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Abstract Concept Lattices
Henry Soldano, Veronique Ventos
To cite this version:
Henry Soldano, Veronique Ventos. Abstract Concept Lattices. International Conference on Formal Concept Analysis, May 2011, Nicosia, Cyprus. pp.235-250, �10.1007/978-3-642-20514-9_18�. �hal- 00642105�
Abstract Concept Lattices
Henry Soldano1, V´eronique Ventos2
1 L.I.P.N, UMR-CNRS 7030, Universit´e Paris-Nord, 93430 Villetaneuse, France
2 LRI, UMR-CNRS 8623, Universit´e Paris-Sud, 91405 Orsay, France
[email protected], [email protected]
Abstract. We present a view of abstraction based on a structure preserving re- duction of the Galois connection between a languageLof terms and the powerset of a set of instancesO. Such a relation is materialized as anextension-intension lattice, namely a concept lattice whenLis the powerset of a setPof attributes.
We define and characterize anabstractionAas some part of either the language or the powerset ofO, defined in such a way that the extension-intension latticial structure is preserved. Such a structure is denoted for short as anabstract lattice.
We discuss the extensional abstract lattices obtained by so reducing the powerset ofO, together together with the corresponding abstract implications, and discuss alpha latticesas particular abstract lattices. Finally we give formal framework allowing to define ageneralized abstract latticewhose language is made of terms mixing abstract and non abstract conjunctions of properties.
1 Introduction
There were in machine learning various attempts to formalize abstraction and charac- terize its desirable properties with respect to induction. An important statement was that abstraction should be order-preserving with respect to the partially ordered language in which hypotheses are searched for [21]. However, the question of what classes of ab- stractions are to be investigated for learning and reasoning is far from being exhausted.
We present a view of abstraction based on a structure preserving reduction of the rela- tion between a termtof a languageL, partially ordered following a general-to-specific partial order, and theextensionofton a set of objects (or instances)O, representing the subset ofOwhose elements satisfy the term.
In Formal Concept Analysis [15] and Galois lattice theory [4]L is a lattice, and the relation betweenLand the powersetP(O)is materialized as anextension-intension lattice. This lattice is the structure of thedefinableelements ofP(O)[1],i.e.the subsets ofOthat are each the extension of some term ofL. In such a lattice, a definable sete represents the equivalence class of all the terms whose extension ise, and a node, also called aconcept, is a pair(e, t)wheretis the most specific term of the class, denoted as theintensionof the concept. Formal Concept Analysis is primarily concerned with the relation between the powerset of a set of properties as a language andP(O). The extension-intension lattice is then denoted as aconcept lattice. Various extensions have been recently proposed to ease the representation in more sophisticated languages [6,
18]. In particular pattern structures [16] have been recently introduced to represent complex data, associating such a pattern structure to each object. Logical Concept Analysis (LCA) [13] has been recently introduced as a general formalization in which Lis a logical language and usesobject descriptionsinL. Though we do not use here the LCA formulation and notations, for technical reasons, our construction of a Galois connection on a modal language is very similar to the construction presented in [12]. In a recent paper [20] particular mappings, denoted asprojections, are used to reduceLor P(O)in such a way that the relation between the language and the extensional space is still materialized as a lattice. In other words, projections ensure that we have a coarser, yet structure preserving, view of concepts representatives of the universe we deal with.
Independently [16] also uses projections on pattern structures.
In this paper we show that projections of a lattice are in a one to one correspondence withabstractionsdefined as parts of the lattice that are closed under least upper bound.
We call the corresponding structuresabstract extension-intension latticesandabstract concept lattice, and for shortabstract lattices. We first briefly discuss intensional ab- stractions and investigate then more specifically extensional abstractions, i.e. parts of P(O)that are closed under set theoretic union.
More precisely, applying an extensional abstraction means that we will no longer consider instances ofOas elements of the extensional space, but rather consider subsets ofOgivena priori. As an example considerO ={o1, o2, o3, o4}, andAbe obtained by closing under union the part{{o1, o2},{o1, o3}}. As a result,{o1, o2, o3}belongs toAbut{o2, o3} does not. Now, consider the smallest elements ofAthat contain a given instanceo. We call these elements the minimal abstractionsof oand consider them asabstract instances. In this example,{o1, o2}is the unique minimal abstraction ofo2,{o1, o3}is the unique minimal abstraction ofo3,{o1, o2}and{o1, o3} are the two minimal abstractions ofo1, ando4has no abstraction inA.
We relate then any termtto anabstract extensionextA(t)that turns out to be the union of all the abstract instances included inext(t). Going back to our example, sup- pose thatext(t) ={o1, o2, o3, o4}, then the abstract instances included inext(t)are {o1, o2}and{o1, o3}and thereforeextA(t) ={o1, o2, o3}. Clearly the abstract exten- sion of a given term is always included in its original extension. In other words we re- duce the extension of the term by excluding any instance that has no minimal abstraction included in the original extension. In the current example,o4is such an instance. The intuition here is a change in granularity: the new objects we deal with are the minimal abstractions of the original instances. Furthermore, as an extension-intension lattice rep- resents a set of valid implications (see for instance [4]), ourabstract extension-intension latticealso represents a set ofvalid abstract implications. Going back to our example, supposeext(p) ={o1, o2, o4}andext(q) ={o1, o2, o3}, then the implicationp→qis not valid onO. However, as we haveextA(p) ={o1, o2}andextA(q) ={o1, o2, o3}, the abstract implicationp→Aqis valid. As a matter of fact a general property of exten- sional abstractions is that they preserve validity of implications. Algorithms that extract valid implications [14, 17] from a set of instancesO, can be extended to extractvalid abstract implications. As an example, we can interpretalpha lattices[28] as particular abstract lattices, and alpha implications are straightforwardly extracted by extending the method of N. Pasquier and collaborators [19] to alpha lattices.
Regarding extensional abstractions, the formal work closest to ours regards rough sets logics. Rough sets originally relies on an indiscernibility relation (see for instance [29]). More recently, generalization of rough sets using coverings, corresponding to our extensional abstractions, has been investigated from an algebraic perspective [5, 8].
However these works do not investigate the extension-intension relationship.
To summarize, in this first part of the paper we characterize and discuss the proper- ties of abstractions, and particularly of extensional abstractions, as a structure and order preserving reduction of extension-intension lattices.
We then remark that abstract implications are in fact implications between abstract terms,i.e.t1→t2rewrites as2t1→2t2, where2tmeans ”Abstractlyt“ with respect to the abstractionA. Then, starting from the semantics, we discuss the corresponding abstract modal logics. This leads us to consider a languageL2a term of which is made of a non abstract part together with an abstract part and to define a new, more expressive, extension-intension lattice.
2 Preliminaries
2.1 Galois Connection
We recall here the definitions of a Galois connection and a Galois lattice.
Definition 1 Given two lattices(E,≤E,∧,∨)and(F,≤F,∧,∨), where≤E and≤F
denote the order relations, and∧,∨the meet and the join operations, a Galois connec- tion betweenEandF is a pair of mappings(f :E → F, g :F →E)verifying the following properties:
– For anyxandx0inE, we have thatx≤Ex0impliesf(x)≥F f(x0) – For anyyandy0inF , we have thaty≤F y0impliesg(y)≥Eg(y0) – For anyxinEandyinFwe have thatg◦f(x)≥Exandf◦g(y)≥F y The Galois lattice defined by the Galois connection (f, g) is then the set{(x, y) ∈ E×F |y=f(x)andx=g(y)}ordered by≤E
2.2 Extension-intension lattices
FCA, in a broad sense, investigates the link between the terms of some language L and a universeOof elements of the universe denoted asinstances. We also callinten- sionalrepresentations the terms ofLand extensional representations the elements of the powersetP(O). In this presentation we suppose that we know whenever an instanceo satisfies the termt, and we define accordingly its extension:
Definition 2 extO(t) ={o∈O|osatisfiest}.
In what follows we considerOas a fixed and finite set of instances and we simply write ext(t)the extension inO.
The languageLis partially ordered by a general-to-specific relation. We writet1 t2whenevert1isless specificthant2, or equivalentlyt1is more general thant2. The following proposition relateLandP(O)by a Galois connection.
Proposition 1 LetLbe a finite language,a partial order onLdenoted as specificity, Obe a finite set of instances and ext:L → P(O)be a mapping such thatt1t2 ⇒ ext(t1)⊇ext(t2). We consider the following conditions:
– (condition 2.1) For any instanceo, there is aunique most specific term, denoted as the object descriptiond(o), among all termstsuch thato∈ext(t)
– (condition 2.2)Lhas a greatest element and is a lower semi lattice, i.e. each pair of termst1, t2ofLhas a unique greatest lower boundt1∧Lt2inL, also called theleast general generalisation(for shortlgg) oft1andt2.
Whenever conditions 2.1 and 2.2 are satisfied,(L,)is a lattice,i.e. two termst1and t2also have a least upper bound denotedt1∨Lt2, and the pair(int, ext)where
int(e) = ^
o∈eL
d(o)
is a Galois connection.
In Formal Concept Analysis [15] and Galois Analysis [4],Lis the powerset of a setP of attributes. Recent extensions to various languages have been performed [14, 13]. In particular in [16],Lis defined as a set ofpattern structures,i.e. terms generated by first considering the set of object descriptions, and then closing it under the least general generalization∧L. Independently, E. Diday and R. Emilion start from the same assumptions [9]. Proposition 1 directly follows from, for instance, theorem 2 in [9].
We callextension-intension latticethe Galois latticeGcoresponding to this Galois connection.Gis ordered using the extensional order and each elementGis a pair(e, t) such thatt=int(e)ande=ext(t),i.e.tis the unique most specific term representing the equivalence class of terms whose extension ise, and is also denoted as theintension ofe. Such most specific terms are also referred asclosed termsorclosed motifsin data mining [25]. In Machine Learning, the search spaceLmay be in this way explored by minimally generalizing or specializing such closed terms [3].
FinallyGis considered as the structure ofOas perceived throughLand can be also represented by the setTOof all the implicationsp→qwhich are valid onO, i.e. such thatext(p)⊆ext(q)is true. The equivalence class whoseintensionist=int(e), also contains various minimal (i.e. most general) termstm, also know asgenerators. The elements ofTOcan be generated from the set of all thetm→timplications also called the min-max implicationsbasis ofTO[19].
2.3 Projections and projected extension-intension lattices
Now, consider the following problem; how to reduceP(O),L, or both, in such a way thati)conditions 2.1 and 2.2 are still satisfied,ii)the resulting extension-intension lat- ticeG0 is isomorphic to part ofG. In other words, how to reduce Gby reducing the language or the powerset of the universe of instances in such a way that the structure is preserved, and its size reduced ? An answer to this question is given in [20] through the use of intensional and extensional projections whose images are also lattices:
Definition 3 (Projection) pis a projection of a lattice(M,≤)iff for each pair(x, y) of elements ofM, we have :
Ifx≤ythenp(x)≤p(y)(monotonicity) p(x)≤x(minimality)
p(x)≤p(p(x))(semi-idempotence)
LetM be a lattice andpa projection ofM, thenp(M)is also a lattice with join operator∨and meet operator∧defined as, for any pairm1, m2∈p(M),m1∧m2= p(m1∧Mm2)andm1∨m2=m1∨M m2.
When considering respectivelyM = L andM = P(O)we obtain respectively intensional and extensional projections and both lead toprojected extension-intension lattices[20, 28].
Proposition 2 Let(int, ext)be a Galois connection on(P(O),),(L,⊆)),Gbe the associated Galois lattice, and(e, t)be a node ofG.
– Let pbe a projection on L, then (p◦int, ext) defines a Galois connection on ((p(L),),(P(O),⊆))and(e, t)is projected in the corresponding Galois lattice p(G)on the node(e0, t0)such thatt0=p(t)ande0 =ext(t0).
– Letpbe a projection onP(O), then(int, p◦ext)defines a Galois connection on ((L,),(p(P(O)),⊆))and(e, t)is projected in the corresponding Galois lattice p(G)on the node(e0, t0)such thate0 =p(e)andt0=int(e0).
Note that in partial order theory, projections are known askernel operators orinterior operators. Their properties are well known [11] and are the basis of the next section.
3 Abstractions
3.1 From projection to abstractions
Reducing through projectionsL,P(O)or both results in a reduced extension-intension representation whose latticial structure is preserved. However, while projections are technically useful, they do not always give a simple way to chose simplified represen- tations. In what follows we present an equivalent view of projections.
Definition 4 An abstraction of a latticeMis a subset3ofM, closed under∨M. Building abstractions therefore simply means to chose any subset ofM and close it by
∨M. We note hereunder that abstractions are in a one to one correspondence with pro- jections and soabstract extension-intension latticesare defined as projected extension- intension lattices.
Proposition 3 Let Abe an abstraction of a lattice M, thenpA defined aspA(x) = W
c∈A,c≤xc, is a projection ofM. Letpbe a projection thenA=p(M)is an abstrac- tion ofM, andpis the projectionpAassociated toA.
3Note that⊥M, the smallest element ofM, belongs to all abstractions.
The equivalence betweeninterior systemsp(M)and subsetsAofMclosed under union is a known property of interior operators [11]. As abstractions are closed under∨M, we have that∨M =∨A. From now on we simply write∨when no confusion is possible.
An important point is that we only need the∨-irreducible elements4ofA:
Proposition 4 LetAbe an abstraction ofM, andAI be the set of∨-irreducible ele- ments ofA, the projectionpAis obtained by only considering elements ofAI:
pA(x) = _
c∈AI,c≤x
c
Intensional abstractions Anintensional abstractionis then simply obtained by select- ing part of the languageLand closing it by the join operator∨L. For instance, consider the latticeLwhose elements are intervals[a, b]such thata, b ∈ {1,2,3,4}. We have then[1,3]∨L[3,4] = [3]. Consider first the abstractionAwhose elements are the inter- vals containing3.Ais closed by∨Las intersecting two intervals containing3results in an interval also containing3.
Consider nowL0 = L − {[1],[2],[3],[4]}.L0 is obtained by simply deleting the most specific terms and is clearly also a lattice. However[1,3]∨L0[3,4]is now[]and so clearlyL0is not closed under∨L, and therefore is not an abstraction. Note that there are inL0 two most specific elements satisfied byo = 3, and as a consequenced(o)is no longer defined.
Extensional abstractions They abstract instances rather than abstracting the language of terms. Note that when considering extensional abstractions, ∨is the set theoretic union∪, and that the order relation is the set theoretic inclusion.
Example 1. As an example considerO = {1,2,3,4}, and the abstractionAobtained by closing under union the part{{1,2},{1,3}}ofP(O), so adding{1,2,3}and∅to buildA. The set of∪-irreducible elements ofAisAI ={{1,2},{1,3}}. For instance, pA({1,2,3}) = {1,2} ∪ {1,3} = {1,2,3}as both elements of AI are included in {1,2,3}, andpA({2,3}) =∅because neither{1,2}nor{1,3}is included in{2,3}.
We remark in the next section that abstractions are partially ordered.
3.2 The lattice of abstractions
There is a partial order on projections of a latticeM[20] and therefore, on abstractions:
Definition 5 LetM be a lattice andp1etp2two projections ofM, we will state that p2≤p1, i.e.p2is lessconcrete, and so moreabstractthanp1, iff there is some projection pdefined onp1(M)such that for allcinM,p2(c) =p◦p1(c).
This means thatA2 = p2(M) is more abstract thanA1 = p1(M) iff A2 is an abstraction ofA1. This is also equivalent to saying that any∨-irr´eductible elementi2of A2may be written as a disjunction of∨-irreducible elements ofA1,i.e.i2=i11∨· · ·∨in1
4Irreducible elements ofAare elements that cannot be obtained as a result of applying∨M.
Example 2 (Extensional abstractions).LetM =P({1,2,3,4}),AI1={{1,2},{1,3}}
and A2I = {{1,2},{1,2,3}}, A2 is more abstract than A1 as {1,2} ∈ AI1 and {1,2,3}={1,2} ∪ {1,3}.
Proposition 5 (Lattice of abstractions) LetM be a lattice, andA be the set of the abstractions ofM. Let thenA1=p1(M)andA2=p2(M)be elements ofA:
– The least upper boundA1∨AA2is obtained by closingA1∪A2under∨.
– The greatest lower boundA1∧AA2is simplyA1∩A2.
This partial order, defined through projections, is known to order projected extension- intension lattices [20]. Therefore, abstract extension-intension lattices, where abstrac- tion is performed onLorP(O), are also ordered following the lattice of abstractions.
4 Extensional abstractions and extensional abstract lattices
4.1 Extensional abstract lattices
An extensional abstraction is obtained by considering part of the powerset ofO, and closing it by the union operator∪. This means that we do not consider any more in- stances but rather subsets of instances, calledabstract instances.
Definition 6 LetAbe an extensional abstraction andpAthe associated projection of P(O), then lettbe a term ofL,extA(t) =pA◦ext(t)is theabstract extensionoft.
From Proposition 4 we also have
extA(t) = [
u∈AI,u⊆ext(t)
u
Definition 7 LetAbe an abstraction ofP(O), the projected extension-intension lattice pA(G)is denoted as an extensional abstract lattice, and notedGA.
Hereunder we defineabstract instancesandminimal abstractions of a given instance.
Definition 8 LetAbe an extensional abstraction.
– An element ofAIis called anabstract instance.
– LetAI(o)be the subset ofAIwhose elements contain the instanceo. We denote as minimal abstractions of an instanceothe minimal elements ofAI(o), i.e. Am(o) = {u∈AI |o∈uand ifo∈u0 ⊂u,thenu0 6∈A}.
An interesting point is thatAI is the set of minimal abstractions of the instances:
Proposition 6
AI = [
{o∈O}
Am(o)
Example 3. In this exampleO = {1,2,3,4}andAI ={{1,2},{2,3},{3,4}}. Here L = P(P)whereP = {X, Y, Z}is a set of properties. We give hereunder a table representing the context relating P(P) toP(O). We have added a column for each element ofAI. In Figure 1 we have represented the original concept lattice together with the extensional abstract lattice (here anabstract concept lattice) associated toAI.
InstancesX Y Z {1,2} {2,3} {3,4}
1 1 0 1 1 0 0
2 1 1 1 1 1 0
3 0 1 0 0 1 1
4 1 0 0 0 0 1
Fig. 1.The concept lattice (on the left) and the abstract concept lattice (on the right) correspond- ing to the context and abstraction given in Example 3. On each nodeIis the intension,Eis the extension, andGis the set of most general terms whose extension isE. The nodes3and4of the original concept lattice are merged into the node8of the abstract concept lattice. As a result, the abstract implicationX→Zis now valid asXandZhave the same abstract extension.
We may note that such an extensional abstract lattice is isomorphic to an extension- intension lattice relatingLto the powerset ofAI. For that purpose, we reformulateAI
as a new instance set and we define(ext0:L →AI)as follows: letu={o1, . . . ok}be an element ofAI, we have then thatu, as a new instance, belongs toext0(t)whenever u, as a subset ofO, is included inext(t). As a consequenceextA(t) = S
u∈ext0(t)u and therefore the imagesext0(L)andextA(L)are in one to one correspondence.
As stated before, an extension-intension lattice corresponds to a set of valid im- plications. We define hereunder the abstract implications associated to an extensional abstract lattice.
Definition 9 (Abstract implication) Lett1ett2be terms ofL. WheneverextA(t1)⊆ extA(t2)we say that the abstract implicationt1→At2is valid onA.
ThenGAis represented by the whole set of abstract implications valid onA, or by any generating subset, as themin-max basis of abstract implicationsextending to the
extensional abstract lattices the definition of the min-max basis of implications (see end of Section 2.2).
Now ast1 → t2 means thatext(t1) ⊆ ext(t2), we have that whenever some ab- stract instanceuis included inext(t1), we also have thatuis included inext(t2). This also means, by definition, that extA(t1) is included inextA(t2). As a consequence validity of implications is preserved by extensional abstraction:
Proposition 7 LetAbe an abstraction ofP(O), andt1andt2two terms ofL.
Ift1→t2is valid onO, then the abstract implicationt1→At2is valid onA.
4.2 Alpha lattices as extensional abstract lattices
The partial order of abstractions (see Definition 5), means thatA0is more abstract than Awhenever any element ofA0 (orA0I) may be written as the union of elements ofA (orAI). An interesting case isalpha abstraction. Starting from a subsetCwhose union closure is an initial abstractionA, more concrete abstractionsAα are built.Aαis ob- tained by deriving from each initial categoryCthe set of frequent enough parts ofC, and closing under union the resulting setCα. We have then thatAαis more abstract than Aα0iffα≥α0. In a previous work the correspondingalpha Galois latticeswere defined through projections and experimented in order to extract alpha association rules [28]. As an example, consider the implication ”Animals that fly are oviparous”, it is not valid on Obecause of thebat. When considering the categorizationC={mammal,insect,bird}, the corresponding abstract implication is valid but never applies: none of the initial cat- egories contains only flying animals. However by considering frequent enough parts of each category, the corresponding (α = 0.1)- implication is valid, as the bat is elim- inated from the premise (very few mammals fly), but still applies to flying birds and flying insects, as they represent large enough parts of their categories.
5 The extended abstract latticeG2
We will formalize the nature of abstract implications by interpreting them as classical implications relating modalized terms. So we rewritet1→A t2as2At1 →2At2. We first define a modal logics of abstraction built on a propositional modal languageLmod. Then we restrictLmodto a languageL2using only conjunction and non nested modal connectors. Finally we present an extended abstract latticeG2 where intensions have both modalized and non modalized parts.
5.1 Modal logics of abstraction
Hereunder we discuss the modal logics of abstractions. Note that in this section we denote asworldsthe elements ofO.
A modal logic, in its simplest form, is a propositional logic to which is added at least one unary modal connector2, referred to as anecessityoperator.Classical modal logics are the modal logics in which formulas are given truth values throughneighborhood semantics, also known asminimal models semantics[7]. This wide class of modal logics
includes in particularnormal modal logicsrelying on Kripke possible world semantics.
We will define hereunderabstract modal logicsas particular classical modal logics. We hereunder very informally summarize neighborhood semantics in order to relate such semantics to our extensional abstractions.
The modal languageLmodis obtained by adding the modal connector2to a propo- sitional languageLbuilt on a set of atomic propositions. So for instance,φ=a∧2(b∧ c)belongs toLmod. In order to give a truth value to a modal formula we first consider a set of worldsO, together with avaluationfunctionext, relating each atomic proposition pto the set of worlds in whichpis true. For any formulaφwithout modal connectors, the computation ofext(φ)is then straightforward, for instanceext(a∧b) =ext(a)∩ext(b) and represents the worldsoin whichφis true, or in other words, the worlds that sat- isfies φ. In order to extendext to modal formulas we need a neighboring function N relating each world oto a set of subsets of O and we say that the world o is in ext(2φ)whenever ext(φ) belongs toN(o). It is known that neighboring functions N are in one to one correspondence with mappings m : P(O) → P(O) such that m(e) ={o∈O|e∈ N(o)}. To summarize we have now that
ext(2φ) =m◦ext(φ)
Now recall that we defined abstract extension asp◦extwherepis a projection.
We will so naturally defineabstract modal logicsas classical modal logics in which the mappingmis a projection. To characterize abstract modal logics we have just to trans- late the properties of projection as axioms and inference rules. Detailed results on modal logics of abstraction, including multimodal logics allowing to access to various levels of abstraction, are outside the scope of this paper [22]. Now, considering the abstract ex- tension of a term as the extension of an abstract term leads to define extension-intension built on languages whose terms contain both non abstract and abstract parts. This is the subject of the remaining of this section.
5.2 Mixing abstract and non abstract statements: the languageL2
We consider here a language L2 where classical properties and modalized proper- ties appear simultaneously. TechnicallyL2is a subset ofLmod whose terms contain
∧ and2as connectors, and in which the nesting of2 is not allowed. For instance, (a∧b)∧2(b∧c)∧2(c∧d)∈ L2, but2((a∧b)∧2(b∧c))6∈ L2.
We first give an inductive definition ofL2.
Definition 10 ( Inductive definition ofLP andL2) The inductive definition of L2 according toP a non-empty set of atoms is the following (note that we define and use the languageLPcorresponding to terms without any2connective):
• ∀a∈ P,a∈ LP
•Constants>and⊥ ∈ LP
•if F1 and F2∈ LPthen(F1∧F2)∈ LP
•ifF ∈ LPthenF ∈ L2
•ifF ∈ LPthen2F ∈ L2
•ifF1∧F2∈ L2then(F1∧F2)∈ L2
An intensional semantics forL2
We define an intensional semantics forL2based on an algebraic approach similar to [27], [10] but for description logics. The definition is made in two steps. In the first step, an equational system which highlights the main properties of theL2connectives is given. During the second step, an homomorphism based on the equational system is defined. This homomorphism is used to map terms ofL2 to theirstructural normal form(snf) in the intensional semantics.
Definition 11 ( The equational systemEQ2) ∀F1, F2, F3∈ L2,∀F, F’∈ LP: 1. (F1∧F2)∧F3=F1∧(F2∧F3)
2. F1∧F2=F2∧F1 3. F1∧F1=F1 4. > ∧F1=F1 5. ⊥ ∧F1=⊥ 6. 2F=2F∧F
7. 2(F∧F0) =2(F∧F0)∧2F ∧2F0
The equational systemEQ2fixes the main properties of the connectives and can be used to define an equivalence relation between terms ofL2. Equality modulo axioms ofEQ2is denoted by≡EQ2. We use it to formalize the subsumption relation inL2. Definition 12 (Subsumption inL2) Let F1, and F2 be two terms ofL2, F1F2 (i.e. F1 subsumes or is less specific than F2) iff F1∧F2≡EQ2F2.
EQ2 induces a class of algebras. From this class, a structural algebra can be pro- posed, which providesL2with an intensional semantics calledCL2. The elements of CL2are structures whose definition is given below. These elements arestructural nor- mal formsof terms ofL2allowing us to obtain an unique class representative for each equivalence classes of terms.CL2can be viewed as a normalized subset ofL2where thelggis unique.
The intensional semanticsCL2
An element ofCL2corresponding to a termT ofL2denotedsnf(T)is a pair defined as follows:< Eclassical, E2>.Eclassicalis a set of atoms belonging toP,E2is a set of subsets ofP. Intuitively,Eclassicalcontains every explicit and implicit classical properties ofT (i.e. properties not at reach to a2connective). The data structure used can then be a simple set of atoms.E2contains2properties ofT. Since for instance 2(a∧b)and2(b∧d)cannot be compared, we must keep both of them. The data structure used is a set of sets.
This definition presents the data structure of elements ofCL2but not how to as- sociate to a term ofL2its corresponding element inCL2. To make this computation and then to defineCL2, an homomorphism from the set of terms ofL2and the set of elements ofCL2need to be defined.
The homomorphism fromL2intoCL2is sketched below. It allows us to associate to each termT ofL2itsstructural normal formdenotedsnf(T). This homoporphism takes into account axioms ofEQ2 and the normalization strategy choosen. Indeed, to
obtain a normal form many normalization strategies may be applied (e.g.deletion of redundant information). We chose to add implicit information in the classical part, this strategy is a kind of partial saturation which is a trick largely used to make easier sub- sumption and lgg computation. On the other hand, for complexity reasons we only keep maximal subsets in the2part. The2part is a Sperner family i.e., an antichain in the inclusion lattice over the power set ofLP (for more details see [2]).
Homomorphism snf fromL2intoCL2:
Term ofL2 Element inCL2
> t=<∅,{∅}>
⊥ bo=<P,{P}>
a∈ P <{a},{∅}>
F1∧F2 snf(F1) ∨s snf(F2) 2F <EF,{EF}>
wheresnf(F) =< EF,∅>(the2part is empty since there is no2nesting).
∨sis the join operator inCL2. It uses the classical union set operator and∨awhich represents the union operator in antichains.
LetK1andK2be two antichains:
K1∨aK2 ={x∈K1∪K2|@y∈K1∪K2s.t.x⊂y}
Letsnf(F i)be < EF i, Ki > fori= 1,2:
snf(F1)∨ssnf(F2) =< EF1∪EF2, K1∨aK2>
Example 4. snf((a∧b)∧2(b∧c)∧2b∧2(c∧d)) =<{a, b, c, d},{{b, c},{c, d}}>
canddare added in the classical part according to axiom6and7,{b}is removed from the2part since{b} ⊆ {b, c}.
Definition 13 (Structural Subsumption inCL2) Let S1 and S2 be two elements of CL2,
S1sS2 (i.e. S1 subsumes S2) iff S1∨sS2 = S2
Proposition 8 Subsumption inL2is equivalent to structural Subsumption inCL2: Let F1, and F2 be two terms ofL2,
F1F2 iff snf( F1)ssnf(F2)
5.3 The Galois lattice G2
We consider the two following posets : (P(O),⊆) and (CL2,s). In order to obtain a Galois connection between the two posets, we need to define the least general generali- sation inCL2. Thelgguses the following definition of∧athe intersection operator in
