ (P)) α f( ~ f( ~ x O P f_sol(P) γ α α γ α γ( (P)) α (P))) α γ( (P)) α γ( P x idempotent concrete problems abstract problems Fig. 5. The general abstra tion s heme, with idempotent. Theorem 6. Given an SCSP problem P over S, onsider a fun tion ~ f on (P ) whi h is safe, solutionorre t, and intensive. Then, if is idempotent, Sol(P ) = Sol(P ( ~ f( (P )))). Proof. Take any tuple t with value v in P , whi h is obtained by ombining the values of some subtuples, say two: v = v1 v2. Let us now onsider the abstra t versions of v1 and v2: (v1) and (v2). Fun tion ~ f hanges these values by lowering them, thus we get ~ f( (v1)) = v0 1 and ~ f( (v2)) = v0 2. Sin e ~ f is safe, we have that v0 1 0 v0 2 = (v1) 0 (v2) = v0. Moreover, ~ f is solutionorre t, thus v S (v0). By monotoni ity of , we have that (v0) S (v0 i) for i = 1; 2. This implies that (v0) S ( (v0 1) (v0 2)), sin e is idempotent by assumption and thus it oin ides with the glb. Thus we have that v S ( (v0 1) (v0 2)). To prove that P and P ( ~ f( (P ))) give the same value to ea h tuple, we now have to prove that v = (v1 (v0 1)) (v2 (v0 2)). By ommutativity of , we an write this as (v1 v2) ( (v0 1) (v0 2)). Now, v1 v2 = v by assumption, and we have shown that v S (v0 1) (v0 2). Therefore v ( (v0 1) (v0 2)) = v. 2 This theorem does not say anything about the power of ~ f , whi h ould make many modi ations to (P ), or it ould also not modify anything. In this last ase, ( ~ f( (P ))) = ( (P )) w P (see Figure 6), so P ( ~ f( (P ))) = P , whi h means that we have not gained anything in abstra ting P . However, we an always use the relationship between P and (P ) (see Theorem 4 and 5) to nd an approximation of the optimal solutions and of the in onsisten ies of P . (P) α f( ~ (P)) α f( ~ (P)) α γ abstract problems concrete problems P α γ α = = γ( (P)) α Fig. 6. The s heme when ~ f does not modify anything. If instead ~ f modi es all semiring elements in (P ), then if the order of the on rete semiring is total, we have that P ( ~ f( (P ))) = ( ~ f( (P ))) (see Figure 7), and thus we an work on ( ~ f( (P ))) to nd the solutions of P . In fa t, ( ~ f( (P ))) is lower than P and thus loser to the solution. Theorem 7. Given an SCSP problem P over S, onsider a fun tion ~ f on (P ) whi h is safe, solutionorre t, and intensive. Then, if is idempotent, ~ f modi es every semiring element in (P ), and the order of the on rete semiring is total, we have that P wS ( ~ f( (P ))) wS fsol(P ). Proof. Consider any tuple t in any onstraint of P , and let us all v its semiring value in P and vsol its value in fsol(P ). Obviously, we have that vsol S v. Now take v0 = ( (v)). By monotoni ity of , we annot have v <S v0. Also, by solutionorre tness of ~ f , we annot have v0 <S vsol. Thus we must have vsol S v0 s v, whi h proves the statement of the theorem. 2 Noti e that we need the idempoten e of the operator for Theorem 6 and 7. If instead is not idempotent, then we an prove something weaker. Figure 8 shows this situation. With respe t to Figure 5, we an see that the possible non-idempoten e of hanges the partial order relationship on the on rete side. In parti ular, we don't have the problem P ( ~ f( (P ))) any more, nor the problem fsol(P ), sin e these problems would not have the same solutions as P (P) α f( ~ (P)) α f( ~ t o t a l o r d e r abstract problems concrete problems