A New Framework for Local Belief Revision
A New Framework for Local Belief Revision Omar Doukari, Robert Jeansoulin and Eric Würbel Laboratoire LSIS, UMR CNRS 6168 Domaine Universitaire de Saint-Jérôme 13397 MARSEILLE CEDEX 20 - FRANCE
Abstract. AGM-style revision consists of two suboperations, contrac-
tion followed by expansion. With respect to Hansson, this is called internal revision and an alternative procedure, external revision is proposed. It consists in, rst, expanding the belief base by the new sentence and after that contracting by its negation. In this paper, on the one hand, we propose a new framework for local belief revision by extending the LS-model introduced by Parikh and studied by several authors in the last decade. The new model, called the
C -structure
model, is based
on adapting the containment property in propositional logic, since it has been de ned in a spatial context. On the other hand, we de ne a local external revision operation equivalent to the one de ned by Hansson -in a classical (not-local) framework- by de ning local contraction operation.
1
Introduction
In knowledge representation, a rational agent tries to translate a mix of perceptions and beliefs. As agents often face incomplete, uncertain, and inaccurate information, they need a revision operation in order to manage possible belief changes in presence of new information. The agent's epistemic state represents its reasoning process; belief revision consists of modifying its initial epistemic state in order to maintain consistency, while keeping new information and modifying the least possible previous information. In belief revision, much work takes as its starting point the work of Alchourrón, Gärdenfors and Makinson [1] who proposed and investigated a set of postulates, widely known as the AGM postulates, which appear to capture much of what characterizes rational belief revision. In their framework (developed for belief sets), the revision by
α
consists of two suboperations, contraction by
takes place rst and is then followed by expansion by
α.
¬α
With respect to Hans-
son [2], this is called internal revision and an alternative procedure, external revision is proposed. It consists in, rst, expanding the belief base by after that contracting by
α
and
¬α.
It turns out that in the general case the theoretical complexity of revision is high. More precisely, it belongs to the
Qp
2 class in the framework of propositional
logic [3, 4]. The same problem for the few applications which have been developed for belief revision is posed [5]. In this paper, on the one hand, we propose a new framework for local belief revision by extending the LS-model (Language Splitting Model) de ned, rstly,
by Parikh [6] and studied by several authors in the last decade [711]. The new model, called the
C -structure
model, is based on adapting containment
property in propositional logic since it has been de ned in a spatial context [12]. This property assumes the limitation of the maximal size of eventual existing minimal inconsistencies. on the other hand, since the revision problem is known to be a di cult problem, and there does not exist any revision approach, really eecient, that can treat real applications with a huge amount of data [5, 13, 14, 4, 3], we de ne a local external revision operation equivalent to the one de ned by Hansson -in a classical (not-local) framework- by de ning local contraction operation. After giving some de nition on kernel contraction, in Section 2, we devote Section 3 to the presentation of the LS-model. In Section 4 we de ne the
C-
structure model. In Section 5 and 6, we present our local revision operator and some complexity properties of this operator before concluding in Section 7.
Notation:
In the following,
L
is a propositional language de ned on a nite
V and the usual connectors (¬, ∨, ∧, →, ↔). If α is a sentence then V(α) denotes the set of atoms composing the sentence α. If V is a subset of V then L(V ) represents the propositional sublanguage de ned over V . If X is a set of sentences then Cn(X) is the logical closure of X . In particular, a subset T of L is a belief set (theory) i T = Cn(T ). We shall use 0 letters T, T etc. for theories, and BT is a belief base of the belief set T i BT is a nite subset of T and Cn(BT ) = T . In particular, if BT is an inconsistent belief base, we say that M ⊆ BT is a MIS of BT i.e., a minimal inconsistent subset of 0 0 sentences of BT i M is inconsistent and for all M ⊂ M , M is consistent. In the following, BT denotes an arbitrary belief base of T . T ∗ α is the revision of T by α, T − α is the contraction of T by α and nally, T + α is the operation of expansion of T by α, it is equal to Cn(T ∪ {α}), i.e. the result of a brute addition of α to T (followed by logical closure) without considering the need for consistency. Constructing T ∗ α is contracting T by ¬α then adding α. Formally, this construction is given by the LEVI identity [2]: T ∗ α = (T − ¬α) + α.
set of propositional variables (atoms)
2
Kernel Contraction
The contraction of a set of sentences
A
by
α
according to kernel contraction
introduced by Hansson in [15] consists in selecting among the sentences of a set
A
that contribute
eectively
to imply
α;
then to remove at least one element of
each selected subset (i.e. a Hitting set of the selected subsets) so that the result does not imply
α. A that contribute eectively to imply α α. Formally:
The subsets of a set subsets of
A
implying
are the minimal
De nition 1. [15] Let A be a set in L and α a sentence. Then A⊥α is a set
such that B ∈ A⊥α i: α∈ / Cn(B 0 ).
(i)
B ⊆ A,
(ii)
α ∈ Cn(B),
and
(iii)
If B 0 ⊂ B then
A.
A⊥α
is called the kernel set of A wrt α and its elements are the α-kernels of
To calculate a hitting set of the kernel set of
A we de ne an incision function
which is a function de ned from sets of sets of sentences into sets of sentences, selecting at least one sentence from each set of the argument. Formally:
De nition 2. [15] Let A be a set of sentences. An incisionS function σ is a
function such that for all sentences α we have: ∀B ∈ A⊥α, and B 6= ∅ then B ∩ σ(A⊥α) 6= ∅.
(i)
σ(A⊥α) ⊆
(A⊥α),
and
(ii)
Thus, kernel contraction is de ned as follows:
De nition 3. [15] Let A be a set of sentences and σ an incision function for A.
Kernel contraction −σ for A is de ned as follows: A −σ α = A \ σ(A⊥α). An operator
function
σ
for
A
−
A is a kernel A − α = A −σ α
for a set
such that
contraction i there is an incision for all sentences
α.
Hansson also provided an axiomatic characterisation for kernel contraction.
Theorem 1 [15] The operator − for a set of sentences A is a kernel contraction i for all sentences α, it satis es:
3
Success : If α ∈/ Cn(∅), then α ∈/ Cn(A − α). Inclusion : A − α ⊆ A. Uniformity : If ∀A0 ⊆ A, α ∈ Cn(A0 ) i β ∈ Cn(A0 ) then A − α = A − β . Core-retainment : If β ∈ A and β ∈/ A − α then there is some set A0 such that A0 ⊆ A and α ∈/ Cn(A0 ) but α ∈ Cn(A0 ∪ {β}). The Language Splitting Model
The intuition behind the language splitting model is that our beliefs are subdivided into
disjoint
areas which do not aect each other [6].
De nition 4. Let T be a theory of L and let V1 , ..., Vn be a partition of V . Then {V1 , ..., VS n } is a T -splitting if ∀i ∈ {1 . . . n} there exist sentences αi ∈ L(Vi ), s.t., n T = Cn( i=1 {αi }). The disjoint sublanguages assumption of the LS-model allows us to revise our beliefs
locally
and to minimize the amount of computation we have to do when
we revise a new piece of information, both in checking whether it is consistent with the old set of beliefs and also in revising our beliefs in view of the new information. For example, an agent that is revising his beliefs about planetary motion is unlikely to revise his beliefs about Malaysian politics. This simple intuition is not fully captured in the AGM paradigm. In [6] Parikh introduced a new axiom (P), as a supplement to the AGM postulates, which is de ned as follows.
(P)
T = Cn(X , Y) where X , Y are sentences of disjoint sublanguages L1 , L2 α ∈ L1 , then T ∗ α = (CnL1 (X ) ◦ α) + Y , where ◦ is a revision operator of the sublanguage L1 . : If
respectively, and
The language splitting model requires an agent's beliefs be partitioned into theories which have totally disjoint languages. In practice, however, beliefs in dierent areas do have some overlap in subject matter and so the partition of the main language is not actually strict. An agent's component theories do contain beliefs that are more relevant to one another than to beliefs in other component theories, but they are not totally irrelevant to beliefs in other subtheories. In the following section, we extend the language splitting model by de ning
C -structure
the
model. This model allows some overlap between sublanguages
using containment property de ned in [12], which has been proposed in order to revise spatial information by splitting up space into dierent subspaces and
if a restriction of the belief base to a subspace is consistent, and if such a restriction is consistent with the belief base attached to the q-covering1 of this subspace, then this restriction is consistent with any other information . revising each one separately. The idea of this property is:
Q1
Example 1. we consider a geographical space of two dimen-
B3 . We denote by SB(X) the subset of sentences attached to subspace X . Q1 , Q2 and Q3 are the q -coverings of B1 , B2 and B3 respectively. SB(B3 ) is outside the q -covering of B1 , so there
sions, subdivided into subspaces
B1 , B2
and
Q3
B1
B3
exists some independence between their minimal inconsis-
SB(B1 ∪ Q1 ), SB(B2 ) SB(B3 ) are consistent, then SB(B1 ) is consistent with subset of sentences SB(B2 ) and SB(B3 ).
tent subsets of sentences (MISs). If and the
q
B2 Q2
So containment property is based on the restriction of the ing MISs by
q,
the
thickness
of
q -coverings.
size
of exist-
Thus by generalizing containment
property which was de ned in a spatial context by de ning it according to the
k -relevance
relation between atoms, we extend the language splitting model to
become the
C -structure
4
The
model.
C -structure
Model
Usually inconsistency is due to the accidental presence of a few pieces of contradictory information about a given subject, moreover large globally inconsistent problems are, usually, generated arti cially since they are scarce in real life applications [16] (see e.g. the ooded valley application [5], the pigeons-holes problem [17], Tseitin and Urqhart's formulas [3], etc).
1
A
q -covering of a q . For
is equal to
subspace
B
is the subspace
further detail, see [12].
Q
covering
B
for some distance which
The language splitting model is based on properties of the agent which carries out the revision like principle of minimal change, limited capacities of real agents, etc, and on properties of belief sets (or belief bases) like modularity. However, it does not take into account the properties of the inconsistencies which may exist in real belief bases like those of the eld of spatial information. Among these properties we nd containment property which assumes that real life applications are locally inconsistent, that means the maximal size of existing MISs is limited by a certain distance depending on the application traited. For the same motivations as those given in the language splitting model, namely minimal change, and by taking into account containment property, we de ne a new model called the
C -structure
model, which keeps the principle
of language splitting model (disjoint sublanguages) to de ne a set of a given language, and each core has a
covering
cores
of
of atoms surrounding it. This
concept of covering allows us some degree of overlapping between the dierent sublanguages. Now, using these two concepts (core and covering) we extend the previous model while avoiding the hard assumption put for it (the assumption of the disjoint sublanguages). Our working hypothesis is a weak one with respect to that of the language splitting model. It is related to the maximal size of eventual existing MISs. We begin by de ning a set of cores of
L
as a partition of a set
V.
De nition 5. {V1 , ..., Vn } is a set of cores of L i it is a partition of V .
Example 2.
Let L be the propositional language de ned over V = {a, b, c, d, e, f, g, h, i, j, k, l}. Let T be an arbitrary theory de ned on L and axiomatized by B T the following belief base: a ∨ b, ¬c, b → c, ¬d, c → (d ∨ e), e ↔ f, f → g, ¬g ∨ h, i → h, i, j → i, j, k ∨ l ∨ j The set {{a, b, c}, {d, e, f }, {g, h, i}, {j, k, l}} is a set of cores of L. Now, to order the atoms of the language
L,
we use the following relevance
relation inspired from [18].
De nition 6. Let BT be a belief base of a theory T . We say that two atoms, p
and q, are directly relevant wrt BT , denoted by R(p, q, BT ) (or by R0 (p, q, BT )), i: (i) ∃α ∈ BT s.t., p, q ∈ V(α), or (ii) p = q. Two atoms p, q are k-relevant wrt BT , denoted by Rk (p, q, BT ), if ∃p0 , p1 , ..., pk+1 ∈ V s.t.: p0 = p; pk+1 = q; and ∀i ∈ {0, ..., k}, R(pi , pi+1 , BT ). In Example 2, we have:
R(a, b, BT ), R1 (a, c, BT ), R2 (a, d, BT ),
etc.
We introduce from this de nition the concept of neighborhood as the following.
De nition 7. Let BT be a belief base of a theory T . Two atoms p, q ∈ V are
neighbors wrt BT i: ∃k ≥ 0 such that Rk (p, q, BT ).
To de ne clearly the degree of overlapping that takes place between the various sublanguages and to quantify local inconsistencies (MISs), we need to de ne a distance between variables.
De nition 8. Suppose two atoms p, q ∈ V , BT a belief base of a theory T . The
distance between p, q wrt BT , denoted by dist(p, q, BT ), is de ned as follows. min{k|Rk (p, q, BT )} if p, q are neighbors dist(p, q, BT ) = ∞ otherwise. In Example 2, we have:
2,
dist(a, b, BT ) = 0, dist(a, c, BT ) = 1, dist(a, d, BT ) =
etc. The covering whose thickness is equal to
k
for a core
Vi
is de ned as follows.
De nition 9. Let {V1 , ..., Vn } be a set of cores of L and BT be a belief base of a theory T . We say that Covk (Vi , BT ) is a covering whose thickness is equal to k of Vi wrt BT i: Covk (Vi , BT ) ⊆ V ; and ∀p ∈ V, if ∃q ∈ Vi s.t., dist(p, q, BT ) ≤ k then p ∈ Covk (Vi , BT ). For example, the set of coverings, for k = 0, corresponding to the set of cores {{a, b, c}, {d, e, f }, {g, h, i}, {j, k, l}} wrt BT (Example 2) is: {{a, b, c, d, e}, {c, d, e, f, g}, {f, g, h, i, j}, {i, j, k, l}}. Since our model is based on the size of MISs, we quantify them as follows.
De nition 10. Let BT be an inconsistent belief base. Let M be a MIS of BT .
The size of M , wrt BT , denoted by Size(M, BT ) is such that: Size(M, BT ) = max{dist(a, b, BT )|a, b ∈ V(M )}. For example (Example 2), if A
C -structure
M = {¬c, c ∨ d, ¬d}
then,
Size(M, BT ) = 0.
is a set of structures such that each one of them is composed
of two parts. The rst part called the
sublanguage part which is composed of a subbase part which is a belief
core and its covering, and the second one is the subbase de ned on the sublanguage part. Informally, a
C -structure represents a thematic view of the overall knowledge
of an agent with a good understanding of the interactions between subjects. In this paper, the only hypothesis, that we make, is that the maximal size of eventual existing MISs in a given belief base is known. Hence, when we want to construct a
C -structure C
of a belief base
of coverings of cores (value of of MISs which may exist in
k)
BT .
BT , we only require that the thickness
should be (at least) equal to the maximal size More formally:
De nition 11. Let T be a theory de ned on L and BT an arbitrary belief base
of T . The set {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} is a C structure of BT i: (i) {V1 , ..., Vn } is a set of cores of L, (ii) {Covk (V1 , BT ), ..., Covk (Vn , BT )} is the corresponding set of coverings wrt BT s.t., ∀i ∈ {1, ..., n}∀α ∈ L(Covk (Vi , BT )), if BT ∪ {α} is inconsistent, then ∀M a MIS of BT ∪ {α}, Size(M, BT ) ≤Sk , and (iii) ∀Bi , Bi = L(Covk (Vi , BT )) ∩ BT . We denote by ni=1 Bi , the informational part of the C -structure. We shall use the letter C both for a C -structure C and for the informational part of C . It will be clear from the context which is meant.
We obtain the following
C -structure
corresponding to Example 2 by assum-
BT is 0. {({a, b, c}, {a, b, c, d, e}, {a∨b, ¬c, b → c, ¬d, c → (d∨e)}), ({d, e, f }, {c, d, e, f, g}, {¬c, c → (d ∨ e), ¬d, e ↔ f, f → g}), ({g, h, i}, {f, g, h, i, j}, {f → g, ¬g ∨ h, i → h, i, j → i, j}), ({j, k, l}, {i, j, k, l}, {i, j → i, j, k ∨ l ∨ j})}. Hence, if all subbases part of C are consistent then C is globally consistent. ing that the maximal size of eventual existing MISs in
Formally, we deduce the containment property as follows.
(Containment Property)
: If {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} is a C -structure C of BT , then: if ∀i ∈ {1, ..., n}, Bi is consistent, then C is globally consistent.
5
C -structure
Revision
AGM-style revision by contraction by
¬α.
sets), contraction by
α.
α
consists of two suboperations, expansion by
α
and
In the AGM revision operation ( rst developed for belief
¬α
takes place rst and is then followed by expansion by
In [2] this is called internal revision and an alternative procedure, external
revision is proposed. It consists in rst expanding the belief base by that contracting by
α and after
¬α.
Hence, we will assume that local external revision consists of (local) expansion followed by local contraction (local expansion in this context consists simply, in adding new sentences according to the sublanguage in which they are de ned. For example, if we want to add a sentence to all structures of
C
new sentences can be added to some
5.1
α to a C -structure C of BT , we must add it Vi are such that V(α) ⊆ Covk (Vi , BT ). So, structures of C and not to all structures).
whose their cores
Local Contraction
The idea behind kernel contraction is that, if we remove from the belief base at least one element of each
α-kernel, we obtain a belief base that does not imply α.
In the case of the local operation, we are interested in obtaining a belief subbase de ned over a sublanguage of
C
that does not imply
α.
We consider then only
the kernel set belonging to this belief subbase and use an incision function to select the sentences from the kernel set to be removed.
De nition 12. Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be a
C -structure C of BT and σ be an incision function. The local contraction of C by α, denoted by C −˙ σ α, is de ned as follows: C −˙ σ α = C \ σ(Bi ⊥α), such that Vi ∩ V(α) 6= ∅. The following theorem characterizes the operation of local contraction with respect to the postulates of kernel contraction de ned in [15]:
Theorem 2 The operation of local contraction −˙ σ , de ned above, for a C -
structure C and some incision function satis es the following postulates:
Success : If α ∈/ Cn(∅), then α ∈/ Cn(C −˙ σ α). Inclusion : C −˙ σ α ⊆ C . Core-retainment : If β ∈ (C \ C −˙ σ α), then there is some C 0 ⊆ C such that
and α ∈ Cn(C 0 ∪ {β}). Uniformity : If ∀C 0 ⊆ C, α ∈ Cn(C 0 ) i β ∈ Cn(C 0 ) then C −˙ σ α = C −˙ σ β . α∈ / Cn(C 0 )
As a special case, consider a sentence with several cores of
L.
α
to contract from
C
which intersects
How do we choose the sublanguage of
C
in order to
perform the local contraction? The following proposition shows that, in such cases, we do not need to make a local contraction for all the sublanguages whose core
Vi
is such as:
Vi ∩ V(α) 6= ∅,
but just for one of them.
Proposition 1. Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be
a C -structure C of BT , and −˙ σ be the operator of local contraction determined by an incision function σ. Let α be a sentence and {Vi , ..., Vj } be a subset of ˙ σα = C \ {V1 , ..., Vn } such that: ∀Vi0 ∈ {Vi , ..., Vj }, Vi0 ∩ V(α) 6= ∅. Then: C − σ(Bi ⊥α) = ... = C \ σ(Bj ⊥α). (¬c∧¬d), we {c, d} intersects the two cores of the rst two structures. In this case,
In Example 2, if we want to apply local contraction by the sentence notice that
it su ces to proceed only one of the two structures. We take the rst structure, for example. We calculate
(¬c ∧ ¬d)-kernel
set, which is equal to
an incision function, we choose the hitting set
{¬c}
{{¬c, ¬d}}.
By
(an arbitrary choice), and
we remove it from the rst structure, as well as from any structure containing
C -structure is equal to: {({a, b, c}, {a, b, c, d, e}, {a ∨ b, b → c, ¬d, c → (d ∨ e)}), ({d, e, f }, {c, d, e, f, g}, {c → (d ∨ e), ¬d, e ↔ f, f → g}), ({g, h, i}, {f, g, h, i, j}, {f → g, ¬g ∨ h, i → h, i, j → i, j}), ({j, k, l}, {i, j, k, l}, {i, j → i, j, k ∨ l ∨ j})}.
it. Finally, the resulting
The following results show that our local contraction is equivalent to kernel contraction.
Lemma 1 Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be a C -
structure C of BT . If C is consistent, then: BT ⊥α = Bi ⊥α such that Vi ∩V(α) 6= ∅. Corollary 1 Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be a C -
structure C of BT , −σ be the operator of kernel contraction, −˙ σ be the operator of local contraction, and let σ be an incision function. If C is consistent, then: ˙ σ α = BT −σ α. C−
BT , of a T , by a sentence α we can disconsider all sublanguages of a C -structure C of BT that are not relevant for α2 , since we are sure that the size of the α-kernels of BT is limited by k . Thus we cannot not see elements of the kernel set of BT when our research is limited locally (in L(Covk (Vi , BT )) ∩ BT ) on one This Corollary shows that when contracting a consistent belief base
theory
2
a sublanguage
Li = (Vi , Covk (Vi )) of C is not relevant for a sentence α if Vi ∩V(α) = ∅
of the sublanguages of
C
whose core
Vi
is such as
Vi ∩ V(α) 6= ∅,
because the
kernel set calculated locally is complete. Hence, the
C -structure
model provides us with substantional gains since the
size of the set to be updated is always a fraction of the size of the original set and ensures the following result: When we want to contract a piece of information which lies in one of the sublanguage (or straddles only two or three of them) then we can leave most of the
C -structure
unchanged and contract only one of
the aected sublanguages.
5.2
Local External Revision
Finally, our local external revision consists in expanding (locally) the
C -structure
then the result is contracted by the local contraction operator.
De nition 13. Let C be a C -structure of a belief base BT , and −˙ be our local
contraction operator. Local external revision (±) consists in extending C by α ˙ then to contract the result by ¬α: C ± α = (C ∪ {α})−¬α.
The following theorem characterizes our local external revision operator with respect to the postulates which should be satis ed by all external revision operators de ned in [15]:
Theorem 3 The operator ± of local external revision satis es the following
properties for all C -structures C and all sentences α:
If ¬α ∈/ Cn(∅), then ¬α ∈/ Cn(C ± α) (no-contradiction). C ± α ⊆ C ∪ {α} (Inclusion). If β ∈ (C \ C ± α), then there is some C 0 ⊆ C ∪ {α} such that ¬α ∈/ Cn(C 0 )
and ¬α ∈ Cn(C 0 ∪ {β}) (Core-retainment). α ∈ C ± α (Success). If α and β are element of C and ∀C 0 ⊆ C , ¬α ∈ Cn(C 0 ) i ¬β ∈ Cn(C 0 ), then C ∩ (C ± α) = C ∩ (C ± β) (Weak Uniformity). C + α ± α = C ± α (Pre-expansion). Corollary 2 Let ∗ be an operation of kernel external revision (not-local), let C
be a C -structure of BT , then: ∀α ∈ L, C ± α = BT ∗ α.
This last Corollary shows that our local revision operator is equivalent to the one de ned by Hansson, in [15], which is a
6
global
revision operator.
Complexity Results
Local external revision operator is computationally e cient. Generally we assume that each xed
p,
L(Covk (Vi , BT )) ∩ BT has relatively small size, say under some BT (i.e, |BT |) might be quite large. Then given
while the cardinality of
a sentence eration by
α with certain number of distinct atoms, local external revision opα runs in time which is exponential in p. Thus if p is small compared
to
|BT |, as is usually the case, the computational cost will be much smaller than |BT |.
that of usual revision operators which are exponential in
In [12], the authors were interested in geographical applications revision. They have proposed a revision strategy based on containment property. Indeed, they have opted for a geographical space decomposition into subspaces called blocks, and each block is included in another neighborhood called covering of the block. In this application, the distance for constructing blocks (cores) and coverings of a given geographical space is de ned spatially, so it is a real distance. The geographical space is segmented on a set of parcels, and a block is a subset of parcels such that the set of blocks constitutes a partition of the space. Covering of a given block is a subset of parcels surrounding this block for a certain distance which represents the thickness of the covering. To each part, composed of a block and its covering, is attached a spatial knowledge subbase which represents the union of all sentences attached to each parcel belonging to this part. The theoretical results obtained are encouraging such that the complexity of
3 is proved equals to:
O(|BT |3 × 22×|BT | ). 3 However, that of the strategy based on containment property is equal to: O(p × 2×p 2 ). one of the classical revision operation
7
Conclusion
There are two principal ways to split up a belief base into several belief subbases. First, it may be done as an addition to the logic by using external information, so that one and the same belief base can be divided into subbases in dierent ways with keeping classical logic. Secondly, that may be done by deriving the several subbases from the logic, therefore we should de ne a non-classical inference operation. This case corresponds to what Wassermann and Hansson have done in [19] where they de ned an operation of local inference before de ning their operators of belief change. In this paper, we have de ned a cutting according to the rst approach such as external information was the maximal size of MISs in the real life applications which we have called containment property. Thus we can de ne an operator of local revision which is at the same time complete and correct. However, if it is rather simple to construct a
C -structure
corresponding to a given belief base, it
is not always possible to make an assumption on the maximal size of MISs. This question is staying open. The contribution of this paper is twofold. On the one hand, an e cient model has been proposed, allowing one to locate and quantify local inconsistencies (MISs) in sets of propositional sentences (belief bases). We think that such a model should be useful with respect to many domains. For example, this should make the handling of local inconsistencies in the diagnosis domain possible. On
3
The corresponding revision operator is based on the adaptation of Reiter algorithm. For further detail, either for Reiter algorithm or the corresponding revision operator, see [12].
the other hand, using the concept of kernel contraction, an operation of local revision has been de ned by a new operation of local contraction. In future work we intend to carry out a thorough study of this interesting new model for belief representation and belief revision by extending our framework to belief fusion.
References 1. Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: Partial meet contraction and revision functions. J. Symb. Logic 50 (1985) 510530 2. Hansson, S.O.: Reversing the levi identity. Philosophical Logic 22 (1992) 637639 3. Eiter, T., Gottlob, G.: On the complexity of propositional knowledge base revision, updates, and counterfactuals. (1992) 261273 4. Liberatore, P., Schaerf, M.: The complexity of model checking for belief revision and update. In: AAAI/IAAI, Vol. 1. (1996) 556561 5. Würbel, E., Papini, O., Jeansoulin, R.: Revision: an application in the framework of gis. In: 7th International Conference on Principles of Knowledge Representation and Reasoning, KR'2000., Breckenridge, Colorado, USA (2000) 505516 6. Parikh, R.: Beliefs, belief revision, and splitting languages. Logic, language and computation, vol. 2 (1999) 266278 7. Chopra, S., Parikh, R.: An inconsistency tolerant model for belief representation and belief revision. In: IJCAI '99: Proc. of the 16th International Joint Conference on Arti cial Intelligence, Morgan Kaufmann Publishers Inc. (1999) 192199 8. Chopra, S., Parikh, R.: Relevance sensitive belief structures. Annals of Mathematics and Arti cial Intelligence 28 (2000) 259285 9. Chopra, S., Georgatos, K., Parikh, R.: Relevance sensitive non-monotonic inference on belief sequences. J. of Applied Non-Classical Logics 11 (2001) 131150 10. Kourousias, G., Makinson, D.: Parallel interpolation, splitting, and relevance in belief change . J. Symb. Logic 72 (2007) 9941002 11. Peppas, P., Chopra, S., Foo, N.Y.: Distance semantics for relevance-sensitive belief revision. In: 9th International Conference on Principles of Knowledge Representation and Reasoning, KR'2004., Canada (2004) 319328 12. Doukari, O., Jeansoulin, R.: Space-contained con ict revision, for geographic information. In: 10th AGILE International Conference on Geographic Information Science, Aalborg (Danmark) (2007) 13. Williams, M.A.: Applications of belief revision. In: ILPS '97: International Seminar on Logic Databases and the Meaning of Change, Transactions and Change in Logic Databases, London, UK, Springer-Verlag (1998) 287316 14. Nebel, B.: How hard is it to revise a belief base? In Dubois, D., Prade, H., eds.: Handbook of Defeasible Reasoning and Uncertainty Management Systems, Volume 3: Belief Change. Kluwer Academic Publishers, Dordrecht (1998) 77145 15. Hansson, S.O.: Kernel contraction. J. Symb. Logic 59 (1994) 845859 16. Mazure, B., Saïs, L., Grégoire, E.: Boosting complete techniques thanks to local search methods. Annals of Mathematics and Arti cial Intelligence 22 (1998) 319 331 17. Cook, S.: A short proof of the pigeon hole principle using extended resolution. In: SIGACT News, Vol. 8. (1976) 2832 18. Chopra, S., Parikh, R., Wassermann, R.:
Approximate belief revision.
Logic J.
IGPL 9 (2001) 755768 19. Hansson, S.O., Wassermann, R.: Local change. Studia Logica 70 (2002) 4976
Abstract. AGM-style revision consists of two suboperations, contrac-
tion followed by expansion. With respect to Hansson, this is called internal revision and an alternative procedure, external revision is proposed. It consists in, rst, expanding the belief base by the new sentence and after that contracting by its negation. In this paper, on the one hand, we propose a new framework for local belief revision by extending the LS-model introduced by Parikh and studied by several authors in the last decade. The new model, called the
C -structure
model, is based
on adapting the containment property in propositional logic, since it has been de ned in a spatial context. On the other hand, we de ne a local external revision operation equivalent to the one de ned by Hansson -in a classical (not-local) framework- by de ning local contraction operation.
1
Introduction
In knowledge representation, a rational agent tries to translate a mix of perceptions and beliefs. As agents often face incomplete, uncertain, and inaccurate information, they need a revision operation in order to manage possible belief changes in presence of new information. The agent's epistemic state represents its reasoning process; belief revision consists of modifying its initial epistemic state in order to maintain consistency, while keeping new information and modifying the least possible previous information. In belief revision, much work takes as its starting point the work of Alchourrón, Gärdenfors and Makinson [1] who proposed and investigated a set of postulates, widely known as the AGM postulates, which appear to capture much of what characterizes rational belief revision. In their framework (developed for belief sets), the revision by
α
consists of two suboperations, contraction by
takes place rst and is then followed by expansion by
α.
¬α
With respect to Hans-
son [2], this is called internal revision and an alternative procedure, external revision is proposed. It consists in, rst, expanding the belief base by after that contracting by
α
and
¬α.
It turns out that in the general case the theoretical complexity of revision is high. More precisely, it belongs to the
Qp
2 class in the framework of propositional
logic [3, 4]. The same problem for the few applications which have been developed for belief revision is posed [5]. In this paper, on the one hand, we propose a new framework for local belief revision by extending the LS-model (Language Splitting Model) de ned, rstly,
by Parikh [6] and studied by several authors in the last decade [711]. The new model, called the
C -structure
model, is based on adapting containment
property in propositional logic since it has been de ned in a spatial context [12]. This property assumes the limitation of the maximal size of eventual existing minimal inconsistencies. on the other hand, since the revision problem is known to be a di cult problem, and there does not exist any revision approach, really eecient, that can treat real applications with a huge amount of data [5, 13, 14, 4, 3], we de ne a local external revision operation equivalent to the one de ned by Hansson -in a classical (not-local) framework- by de ning local contraction operation. After giving some de nition on kernel contraction, in Section 2, we devote Section 3 to the presentation of the LS-model. In Section 4 we de ne the
C-
structure model. In Section 5 and 6, we present our local revision operator and some complexity properties of this operator before concluding in Section 7.
Notation:
In the following,
L
is a propositional language de ned on a nite
V and the usual connectors (¬, ∨, ∧, →, ↔). If α is a sentence then V(α) denotes the set of atoms composing the sentence α. If V is a subset of V then L(V ) represents the propositional sublanguage de ned over V . If X is a set of sentences then Cn(X) is the logical closure of X . In particular, a subset T of L is a belief set (theory) i T = Cn(T ). We shall use 0 letters T, T etc. for theories, and BT is a belief base of the belief set T i BT is a nite subset of T and Cn(BT ) = T . In particular, if BT is an inconsistent belief base, we say that M ⊆ BT is a MIS of BT i.e., a minimal inconsistent subset of 0 0 sentences of BT i M is inconsistent and for all M ⊂ M , M is consistent. In the following, BT denotes an arbitrary belief base of T . T ∗ α is the revision of T by α, T − α is the contraction of T by α and nally, T + α is the operation of expansion of T by α, it is equal to Cn(T ∪ {α}), i.e. the result of a brute addition of α to T (followed by logical closure) without considering the need for consistency. Constructing T ∗ α is contracting T by ¬α then adding α. Formally, this construction is given by the LEVI identity [2]: T ∗ α = (T − ¬α) + α.
set of propositional variables (atoms)
2
Kernel Contraction
The contraction of a set of sentences
A
by
α
according to kernel contraction
introduced by Hansson in [15] consists in selecting among the sentences of a set
A
that contribute
eectively
to imply
α;
then to remove at least one element of
each selected subset (i.e. a Hitting set of the selected subsets) so that the result does not imply
α. A that contribute eectively to imply α α. Formally:
The subsets of a set subsets of
A
implying
are the minimal
De nition 1. [15] Let A be a set in L and α a sentence. Then A⊥α is a set
such that B ∈ A⊥α i: α∈ / Cn(B 0 ).
(i)
B ⊆ A,
(ii)
α ∈ Cn(B),
and
(iii)
If B 0 ⊂ B then
A.
A⊥α
is called the kernel set of A wrt α and its elements are the α-kernels of
To calculate a hitting set of the kernel set of
A we de ne an incision function
which is a function de ned from sets of sets of sentences into sets of sentences, selecting at least one sentence from each set of the argument. Formally:
De nition 2. [15] Let A be a set of sentences. An incisionS function σ is a
function such that for all sentences α we have: ∀B ∈ A⊥α, and B 6= ∅ then B ∩ σ(A⊥α) 6= ∅.
(i)
σ(A⊥α) ⊆
(A⊥α),
and
(ii)
Thus, kernel contraction is de ned as follows:
De nition 3. [15] Let A be a set of sentences and σ an incision function for A.
Kernel contraction −σ for A is de ned as follows: A −σ α = A \ σ(A⊥α). An operator
function
σ
for
A
−
A is a kernel A − α = A −σ α
for a set
such that
contraction i there is an incision for all sentences
α.
Hansson also provided an axiomatic characterisation for kernel contraction.
Theorem 1 [15] The operator − for a set of sentences A is a kernel contraction i for all sentences α, it satis es:
3
Success : If α ∈/ Cn(∅), then α ∈/ Cn(A − α). Inclusion : A − α ⊆ A. Uniformity : If ∀A0 ⊆ A, α ∈ Cn(A0 ) i β ∈ Cn(A0 ) then A − α = A − β . Core-retainment : If β ∈ A and β ∈/ A − α then there is some set A0 such that A0 ⊆ A and α ∈/ Cn(A0 ) but α ∈ Cn(A0 ∪ {β}). The Language Splitting Model
The intuition behind the language splitting model is that our beliefs are subdivided into
disjoint
areas which do not aect each other [6].
De nition 4. Let T be a theory of L and let V1 , ..., Vn be a partition of V . Then {V1 , ..., VS n } is a T -splitting if ∀i ∈ {1 . . . n} there exist sentences αi ∈ L(Vi ), s.t., n T = Cn( i=1 {αi }). The disjoint sublanguages assumption of the LS-model allows us to revise our beliefs
locally
and to minimize the amount of computation we have to do when
we revise a new piece of information, both in checking whether it is consistent with the old set of beliefs and also in revising our beliefs in view of the new information. For example, an agent that is revising his beliefs about planetary motion is unlikely to revise his beliefs about Malaysian politics. This simple intuition is not fully captured in the AGM paradigm. In [6] Parikh introduced a new axiom (P), as a supplement to the AGM postulates, which is de ned as follows.
(P)
T = Cn(X , Y) where X , Y are sentences of disjoint sublanguages L1 , L2 α ∈ L1 , then T ∗ α = (CnL1 (X ) ◦ α) + Y , where ◦ is a revision operator of the sublanguage L1 . : If
respectively, and
The language splitting model requires an agent's beliefs be partitioned into theories which have totally disjoint languages. In practice, however, beliefs in dierent areas do have some overlap in subject matter and so the partition of the main language is not actually strict. An agent's component theories do contain beliefs that are more relevant to one another than to beliefs in other component theories, but they are not totally irrelevant to beliefs in other subtheories. In the following section, we extend the language splitting model by de ning
C -structure
the
model. This model allows some overlap between sublanguages
using containment property de ned in [12], which has been proposed in order to revise spatial information by splitting up space into dierent subspaces and
if a restriction of the belief base to a subspace is consistent, and if such a restriction is consistent with the belief base attached to the q-covering1 of this subspace, then this restriction is consistent with any other information . revising each one separately. The idea of this property is:
Q1
Example 1. we consider a geographical space of two dimen-
B3 . We denote by SB(X) the subset of sentences attached to subspace X . Q1 , Q2 and Q3 are the q -coverings of B1 , B2 and B3 respectively. SB(B3 ) is outside the q -covering of B1 , so there
sions, subdivided into subspaces
B1 , B2
and
Q3
B1
B3
exists some independence between their minimal inconsis-
SB(B1 ∪ Q1 ), SB(B2 ) SB(B3 ) are consistent, then SB(B1 ) is consistent with subset of sentences SB(B2 ) and SB(B3 ).
tent subsets of sentences (MISs). If and the
q
B2 Q2
So containment property is based on the restriction of the ing MISs by
q,
the
thickness
of
q -coverings.
size
of exist-
Thus by generalizing containment
property which was de ned in a spatial context by de ning it according to the
k -relevance
relation between atoms, we extend the language splitting model to
become the
C -structure
4
The
model.
C -structure
Model
Usually inconsistency is due to the accidental presence of a few pieces of contradictory information about a given subject, moreover large globally inconsistent problems are, usually, generated arti cially since they are scarce in real life applications [16] (see e.g. the ooded valley application [5], the pigeons-holes problem [17], Tseitin and Urqhart's formulas [3], etc).
1
A
q -covering of a q . For
is equal to
subspace
B
is the subspace
further detail, see [12].
Q
covering
B
for some distance which
The language splitting model is based on properties of the agent which carries out the revision like principle of minimal change, limited capacities of real agents, etc, and on properties of belief sets (or belief bases) like modularity. However, it does not take into account the properties of the inconsistencies which may exist in real belief bases like those of the eld of spatial information. Among these properties we nd containment property which assumes that real life applications are locally inconsistent, that means the maximal size of existing MISs is limited by a certain distance depending on the application traited. For the same motivations as those given in the language splitting model, namely minimal change, and by taking into account containment property, we de ne a new model called the
C -structure
model, which keeps the principle
of language splitting model (disjoint sublanguages) to de ne a set of a given language, and each core has a
covering
cores
of
of atoms surrounding it. This
concept of covering allows us some degree of overlapping between the dierent sublanguages. Now, using these two concepts (core and covering) we extend the previous model while avoiding the hard assumption put for it (the assumption of the disjoint sublanguages). Our working hypothesis is a weak one with respect to that of the language splitting model. It is related to the maximal size of eventual existing MISs. We begin by de ning a set of cores of
L
as a partition of a set
V.
De nition 5. {V1 , ..., Vn } is a set of cores of L i it is a partition of V .
Example 2.
Let L be the propositional language de ned over V = {a, b, c, d, e, f, g, h, i, j, k, l}. Let T be an arbitrary theory de ned on L and axiomatized by B T the following belief base: a ∨ b, ¬c, b → c, ¬d, c → (d ∨ e), e ↔ f, f → g, ¬g ∨ h, i → h, i, j → i, j, k ∨ l ∨ j The set {{a, b, c}, {d, e, f }, {g, h, i}, {j, k, l}} is a set of cores of L. Now, to order the atoms of the language
L,
we use the following relevance
relation inspired from [18].
De nition 6. Let BT be a belief base of a theory T . We say that two atoms, p
and q, are directly relevant wrt BT , denoted by R(p, q, BT ) (or by R0 (p, q, BT )), i: (i) ∃α ∈ BT s.t., p, q ∈ V(α), or (ii) p = q. Two atoms p, q are k-relevant wrt BT , denoted by Rk (p, q, BT ), if ∃p0 , p1 , ..., pk+1 ∈ V s.t.: p0 = p; pk+1 = q; and ∀i ∈ {0, ..., k}, R(pi , pi+1 , BT ). In Example 2, we have:
R(a, b, BT ), R1 (a, c, BT ), R2 (a, d, BT ),
etc.
We introduce from this de nition the concept of neighborhood as the following.
De nition 7. Let BT be a belief base of a theory T . Two atoms p, q ∈ V are
neighbors wrt BT i: ∃k ≥ 0 such that Rk (p, q, BT ).
To de ne clearly the degree of overlapping that takes place between the various sublanguages and to quantify local inconsistencies (MISs), we need to de ne a distance between variables.
De nition 8. Suppose two atoms p, q ∈ V , BT a belief base of a theory T . The
distance between p, q wrt BT , denoted by dist(p, q, BT ), is de ned as follows. min{k|Rk (p, q, BT )} if p, q are neighbors dist(p, q, BT ) = ∞ otherwise. In Example 2, we have:
2,
dist(a, b, BT ) = 0, dist(a, c, BT ) = 1, dist(a, d, BT ) =
etc. The covering whose thickness is equal to
k
for a core
Vi
is de ned as follows.
De nition 9. Let {V1 , ..., Vn } be a set of cores of L and BT be a belief base of a theory T . We say that Covk (Vi , BT ) is a covering whose thickness is equal to k of Vi wrt BT i: Covk (Vi , BT ) ⊆ V ; and ∀p ∈ V, if ∃q ∈ Vi s.t., dist(p, q, BT ) ≤ k then p ∈ Covk (Vi , BT ). For example, the set of coverings, for k = 0, corresponding to the set of cores {{a, b, c}, {d, e, f }, {g, h, i}, {j, k, l}} wrt BT (Example 2) is: {{a, b, c, d, e}, {c, d, e, f, g}, {f, g, h, i, j}, {i, j, k, l}}. Since our model is based on the size of MISs, we quantify them as follows.
De nition 10. Let BT be an inconsistent belief base. Let M be a MIS of BT .
The size of M , wrt BT , denoted by Size(M, BT ) is such that: Size(M, BT ) = max{dist(a, b, BT )|a, b ∈ V(M )}. For example (Example 2), if A
C -structure
M = {¬c, c ∨ d, ¬d}
then,
Size(M, BT ) = 0.
is a set of structures such that each one of them is composed
of two parts. The rst part called the
sublanguage part which is composed of a subbase part which is a belief
core and its covering, and the second one is the subbase de ned on the sublanguage part. Informally, a
C -structure represents a thematic view of the overall knowledge
of an agent with a good understanding of the interactions between subjects. In this paper, the only hypothesis, that we make, is that the maximal size of eventual existing MISs in a given belief base is known. Hence, when we want to construct a
C -structure C
of a belief base
of coverings of cores (value of of MISs which may exist in
k)
BT .
BT , we only require that the thickness
should be (at least) equal to the maximal size More formally:
De nition 11. Let T be a theory de ned on L and BT an arbitrary belief base
of T . The set {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} is a C structure of BT i: (i) {V1 , ..., Vn } is a set of cores of L, (ii) {Covk (V1 , BT ), ..., Covk (Vn , BT )} is the corresponding set of coverings wrt BT s.t., ∀i ∈ {1, ..., n}∀α ∈ L(Covk (Vi , BT )), if BT ∪ {α} is inconsistent, then ∀M a MIS of BT ∪ {α}, Size(M, BT ) ≤Sk , and (iii) ∀Bi , Bi = L(Covk (Vi , BT )) ∩ BT . We denote by ni=1 Bi , the informational part of the C -structure. We shall use the letter C both for a C -structure C and for the informational part of C . It will be clear from the context which is meant.
We obtain the following
C -structure
corresponding to Example 2 by assum-
BT is 0. {({a, b, c}, {a, b, c, d, e}, {a∨b, ¬c, b → c, ¬d, c → (d∨e)}), ({d, e, f }, {c, d, e, f, g}, {¬c, c → (d ∨ e), ¬d, e ↔ f, f → g}), ({g, h, i}, {f, g, h, i, j}, {f → g, ¬g ∨ h, i → h, i, j → i, j}), ({j, k, l}, {i, j, k, l}, {i, j → i, j, k ∨ l ∨ j})}. Hence, if all subbases part of C are consistent then C is globally consistent. ing that the maximal size of eventual existing MISs in
Formally, we deduce the containment property as follows.
(Containment Property)
: If {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} is a C -structure C of BT , then: if ∀i ∈ {1, ..., n}, Bi is consistent, then C is globally consistent.
5
C -structure
Revision
AGM-style revision by contraction by
¬α.
sets), contraction by
α.
α
consists of two suboperations, expansion by
α
and
In the AGM revision operation ( rst developed for belief
¬α
takes place rst and is then followed by expansion by
In [2] this is called internal revision and an alternative procedure, external
revision is proposed. It consists in rst expanding the belief base by that contracting by
α and after
¬α.
Hence, we will assume that local external revision consists of (local) expansion followed by local contraction (local expansion in this context consists simply, in adding new sentences according to the sublanguage in which they are de ned. For example, if we want to add a sentence to all structures of
C
new sentences can be added to some
5.1
α to a C -structure C of BT , we must add it Vi are such that V(α) ⊆ Covk (Vi , BT ). So, structures of C and not to all structures).
whose their cores
Local Contraction
The idea behind kernel contraction is that, if we remove from the belief base at least one element of each
α-kernel, we obtain a belief base that does not imply α.
In the case of the local operation, we are interested in obtaining a belief subbase de ned over a sublanguage of
C
that does not imply
α.
We consider then only
the kernel set belonging to this belief subbase and use an incision function to select the sentences from the kernel set to be removed.
De nition 12. Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be a
C -structure C of BT and σ be an incision function. The local contraction of C by α, denoted by C −˙ σ α, is de ned as follows: C −˙ σ α = C \ σ(Bi ⊥α), such that Vi ∩ V(α) 6= ∅. The following theorem characterizes the operation of local contraction with respect to the postulates of kernel contraction de ned in [15]:
Theorem 2 The operation of local contraction −˙ σ , de ned above, for a C -
structure C and some incision function satis es the following postulates:
Success : If α ∈/ Cn(∅), then α ∈/ Cn(C −˙ σ α). Inclusion : C −˙ σ α ⊆ C . Core-retainment : If β ∈ (C \ C −˙ σ α), then there is some C 0 ⊆ C such that
and α ∈ Cn(C 0 ∪ {β}). Uniformity : If ∀C 0 ⊆ C, α ∈ Cn(C 0 ) i β ∈ Cn(C 0 ) then C −˙ σ α = C −˙ σ β . α∈ / Cn(C 0 )
As a special case, consider a sentence with several cores of
L.
α
to contract from
C
which intersects
How do we choose the sublanguage of
C
in order to
perform the local contraction? The following proposition shows that, in such cases, we do not need to make a local contraction for all the sublanguages whose core
Vi
is such as:
Vi ∩ V(α) 6= ∅,
but just for one of them.
Proposition 1. Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be
a C -structure C of BT , and −˙ σ be the operator of local contraction determined by an incision function σ. Let α be a sentence and {Vi , ..., Vj } be a subset of ˙ σα = C \ {V1 , ..., Vn } such that: ∀Vi0 ∈ {Vi , ..., Vj }, Vi0 ∩ V(α) 6= ∅. Then: C − σ(Bi ⊥α) = ... = C \ σ(Bj ⊥α). (¬c∧¬d), we {c, d} intersects the two cores of the rst two structures. In this case,
In Example 2, if we want to apply local contraction by the sentence notice that
it su ces to proceed only one of the two structures. We take the rst structure, for example. We calculate
(¬c ∧ ¬d)-kernel
set, which is equal to
an incision function, we choose the hitting set
{¬c}
{{¬c, ¬d}}.
By
(an arbitrary choice), and
we remove it from the rst structure, as well as from any structure containing
C -structure is equal to: {({a, b, c}, {a, b, c, d, e}, {a ∨ b, b → c, ¬d, c → (d ∨ e)}), ({d, e, f }, {c, d, e, f, g}, {c → (d ∨ e), ¬d, e ↔ f, f → g}), ({g, h, i}, {f, g, h, i, j}, {f → g, ¬g ∨ h, i → h, i, j → i, j}), ({j, k, l}, {i, j, k, l}, {i, j → i, j, k ∨ l ∨ j})}.
it. Finally, the resulting
The following results show that our local contraction is equivalent to kernel contraction.
Lemma 1 Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be a C -
structure C of BT . If C is consistent, then: BT ⊥α = Bi ⊥α such that Vi ∩V(α) 6= ∅. Corollary 1 Let {(V1 , Covk (V1 , BT ), B1 ), ..., (Vn , Covk (Vn , BT ), Bn )} be a C -
structure C of BT , −σ be the operator of kernel contraction, −˙ σ be the operator of local contraction, and let σ be an incision function. If C is consistent, then: ˙ σ α = BT −σ α. C−
BT , of a T , by a sentence α we can disconsider all sublanguages of a C -structure C of BT that are not relevant for α2 , since we are sure that the size of the α-kernels of BT is limited by k . Thus we cannot not see elements of the kernel set of BT when our research is limited locally (in L(Covk (Vi , BT )) ∩ BT ) on one This Corollary shows that when contracting a consistent belief base
theory
2
a sublanguage
Li = (Vi , Covk (Vi )) of C is not relevant for a sentence α if Vi ∩V(α) = ∅
of the sublanguages of
C
whose core
Vi
is such as
Vi ∩ V(α) 6= ∅,
because the
kernel set calculated locally is complete. Hence, the
C -structure
model provides us with substantional gains since the
size of the set to be updated is always a fraction of the size of the original set and ensures the following result: When we want to contract a piece of information which lies in one of the sublanguage (or straddles only two or three of them) then we can leave most of the
C -structure
unchanged and contract only one of
the aected sublanguages.
5.2
Local External Revision
Finally, our local external revision consists in expanding (locally) the
C -structure
then the result is contracted by the local contraction operator.
De nition 13. Let C be a C -structure of a belief base BT , and −˙ be our local
contraction operator. Local external revision (±) consists in extending C by α ˙ then to contract the result by ¬α: C ± α = (C ∪ {α})−¬α.
The following theorem characterizes our local external revision operator with respect to the postulates which should be satis ed by all external revision operators de ned in [15]:
Theorem 3 The operator ± of local external revision satis es the following
properties for all C -structures C and all sentences α:
If ¬α ∈/ Cn(∅), then ¬α ∈/ Cn(C ± α) (no-contradiction). C ± α ⊆ C ∪ {α} (Inclusion). If β ∈ (C \ C ± α), then there is some C 0 ⊆ C ∪ {α} such that ¬α ∈/ Cn(C 0 )
and ¬α ∈ Cn(C 0 ∪ {β}) (Core-retainment). α ∈ C ± α (Success). If α and β are element of C and ∀C 0 ⊆ C , ¬α ∈ Cn(C 0 ) i ¬β ∈ Cn(C 0 ), then C ∩ (C ± α) = C ∩ (C ± β) (Weak Uniformity). C + α ± α = C ± α (Pre-expansion). Corollary 2 Let ∗ be an operation of kernel external revision (not-local), let C
be a C -structure of BT , then: ∀α ∈ L, C ± α = BT ∗ α.
This last Corollary shows that our local revision operator is equivalent to the one de ned by Hansson, in [15], which is a
6
global
revision operator.
Complexity Results
Local external revision operator is computationally e cient. Generally we assume that each xed
p,
L(Covk (Vi , BT )) ∩ BT has relatively small size, say under some BT (i.e, |BT |) might be quite large. Then given
while the cardinality of
a sentence eration by
α with certain number of distinct atoms, local external revision opα runs in time which is exponential in p. Thus if p is small compared
to
|BT |, as is usually the case, the computational cost will be much smaller than |BT |.
that of usual revision operators which are exponential in
In [12], the authors were interested in geographical applications revision. They have proposed a revision strategy based on containment property. Indeed, they have opted for a geographical space decomposition into subspaces called blocks, and each block is included in another neighborhood called covering of the block. In this application, the distance for constructing blocks (cores) and coverings of a given geographical space is de ned spatially, so it is a real distance. The geographical space is segmented on a set of parcels, and a block is a subset of parcels such that the set of blocks constitutes a partition of the space. Covering of a given block is a subset of parcels surrounding this block for a certain distance which represents the thickness of the covering. To each part, composed of a block and its covering, is attached a spatial knowledge subbase which represents the union of all sentences attached to each parcel belonging to this part. The theoretical results obtained are encouraging such that the complexity of
3 is proved equals to:
O(|BT |3 × 22×|BT | ). 3 However, that of the strategy based on containment property is equal to: O(p × 2×p 2 ). one of the classical revision operation
7
Conclusion
There are two principal ways to split up a belief base into several belief subbases. First, it may be done as an addition to the logic by using external information, so that one and the same belief base can be divided into subbases in dierent ways with keeping classical logic. Secondly, that may be done by deriving the several subbases from the logic, therefore we should de ne a non-classical inference operation. This case corresponds to what Wassermann and Hansson have done in [19] where they de ned an operation of local inference before de ning their operators of belief change. In this paper, we have de ned a cutting according to the rst approach such as external information was the maximal size of MISs in the real life applications which we have called containment property. Thus we can de ne an operator of local revision which is at the same time complete and correct. However, if it is rather simple to construct a
C -structure
corresponding to a given belief base, it
is not always possible to make an assumption on the maximal size of MISs. This question is staying open. The contribution of this paper is twofold. On the one hand, an e cient model has been proposed, allowing one to locate and quantify local inconsistencies (MISs) in sets of propositional sentences (belief bases). We think that such a model should be useful with respect to many domains. For example, this should make the handling of local inconsistencies in the diagnosis domain possible. On
3
The corresponding revision operator is based on the adaptation of Reiter algorithm. For further detail, either for Reiter algorithm or the corresponding revision operator, see [12].
the other hand, using the concept of kernel contraction, an operation of local revision has been de ned by a new operation of local contraction. In future work we intend to carry out a thorough study of this interesting new model for belief representation and belief revision by extending our framework to belief fusion.
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