Proceedings of the Ninth Symposium on Abstraction, Reformulation and Approximation
A Modal View on Abstract Learning and Reasoning
Henry Soldano
L.I.P.N, Université Paris-Nord, UMR-CNRS 7030
99 Avenue J.-B. Clément, 93430, Villetaneuse, France
sentence. The starting point has to be found in Formal Concept Analysis (Ganter and Wille 1999) and Galois lattice
theory (Caspard and Monjardet 2003). In such approaches,
both L and P(W ) are lattices, and the relation between
L and P(W ) is materialized as an extension-intension lattice. This lattice is the structure of the definable elements of
P(W )(Ben-David and Ben-Eliyahu-Zohary 2000), i.e. the
subsets of W that each are the extension of some sentence
of L. In such a lattice each definable set e represents the
equivalence class of all the sentences whose extension is e,
and a node, also called a concept, is made of a pair (e, t)
where t is the most specific sentence of the class, denoted
as the intension of the concept. Note that from a Machine
Learning perspective, we only have access to a learning set
of instances W , and so the definable sets, with respect to W
are arguably the only subsets of interest: searching hypotheses in L comes down to searching the space of definable sets.
In (Pernelle et al. 2002) particular self-maps, denoted as
projections, are used to reduce L or P(W ) in such a way
that the relation between the language and the extensional
space is still materialized as a lattice, denoted as a projected
extension/intension lattice. In other words, projections ensure that we have a coarser, still structure preserving, view
of concepts representatives of the universe we deal with, or
at least on the data we have on this universe. More recently,
projections on a lattice have been shown to be in one to one
correspondence with parts of the lattice that are closed under
the least upper bound operator. These structures are denoted
as abstractions and so to each abstraction A is associated the
corresponding projection pA . As a consequence projected
extension/intension lattices are also denoted as abstract extension/intension lattices (Soldano and Ventos 2011).
To summarize, the set of the definable subsets of P(W ),
i.e the extensional space with respect to the language L,
has, under some conditions, a complete lattice structure. Abstractions of either L or P(W ), preserve the lattice structure
while reducing the size of the structure, i.e. while simplifying our representations.
We investigate here more specifically abstractions of
P(W), denoted as extensional abstractions i.e. parts of
P(W ) that are closed under set theoretic union, and simply
call them abstractions of W in what follows.. This leads us
to a class of modal logics whose semantics relies on extensional abstractions, and we will denote them modal logics
Abstract
We present here a view on abstraction originating from the relation between formulas in a partially ordered language L and
their extension on a set of instances W . In Formal Concept
Analysis, this relation is materialized as a lattice G. Particular self-maps on either L or the powerset P(W ) are known
to ensure structure-preserving reductions of the lattice G and
have been shown to be in one to one correspondence with
abstractions, defined as subsets of either L or P(W ) closed
under union. We investigate specifically extensional abstractions (subsets of P(W )). Such an abstraction, a special case
of domain abstraction, comes down to a change in granularity: extensions are now considered as union of abstract instances, i.e. union of predefined subsets of instances.
The main contribution of the paper is the investigation of
the class of (non normal) monotonic modal logics whose semantics relies on such abstractions, and that we call abstract
modal logics.
1
Introduction
Abstraction has been extensively investigated in Artificial
Intelligence, and was demonstrated as useful in particular in
Problem Solving (see for instance (Yang et al. 2008)). There
were various attempts to formalize abstraction and characterize desirable properties with respect to induction(Zucker
2003). An early statement about abstraction in Machine
Learning was that abstraction should be order-preserving
with respect to the partially ordered language in which hypotheses are searched for. One of the purpose of the program
INTHELEX (Esposito et al. 2004) is to induce such semantic abstractions. The model KRA, proposed by J-D Zucker
and L. Saitta (Zucker 2003; Saitta and Zucker 2001) starts
from a grounded representation of a perceived world, using
various representation layers, from a universe of configurations to the highest level language. However, the question of
what classes of abstractions are to be investigated for learning and reasoning is far of being exhausted.
We present here a view on abstraction that originates from
the relation between a sentence t of a partially ordered language L and the extension of the sentence t on a set of instances W , i.e. the subset of W whose elements satisfy the
c 2011, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
99
where 2A t means ”Abstractly t“ with respect to the abstraction A. Then, so starting from the semantics, we show that
the corresponding modal logics are particular cases of classical modal logics (Chellas 1980) and relate properties of the
abstractions A to axioms and inference rules of such modal
logics, further denoted as abstract modal logics. This also
leads to propose an abstract worlds semantics, relying on
the abstraction A, of abstract modal logics as a particular
and intuitive case of neighborhood semantics.
Furthermore, consider the set A of all the abstractions of
W . We will see that A is a lattice with respect to the following abstraction order: if an abstraction A is a subset of
an abstraction A, then A is more abstract than A. As we
will see whenever p →A q is valid and the abstraction A
is more abstract than A, then p →A q is also valid. This
ordering of abstractions leads to introduce multimodal logics of abstraction, in which a modal connector is associated
to each abstraction Ai . In such a multimodal logic we can
reason with different abstractions.
of abstraction. The main purpose of this paper is to motivate, introduce and investigate these logics that turn to be in
general non normal monotonic modal logics.
Defining an extensional abstraction consists in restricting
the extensional space to a part A of P(W ). This means that
we will then no more consider instances as unitary elements
of the extensional space, but rather particular subsets of W ,
denoted as abstract instances. The extension of a given sentence t is then transformed in an abstract extension and is
written as extA (t) = pA ◦ ext(t) that turns to be the union
of all the abstract instances whose elements all satisfy t.
As an example consider W = {w1 , w2 , w3 , w4 },
and let A be obtained by closing under union the part
{{w1 , w2 }, {w1 , w3 }}. As a result, {w1 , w2 , w3 } belongs to
A but {w2 , w3 } does not. Now, consider the smallest elements of A that contain a given instance w. We call these
elements the minimal abstractions of w and denote them as
abstract instances. So such an abstraction corresponds to a
change in the granularity of the instance space. In the previous example, {w1 , w2 } is the unique minimal abstraction
of w2 , {w1 , w3 } is the unique minimal abstraction of w3 ,
{w1 , w2 }, {w1 , w3 } are the two minimal abstractions of w1 ,
and w4 has no abstraction in A.
Suppose furthermore that ext(t) = {w1 , w2 , w3 , w4 },
then the abstract instances included in ext(t) are {w1 , w2 }
and {w1 , w3 } and therefore extA (t) = {w1 , w2 , w3 }.
Clearly the abstract extension of a given term is always included in its original extension. In other words, we reduce
the extension of the term by excluding any instance that has
no minimal abstraction included in the original extension. In
the current example, w4 is 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.
Consequently we define an abstract implication as valid
on an abstraction A whenever the abstract extension of the
antecedent is included in the abstract extension of the consequent. Going back to our example, suppose ext(p) =
{w1 , w2 , w4 } and ext(q) = {w1 , w2 , w3 }, then the implication p → q is not valid on W . Consider now the
corresponding abstract implication p →A q. As we have
extA (p) = {w1 , w2 } and extA (q) = {w1 , w2 , w3 }, p →A q
is valid. As a matter of fact, this turns to be a general property of abstractions: they preserve validity of implications.
In other words we have that whenever p → q is valid, then
p →A q is also valid.
An interesting case is the Alpha abstraction that corresponds to Alpha Lattices (Ventos and Soldano 2005). Such
an abstraction is obtained by considering an a priori categorization of the instances together with a parameter α
that corresponds to some degree of influence of the categorization. As we will see in section 4, this allows for instance, when considering animal species divided into mammals, insects and birds, to obtain valid abstract implications
as apt to fly →A oviparous, when the corresponding logical implication apt to fly → oviparous is falsified by few
mammals as bats.
We then define modal logics of abstraction by stating that
abstract implications are in fact implications between abstract sentences, i.e. rewriting t1 →A t2 as 2A t1 → 2A t2 ,
2
Preliminaries
In a recent paper, extensional and intensional abstractions
have been introduced as reductions of either the language
L or the extensional space P(W ) that preserve the original latticial structure of the relation between the language
and the instance universe (Soldano and Ventos 2011). We
present in this section the necessary definitions and various
results about abstractions, as intended here. We introduce
the sufficient conditions to obtain a Galois connection relating L and P(W ) and discuss the corresponding extension/intension lattice G. The question of a structure preserving way to reduce G is first answered using self-maps on
either P(W ) or L, and then in showing the one to one correspondence between these self-maps and particular subsets
of the domain, further denoted as abstractions.
Extension-intension lattices
We call here instances the elements of W but later denote
them as worlds in the section discussing modal logics of abstraction. We also call intensional representations the terms
of L and extensional representations the elements of the
powerset P(W ). In this presentation we suppose that we
know whenever an instance w satisfies the term t, and we
define accordingly its extension:
Definition 1 extW (t) = {w ∈ W | w satisfies t}.
In what follows we consider W as a fixed set of instances
and we simply write ext(t) the extension in W .
The language L is partially ordered by a general-tospecific relation. We write t1 t2 whenever t1 is less specific than t2 , or equivalently t1 is more general than t2 . Conversely, there is also a semantic order relation representing
the inclusion order of extensions. We further consider that
t1 t2 ⇒ extW (t1 ) ⊆ extW (t2 ) is always true but that
the converse is false. This simply means that some representable instances do not belong to the universe we modelise. For instance, as all birds are blue in our universe, the
100
Let M be a lattice and p a projection of M , then p(M )
is also a lattice with join operator ∨ and meet operator ∧
defined as, for any pair m1 , m2 ∈ p(M ), m1 ∧ m2 =
p(m1 ∧M m2 ) and m1 ∨ m2 = m1 ∨M m2 .
When considering respectively M = L and M = P(W )
we obtain respectively intensional and extensional projections and both lead to projected extension-intension lattices
(Pernelle et al. 2002).
Recently, abstractions, as defined hereunder, have been
proposed as an equivalent, and more intuitive, view of projections (Soldano and Ventos 2011).
terms blue bird and bird have the same extension in our universe of worlds. As a consequence there is an equivalence
relation on terms related to the extensional order:
Definition 2 t1 ≡ t2 iff ext(t1 ) = ext(t2 )
The following proposition relates L and P(W ) by a Galois connection. The proposition 1 directly follows from, for
instance, theorem 2 in (Diday and Emilion 2003).
Proposition 1 Let L be a finite language, a partial order
on L denoted as specificity, W be a finite set of instances
and ext : L → P(W ) be a mapping such that t1 t2 ⇒
ext(t1 ) ⊇ ext(t2 ). We consider the following conditions:
Definition 4 An abstraction of a lattice M is a subset1 of M ,
closed under ∨M .
• (condition 2.1) For any instance w, there is a unique most
specific term, denoted as the instance description d(w),
among all terms t such that w ∈ ext(t)
• (condition 2.2) L has a greatest element and is a lower
semi lattice, i.e. each pair of terms t1 , t2 of L has a
unique greatest lower bound t1 ∧L t2 in L , also called
the least general generalisation (for short lgg) of t1 and
t2 .
Building abstractions therefore simply means to choose any
subset of M and close it by ∨M . We note hereunder that
abstractions are in a one to one correspondence with projections.
Proposition 2 Let A be an abstraction of a lattice M , then
pA defined as pA (x) = c∈A,c≤x c , is a projection of M .
Let p be a projection then A = p(M ) is an abstraction of
M , and p is the projection pA associated to A.
Whenever conditions 2.1 and 2.2 are satisfied, (L, ) is a
lattice, i.e. two terms t1 and t2 also have a least upper
bound denoted t1 ∨L t2 , and the pair (int, ext) where
d(w)
int(e) =
The equivalence between interior systems p(M ) and subsets
A of M closed under union is a known property of interior
operators (Erné et al. 1993) i.e projections. 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 elements of
A. These are defined as the elements of A that cannot be
obtained by applying ∨M to other elements.
w∈eL
is a Galois connection.
This means that whenever the previous conditions are satisfied, i) an equivalence class of L with respect to W corresponds to a given subset e of W and is represented by a
unique most specific term t such that e = ext(t), ii) the set
G of the pairs (e, t) is structured as a lattice, denoted as an
extension-intension lattice,
Finally G is considered as the structure of W as perceived
through L and can be also represented by the set TW of all
the implications p → q which are valid on W , i.e. such that
ext(p) ⊆ ext(q) is true.
Proposition 3 Let A be an abstraction of M , and AI be
the set of ∨-irreducible elements of A, the projection pA is
obtained by only considering elements of AI :
pA (x) =
c
c∈AI ,c≤x
3
Ordered abstractions
We discuss here both intensional and extensional abstractions, and define an order on abstractions.
Projections and Abstractions
Now, consider the following problem; how to reduce P(w),
L, or both, in such a way that the extension-intension lattice is smaller that G ? An answer to this question is given
in (Pernelle et al. 2002) through the use of intensional and
extensional projections whose images are also lattices:
Intensional abstractions An intensional abstraction is
simply obtained by selecting part of the language L and closing it by the join operator ∨L . Therefore, given a pair of
terms (t1 , t2 ) belonging to an abstraction A of the language
L, the minimal specialization in L of t1 and t2 , namely
t1 ∨L t2 , also belongs to A.
For instance, consider the lattice L whose elements are intervals [a, b] such that a, b ∈ {1, 2, 3, 4}. Here applying the
minimal specialization operator we obtain [1, 3] ∨L [2, 4] =
[2, 3]. The ∨L -irreducibles are then [1, 3], [1, 2], [1], [2, 4],[3, 4], [4]. Consider first the abstraction A whose elements
Definition 3 (Projection) p is a projection of a lattice
(M, ≤) iff for each pair (x, y) of elements of M , we have
the following properties:
• If x ≤ y then p(x) ≤ p(y) (monotonicity)
• p(x) ≤ x (minimality)
• p(x) ≤ p(p(x)) (semi-idempotence)
1
Note that ⊥M , the smallest element of M , belongs to all abstractions.
101
• The least upper bound A1 ∨A A2 is obtained by closing
A1 ∪ A2 under ∨M .
• The greatest lower bound A1 ∧A A2 simply is A1 ∩ A2 .
are the intervals containing 3. A is closed by ∨L as intersecting two intervals containing 3 results in an interval also
containing 3.
As a counterexample, consider now L = L −
{[1], [2], [3], [4]}. L is obtained by simply deleting the most
specific terms and clearly still is a lattice. However the minimal specialization in L of [2, 3] and [3, 4] is now ∅ and so
clearly L is not closed by ∨L , and therefore is not an abstraction.
Note that such a restricted view of language reduction
has nice properties. For instance, the possibility to check
whether an instance w satisfies a term through a simple
subsumption test is preserved when reducing the language
through intensional abstractions:
The ordering of extensional abstractions will further be
the basis for the definition of multimodal logics of abstraction.
4
Extensional abstractions
An extensional abstraction is obtained by considering part
of the powerset of W , and closing it by the union operator
∪. This means that we do not consider any more instances
but rather sets of instances, called abstract instances.
Proposition 4 Let L and extW satisfy condition 2.1 and
2.2, and A = pA (L) be an intensional abstraction, then for
all h in pA (L), {x} ⊆ ext(h ) iff h pA (d(x))
Definition 6 Let A be an extensional abstraction A and pA
the associated projection on P(W ), then let t be a term of
L, extA (t) = pA ◦ ext(t) is called the abstract extension of
t.
Let AI be the set of ∪-irreducibles elements of A, from
Proposition 3 we also have
u
extA (t) =
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.
u∈AI ,u⊆ext(t)
Example 1 As an example consider W = {1, 2, 3, 4},
and the abstraction A obtained by closing under union
the part {{1, 2}, {1, 3}} of P(W ), so adding {1, 2, 3}and
∅ to build A. The set of ∪-irreducible elements of A is
AI = {{1, 2}, {1, 3}}. From Proposition 3, pA ({1, 2, 3}) =
{1, 2} ∪ {1, 3} = {1, 2, 3} as both elements of AI are included in {1, 2, 3}, and pA ({2, 3}) = ∅ because neither
{1, 2} nor {1, 3} is included in {2, 3}.
Hereunder we define abstract instances and minimal abstractions of a given instance.
Definition 7 Let A be an extensional abstraction and AI
the set of the ∪-irreducible elements of A.
• An element of AI is called an abstract instance.
• We denote as minimal abstractions of w the minimal
elements of A that contains w. More precisely, the
set Am (w) of minimal abstractions of w is defined as
Am (w) = {u ∈ AI | w ∈ u and if w ∈ u ⊂
u, then u ∈ A}.
In section 4 we will investigate that particular kind of abstraction.
The lattice of abstractions
There is a partial order on projections of a lattice M (Pernelle et al. 2002):
An interesting point is that AI is the set of all the minimal
abstractions of instances in W :
Definition 5 Let M be a lattice and p1 et p2 two projections
on M , we will state that p2 ≤ p1 , i.e. p2 is less concrete,
and so more abstract than p1 , iff there is some projection p
defined on p1 (M ) such that for all c in M , p2 (c) = p◦p1 (c).
Proposition 6
AI =
Am (w)
{w∈W }
So A2 is more abstract than A1 iff A2 is an abstraction
of A1 . This is also equivalent to say that any ∨-irréductible
element i2 of A2 may be written as a join of ∨-irreducible
elements of A1 , i.e. i2 = i11 ∨ · · · ∨ in1
In the particular case where A is closed under intersection,
each element has at most one minimal abstraction. In the
example hereunder we consider instances that may belong
to more than one minimal abstract instance.
Example 2 (Extensional abstractions)
Let M = P({1, 2, 3, 4}), AI1 = {{1, 2}, {1, 3}} and
AI2 = {{1, 2}, {1, 2, 3}}, A2 is more abstract than A1 as
{1, 2} ∈ AI1 and {1, 2, 3} = {1, 2} ∪ {1, 3}.
Example 3 Let W ={plastic chair, wood table, wooden
box} and AI = {{plastic chair, wood table}, {wood table,
wooden box}}. Here the objects are abstracted as furnitures
or wooden objects. The wood table has here two minimal
abstractions, first as a furniture, second as a wooden object.
When closing AI under intersection we add {wood table}
as an abstract instance, and all objects have now a unique
minimal abstraction in the resulting abstraction.
The set A of abstractions on M is a lattice.
Proposition 5 (Lattice of abstractions) Let M be a lattice, the set A of the abstractions of M is a lattice. Let then
A1 = p1 (M ) and A2 = p2 (M ) be elements of A:
102
5
Definition 8 (Abstract implication) Let t1 et t2 be terms of
L: whenever extA (t1 ) ⊆ extA (t2 ) we say that the abstract
implication t1 →A t2 is valid on A.
We will formalize the nature of abstract implications by interpreting them as classical implications relating modalized
terms. So we rewrite t1 →A t2 as 2A t1 → 2A t2 , and
consider that from an observed set of worlds W , in some
model M , we extract a set of modal valid implications that
we translate as modal axioms M 2A t1 → 2A t2 for further
reasoning. Hereunder we discuss the corresponding modal
logics of abstraction, denoted as abstract modal logics.
Now as t1 → t2 means that ext(t1 ) ⊆ ext(t2 ), we
have that whenever some abstract instance u is included in
ext(t1 ), we also have that u is included in ext(t2 ). This also
means, by definition, that extA (t1 ) is included in extA (t2 ).
Proposition 7
• Let A be an abstraction, then t1 → t2 ⇒ t1 →A t2
• Let A be an abstraction more abstract than A, then
t1 →A t2 ⇒ t1 →A t2
Neighborhood semantics
A modal logic, in its simplest form, is a propositional logic
to which is added at least one unary modal connector 2,
generally referred to as a necessity operator. They differ following the axiom schemata and inference rules in which
appear the modal connectors. Axiom schemata are written
as φ where φ is a modal formula, Axioms are theorems,
and inference rules states that whenever we have some antecedent theorem, we can infer a consequent theorem. Classical modal logics are the modal logics in which formulas
are given truth values through neighborhood semantics, also
known as minimal models semantics (Chellas 1980). This
wide class of modal logics includes in particular normal
modal logics relying on Kripke possible world semantics.
We will define hereunder abstract modal logics as particular
classical modal logics, usually non normal, and discuss their
properties. We hereunder recall neighborhood semantics.
The modal language Lmod is obtained by adding the
modal connector 2 to a propositional language L. A model
M = (W, N , ext) is defined through i) a set of worlds W ,
ii) a neighboring function N relating each world w to a set
of subsets of W , iii) a valuation function ext, relating each
atomic proposition P to the set of worlds in which P is true,
and we write for such a world w that M, w |= P . The valuation function is then straightforwardly extended to L (for
instance, for any pair of formulas φ and ψ, we have that
M, w |= φ ∧ ψ iff M, w |= φ and M, w |= ψ) and is then
extended to Lmod , in the following way :
The partial order of abstractions, as defined above, means
that A is more abstract than A whenever any element of
AI may be written as the union of elements of AI . This
means in particular that considering a categorization of animal species on mammals, birds and insects is more abstract
than also considering that mammals have subclasses made
of terrestrial and non terrestrial mammals.
Alpha abstractions
An interesting case is Alpha abstractions. Starting from a
subset C whose union closure is an initial abstraction A,
more concrete abstractions Aα are built, where α represents some degree of influence of the initial categorization
C. More precisely Aα is obtain by deriving from each initial
category C the set C α of frequent enough parts of C 2 , joining the C α ’s of all the categories, and closing under union
the resulting set C α . We have then that Aα is more abstract
than Aα iff α ≥ α . In a previous work the corresponding
Alpha Galois lattices were defined through projections and
experimented in order to extract abstract implications (Ventos and Soldano 2005). As an example, consider the categorization of species in ”mammals“, ”insects“ and ”birds“
and a value α = 0.1. The implication ”Animals that fly are
oviparous” is not valid on W because of the bat that belongs to extW (flies) but not to extW (oviparous). However
consider the corresponding 0.1-abstract implication. The bat
belongs to the category ” mammals“ and the proportion of
mammals that fly is less than 0.1, and so there is no frequent enough part of the category ”mammals“ included in
extW (flies), and as a consequence the bat does not belong
to ext0.1 (flies). Therefore, the abstract implication is not falsified. Note that flying birds and flying insects do belong to
ext0.1 (flies) and also to ext0.1 (oviparous), and as a consequence do not falsify the abstract implication.
Now, what is the use of abstract implications as t1 →A
t2 ? The basic idea, is that when considering some particular
instance w, whether w is in extA (t1 ), i.e. t1 is abstractly
true, we may deduce that t2 is also abstractly true. So such an
implication is useful as far as there are instances in which the
premise t1 is abstractly true. When abstraction increases, the
following effects occur: i) There are more valid implications,
ii) as extA (t1 ) decreases with abstraction (second property
of projections) each implication tends to be less useful.
2
Modal logics of abstraction
∀φ ∈ Lmod , M, w |= 2φ iff ext(φ) ∈ N (w)
(1)
We have then that M, w |= φ holds for all models M and
worlds w, iff φ is a theorem of the corresponding classical
modal logic.
A mapping m from P(W ) to P(W ) is associated to the
neighborhood function N and defined as m(E) = {w ∈
W | E ∈ N (w)}. As a consequence we have
ext(2φ) = {w ∈ W | ext(φ) ∈ N (w)} = m ◦ ext(φ)
and as a consequence Equation 1 rewrites as:
M, w |= 2φ iff w ∈ m ◦ ext(φ)
(2)
These mappings m are in one-to-one correspondence with
the neighborhood functions N as we have E ∈ N (w) iff
w ∈ m(E). Properties of classical modal logics, in terms of
axioms and valid inference rules, follow then from properties of the mapping m.
Let c ⊆ C, then c ∈ C α iff |c| ≥ α ∗ |C|
103
Abstract modal logics
Proposition 8
Now recall that we defined abstract extension extA as p◦ext
where p is a projection. We will so naturally define abstract
modal logics as classical modal logics in which the mapping
m is a projection. To characterize abstract modal logics we
have so to translate the properties of projections as axioms
and inference rules. We have the following correspondence:
1. X ⊆ Y ⇒ p(X) ⊆ p(Y ) corresponds to:
• 2True (axiom N ) iff the abstraction A is complete,
i.e. every world w has at least one minimal abstraction
in AI . The smallest logic for complete abstractions is then
EMT4N .
• (2P ∧ 2Q) → 2(P ∧ Q) (axiom C) iff A is closed
under ∩, i.e every world w has at most one minimal abstraction in AI . The smallest regular abstract logic is then
EMCT4 .
• Normal abstract modal logics, i.e. abstract modal logics
with N and C, are at least EMCNT4 , known as S4 . S4
obeys to the Kripke possible worlds semantics relying on
a reflexive and transitive accessibility relation R (Chellas
1980).
• Normal abstract logic with ¬2¬P → 2¬2¬P , (axiom
5) is EMCNT5 , known as S5 in which R is an equivalence relation and AI is a partition of W .
From (P → Q) infer (2P → 2Q)
the inference rule (RM ) that characterizes monotonic
modal logics (Hansen and Kupke 2004)
2. p(X) ⊆ X corresponds to the axiom schema T :
2P → P
3. p(X) ⊆ p ◦ p(X) corresponds to the axiom schema 4 :
2P → 22P
The smallest3 monotonic modal logic EM is the smallest
classical logic E equipped with RM as an inference rule.
To define the class of abstract modal logics we have to add
axioms T and 4, and so the smallest abstract modal logic is
EMT4 .
Note that the smallest normal abstract logic S4 is obtained
when each world have exactly one minimal abstraction. See
Example 3 how closing by intersection the set of minimal
abstractions changes the nature of the underlying modal
logic, at the price that additional abstract instances are imposed. However is it practically useful to consider nonnormal abstract logics? Or, in other words, is it useful to consider abstractions that are not closed under intersection ? As
an example, alpha abstractions discussed at the end of section 4 are not closed under intersection. Consider the alpha
abstraction Aα on animal species previously described and
two different minimal abstractions c1 and c2 in the category
mammal. Each contains α ∗ |mammal | elements, but clearly
(c1 ∩c2 ) has strictly less than α∗|mammal | elements4 and so
does not belong to Aα . When closing Aα under intersection
we obtain as minimal abstractions all the single species, and
so there is no more abstraction. More generally, as soon has
we want to express a notion of granularity in which granules
overlap, as in example 3, the underlying abstract modal logic
is non-normal.
Definition 9 The class of abstract modal logic is defined as
the class of monotonic modal logics with T and 4 as axioms.
The abstract worlds semantics
Considering the abstraction A associated to a projection pA ,
we have then that:
ext(2φ) = {w ∈ W | ∃c ∈ Am (w), s.t c ⊆ ext(φ)}
and Equation 1 and 2 both rewrites also as:
M, w |= 2φ iff ∃c ∈ Am (w), s.t c ⊆ ext(φ)
(3)
and we say that φ is abstractly true in the world w.
This new semantics, simply obtained by replacing the
neighborood function N by the abstraction A, and equation
1 by equation 3, is denoted as the abstract worlds semantics,
and (W, A, ext) is denoted as an abstract model. To summarize, the abstract worlds semantics states that φ is abstractly
true in w iff there is some minimal abstraction c of w whose
worlds all satisfy φ.
Note that, any element φ of Lmod can then be stated as
either true or false. For instance, consider a model M whose
worlds represents animal species, and let A0.1 be the alpha
abstraction described at the end of Section 4. Consider then
the formula φ = viviparous ∧ 2 flies. This sentence is intended to be true for all animals that both are viviparous
and abstractly fly. Let w represent the bat. We have then
that both viviparous and flies are true in w, and therefore
the formula viviparous ∧ flies is also true in w. However,
as noticed before, 2flies is false in w as few mammals (i.e.
the category to which w belongs) do fly. As a consequence
φ = viviparous ∧ 2flies is also false in w.
Now we relate the properties of the abstraction A to properties of modal logics (omitting proofs). We summarize the
derived classification in table 1.
3
Abstract reasoning
Modeling We consider that the worlds (or instances) are
objects of a universe from which we intend to extract knowledge using some modeling or learning process. We observe
then a finite set of objects W , considering some a priori abstraction A. We have also a set of atomic propositions (properties) whose truth values are observed on each object, and
therefore their extensions on W are known.
Suppose for instance that we have observed that on the
learning set W the implication φ1 → φ2 , where φ1 and φ2
are formulas of Lmod , holds. We write then M φ1 → φ2
to state that this implication is supposed to be true in all the
worlds of some intended model M . This means that we consider now that the implication holds for all objects of the
universe and that for any object w of the universe, we can
always state for any property P whether one of its minimal
4
Except when α = 0 or when α = 1 and that the categorization
is a partition of the universe W .
i.e. with the minimal set of axiom schemata and inference rules
104
• p2 (X) ⊆ p1 (X)
• p2 (X) ⊆ p2 ◦ p1 (X)
From these properties it follows that in our multimodal logic
RM , T and 4 are generalized whenever A2 is more abstract
than A1 in:
• From 2A1 φ → 2A1 ψ infer 2A2 φ → 2A2 ψ (RM’)
• 2A2 φ → 2A1 φ (T’)
• 2A2 φ → 2A2 2A1 φ (4’)
Furthermore as A is a sublattice the greatest lower bound
∧ and least upper bound ∨ are as defined in section 3 and
so for any pair of abstractions (A, A ) in A, A ∧ A is more
abstract than both A and A . As a result, multiple abstract
reasoning is possible.
As an example, consider two abstractions A and A and
suppose that we have both 2A φ → 2A ψ and 2A ψ →
2A λ. Then, from 2A φ → 2A ψ we infer by (RM )
2A∧A φ → 2A∧A ψ. In the same way from 2A ψ →
2A λ, we infer 2A∧A ψ → 2A∧A λ. From these we deduce 2A∧A φ → 2A∧A λ.
This means that we may compose two abstract implications associated to two distinct and possibly incomparable
abstractions, and obtain an abstract implication.
This may be useful for instance when modeling an agent
that have multiple points of view, each corresponding to an
abstraction, on the universe. Another interesting case is the
multi-agent one, where each agent has its own point of view
on the universe. In the later case, we may interpret the previous multimodal example as a way for two agents to find
a sufficiently high level of abstraction to perform a kind of
cooperative reasoning.
Table 1: Properties of the set of abstraction A versus classical modal logic classification
A is a partial covering of W
closed by ∪
A is complete
|Am (w)| ≥ 1
A is closed under intersection
|Am (w)| ≤ 1
A is closed under intersection
and complete
|Am (w)| = 1
AI is a partition
EM T 4 :
monotonic + T 4
EM N T 4 :
monotonic +T 4N
EM CT 4 :
regular +T 4
EM CN T 4 = S4 :
normal+T 4
(R reflexive and transitive)
EM CN T 5 = S5 :
normal+T 5
(R equivalence relation))
abstraction have this property. For instance in the previous
examples on alpha abstraction about animal species, we suppose that we will always be able to categorize an animal as a
mammal, a bird or an insect, and that we can always state for
each property P the proportion of animals in each category
that possess the property. Of course, these proportions are
estimated on the objects of W during the modeling process.
From a practical point of view, abstract implications as
defined in Section 4 are of the form 2φ1 → 2φ2 where
φ1 and φ2 are conjunction of properties, and are straighforwardly extracted from abstract concept lattices.
Simple abstract reasoning We consider here reasoning
in a modal logic as drawing consequences from a set Γ of
antecedents, and from a set of axioms and inference rules.
Inference is performed as in propositional logic, as far as
no modal inference rules are used, and the modal inference
rules, as RM , are used only on axioms. For sake of clarity
we write then the modus ponens rule MP as: from Γ φ and
Γ φ → ψ infer Γ ψ, and the modal rule RM as : from
φ → ψ infer 2φ → 2ψ.
Now let us suppose that we intend to use M φ1 → φ2 .
In our reasoning process such statements act as axioms and
we omit the subscript M .
Now consider 2φ1 as an antecedent. We have then the
following reasoning:
From φ1 → φ2 and RM we infer 2φ1 → 2φ2 ,
then from Γ 2φ1 and 2φ1 → 2φ2 we infer Γ 2φ2 ,
and so to summarize, from abstractly φ1 we have deduced
abstractly φ2 .
6
Related work and conclusion
In Imielisnski(Imielinski 1987) domain abstraction was proposed to query databases. Domain abstractions, that corresponds to our extensional extensions, are also used in reinforcement learning: in order to find good policies with a reasonable amount of exploration it is necessary to reduce the
state space. Loscalzo, Lin and Wright (Loscalzo and Wright
2010; Lin and Wright 2010) investigate in this context tile
coding as an abstraction method: the space state is dynamically discretized into variable size tiles. Each tile covers
range of values for each feature in the state space.
Regarding extensional abstractions, the formal work closest to ours concerns rough sets logics. Rough sets basically
relies on an indiscernability relation, first considered as an
equivalence relation, and further relaxed to any binary relation in generalized rough sets theory (see (Yao and Lin
1996) for a review relating rough sets to normal modal
logics). More recently generalization of rough sets using
coverings, corresponding to our complete abstractions, has
been investigated from an algebraic perspective (Cattaneo
and Ciucci 2009; Davvaz and Mahdavipour 2008). A recent
overview of rough set semantics in a modal logic perspective
goes beyond our abstractions by considering any neighborhood system (Liau 2000).
Our main purpose here, though formal, was to investigate
which restriction in abstracting intensional or extensional
A multimodal abstract logic
We consider here a sublattice A = {A1 , . . . } of abstractions
and we define a multimodal logics using the set of modalities
{2A | A ∈ A}. Now, recall that the partial order states that
A2 is more abstract than A1 whenever there exists a projection p such that A2 = p(A1 ). As a consequence, given any
subset of worlds X, we have that p2 (X) = p ◦ p1 (X). As p
is a projection, the following properties hold:
• p1 (X) ⊆ p1 (Y ) ⇒ p2 (X) ⊆ p2 (Y )
105
spaces were of interest in a learning and reasoning perspective, and to propose practical guidelines to incorporate abstraction, in the sense given here, to learning and reasoning
methods in AI. We have here introduced, starting from semantics, modal logics of abstraction, by considering minimal abstractions of worlds. and shown how a multimodal
logic allows to formalize reasoning at various levels of abstractions. An interesting result is that these logics are non
normal when some world may have various minimal abstractions. Furthermore we have defined a new semantics for
these abstract modal logics, i.e. for classical modal logics
with at least EMT4. Because it simply relies on a covering of
the universe, this semantics is more intuitive than the neighborhood semantics of classical modal logics. In the normal
case, i.e. S4, the abstract models introduced here are topological models as proposed by J. C. C. Mc Kinsey and A.
Tarski in the early 40’s (McKinsey and Tarski 1944) (see for
instance (Parikh, Moss, and Steinsvold 2007) for recent developments). Interestingly, a recent line of work started by
Á. Czászár (Császár 2002) introduces a notion of generalized topology exactly defined as our notion of abstraction.
From a practical point of view, we have tools to learn abstract implications and association rules from instances (Soldano et al. 2010), and we are currently investigating supervised abstract concept learning.
Hansen, H. H., and Kupke, C. 2004. A coalgebraic perspective on monotone modal logic. Electronic Notes in Theoretical Computer Science 106:121–143.
Imielinski, T. 1987. Domain abstraction and limited reasoning. In IJCAI, 997–1003.
Liau, C.-J. 2000. An overview of rough set semantics for
modal and quantifier logics. Int. J. Uncertain. Fuzziness
Knowl.-Based Syst. 8(1):93–118.
Lin, S., and Wright, R. 2010. Evolutionary tile coding:
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Loscalzo, S., and Wright, R. 2010. Automatic methods for
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