Conducta ante el diagnostico histologico de LSIL/CIN1

In Section 3.5, we have seen that arbitrary pattern setups may have some issues regarding the maximal common descriptions and their expressiveness with regard the extents they induce (see Section 3.5.1.3). This is due to the fact that pattern setups are too permissive. On the other hand, while pattern structures provide rather strong and useful properties, they are somehow too restrictive since they cannot model some usual pattern spaces as the one of sequential patterns. We propose a new structure, dubbed Pattern Multistructure, that lies between pattern setups which rely only on arbitrary posets and pattern structures which rely on lattices. We will see that pattern multistructures solve many of the aforementioned issues including the ones presented in section 3.5.1.3 and section 3.5.3.

3.6.1 Basic Definitions and Properties

Definition 3.25. A pattern setupP = (_G, (_{D, v),}δ) is said to be a pattern multistructure if for any A ⊆G, cov(A) is maximal-handle (cf. definition 2.10). In other words:

(∀A ⊆G) cov(A) =↓ cov∗(A)

Note 3.8. _{Note that in a pattern multistructure, we have cov(;) = ;}`=D is maximal-handle. Hence, we have_{D=↓ max(D}) or equivalently every chain in (_{D, v) is upper-bounded.}

A pattern multistructure adds an additional condition on a pattern setup which is the following: knowing maximal common descriptions covering all elements of a set of objects A allows us to deduce using the order v every single covering description.

It is clear that all pattern structures are by definition pattern multistructures. Graphs ordered by subgraph isomorphism relation introduced in [110] induce a pattern multistructure on the set of graphs, but not a pattern structure (a pattern structure is induced on sets of graphs incomparable w.r.t. subgraph isomorphism). Same remark holds for sequential patterns [39, 48]. This is under the assumption of the existence of a largest element > subsumed by all sequences/graphs (see Example 3.27).

Example 3.27. Reconsider the pattern setup presented in Example 3.12. Since only finite sequences are considered in the description space, we have: cov∗_{(;) = ; even if cov(;) =}D. Thus, the considered pattern setup in Example 3.12 is not a pattern multistructure due to the empty set. The common trick to handle the empty set is to enrich_{D with a largest element >,}W

Dif it does not exist, i.e. apply a dual lifting ofD(cf. Section 2.3.1). In such a case, we have cov∗_{(;) = {>}.}

Another less common pattern language is presented below.

Example 3.28. ConsiderDs:=©[a, a + l] × [b, b + l] | a, b ∈ R, l ∈ R+ª ∪ R2 the pattern language

of closed squares in R2. This language can be seen as a neighborhood pattern language [86] where the used norm is the ∞-norm. Let P :=¡{g1, g2}, (Ds, ⊇),δ¢ be a pattern setup where

δ(g1) = [2,5] × [1,4] andδ(g2) = [3,6] × [3,6] (i.e. the dark gray rectangles depicted in Fig. 1.4).

We have, cov∗({g1, g2}) = {[1 + l,6 + l] × [1,6] | l ∈ [0,1]} is infinite (see Fig. 1.4). Still, for any

d ∈ cov({g1, g2}) one can always find d∗∈ cov∗({g1, g2}) s.t. d ⊇ d∗ making pattern setupP a

pattern multistructure.

Let us reconsider the question: “What is the link between maximal covering descriptions and upper-approximations extents in a pattern multistructure?”. Before answering this question, we shall start by stating the following Lemma.

Lemma 3.2. Let (P, ≤) and (Q,≤) be two posets and let f : P → Q be an order-reversing mapping. We have for any S ⊆ P that ↑ f [↓ S] =↑ f [S].

Proof. _{Recall that ↑ and ↓ are closure operators (cf. Proposition 2.7). Let us start by showing} that ↑ f [S] ⊆↑ f [↓ S]. Since S ⊆↓ S, we have f [S] ⊆ f [↓ S]. Since ↑ is monotone, we have ↑ f [S] ⊆↑ f [↓ S]. It remains to show that ↑ f [↓ S] ⊆↑ f [S]. Let u ∈↑ f [↓ S], that is ∃v ∈ f [↓ S] s.t. v v u. Since v ∈ f [↓ S], then ∃x ∈↓ S s.t. v = f (x). Hence ∃y ∈ S s.t. x ≤ y. Using the fact that f is an anti-embedding, we obtain that f ( y) v f (x) v u. In other words, ∃w ∈ f [S] s.t. w v u. This is equivalent to say that u ∈↑ f [S]. We conclude hence that ↑ f [↓ S] ⊆↑ f [S].

Theorem 3.4 links between maximal common descriptions and upper-approximation extents in pattern multistructures.

Theorem 3.4. For any pattern multistructureP we have: (∀A ⊆_{G) A = min(ext[cov}∗_(A)])

Proof. The proof of the theorem is a straightforward application of Lemma 2.1 and Lemma 3.2. Let A ⊆G, sinceP is a pattern multistructure, then:

cov(A) =↓ max(cov(A)) ⇒ ext[cov(A)] = ext[↓ max(cov(A))] ⇒↑ ext[cov(A)] =↑ ext[↓ max(cov(A))] Since ext :D→℘(G) is an order reversing, then using Lemma 3.2 we have: ↑ ext[↓ max(cov(A))] =↑ ext[max(cov(A))] ⇒↑ ext[cov(A)] =↑ ext[max(cov(A)) ⇒

min(↑ ext[cov(A)]) = min(↑ ext[max(cov(A)]) Using Lemma 2.1 we obtain A = min(ext[cov(A)]) = min(ext[max(cov(A))).

Another important observation related to Example 3.25 is the fact that the support-closed patterns in a pattern setup does not hold all the information about the definable sets. Theorem 3.5 states that this is no longer the case for pattern multistructures.

Theorem 3.5. Given a pattern multistructureP for which the set of support-closed patterns isPint(cf. Theorem 3.3), we have:Pext= ext[Pint].

Proof. Recall thatPint=S_B⊆Gext[cov∗(B)]. SincePint⊆Dand by definitionPext= ext[D]. It

is clear that ext[Pint] ⊆ Pext. It remains to show thatPext⊆ ext[Pint]. Let A ∈ Pext, then ∃d ∈

cov(A) s.t. A = ext(d). Since P is a pattern multistructure, we have cov(A) =↓ cov∗_{(A). Then,}

we have a support-closed pattern d∗_{∈ cov}∗_{(A) ⊆ P}int s.t. d v d∗. Hence, ext(d∗) ⊆ ext(d).

Moreover, since cov∗_{(A) ⊆ cov(A), we have d}∗∈ cov(A). Therefore, A = ext(d) ⊆ ext(d∗). We obtain thus A = ext(d) = ext(d∗_{), that is A ∈ ext[Pint].}

Similarly to Theorem 3.2 for pattern structures, Theorem 3.6 connects multilattices (see Definition 2.18) with pattern multistructures. It states that (complete) meet-multisemilattices are to pattern multistructures what (complete) lattices are to pattern structures.

Theorem 3.6. Let_{D= (D, v) be a poset, the following properties are equivalent:} • For any finite set_{G6= ; and any}δ_∈DG, (_G,_D,δ) is a pattern multistructure. • D is a meet-multisemilattice having all its maximal elements (i.e.D=↓ max(D)) More generally, the following properties are equivalent:

• For any set_{G6= ; and any}δ_∈DG, (_G,_D,δ) is a pattern multistructure. • D is a complete meet-multisemilattice.

Proof. Recall that P = (_G,_D,δ_{) is a pattern multistructure iff for any subset S ⊆}δ[_G], S` is maximal-handle. The proof of this theorem follows the same spirit of Theorem 3.2’s proof where the existence of the meet (i.e. S` is maximum-handle) is replaced by S` is maximal-handle.

Note 3.9. Pattern multistructures are named so since they rely on meet-multisemilattices.

3.6.2 Some Issues with Pattern Multistructures

We have seen earlier in this section that some issues related to pattern setups no longer exists for pattern multistructures. However, it should be noticed that the two issues presented in section 3.5.1.1 and section 3.5.1.2 still exist as the examples used rely on the pattern setup presented example 3.12 which induce a pattern multistructure (see example 3.27). We will investigate now two additional issues that pattern multistructures may have.

3.6.2.1 Maximal covering descriptions of an extent could be infinite

The following Example shows that even with pattern multistructures with finite set of objects, upper-approximations extents are not computable using Theorem 3.4 as the set of maximal covering descriptions may be infinite.

Example 3.29. Consider the description space (D, v) depicted in Fig. 3.11 and let be the pattern multistructure (_G, (_{D, v),δ}) where_{G= {g1}, g₂},_δ(g_{1) = aα}and_δ(g_{2) = aβ}. We have cov({g₁, g₂}) = {bi| i ∈ N} and since cov({g1, g2}) is an antichain, we conclude that cov∗({g1, g2}) = {bi| i ∈ N}.

a₀ a₁ . . . a_n . . . a_α a_β b0 b1 . . . bn . . .

Figure 3.11: A complete multilattice with an infinite antichain. We have (∀i, j ∈ N) i ≤ j ⇔ biv aj

and (∀i ∈ N) biv aαand biv aβ

3.6.2.2 Definable sets do not form a join-multisemilattice

Last but not least, we have seen in Section 3.4 that in the case of a pattern structure, (Pext, ⊆) is

a complete lattice since it is closed under arbitrary intersections. One can say that the property of having the infimum in the description space has been transferred to the poset of definable sets thanks to extent operator. When it comes to a pattern setup on finite set of objects, it is clear that (Pext, ⊆) is a complete multilattice since it is finite. However, does this property still holds for the

case of infinite set of objects? Unfortunately, the answer is negative as stated in Proposition 3.9. This proposition tells also that not all definable sets above A in a pattern multistructure are above at least one upper-approximation of A.

Proposition 3.9. There exists a pattern multistructureP = (G, (D, v),δ) such that (Pext, ⊆) is

not a join-multisemilattice and in which (∃A ⊆G) ↑ A ∩ Pext6=↑ A ∩ Pext.

Proof. Consider the pattern setupP = (G, (D, v),δ) where (D, v) is the complete multilattice depicted in Fig. 3.11. Since (_{D, v) is a complete multilattice (i.e. it is chain-finite), then P is a} pattern multistructure. Consider now an infinite set_{G= {g}i| i ∈ N} ∪ {gα, gβ}. The mappingδ

is given by:δ(g_{α) = aα},δ(g_{β) = aβ and (∀i ∈ N)}δ(gi) = ai. To show that the poset (Pext, ⊆)

is not a join-multisemilattice we need to consider two definable sets inP_extand show that the set of their common upper-bounds inPext does not have all its minimal elements. Let us

compute ext for every d ∈D:

• ext(a_{α) = {gα}}and ext(a_{β) = {gβ}}.

• (∀i ∈ N) ext(ai) = {gi}and (∀i ∈ N) ext(bi) = {gα, gβ} ∪ {gj| j ≥ i}.

Consider now the set of definable sets {{g_α}, {g_β}}, it is clear that the set of their common upper-bounds (inPext) is given by: {{gα}, {gβ}}u=© © gα, gβª ∪ ©gj| j ≥ iª | i ∈ Nª.

The set of upper bounds is hence an infinitely descending chain and hence does not have a minimal element, in other words: min({{g_α}, {g_β}}u_{) = ;. Hence, (P}ext, ⊆) is not a

join-multisemilattice. The proof of the second part of the proposition is straightforward. Indeed, consider the non-definable set A = {gα, gβ}. We do have: ↑ A ∩ Pext= ext[cov(A)] =