Conflicts between relevance-sensitive revision

Conflicts between Relevance-Sensitive and
Iterated Belief Revision
Pavlos Peppas and Anastasios Michael Fotinopoulos and Stella Seremetaki1
Abstract. The original AGM paradigm focuses only on one-step
belief revision and leaves open the problem of revising a belief state
with whole sequences of evidence. Darwiche and Pearl later ad-
dressed this problem by introducing extra (intuitive) postulates as
a supplement to the AGM ones. A second shortcoming of the AGM
paradigm, seemingly unrelated to iterated revision, is that it is too
liberal in its treatment of the notion of relevance. Once again this
problem was addressed with the introduction of an extra (also very
intuitive) postulate by Parikh. The main result of this paper is that
Parikh postulate for relevance-sensitive belief revision is inconsis-
tent with each of the Darwiche and Pearl postulates for iterated belief
revision.
1 INTRODUCTION
The original AGM paradigm for belief revision, [1, 3, 11], focuses
only on one-step transitions leaving open the problem of how to re-
vise a belief state with a whole sequence of evidence. This problem
was later addressed by Darwiche and Pearl who formulated four intu-
itive new postulates (known as the DP postulates) to regulate iterated
revisions. Possible world semantics were introduced to characterize
the new postulates, and with some adjustments (see section 4) the
DP postulates were shown to be compatible with the original AGM
ones.2
Although Darwiche and Pearl’s work has received some criticism,
it remains very influential in the literature of iterated belief revision
and has served as a basis for further developments in the area [7, 5].
A shortcoming of a different nature of the original AGM paradigm
is that it neglects the important role of relevance in belief revision.
As noted by Parikh, [8], when a belief state ψ is revised by new in-
formation µ, only the part of ψ that is related to µ should be effected;
the rest of ψ should remain the same. Parikh proceeded to formulate
a postulate, called (P), that captures this intuition (albeit in limited
cases). Postulate (P) was later shown to be consistent with the AGM
postulates and possible-world semantics were introduced to charac-
terize it, [10].
The main contribution of this paper is to show that, in the pres-
ence of the AGM postulates, Parikh postulate for relevance-sensitive
belief revision is inconsistent with each of the (seemingly unrelated)
Darwiche and Pearl postulates for iterated belief revision.
This of course is quite disturbing. Both the concept of relevance
and the process of iteration are key notions in Belief Revision and
1 Dept of Business Administration, University of Patras, Patras
26500, Greece, emails: pavlos@upatras.gr, anfotino@upatras.gr, sers-
tel@master.math.upatras.gr
2 To be precise, the DP postulates were shown to be compatible with the re-
formalization of the AGM postulates introduced by Katsuno and Mendel-
zon in [6].
we can do away with neither. Moreover, the encoding of these no-
tions proposed by Parikh, Darwiche, and Pearl appears quite natural
and it is not obvious how one should massage the postulates in order
to reconcile them. Further to this point, subsequent postulates intro-
duced to remedy problems with the (DP) ones, [7, 5], are also shown
to be incompatible with postulate (P) (see section 6). On the positive
side, these incompatibility results reveal a hitherto unknown connec-
tion between relevance and iteration that deepens our understanding
of the belief revision process.
The paper is structured as follows. The next section introduces
some notation and terminology. Following that we briefly review
(Katsuno and Mendelzon’s re-formalization of) the AGM postulates,
Darwiche and Pearl’s approach for iterated revisions, and Parikh’s
proposal for relevance-sensitive belief revision (sections 3, 4, and 5).
Section 6 contains our main incompatibility results. Finally in sec-
tion 7 we make some concluding remarks.
2 PRELIMINARIES
Throughout this paper we shall be working with a finitary proposi-
tional language L. We shall denote the (finite) set of all propositional
variables of L by A. For a set of sentences Γ of L, we denote by
Cn(Γ) the set of all logical consequences of Γ, i.e. Cn(Γ) = {ϕ ∈
L: Γ ϕ}. A theory K of L is any set of sentences of L closed
under , i.e. K = Cn(K). We shall denote the set of all theories of
L by TL. A theory K of L is complete iff for all sentences ϕ ∈ L,
ϕ ∈ K or ¬ϕ ∈ K. As it is customary in Belief Revision, herein
we shall identify consistent complete theories with possible worlds.
We shall denote the set of all consistent complete theories of L by
ML. If for a sentence ϕ, Cn(ϕ) is complete, we shall also call ϕ
complete. For a set of sentences Γ of L, [Γ] denotes the set of all
consistent complete theories of L that contain Γ. Often we shall use
the notation [ϕ] for a sentence ϕ ∈ L, as an abbreviation of [{ϕ}].
When two sentences ϕ and χ are logically equivalent we shall often
write ϕ ≡ χ as an abbreviation of ϕ ↔ χ. Finally, the symbols
and ⊥ will be used to denote an arbitrary (but fixed) tautology and
contradiction of L respectively.
3 THE KM POSTULATES
In the AGM paradigm belief sets are represented as logical theories,
new evidence as sentences of L, and the process of belief revision is
modeled as a function mapping a theory K and a sentence µ to a new
theory K ∗ µ. Moreover, eight postulates for ∗ are proposed, known
as the AGM postulates, that aim to capture the notion of rationality
in belief revision.
Katsuno and Mendelzon in [6] slightly reshaped the AGM con-
stituents to make the formalization more amendable to implementa-

tion. In particular, the object language L is set to be a finitary propo-
sitional one,3
and belief sets are defined as finite sets of sentences of
L. Since a belief set contains only finitely many elements, one can
in fact represent it as a single sentence ψ; namely the conjunction of
all its elements. This is the representation eventually adopted in [6]
and the one we shall use herein. To emphasize the finiteness of the
new representation we shall call ψ a belief base and reserve the term
belief set for the closure of ψ, i.e. the theory Cn(ψ).
With the above reformulation, a revision function becomes a func-
tion ∗ mapping a sentence ψ and a sentence µ to a new sentence
ψ ∗ µ; i.e. ∗ : L × L → L. Moreover in the new formalization the
AGM postulates are equivalent to the following six, known as the
KM postulates:
(KM1) ψ ∗ µ µ.
(KM2) If ψ ∧ µ is satisfiable then ψ ∗ µ ≡ ψ ∧ µ.
(KM3) If µ is satisfiable then ψ ∗ µ is also satisfiable.
(KM4) If ψ1 ≡ ψ2 and µ1 ≡ µ2 then ψ1 ∗ µ1 ≡ ψ2 ∗ µ2.
(KM5) (ψ ∗ µ) ∧ ϕ ψ ∗ (µ ∧ ϕ).
(KM6) If (ψ∗µ)∧ϕ is satisfiable then ψ∗(µ∧ϕ) (ψ∗µ)∧ϕ.
4 ITERATED BELIEF REVISION
One thing to notice about the KM postulates is that they all refer to
single-step revisions; no constraints are placed on how the revision
policy at the initial belief base ψ may relate to the revision policies at
it descendants (i.e. at the belief bases resulting from ψ via a sequence
of revisions).
Darwiche and Pearl’s solution to this problem came in the form of
four additional postulates, known as the DP postulates, listed below,
[2]:
(DP1) If ϕ µ then (ψ ∗ µ) ∗ ϕ = ψ ∗ ϕ.
(DP2) If ϕ ¬µ then (ψ ∗ µ) ∗ ϕ = ψ ∗ ϕ.
(DP3) If ψ ∗ ϕ µ then (ψ ∗ µ) ∗ ϕ µ.
(DP4) If ψ ∗ ϕ ¬µ then (ψ ∗ µ) ∗ ϕ ¬µ.
The DP postulates are very intuitive and their intended interpreta-
tion, is loosely speaking as follows (see [2, 7] for details). Postulate
(DP1) says that if the subsequent evidence ϕ is logically stronger
than the initial evidence µ then ϕ overrides whatever changes µ may
have made. (DP2) says that if two contradictory pieces of evidence
arrive sequentially one after the other, it is the later that will prevail.
(DP3) says that if revising ψ by ϕ causes µ to be accepted in the new
belief base, then revising first by µ and then by ϕ can not possibly
block the acceptance of µ. Finally, (DP4) captures the intuition that
“no evidence can contribute to its own demise” [2]; if the revision of
ψ by ϕ does not cause the acceptance of ¬µ, then surely this should
still be the case if ψ is first revised by µ before revised by ψ.
An initial problem with the DP postulates (more precisely with
(DP2)) was that they were inconsistent with the KM postulates. How-
ever this inconsistency was not deeply rooted and was subsequently
resolved. There are in fact (at least) two ways of removing it. The
first, proposed by Darwiche and Pearl themselves, [2], involves sub-
stituting belief bases with belief states, and adjusting the KM pos-
tulates accordingly. The second, proposed by Nayak, Pagnucco, and
Peppas, [7], keeps belief bases as the primary objects of change, but
modifies the underlying assumptions about the nature of ∗.
3 In the original AGM paradigm, the object language L is not necessarily
finitary nor propositional. The details of L are left open and only a small
number of structural constraints are assumed of L and its associated entail-
ment relation (see [3, 11]).
In particular, notice that, with the exception of (KM4), the KM
postulates apply only to a single initial belief base ψ; no reference to
other belief bases are made. Even postulate (KM4) can be weakened
to comply with this policy:
(KM4)’ If µ1 ≡ µ2 then ψ ∗ µ1 ≡ ψ ∗ µ2.
With the new version of (KM4), the KM postulates allow us to
define a revision function as a unary function ∗ : L → L, mapping
the new evidence µ to a new belief base ∗ψ(µ), given the initial belief
base ψ as background. This is the first modification made by Nayak
et al. The second is to make revision functions dynamic. That is,
revision functions may change as new evidence arrives. With this
relaxation it is possible for example to have one revision function
∗ψ associated initially with ψ, and a totally different one after the
revision of ψ by a sequence of evidence that have made the full circle
and have converted ψ back to itself.4
Notice that the weakening of
(KM4) to (KM4)’ is consistent with such dynamic behavior.
As shown by Nayak et al., these two modifications suffice to recon-
cile the DP postulates with the KM ones, and it is these modifications
we shall adopt for the rest of the paper.5
Hence for the rest of the pa-
per, unless specifically mentioned otherwise, we shall use the term
“KM postulates” to refer to (KM1)-(KM6) with (KM4) replaced by
(KM4)’, and we shall assume that revision function are unary and
dynamic.
We close this section with a remark on notation. Although we as-
sume that revision functions are unary (relative to some background
belief base ψ), for ease of presentation we shall keep the original no-
tation and denote the revision of ψ by µ as ψ ∗ µ rather than ∗ψ(µ).
5 RELEVANCE-SENSITIVE BELIEF REVISION
Leaving temporarily aside the issue of iterated belief revision, we
shall now turn back to one-step revisions to review the role of rele-
vance in this process.
As already mentioned in the introduction, Parikh in [8] pointed out
that the AGM/KM postulates fail to capture the intuition that during
the revision of a belief base ψ by µ, only the part of ψ that is related
to µ should be effected, while everything else should stay the same.
Of course determining the part of ψ that is relevant to some new
evidence µ is not a simple matter. There is however at least one spe-
cial case where the role of relevance can be adequately formalized;
namely, when it is possible to break down ψ into two (syntactically)
independent parts such that only the fist of the two parts is syntacti-
cally related to the new evidence µ.
More precisely, for a sentence ϕ of L, we shall denote by Aϕ
the smallest set of propositional variables, through which a sentence
that is logically equivalent to ϕ can be expressed. For example, if
ϕ is the sentence (p ∨ q ∨ z) ∧ (p ∨ q ∨ ¬z), then Aϕ = {p, q},
since ϕ is logically equivalent to p ∨ q, and no sentence with fewer
propositional variables is logically equivalent to ϕ. We shall denote
by Lϕ the propositional sublanguage built from Aϕ via the usual
boolean connectives. By Lϕ we shall denote the sublanguage built
from the complement of Aϕ, i.e. from A − Aϕ. Parikh proposed the
following postulate to capture the role of relevance in belief revision
(at least for the special case mentioned above):6
4 In such cases, although the sequence of evidence does not effect the beliefs
of the agent, it does however change the way the agent reacts to new input.
5 It should be noted though that our results still hold even if Darwiche and
Pearl’s proposal of switching to belief states was adopted.
6 The formulation of (P) in [8] is slightly different from the one presented
below since Parikh was working with theories rather than belief bases. The
two version are of course equivalent.

(P) If ψ ≡ χ ∧ ϕ where χ, ϕ are sentences of disjoint sublan-
guages Lχ, Lϕ respectively, and Lµ ⊆ Lχ, then ψ ∗ µ ≡
(χ ◦ µ) ∧ ϕ, where ◦ is a revision function of the sublan-
guage Lχ.
According to postulate (P), whenever it is possible to break down
the initial belief base ψ into two independent parts χ and ϕ, and
moreover it so happens that the new evidence µ can be expressed
entirely in the language of the first part, then during the revision of
ψ by µ, it is only the first part that is effected; the unrelated part ϕ
crosses over to the new belief base verbatim.
Notice that of the nature of the “local” revision operator ◦ and
its relationship with the “global” revision operator ∗ is not clearly
stated in (P). Peppas et al., [10], therefore proposed a re-formulation
of axiom (P) in terms of two new conditions (R1) and (R2) that do
not refer to a “local” revision operator ◦. Only the first of these two
conditions will be needed herein:
(R1) If ψ ≡ χ∧ϕ, Lχ∩Lϕ = ∅, and Lµ ⊆ Lχ, then Cn(ψ)∩Lχ
= Cn(ψ ∗ µ) ∩ Lχ.
At first (R1) looks almost identical to (P) but it is in fact strictly
weaker than it (see [10] for details). It is essentially condition (R1)
that we will be using to derive our incompatibility results.
6 INCOMPATIBILITY RESULTS
As already announced in the introduction, we shall now prove that
in the presence of the KM postulates, (R1) – and therefore (P) – is
inconsistent with each of the postulates (DP1)-(DP4). The proof re-
lies on the semantics characterization of these postulates so we shall
briefly review it before presenting our results.
We start with Grove’s seminal representation result [4] and its sub-
sequent reformulation by Katsuno and Mendelzon [6].
Let ψ be a belief base and ≤ψ a total preorder in ML. We denote
the strict part of ≤ψ by <ψ. We shall say that ≤ψ is faithful iff the
minimal elements of ≤ψ are all the ψ-worlds:7
(SM1) If r ∈ [ψ] then r ≤ψ r for all r ∈ ML.
(SM2) If r ∈ [ψ] and r ∈ [ψ] then r <ψ r .
Given a belief base ψ and a faithful preorder ≤ψ associated with
it, one can define a revision function ∗ : L → L as follows:
(S*) ψ ∗ µ = γ(min([µ], ≤ψ).
In the above definition min([µ], ≤ψ) is the set of minimal µ-
world with respect to ≤ψ, while γ is a function that maps a set of
possible worlds S to a sentence γ(S) such that [γ(S)] = S.
The preorder ≤ψ essentially represents comparative plausibility:
the closer a world is to the initial worlds [ψ], the more plausible it is.
Then according to (S*), the revision of ψ by µ is defined as the belief
base corresponding to the most plausible µ-worlds.
In [4, 6] it was shown that the function induced from (S*) satisfies
the KM postulates and conversely, every revision function ∗ that sat-
isfies the KM postulates can be constructed from a faithful preorder
by means of (S*).8
7 In [6] a third constraint was required for faithfulness, namely that logically
equivalent sentences are assigned the same preorders. This is no longer
necessary given the new version of (KM4).
8 We note that the weakening of (KM4) to (KM4)’ does not effect these re-
sults since it is accommodated by a corresponding weakening of the notion
of faithfulness.
This correspondence between revision functions and faithful pre-
orders can be preserved even if extra postulates for belief revision are
introduced, as long as appropriate constraints are also imposed on
the preorders. In particular, Darwiche and Pearl proved that the fol-
lowing four constraints (SI1)-(SI4) on faithful preorders correspond
respectively to the four postulates (DP1)-(DP4).
(SI1) If r, r ∈ [µ] then r ≤ψ r iff r ≤ψ∗µ r .
(SI2) If r, r ∈ [¬µ] then r ≤ψ r iff r ≤ψ∗µ r .
(SI3) If r ∈ [µ] and r ∈ [¬µ] then r <ψ r entails r <ψ∗µ r .
(SI4) If r ∈ [µ] and r ∈ [¬µ] then r ≤ψ r entails r ≤ψ∗µ r .
Notice that all of the above constraints make associations between
the preorder ≤ψ related to the initial belief base ψ and the preorder
≤ψ∗µ related to the belief base that results from the revision of ψ by
µ.
The semantic constraint(s) corresponding to postulate (P) have
also been fully investigated in [10].9
Herein however we shall focus
only on condition (R1); in fact we shall be even more restrictive and
consider only the semantic counterpart of (R1) in the special case of
consistent and complete belief bases:
(PS) If Diff(ψ, r) ⊂ Diff(ψ, r ) then r <ψ r .
In the above condition, for any two worlds w, w , Diff(w, w ) rep-
resents the set of propositional variables that have different truth val-
ues in the two worlds; in symbols, Diff(w, w ) = {p ∈ A : w p
and w p} ∪ {p ∈ A : w p and w p}. Whenever a sentence
ψ is consistent and complete, we use Diff(ψ, w ) as an abbreviation
of Diff(Cn(ψ), w ).
Intuitively, (PS) says that the plausibility of a world r depends
on the propositional variables in which it differs from the initial
(complete) belief base ψ: the more the propositional variables in
Diff(ψ, r) the less plausible r is. In [10] it was shown that, for the
special case of consistent and complete belief bases, (PS) is the se-
mantic counterpart of (R1); i.e. given a consistent and complete be-
lief base ψ and a faithful preorder ≤ψ, the revision function ∗ pro-
duced from ≤ψ via (S*) satisfies (R1) iff ≤ψ satisfies (PS).
Although it is possible to obtain a fully-fledged semantic charac-
terization of postulate (P) by generalizing (PS) accordingly (see [10])
the above restricted version suffices to establish the promised results:
Theorem 1 In the presence of the KM postulates, postulate (P) is
inconsistent with each of the postulates (DP1)-(DP4).
Proof. Since (P) entails (R1) it suffices to show that (R1) is incon-
sistent with each of (DP1)-(DP4).
Assume that the object language L is built from the propositional
variable p, q, and z. Moreover let ψ be the complete sentence p∧q∧z
and let ≤ψ be the following preorder in ML:
pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz
In the above definition of ≤ψ, and in order to increase readabil-
ity, we have used sequences of literals to represent possible worlds
(namely the literals satisfied by a world), and we have represented
the negation of a propositional variable v by v.
Notice that ≤ψ satisfies (PS). In what follows we shall construct
sentences µ1, µ2, µ3, and µ4, such that no preorder satisfying (PS)
and related to ψ ∗µ1 (respectively to ψ ∗µ2, ψ ∗µ3, ψ ∗µ4) can also
9 In the same paper axiom (P) was shown to be consistent with all the
AGM/KM postulates.

satisfy (SI1) (respectively (SI2), (SI3), (SI4)). Given the correspon-
dence between (R1) and (PS) on one hand, and the correspondence
between (DP1)-(DP4) and (SI1)-(SI4) on the other, this will suffice
to prove the theorem.
Inconsistency of (PS) and (SI1):
Let µ1 be the sentence q∨z. According to the definition of ≤ψ, there
is only one minimal µ1-world, namely pqz, and therefore by (S*),
ψ∗µ1 ≡ p∧q∧z. Consider now the possible worlds w = pqz and w
= pqz. Clearly, Diff(ψ ∗ µ1, w ) = {q} ⊂ {q, z} = Diff(ψ ∗ µ1, w).
Consequently, no matter what the new preorder ≤ψ∗µ1 is, as long
as it satisfies (PS) it holds that w <ψ∗µ1 w. On the other hand,
since w, w ∈ [µ1] and w ≤ψ w , (SI1) entails that w ≤ψ∗µ1 w .
Contradiction.
Let µ2 be the sentence p∧q∧z. Once again there is only one minimal
µ2-world, namely pqz, and therefore ψ ∗ µ2 ≡ p ∧ q ∧ z. Let w
and w be the possible worlds pqz and pqz respectively. It is not
hard to verify that Diff(ψ ∗ µ2, w ) ⊂ Diff(ψ ∗ µ2, w) and therefore
(PS) entails w <ψ∗µ1 w. On the other hand, given that w, w ∈
[¬µ2] and w ≤ψ w , (SI2) entails that w ≤ψ∗µ1 w , leading us to a
contradiction.
Let µ3 be the sentence (p ∧ q ∧ z) ∨ (p ∧ q ∧ z). Given the above
definition of ≤ψ it is not hard to verify that ψ ∗ µ3 ≡ p ∧ q ∧ z.
Once again, define w and w to be the possible worlds pqz and pqz
respectively. Clearly Diff(ψ∗µ3, w ) ⊂ Diff(ψ∗µ3, w) and therefore
(PS) entails w <ψ∗µ3 w. On the other hand notice that w ∈ [µ3],
w ∈ [¬µ3] and w <ψ w . Hence (SI3) entails that w <ψ∗µ3 w .
Contradiction.
Let µ4 = µ3 = (p ∧ q ∧ z) ∨ (p ∧ q ∧ z) and assume that w, w are as
previously defined. Then clearly, like before, (PS) entails w <ψ∗µ4
w. On the other hand, since w ∈ [µ4], w ∈ [¬µ4] and w ≤ψ w ,
(SI4) entails that w ≤ψ∗µ4 w . A contradiction once again. 2
The above inconsistencies extend to subsequent developments of
the Darwiche and Pearl approach as well. Herein we consider two
such extensions in the form of two new postulates introduced to rec-
tify anomalies with the original DP approach.
The first postulate, called the Conjunction Postulate, was intro-
duced by Nayak, Pagnucco, and Peppas in [7]:
(CNJ) If µ ∧ ϕ ⊥, then ψ ∗ µ ∗ ϕ ≡ ψ ∗ (µ ∧ ϕ).
As shown in [7], in the presence of the KM postulates, (CNJ) en-
tails (DP1), (DP3), and (DP4). Hence the following result is a direct
consequence of Theorem 1:
Corollary 1 In the presence of the KM postulates, postulate (P) is
inconsistent with postulate (CNJ).
The last postulate we shall consider herein is the Independence
Postulate proposed by Jin and Thielscher in [5]:
(Ind) If ¬µ ∈ ψ ∗ ϕ then µ ∈ ψ ∗ µ ∗ ϕ.
Jin and Thielscher proved that, although weaker than (CNJ), (Ind)
still entails (DP3) and (DP4). Hence, from Theorem 1, it follows:
Corollary 2 In the presence of the KM postulates, postulate (P) is
inconsistent with postulate (Ind).
7 CONCLUSION
In this paper we have proved the inconsistency of Parikh’s postulate
(P) for relevance-sensitive belief revision, with each of the four pos-
tulates (DP1)-(DP4) proposed by Darwiche and Pearl for iterated be-
lief revision. This result suggests that a major refurbishment may be
due in our formal models for belief revision. Both relevance and iter-
ation are central to the process of belief revision and neither of them
can be sacrificed. Moreover, the formalizations of these notions by
postulates (P) and (DP1)-(DP2) respectively seem quite intuitive and
it is not clear what amendments should be made to reconcile them.
On a more positive note, the inconsistencies proved herein reveal a
hitherto unknown connection between relevance and iteration, which
will eventually lead to a deeper understanding of the intricacies of
belief revision.
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