Controlled Revision - An algorithmic
approach for belief revision
DOV GABBAY, Department of Computer Science, King’s College London,
Strand, London WC2R 2LS.
e-mail: dg@dcs.kcl.ac.uk
GABRIELLA PIGOZZI, Department of Philosophy, University of Konstanz,
78457 Konstanz, Germany.
e-mail: gabriella.pigozzi@uni-konstanz.de
JOHN WOODS, The Abductive Systems Group, University of British
Columbia, Vancouver, BC, Canada V6T 1Z1, and Department of Computer
Science, King’s College, London, Strand, London WC2R 2LS
e-mail: woods@uleth.ca
Abstract
This paper provides algorithmic options for belief revision of a database receiving an infinite stream of inputs. At
, receiving the input
. The revision algorithms for moving to the new database
stage , the database is
take into account the history of previous revisions actually executed as well as possible revision
options which were discarded at the time but may now be pursued. The appropriate methodology for carrying this
out is Labelled Deductive Systems.
1 Background
The present paper offers an algorithmic dynamic model of revision, as part of our dynamic
approach to practical reasoning. This model contains the revision algorithm as part of the
logic. The methodology that an essential part of a logical system is an algorithm for the logic
as well as for additional mechanisms such as abduction and revision, has been strongly put
forward in [8, 9, 11] and seriously developed in [13, 14]. The idea that formal modelling of
an epistemic change cannot ignore the algorithmic approach, or other computational notions,
is recognized also by other active researchers in the community (see for example, Shoham
and Cousins [31]).
A major component of any practical reasoning system is the representation of the epistemic
change occasioned by the addition of new information to an existing set of data . When
a consistent theory receives
a non-contradictory wff as input, the new theory will simply be if it is consistent. A more interesting case is when the new
piece
of information causes inconsistency, in which case needs to be revised to a which
consistently includes . The main application area for revision theory is commonsense
"!#%rea$&$%$
soning, which involves a current database and a sequence of different inputs
.
The seminal work in this area is due to Alchourrón, Gärdenfors and Makinson (see [15] for
an extended introduction to belief revision). The AGM approach does not, however, provide
specialized algorithms for revision. Moreover, no special attention (in the form of properties)
J. Logic Computat., Vol. 12 No. DMC1, pp. 1–21 2002
'c
Oxford University Press
2 Controlled Revision - An algorithmic approach for belief revision
is accorded to the iterated revision process. In order to choose which propositions to remove a
system called epistemic entrenchment has been proposed in [17]. Intuitively, a total ordering
is defined over formulas, so that less entrenched beliefs are retracted in preference to those
more entrenched. The system, however, does not provide a mechanism which defines how the
ordering should change after a revision takes place; nor does it indicate where the ordering
itself comes from.
The postulates for epistemic change proposed by Alchourrón, Gärdenfors and Makinson
had the undeniable merit of having provoked formal reflections in an area which has been active in both philosophy and artificial intelligence since the early 1980s. It should be noticed
that the AGM approach cannot be applied in computational contexts, since belief sets (which
are deductively closed) are too large. For this reason, belief bases have been considered instead of belief sets. We share Friedman and Halpern’s view [7] that it is no longer sufficient
simply to propose a new set of rational axioms for belief revision and to prove some consequences from it. Our approach attempts instead to fill the gap in AGM theory, providing
a reasoned mechanism which selects the data to be removed. We suggest that, even in the
simplest cases, it is hard to imagine that we can revise our knowledge set in one way only. Selection among several revision options is assigned to specific policies, some of which can be
specified by the application area. Time is also encoded in our model, and the ordering among
options derives in part from the history of past revisions, and partly from consideration of
economic factors or the nature of the inputs (as, for example, when we receive the same input
several times). This means that we need a richer structure for the set of data: and we find
that Labelled Deductive Systems (LDS) theory [8] proves to be a framework in which these
features can be implemented attractively, and an appropriate revision algorithm defined.
We call the algorithm which takes account of all the above components Controlled Revision
( ). One of the attractions of a Labelled Deductive Systems approach to Controlled Revision
is that the labels give us a logical means of control and representation, which facilitates the
development of principles of Controlled Revision with relative ease and generality. The labels
have multiple roles. One is to help define the properties of the selected logic, and the second
to annotate and control the iterated revision process. In particular, when data are rejected
from the database, labels allow us to retrace and to retract the consequences of the data which
have been just rejected and which are not otherwise supported.
Another notable
feature of Controlled Revision can be seen as follows: In the literature,
when is inconsistent, non-tolerant approaches to inconsistency assume two main
forms:
1. reject the input which causes inconsistencies (non-insistent policy);
2. maintain , identifying the subset rejected, the consistency is
such that, when
restored (insistent policy). Then .
We will show that in Controlled Revision both the options are possible and that the algorithm chooses the best option according to the parameters recorded in labels and the defined
policies.
Some work compatible with our approach to revision has already been done. We refer, in
particular, to the Truth Maintenance System (TMS) approach, proposed by Doyle in [4]. The
main idea of this orientation (and, at the same time, the main difference from AGM theory
— see [5]) is that the set of data to be revised exhibits additional structure in which certain
beliefs (i.e. wffs) are annotated as justifying other beliefs. Our own proposal shares with
the foundations approach the aim of defining a procedure for identifying candidate wffs for
Controlled Revision - An algorithmic approach for belief revision 3
consistency-restoring revision. As Gärdenfors and Rott have noticed [18], TMS performs
belief changes differently from the way they are achieved in AGM. TMS executes a direct
and procedural belief revision, in which no role is played by rationality assumptions and
revision reduces to inference relations which are usually non-classical (e.g., paraconsistent
and non-monotonic).
One of the main difficulties with TMS is the handling of beliefs responsible for contradictions, known as dependency-directed backtracking. It has been criticized for insufficient
control of identifications of nodes responsible for contradictions. One of the main advantages
of our model is that it is able to find the best solution for the management of inconsistency
problems, as they arise in their considerable variety.
2 Introduction
This section explains the ideas and concepts of this paper through an example.
Imagine a police inspector investigating some serious fraud case which has made headline
news. He is expected to deliver a report outlining the details of the fraud. His investigations stretch over a period
he accumulates
the information sequence
of time
! &during
$ $ $ . Thewhich
denotes
which we
denote
by
expression
schematically the se &$ $ $ which is the batch of information he gets at stage . We
quence
allow for the same
information to come from different sources, so
may be equal to . However the news
is so hot that, each time a new piece of information is uncovered, a new interim report is
expected. In order to do this, the inspector has to perform three operations:
1. Evaluate the information at each stage to try and build the current coherent picture of
what has happened. (For simplicity, let us assume the information arrives one well formed
formula (wff) at a time.)
2. Send interim report to describe what he currently thinks has happened.
3. Keep full record of what piece of information came from where and at what time.
therefore labels his information by double labels. Each item is labelled
The
inspector
. identifies the piece of information with respect to identification tag of source,
reliability, etc. and is the time it has arrived. At each time , a theory is formed (the
interim report) being formally a consistent subset of all the information received up to time
. The flow of information is then represented by a sequence of evolving databases. The
initial state is called and it can be the empty set or it contains the set of fixed integrity
constraints (a consistent set, not to be revised). In our example, contains some rules
of commonsense knowledge the inspector needs to use in order to manage the information
he collects. Let us assume the interim report is . Of course, information arriving at
time may be rejected as unreliable (because of inconsistency with other sources) but a new
information uncovered at time may not only be accepted but cause the acceptance
of the previous
information
.
Therefore,
inspector
labels his information
as
at stage" ! , the
&$ $ $ .
follows:
,
where
is
the
stage
when
arrived
(i.e.
$#
$%
&!'%(!
%
%
, indicates whether it was accepted ( ) and
, for
) or rejected (
)
at *) .
The above is the model of belief revision we are using. Since the inspector needs to keep
a full record of the past and present information, the database will list all the wffs, both
in and out at stage . When the investigator receives a new information at stage +
, he
updates the history labels of the formulas in the new database. An interesting consequence of
4 Controlled Revision - An algorithmic approach for belief revision
the record of the history of past revisions, is that in our model we can allow the reinstatement
of a formula. Consider the following case:
. We get
.
1. Start
with input . Name it and
let 2.
comes in. Name it . It is inconsistent with ; so it excludes it. This gives:
!
3.
!
$
enters. Name it . It is inconsistent with , but we want to restore . Thus:
! ! ! $
Here
is reinstated at stage . The labelling allows us to remember what was rejected
and when.
Consider
are we going to present it? If the inputs
&$%$&$&again
&$&the
$%$& interim
$ report
$ $ , then . How
were
is a subsequence of formulas
which
&$ $ $ are in the
database
at
stage
in
the
history
labels,
say,
, i.e. have $
,
!
!
$
$
$
!
+
!
&
$
$
$
. We can present as , but we prefer to include the
full
full labelling with the
all the wffs,
both
&$ $ $ Thus
is
presented
, where as the
list
ofmeans
information.
in and out : that
is
the
.
information label of and gives
the
revision
history
of
&$ $ $ %
%
%
We call the label
active at
if (as opposed to
) is in the
sequence.
If we use our labels in the above form, we need to indicate how to propagate labels when
we do deduction. The general modus ponens rule for LDS has the form:
is a compatibility
between labels (if it does not hold we cannot
execute
the
predicate
.
modus ponens).
gives the new label of
as
a
function
of
the
labels
of
and
To simplify, we assume our tagging labels are atomic and so
they
accumulate
and
always holds in this case. We accumulate also our history labels in
. means being active at the same time. Thus we let:
" ! $ # %
& ! '# # tagging labels by concatenation and accumulate history labels as pairs
#We
. accumulate
This
is
just
a convenient notation. We now need to explain . We already said that
&$ $ $ is active
all points
for which
is in the
sequence.
Let us define by
' # is atactive
# and
are active.
induction
that
at
all
times
such
that
both
We define
" () to mean () have some common active time.
%
%
%
We now need to explain how this labelling changes facing a new input. We explain it by
example and, simplifying, we
assume we&get
$ $ $ one
input
formula
at each stage. Suppose the
database has the form . Under the assumption that at
each stage only one well formed
formula
comes
as
input,
and
formulas&$ are
# #
# has the form # that
$ $ tagged
by# .
different
atomic
names,
then
# is atomic name and $% , "! % ! , indicates whether # was accepted in ) or not.
Controlled Revision - An algorithmic approach for belief revision 5
Suppose
that !
comes
Assume now that our revision policy accepts
$ $ $ in
as! a new input.
&$ $ $ now
, with as
the
report).
# "new
# consistent
database
# ,(interim
!
!
Then
in
the
new
database
we
have
wffs
&$ $ $ , and
, where is used for the accepted cases of
and
&
to give us the new label, i.e. to take us from
otherwise
.
We
will
use
a
function
# # to the appropriate # " # .
3 Formal presentation of the model
%$ $ $
Let be a finitary propositional language with
and .
are wffs, i.e., data
or
and
beliefs. We can base our logic on Johansson’s minimal logic, with an intuitionistic
with no special axioms for it. This means that does
with an arbitrary absurd constant
. If weadd
not satisfy the axiom , for every
this
it would become an
axiom
intuitionistic negation. Negation is instead defined .
Here does not necessarily mean falsity, but rather is an atomic symbol which
not
we do
want to derive.
We
use
it
to
indicate
inconsistency
conditions
in
the
form
which
we indicate that is inconsismeans that should not be derivable. Similarly, with &$ $ $ , meaning
tent.
constraints of the form
also
We can
$ $ $ specify
integrity
. We
assume that integrity constraints are not to be revised
in time and therefore they are specified in the initial theory .
D EFINITION 3.1 (Labels and histories)
1. Let be an infinite set of atomic labels. The( elements
( %$ $ $ of are referred to as atomic
resource labels. We also allow variable labels
.
# # &$ $ $ !
'!
2. Let
where
and
be a set of sequences of the form
!
%
!
%
%
)
(
or
. We say is the%starting
time
of
the
sequence,
) is either %
is the set of active
is the current time of the sequence and the set )
moments
of . It will be
convenient for the proof theory also to allow for variable labels
#
, which are active at only.
, , . Let "! # .
3. An atomic label has the form
4. Complex labels are defined by induction:
(i) Atomic labels are complex.
"( ( and ') ) are complex labels, then "( ) '( ) is a complex la(ii) If
bel. Assuming $% % ! and $& & ! are defined, let $% & '% & !!
(% % !*) $& & ! .
,+-/.102+ gives the times in which the data with these labels are active.
D EFINITION 3.2 (Databases)
& &$ $ $ , where
1. A database at time has the form # #
# is built up from atomic labels # , # and # is a wff of . Note
! !43
#
is and
that ,
, are the inputs at stage . We assume the starting time of
#
the end time is . The labels are all pairwise different.
2. We say
is consistent if for each ! % ! , 3 %$ $ $ 3 ) , the set ) )
is active
at m is consistent in with the integrity constraints of .
65
3. For the initial database contains integrity constraints.
are always
active and
17 They
87 , where
7
we shall see later that it is convenient to label them by
is a special
symbol, the empty label.
6 Controlled Revision - An algorithmic approach for belief revision
D EFINITION 3.3 (Input and revision)
&$ $ $ be all the new input formulas.
Let be
a
consistent
database
and
let
%$ $ $ , ! ! ! $ $ $! !
Let , be a consistent selection
&$ $ $ from
. The selection is done by our revision policy to
be described in a
later section (see beginning of Section 4). We define a new database as follows ( depends on ):
' &$ $ $ ' &$ $ $ where the following holds:
atom from and
if is in and is a new
3
otherwise, for
atoms
are pairwise
3 , ."All thenew
different.
2. For
each
,
if
and
is in % "
and
depends
otherwise. Note that depends on
1.
)
!
!
%
!
&
)
!
!
!
)
)
)
)
on our controlled revision policy.
D EFINITION 3.4 (Hypothetical input)
follows at stage .
Suppose we want to prove from it that
Let be a database.
This
that
can be proved from all the data which are active at . To show
means
, we need to add (input) A into the database and prove . We therefore need to define
# (
the hypothetical
input , using a new variable label to name the input . We let
# be defined as:
( It is convenient to introduce variables
#
%$ $ $ which are active only at . Thus we can write:
'(
# and thus we need not to know what is the stage of .
#
R EMARK 3.5
1. Note
the result of the hypothetical input of Definition
3.4 may be inconsistent.
Given
# that
( and a variable
#
simply puts in
with
a
new
name
label
active
at
. So
the set of all wffs active at may now be inconsistent.
2. This same procedure
be used
for testing inconsistency. Suppose we are at stage %may
&
$
%
$
&
$
and new inputs come to us. How can we tell whether they are consistent
making them all active
with ? We can put them in with new test labels at time and using the proof theory to test for consistency.
If they are consistent we can
input them properly. Otherwise we define using our revision algorithm. All of this
will be explained later.
D EFINITION 3.6 ( rules)
Let be a database.
We want to define the notion of . First we need
depends. Assume to define the
elimination and introduction
rules
on
which
and with compatibility relation . Assumeisana
labelling algebra for the language
with
(( "(
! '( and a binary function . Then
Exit function on the algebra the elimination and introduction rules for this algebra are as follows:
Controlled Revision - An algorithmic approach for belief revision 7
" " ' ( ( and prove
# , the Exit
In our example,
since
our
labelling
is a resource labelling and
"
(
!
'
(
!
"
(
(
(
function
is Exit
# ! # which
is equal
(by definition) the result of deleting
!
"
(
!
#
# 1
from
. Similarly Exit
. Assuming the above, here are the rules
! " ( : '( To show ! for a label
(
, assume ( and
'(
! for
'( a new
. variable
depending on such that Exit
for our Controlled Revision:
# ' " #
' # says that both are active at .
: To show , assume "( with new variables (
"! '( $ # with ( "( ! "( and # .
Where #
#
#
#
#
#
and
#
and prove
D EFINITION 3.7 (Proof Theory)
%
We now specify when (at a certain stage ) a set of wffs proves
: with maximal
number of applications of -introduction and maximal 7 -eliminations.
For a wff or .
87 with the
We give
the
integrity
constraints
in
the empty labels
7 17 . The members of * are always active. We useunderstanding
that and
the notation
5
(for
:
1.
#
: 6 #
if
)
and
#
is active at # .
and both
" .
hold, then ! '
!
"
(
(
"
"( 3. -introduction rule: if (
'( where
are new variables and we assume and hold as
2.
#
-elimination rule: If
" and #
)
#
#
)
#
#
)
)
)
)
#
#
#
#
#
required.
The logic we have defined so far is a logic with only
and . However, the mechanism
of Controlled Revision that we will soon define is applicable to any logic defined by LDS.
! " 1 The
1.
2.
3.
deletion process can be formally defined as follows. Let
. Then:
denote the empty label, satisfying
,
8 Controlled Revision - An algorithmic approach for belief revision
D EFINITION 3.8 (Inconsistent Database) Let
. We say that
# active at some
%
be
a set of labels
we have: ) with
.2
is
-inconsistent if for some
Assume we are at stage and we have a consistent theory . We get an input . What
do we do? First we perform
a consistency check: We insert the input as
. is
an
label at and a new temporary name for it. Call the result active
. Because
of
future
references
to
this
construction,
we
shall
call
it
temporary
input
into , forming , to test for consistency. See item 2 of Remark
of a formula 3.5. If this is consistent, then we proceed
as in Definition
3.3, with equal the set of
active elements of together with (of course will get a new
label
in as
is inconsistent, this means specified
in
Definition
3.3).
If
for
various
. The labels tell us the possible candidate
subsets of to be rejected (i.e. to be
made inactive in the newly revised ) in order to make not derivable. The task of the
Controlled Revision algorithm is to select the best option. We can use the labels to identify
and choose these options. For simplicity assume inconsistency means proving with any
label.
First let us note that it is possible to have different proofs for a formula . For example,
might be derived by modus ponens from different formulas, and it might
also be obtained as
a past input. Labels contain all such information: If the same formula can be proved with
different complex labels, this means that it is proved from possibly different data
and/or in
different ways. Rejecting one of these proofs may not block other derivations of , in which
the de-activated data do not appear. The labels used in Controlled Revision keep track of the
history of each proof. Suppose, for example, that at a certain
stage , can be obtained in
) , and the result of a rule with
several ways
(
is
both
an
independent
input
with
label
, then we will have
label from and
), and let that:
)
' " )
D EFINITION 3.9 (Inconsistent set of labels)
be a set of labels active
, where
are labels
for names and
Let
atwith
are labels
for
the
history.
If
for
some
,
and
we have that
* , we say that is the set of labels showing the inconsistency.
To maintain consistency we must block all proofs indicated by these labels. This means
throwing out some of these data. The labels tell us exactly how many time and in what form
this data was used to derive . Also we know the history of the data. This information can
be used by our revision algorithm to decide which item to reject.
To this purpose we define an option set of labels as follows:3
2 Note that the notion of inconsistency so far discussed was that
is provable with any label. Definition 3.8
with only certain
allows us to refine this notion. We may consider a database inconsistent provided it proves
labels but not any label; for example from active labels which have always been active, but not from active labels
which have been inactive in the past! In fact we can make
dependent on the stage .
3 This definition calls to mind Reiter’s hitting sets [29]. See also [32] for the relation between hitting sets and
Hansson’s kernel sets.
Controlled Revision - An algorithmic approach for belief revision 9
D EFINITION
3.10 (Option sets of labels)
For
, we know
that for some , * . Let
be the set of all atomic
names from in . Let be the collection
of all sets of
. An option set for is a
labels
and such that this condition does
set such that for every
not hold for any other proper subsets of . If we deactivate all data with labels in , then all
proofs of at will be blocked.
,
3 ) %$ $ $and
are: ! In the above
example, the option sets of labels
3 %$ $ $ )
. Every option set of labels (
) identifies a set (
)
. We then have that whose
elements
are
labelled
formulas
with
names
in
, ) ) and ! ) ) .
Having chosen what set to reject from the active part of , we can now try and
reinstate some formulas rejected at previous stages. Here our Controlled Revision policy can
again have a say. We may decide to reinstate first from rejections at , then at ,
etc or we can use some other criteria on all rejected formulas at all previous stage. The next
section gives us some options.
4 Policies for inconsistency
In the discussion at the end of the previous section, we have suggested the following outline
for an algorithm.
, consider and test for consisGiven and an input into as described in Definition 3.3 for being
tency. If is
consistent,
put
then use a selection
is active in . If is not consistent,
be what
algorithm to reject some of the active elements of and let remains.
some of the previously rejected formulas using
We can now add to a reinstating
algorithm to form a new which we will now use to define as done in Definition
3.3.
We see that we need to specify two algorithms: The selection algorithm and the reinstating
algorithm. This section will give an example to illustrate some of our options for these
algorithms, as well as a proposal for such algorithms.4
It is important to emphasize that, depending on the applications, different policies can be
adopted. In the following example we will show that it is often desirable to have a combination of different pure options. The first policy we want to consider is a very common principle
in belief revision, namely, that new information receives the highest priority. This principle
affect both selection and reinstatement algorithms.
The
selection must not throw away &
$
$
$
and the reinstatement works backwards . We call this policy option the input
priority option. If we accept this policy, the record of the history of every formula gives a
priority ordering on the information in the database. We will show that this gives undesirable
(in the sense of arbitrary) results.5
If we accept this, we can reject one of two labelled conflicting
formulas
and just comparing when
just in case that
is
they first came into : we say that more recent than . To show that, using only this pure policy,
we
can
end
in
counterintuitive
for every couple of natural
revisions,
we
assume that
a3 linear ordering exists such that
3
number , such that
. Furthermore, the problem to address is how we can select a
4 Inconsistency-management
is also discussed in Gabbay and Woods [14, 13] and Woods [33].
a future paper we will show how this requires a deeper consideration of the AGM success postulate. The
success postulate is problematic and has been discussed also in [2] and [7].
5 In
10 Controlled Revision - An algorithmic approach for belief revision
revision over another when the criteria are all partial. We then need a way of combining
all partial options in a unique algorithm. Controlled Revision can do exactly this: Take into
consideration several parameters and provide a unique tool for them.
In this section we assume for simplicity to receive one input per time.
E XAMPLE 4.1
Suppose that the initial knowledge base is:
and the input stream is:
After the first input we get:
! % # , consider Given and the new input
to test for consistency. Since it is consistent we obtain:
can prove:
and therefore:
We now address the third input. Let:
"
If we accept the principle that the highest priority has to be assigned
to the most recent
3
, for every
input, we should assume a linear ordering among
labels
such
that
.
Therefore to
block
the
derivability
of
we
should
reject
and
maintain
and the new input. The labels of the history would have been updated,
the new input is provided with new labels and the new database would then become:
! ! Note that we can still formally prove , but that there is no active proof of
6 It is interesting
at !
:6
to compare this result from the perspective of a different revision policy: the compromise revision
[9]. If
gives inconsistency, we try to maintain
and all the formulas whose presence does not imply
is defined. Compromise
inconsistency (even those whose proofs depend on ), and a new consistent set
revision (which will not be used in the general model for Controlled Revision we want to introduce here, but can
become one of the policies in future developments of the model) allows us to restore consistency by rejecting an
inconsistency-causing formula
. At the same time it compromises and keeps some of the consequences derived
, namely, all those formulas that require
in their proof, but do not lead to inconsistency.
from
&
$
Controlled Revision - An algorithmic approach for belief revision 11
At the fourth stage the new input
does not causes inconsistency, so we
provide the new information of a name and we update the history labels of all the other data
in the database to move to :
! 1 is ready to receive new data.
After the inputs
and , which are consistent, we
get and . Let us write in detail:
! can prove:
Therefore,
is provable with different labels, depending on how it was proved.
says
that
was obtained independently as an input, while from
we know that
or
is rejected,
was obtained via modus ponens. In this way, if in a future stage
we can nevertheless maintain
.
To show that the input priority option (common in the literature on belief revision) is not
as such the most desirable, and why we should
into consideration the history of past
also
take
revisions, let us consider now theinput
comes
the set of data. We test for
with # . We find itinto
consistency by forming to be inconsistent, so we have
that can prove (to simplify, we don’t write the history labels):
! We have several options for removing the inconsistency:
1. Insistent input policy (which is the most natural for input priority input policies.)
. To maintain consistency four options are available: We keep the input or . Other options are to
could be rejected together with
de-activate , or so that the proof of
is blocked. Therefore,
the options for the subsets of to reject are:
! 1 12 Controlled Revision - An algorithmic approach for belief revision
We want to define a preference ordering among the different options.
intu
itively means that we prefer to refuse
than . Which revision should we select?
! we also want to maintain
contains formulas with lower priorities than the others but, since
as many data as possible, it might be preferable to reject
or
rather than
or .
It is clear then that since policies such as input priority or minimal change do not receive
a uniform treatment, it is not possible to decide on an ordering among the options. In
our algorithm parameters, such as the persistence of a formula (how long a formula has
been in the database), the number of changes +/- in the history
each formula (in other
! for ofeach
formula) determine
developments of the model [27], a degree of reliability
the best subset of data to de-activate.
Depending
on
which
set
we choose to reject,
(we adopt the principle to reinstate as much as
we may be able to reinstate or will allow us to reinstate . It is important
possible). In fact, choosing
to observe that this intuitive principle is not considered in the traditional approaches to
revision, since they do not record the history of beliefs.
2. Not-insistent input policy. Among the options
to refuse,
Revision algorithm
Controlled
considers also rejecting the new input
, so that the choice between
insistent and not-insistent input policy is automatized.
Inputs conflicting with integrity constraints
We now discuss the case where the input is inconsistent with the integrity constraints through
an example.
E XAMPLE 4.2
Let be a stage of the evolving database. There are basically two different kinds of conflicting inputs (let assume that an input cannot be itself contradictory):
(a) A new input directly conflicts with some integrity constraints in conflict is the following:
integrity
input . An example of direct
(b) A new input indirectly conflicts with some of the integrity constraints in is:
integrity input
. An example
Resolution of the Conflict
We now propose a resolution strategy for the inconsistency arising from conflicting inputs in
the above cases.
(a) When an input directly conflicts with an integrity constraint, we reject the input (it is
commonly accepted that integrity constraints have the highest priority, and do not change
over time), even if we have an insistent input policy.
Controlled Revision - An algorithmic approach for belief revision 13
(b) In the case of indirect conflict, there are other formulas involved in the inconsistency
besides the integrity constraints. Let the Controlled Revision algorithm decide what to
reject.
Going %back
$ $ $ to the general case, where is inconsistent and we have a choice of ,
, to reject. How do we choose between them? The following are some policy
3
considerations.
1. The history
was in or out
allows us to know how many times
$&$&of$% a formula
. This is an important element to evaluate what we call the persistence
of priority: The longer a formula has been in, the more reliable it is. In Example 4.1 we
have shown that the input priority relation alone gives inconclusive results, which can be
corrected by considering the whole history of a formula.7
2. The number of formulas one should give up in each option (economic policy).
3. The number of changes +/- in the history of data. It is clear that it is preferable to deactivate the data with higher number of changes, because less ‘stable’.
R EMARK 4.3
Let us suppose that one
revision and that is inconsistent
of the policies is the compromise
.
Assume
,
and
suppose
that (for some reasons) we
because of an input
want to retain . Assume that we are using the insistent input policy. Then we maintain
and look for a such that is consistent. Let assume now that does not
prove
. We could decide to keep
defining a . The label tells us
. In abyrevision
that is derived from which accepts also the compromise revision
(which we do not take into consideration in this general model for Controlled Revision), a
further consideration would then be:
5. Whether
is an earlier compromise.
5 The algorithms for
Let us suppose now that, at stage , is inconsistent. We want
to select a
the
algorithm
,3
%$ $ $ and
among the possible subsets
of , where
&$ $ $ (recall that sets of labelled formulas are determined by option sets of name
labels ). We use &$ $ $ as%$ $ the
set of indexes of . Clearly, the data in each $
are only some of the formulas in + : we then call the set of indexes in , where
. Among the there are sets containing data already in and a set with the
new input. The algorithm decides to accept or to refuse the new input, comparing with the
other available options. It is not an a priori principle, as in AGM, where the new information
always receives the highest priority.
The first step of the Controlled Revision algorithm sums up the number of stages the
in
each were in. We call this number the persistence of and we use :
7 It might be objected that such a policy can lead to counterintuitive results. Consider, for example, the case where
is active in the database since the initial stage and it has always been active until a new input is obtained
at stage . Why should we give higher priority to than to ? A possible solution is to assign a reliability
, as in [27].
degree to each formula, i.e., a real number from the interval 14 Controlled Revision - An algorithmic approach for belief revision
!
In the above example, the persistence of is ,
is ,
is while
contains the new
input (so we can assume
that,
in
virtue
of
its
‘hypothetical’
persistence
is
just ).
&$ $ $ Let be
, the algorithm rejects the such that:
In the above example, our selection algorithm would then choose to reject the new input.
If there is not a unique , the algorithm rejects the set with the lowest cardinality.8
If two options have also the same cardinality, the algorithm would then look into the history
of the data of each option and would count the number of changes +/-, keeping the option
with the least number of changes. In case two or more options are equivalent, reject one of
the ,9 or use the tree-revision, which will be introduced in the next section.
Summarizing:
1. When at stage a database receives a new input , add to with labels
and obtain .
2. If , then moves to providing the new input of appropriate labels and
updating the history labels of all the other data in the database.
3. If the system verifies that the selection algorithm is activated.
Algorithm of Controlled Revision
1. The option sets of labels are determined as defined
in Definition 3.10, and from these
.
the sets of labelled formulas such that 2. For each calculate .
.
such that 4. If there is not a unique satisfying point , refuse the set such that the index set has
the lowest cardinality .
5. If two or more have the same cardinality, reject the whose formulas have the greatest number of changes +/-.
6. If again there is not a unique such that the above condition is satisfied, reject one of
the , or use the tree-revision.
3. Among all the
de-activate the
Once the consistency is restored, the system updates the history labels and, finally, reinstates as many data as possible in as described in the following reinstatement algorithm.
Reinstatement algorithm
&$ $ $ )
Assume we have . At each stage , , we call
the set of the
) . Let
formulas which
was
rejected
from
be
an
input.
If
it
is
consistent
with ,
then is defined and
. In this case the formulas active at are
%
8 Which
!
%
!
is the economic principle that motivates most of the postulates in AGM theory.
our algorithm the arbitrary rejection of a set appears only in few cases or at the initial stages of a database.
This was one of the most serious critique to TMS where, in case of inconsistency, the process of dependencydirected backtracking could not exhibit a satisfactory control in choosing the data to de-activate. Depending on the
applications, the last step can be substituted with the non-insistent policy.
9 With
Controlled Revision - An algorithmic approach for belief revision 15
together with the formulas active at . If it inconsistent (i.e. is inconsistent), let be the selection to be rejected (we do not care what selection
was used). Recall that
&$ $ $ ).algorithm
the selection algorithm
chooses
one
of
the
sets
We
now
a consistent
set
comprised of and the set of active wffs at have
which we call .
There
is
&$&$%$& now hope to reinstate some previously rejected formulas from . Now we
can define our algorithm to reinstate. We first look at and we order the elements of according to some principle taking into account their history, for example according to when
they came as input. Now we try to reinstate the elements of one by one after checking for
as if they were new input to temporary add to the database to
consistency (by forming test for consistency). Then we proceed
with the data in and so on until
. When we
finish we get a consistent set which we will use to define 6 The tree-revision
The tree-revision is activated when the algorithm cannot find a unique satisfying the
conditions at steps , and of the selection algorithm. Imposing a linear ordering relation
on formulas (common in the literature of belief revision) sounds too strong an assumption.
Tree-revision is an attractive proposal for a revision which, in case of ‘equivalent’ , waits
for incoming inputs. Every option is then associated with a possible revision, waiting for new
inputs. The elements of such a hierarchical structure are called nodes. The root node (the
initial database ) is the node occupying the top level of the hierarchy. Underneath the root,
there can be multiple children. Every child may in turn be the parent of other children, thus
branching like a tree. Nodes without children are leaf nodes. No node can have more than
one parent.
#&$%$&$ We have previously defined a database under revision as a sequence like .
Tree-revision makes the structure more complex, since from a step multiple possible
revisions can be considered, and we want to provide each node with its own name. The
initial database is , where is the empty sequence. The databases after the -th input
are indexed by sequences of numbers % % . When comes in, if there are several
options, they become % % , % % , etc. Thus, if in our notation there is always one
option, the sequence becomes , ! , ! , ! , ... which, in our old notation
we
$ $ $ . To make the two notations similar, let would write denote
.
For
!
becomes , or even if there is no possibility of confusion.
example, !
two
and . They are consistent,
Suppose that at the first stage
we receive
inputs:
so our knowledge is then . At the next step, we
. If we choose the insistent input policy, we
receive a new piece of evidence
:
have two possible revised sets which consistently include The data in have the same history, so the algorithm cannot select a unique solution
to resolve inconsistency. In case of equivalent revision options, we can either reject one of
the candidates, or we can wait for
If %$ $ $ the
) next one or two inputs (to avoid complexity).
, we represent the
is inconsistent and there
are
option
sets
of
labels
to
revise
#%$ $ $ )
5 ! %
databases options ! 3 ! %
as branches of a tree. Each ) (
)
corresponds to a (
), i.e., reject the wffs in . We recall that each contains
a set of labels of wffs to de-activate in order to restore consistency. We call these wffs option
formulas. When a new input comes, it is added to each branch, and the revision machinery
16 Controlled Revision - An algorithmic approach for belief revision
starts checking for inconsistencies.
We explain this using the above example. Our set is:
comes as a new input and we want to accept it. Suppose that
is inconsistent and we have two options to restore consistency (maintaining the new information).
We assign a new name to the input, we make it active at stage and we consider the following
two options:
and $ $ $ $ $
$ mean that the input added at stage caused inconsistency and the algorithm
could not decide a priority
over the two option
sets. So at we have two possible revised
is rejected, while in is% and
database, one
the
other
is
.
In
the
first one
%
.
we reject
and$
are option formulas. We keep record of the branching
also in the history labels:
means
the
formula is rejected in the first branch of the second
$
step. Similarly, means the formula is active in the first branch of the second step. We
also need to specify that
a$ database ) actively prove when all the formulas involved in
%
the proof are active at .
Let us consider the following example to explain these notions.
E XAMPLE 6.1
Let us recall the scenario in [26]:
Assume that a suspect tells you that he went to the beach for swimming and assume
that you have observed that the sun was shining. Further, you firmly believe that
going to the beach for swimming when the sun is shining implies a sun tan. If you
then discover that the suspect is not tanned, there is an inconsistency to resolve. ([26],
p.57)
Let:
B = “going to the beach for swimming”
S = “the sun is shining”
T = “sun tan”
After the first step, our database would then be:
! In his example, Nebel has already in mind a prioritized base
! !
:
Then, revising
by
we will infer that the suspect is a liar.
Controlled Revision - An algorithmic approach for belief revision 17
But do the prioritized bases approaches provide a mechanism to modify the ordering after
a revision process has been run? Unfortunately, not. As noticed also in [7], the effect of a
revision operator may change over time. Friedman and Halpern also claim that a revision
operator can be defined as a function only if the language is “rich enough to express relative
degrees of strength in beliefs”. That is something we can certainly agree on, but how can the
degrees of strength in our beliefs be decided? In this example we are a detective who wants
to find the person guilty of a terrible murder. The database we are considering is only a piece
of information regarding one of the suspects (Mr X) and we are assessing his alibi. Labelling
the data as usual:
! Then:
! ), we have:
When we observe that Mr X is not tanned (
!
!
! ! In order to restore
consistency,
the
option
sets
are
. Since we have no strong evidence to the, guilt of Mr ,X, we decide
and, finally, to keep our mind open and
search for
information. Formally, this is equivalent to say
other
that in order to accept in (we are in the insistent input policy) we are
prepared to remove
, or , but we still do not know which one of these. We
accept the new input, provide it with a new name and update the labels of the other formulas
making explicit the options. We can think of this as three branches departing from , each
branch being one different option:
!
! $ $ $ $
$ $ ! $ $ $ $ ! $ $ 18 Controlled Revision - An algorithmic approach for belief revision
! by some authoritative source that the sun
After further careful searches, we are persuaded
was really shining the day of the murder. Since
is already in the database, we do not
need a new name
for it. But we want to mark that
we got again the same information as an
input at the step of the database, so we enter into the labels. This provides a way to
deal with iterated inputs of the same information . Intuitively, the repeated observation of
the same fact increases the reliability of the fact itself. Also, when an inconsistency between
two formulas (which have been active for the same length of time) arises, the one with more
occurrences of in its history will be retained. This is called repeated input policy and is a
special feature of our model. It shows also that even a “beyond controversy” [30] axiom such
as the vacuity postulate may not catch some important aspects of a revision
! ! process. The new
evidence about the fact that the sun was shining
leads
to
close
the
branch
(because
! !
of the re-activation of the option formula in :
! ! ! !
! $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ ! !
now that in talking to Mr X’s doctor we discover that he is affected with a rare
Suppose
!
skin
and,
after some
that this kind of illness prevents him
research,
we also discover
from tanning
. Let:
!
I = “skin illness”
But, when we add to our knowledge base
does
not give
inconsistency.
, and become respectively :
$ $ $ $ ! $ $ $ $ ! $ $ $ $ $ $ $ Controlled Revision - An algorithmic approach for belief revision 19
! $ $ ! $ is inconsistent:
! ! $ $ $ $
The Controlled Revision algorithm rejects
and reinstates
. The new inputs will
determining the closure of be then added only to the remaining
. The new database will be obtained providing the last input with appropriate labels
and updating the other history labels.
Whenever the Controlled Revision algorithm cannot find a unique solution for the inconsistency in , tree-revision is an alternative to random choice rejection (as in TMS). Whether
to opt for one of the option sets or to wait for new inputs, depends on factors such as the
number of . This kind of revision seems to capture a typical feature of practical reasoning
according to which, when undecided, we keep our mind open while waiting for new information. In particular, the implicit risk in refusing any of the sets responsible for inconsistency is
reflected in the idea of iterated revision. If previous revisions have some influence on future
states of belief, rejecting the wrong set might compromise future revisions. This nicely captures Gäardenfors pithy observation that information costs. Therefore, we should not only
give up the least possible information, but it should also be the right information to give up!
7 Conclusion
We have proposed an algorithmic approach to the revision of inconsistent databases. We have
shown the special role the history of past revisions plays in devising a specialized algorithm
. In future work we explore the idea that aspects of non-monotonicity arises from
for the history of each formula and from the history of past revisions. We bring the present paper
to a close with a brief explanatory sketch.
Imagine a revision policy allowing for compromise revision and reinstatement priority. At
time , the current database is . It is necessary to recognize the core part of ,
which is composed of all data except the compromise and reinstated wffs. We assume that
in cases
of inconsistency in the face of a new input, these are the first to be rejected. Write
if logically follows from . Write if (i.e. the are the
non-monotonic consequences of ).
We now check to see whether the axioms for the non-monotonic consequence relation hold.
Assume input . is updated to , in the manner
discussed in the paper. 6
The
core of
input
insistence,
we
get
.
Clearly
we
may
have
is . If we opt for
but
hence , but . However, if , this means that
,
hence that
is
consistent.
Whereupon
(
,
and
for
all
iff
(which is restricted monotonicity). This is reminiscent of the Gärdenfors and
Makinson
observation
[24]
that given a theory we can get non-monotonic consequence by
iff
.
Our conviction is that the dynamic dimension has been disregarded not only in belief revision, but also in other areas of knowledge representation. For example, one of the actual
20 Controlled Revision - An algorithmic approach for belief revision
major problems in artificial intelligence is the aggregation of several databases, i.e., the combination of pieces of knowledge from different sources. This is a distinct issue, but related,
to belief revision. In belief revision, we face the question of accepting or rejecting a new
information which can contradict our knowledge base. In the case of aggregation, we want
to merge information from different sources. The difference is that no one of these bases is
the representation of our actual theory, but we need to fuse all the bases to obtain a unique
theory (we may think of building an expert system from a group of human experts, for example). Furthermore, the final merged base is not necessarily one of the initial bases. It is
not possible to define a fusion operator from a revision one without specifying how to combine the individual preferences. To collect information from different sources often results in
opposite information from equally reliable sources. Our proposal is to develop an approach
which, in this case, waits for more information. The next inputs will inform us on which
source provides a better picture of the world, so that we can update our confidence degree on
that expert. We leave further investigation of these aspects and their application areas for a
future paper [12].
8 Acknowledgments
We want to thank David Makinson, Peter McBurney and Odinaldo Rodrigues who read a
previous version of the paper for their valuable comments and questions.
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Received November 2001