The aim of this work is to relate two apparently independent concepts as belief revision and relevance. We propose a formal characterization for the notion of (ir)relevance by means of a set of postulates. Then we define (ir)relevance using change operators and we show that the contraction operations of the AGM model can be modeled through (ir)relevance relations and vice versa.Eje: Aspectos teóricos de la inteligencia artificialRed de Universidades con Carreras en Informática (RedUNCI

Falappa, Marcelo Alejandro

Maguitman, Ana Gabriela

English

El Servicio de Difusión de la Creación Intelectual

Belief Revision and Relevance 
Marcelo Alejandro Falappa Ana Gabriela Maguitman 
Departamento de Ciencias de la COlnputación 
Grupo de Investigación en Inteligencia Artificial 
Universidad Nacional del Sur 
Avenida Alem 1253 - (8000) Bahía Blanca - Argentina 
Tel/Fax: (54)(291)4595135 
email: [mfalappa.ccmagit]@criba.edu.ar 
Abstract 
The aim of this work is to relate two apparently independent concepts as belief 
revision and relevance. We propose a formal characterization for the notion of 
(ir)relevance by means of a set of postulates. Then we define (ir)relevance using 
change operators and we show that the contraction operations of the AGM model 
can be modeled through (ir)relevance relations and vice versa. 
Keywords: Belief Revision, Relevance, Theory Change, Knowledge Representation. 
1 Introd uction 
Human cognitive process is geared to achieving as much knowledge as possible. On 
learning new information, a human wiII try to accommodate it in such a way that this 
new knowledge does not do damage to the knowledge he already has. However, in 
many cases, the task of incorporating new information to a knowledge set wiII require 
previously maintained beliefs to be discarded. In other words, when an agent learns 
new information it is possible that this information conflicts with his belief seto When a 
conflict exists, it is necessary to eliminate old beliefs in order to consistently incorporate 
the new information. On the other hand, when no conflict exists, the reasoner can 
simply add the new information to his set of beliefs. This process is known as belief 
revision. 
During the revision process the agent needs to identify the relevant pieces of infor-
rnation that should be incorporated or retracted form the old belief seto This means 
that if a belief set K is revised by a sentence Q and Q is a reason to believe in a sentence 
/3 rt K, {3 wiII be incorporated to the updated belief seto On the other hand, if a belief 
set K is revised by a sentence Q and Q conflicts with a sentence {3 E K, it will be 
necessary to retract {J. According to this, given a belief set K, there are two main 
relevanee relations in whieh two formulas, Q and {J, can be involved: 
1 
• a is positively relevant to (3 if and only if (3 does not belong to K but f3 belongs 
to K revised by a. 
• a is negatively relevant to f3 if and only if f3 belongs to K but f3 does not be long 
to J( revised by a. 
Analogously, we can identify two kinds of irrelevance relations between formulas: 
• a is positively irrelevant to f3 if and only if f3 belongs neither to K nor to K 
revised by a. 
• a is negatively irrelevant to f3 if and only if f3 belongs both to K and to J( revised 
bya. 
Tbe aboye intuitive considerations lead us to propose the following thesis: 
The notions of (ir )relevance and belief revision are interdefinable. 
In other words: 
• It is possible to build the process of belief revision on notions of (ir)relevance. 
• It is possible to identify (ir)relevance relations through the process of belief revi-
sion. 
The driving motivations for this work are, therefore, to analyze the notions of 
positive and negative (ir)relevance and to study the way in which these notions can 
be used to develop a constructive model of belief revision. Then we will show the 
connections existing between beliefrevision defined in terms ofrelevance and the AGM 
postulates. 
2 Preliminaries 
Depending on how beliefs are represented and what kinds of input s are used, different 
kinds of belief changes are possible. In the most common case beliefs are represented 
by sentences in a logical language. We will adopt a propositional language 1:.- with a 
complete set of boolean connectives: " 1\, V, ---., -. Formulre in 1:.- will be denoted 
by lowercase Greek characters: a, (3, 6, . . .. Sets of sentences in 1:.- will be denoted 
by uppercase Latin characters: A, B, e, . . .. The symbol T represents a tautology or 
truth. The symbol ..1 represents a contradiction or falsum. We also use a consequence 
operator Cn. Cn takes sets of sentences in 1:.- and produces new sets of sentences. The 
operator Cn satisfies the following three conditions: 
Inclusion: A ~ Cn(A) for any A ~ 1:.-. 
Iteration: Cn(A) = Cn(Cn(A)) for any A ~ 1:.-. 
Monotony: If A ~ B then Cn(A) ~ Cn(B) for any A, B ~ 1:.-. 
2 
Vve will assurne that tIte consequence operator includes classical consequences and 
verifies the standard properties of deduction and cornpactness: 
Supraclassicality: If a can be derived frarn A by deduction in classical logic, then 
a E Cn(A). 
Deduction: (3 E Cn(A U {Q}) if and only if (a - (3) E Cn(A). 
Compactness: If a E Cn(A) then a: E Cn(.4') fol' sorne finite subset A' of A. 
To sirnplify notation, we write Cn(o) fOl' Cn({o}) where a is any sentence in J:..,. We 
also write a E Cn(A) as A f- o. The episternic states will be represented by belief 
set, i.e., set of sentences closed under logical consequence. This rneans that K is a 
belief set if and only if K = Cn(K). Usually, belief sets will be denoted by boldface 
uppercase Latin characters. 
3 Recalling Belief Revision 
In the AGM approach the episternic states are represented by belief sets. Let K be a 
belief set and Q a sentence in J:..,. The three rnain kinds of changes are the following 
[Ga92]: 
Expansion: A new sentence is added to an episternic state regardless of the 
consequences of the so forrned larger seto If + is an expansion operator, then 
K+o denotes the belief set K expanded byo. 
Contraction: Sorne sentence in the episternic state is retracted without adding 
any new belief. If - is a contraction operator, then K-a denotes the belief set 
K contracted by O. 
Revision: A new sentence is consistently added to an episternic state. In order to 
rnake possible this operation, sorne sentences rnay be retracted frorn the original 
episternic state. If * is a revision operator, then K*o denotes the belief set K 
revised by O. 
E>""Pansions can sirnply be defined as the logical closure of K and Q: 
K+o = Cn(K U {a}) 
It is not possible to give a similar explicit definition of contractions and revisions in 
logical and set-theoretical notions only. These operations can be defined using logical 
notions and sorne selection rnechanisrn. Contractions and Revisions are interdefinable 
by the following identities: 
Levi Identity: K*o = (K--¡a)+o. 
Harper Identity: K-a = K n K*-¡a. 
By giving a definition of one of these operators we can obtain the other one using the 
aboye identities. In this work we will show the relation existing between the notion of 
(ir )relevance and the contraction operator. 
3 
3.1 Postulates for Contractions 
Gardenfors [Ga88] proposed the following rationality postulates for contraction opera-
tors: 
(K-l) Closure: For every belief set K and every sentence a, K-a is a belief seto 
(K-2) Inclusion: K-a ~ K. 
(K-3) Vacuity: If a rt K then K-a = K. 
(K-4) Success: If}L a then a rt K-a. 
(K-5) Recovery: K ~ (K-a)+a. 
(K-6) Extensionality: If 1- a f-t (3 then K-a = K-(3. 
(K-7) Conjunctive Overlap: K-a n K-(3 ~ K-(a 1\ (3). 
(K-S) Conjunctive Inclusion: If a rt K-(a 1\ (3) then K-(a 1\ (3) ~ K-a. 
The following Lemma is a straight consequence of Levi Identity. 
LEMMA 3.1 Given a belief set K and a formula a E Á..J, K-.a ~ K*a. • 
Consider the following postulate for contraction: 
(K-F) Conjunctive Factoring: K-(a 1\ (3) = K-a, or K-(a 1\ (3) 
K-(a 1\ (3) = K-a n K-(3. 
K-(3, or 
The following Lemma states that given postulates (K-l) to (K-6), postulate (K-F) 
is equivalent to the conjunction of postulates (K-7) and (K-S). 
LEMMA 3.2 [AGMS5] If a contraction operator satisfies (K-l) .. (K-6) then it 
satisfies (K-7) and (K-S) if and only if it satisfies (K-F). • 
4 Our approach to (Ir)relevance 
Relevance is a conceptual relation. A piece of information is said to be relevant if it is 
"of importance, of worth, of consequence", "bearing upon, connected with, pertinent 
to the matter in hand."¡ There are many different proposals aimed to characterize the 
notion of relevance in logic systems. For a detailed analysis of different approaches to 
relevance that appear in the literature see [Ma97]. 
We propose a characterization of the notion of relevance as a metalinguistic relation 
between formulas of a propositional language Á..J. In our approach relevance is seen as 
an "interference relation" between two formulas, in the context of a belief seto We will 
take the notion of negative irrelevance as a primitive relation. This notion is formalized 
by the following set of postulates. 
1 The Oxford English Dictionary. 
4 
4.1 Postulates for Irrelevance Relations 
(Irrl) If -'all~a then f- a. 
(Irr2) If all~¡3 and all~(¡3 --+ ó) then all~ó. 
(1 rr3) If f- (a ~ 13) then all~ó if and only if ¡3ll~ó for aH Ó E 1:.-. 
(Irr4) If f- 13 then all~p for al! a E J..,. 
(Irrs) If a 5t K then -'all~¡3 for aH 13 E K. 
(Irr6) If 13 E K then all~(a V 13) for aH a E 1:.-. 
(1 rr7) If aTI~ó and ¡3ll~ó then (a V ¡3)ll~ó. 
(Irra) If (a V ¡3)ll~ó then all~ó or ¡3ll~Ó. 
Postulate (lrrl) establishes that if the negation of a formula a does not interfere 
with a, then a is logically true. Postulate (Irr2) stands for the condition of modus 
ponens in the consequent. The irrelevance of the syntax condition is valid for negative 
irrelevance and this is represented by postulate (Irr3). According to postulate (Irr4) 
no formula of the language can interfere wlth a theorem. Postulate (Irrs) establishes 
that if a formula a is not in the belief set, the negation of a cannot interfere with 
any formula that belongs to the belief seto Vve will assume that a formula a cannot 
interfere with the formula that results from the disjunction of O: and any other formula 
of the belief seto This last condition is represented by postulate (Irr6). According to 
postulate (lrr7), if neither a nor 13 interferes \vith Ó, then the formula that results from 
the disjunction of a and 13 cannot interfere with ó. Final!y, postulate (Irra) establishes 
that if the formula a V 13 do es not interfere with ó then, either a does not interfere 
with ó or 13 does not interfere with ó. 
5 From Belief Revision to (Ir) relevance 
'TVe have already proposed informal definitions of (ir )relevance in terms of belief revi-
sion. The following formalizes those definitions: 
(Def llk) aTIk¡3 = {¡3 13 5t K and 13 5t K*a}. 
(Def ll~) all~¡3 = {¡3 13 E K and 13 E K*a}. 
(Def lRk) alRk¡3 = {¡3 13 5t K and 13 E K*a}. 
(Def lR~) alR~¡3 = {¡3 13 E K and 13 5t K*a}. 
The following schemas are consequences of the definitions presented aboye. Ac-
cording to them, positive irrelevance, positive relevance and negative relevance can be 
described in terms of negative irrelevance: 
5 
(Schema Rk) oJI~¡J if and on1y if (3 rf- K ana it is not the case that aTI~(3 holds. 
(Schema lRk) alR~(3 if and only if (3 rf- K and it is not the case that aTI~¡J holds. 
(Schema lR~) alR~(3 if and only if (3 E K and it is not the case that aTI~(3 holds. 
The following Lemma is a straight consequence of (Def TI~) and Harper Identity. 
LEMMA 5.1 Civen a belief set K and a pair of formulas a,(3 E.e, aTI~rj if and only 
if ¡J E K--'a. • 
LEMMA 5.2 If the postulates for contraction are satisfied then the postulates for 
irrelevance relations are satisfied. 
PROOF IN THE ApPENDIX. • 
6 From (Ir) relevance to Belief Revision 
Now we define ACM change operators in terms of relevance and irrelevance notions: 
(Def K+a) K+a = {(3 K U {a} f- (3}. 
(Def K-a) K-a = {(3 -'aTI~(3}. 
(Def K*a) K*a = {(3 aTI~(3 or alR~(3}. 
LEMMA 6.1 If the postulates for irrelevance relations are satisfied then the postu-
lates for contraction are satisfied. 
PROOF IN THE ApPENDIX. • 
7 Concluding Result 
The following theorem establishes the interrelation between the notion of irrelevance 
and the ACM contraction operators. 
THEOREM 7.1 The postulates for irrelevance relations (Irr¡) .. (lrra) are satisfied if 
and only if the postulates for contraction (K-l) .. (K-S) are satisfied. 
PROOF IN THE ApPENDIX. • 
8 Conclusion 
We have proposed a formal characterization for the notion of (ir )relevance. An intuitive 
relation between the process of contraction and the notion of negative irrelevance has 
been identified. This led us to show that the notions of (ir )relevance and belief revision 
are interdefinable. 
6 
9 Appendix: Proofs 
LEMMA 5.2. If the postulates for contraction are satisfied then the postulates for 
irrelevance relations are satisfied. 
Proof. Suppose that tlle set of postulates for contraction are satisfied. \Ve have to 
sllo\V that the set of postulates for irrelevance relations are satisfied. 
(Irr¡) Assume -'arr~a. \Ve llave to prove f- a. According to Lemma 5.1 -'arr~a is 
equivalent to a E K-a which, by (K-4), implies f- a. This concludes our proof. 
(Irr2) Assume arr~¡3 and alI~(¡3 --+ 8). VVe want to prove arr~8. By Lemma 5.1, from 
arr~¡3 we llave ¡3 E K--,a, and from arr~(¡3 --+ 8) we have ¡3 --+ 8 E K--'a. Then, 
by (K-l), \Ve have 8 E K--,a which, by Lemma 5.1, is equivalent to arr~8. 
(1 rr3) Suppose tllat f- (a f-t 11). \Ve llave to prove arr~8 if and only if ¡31I~8 for aH 
8 E ,c. From f- (a f-t 11) we have f- (-,a f-t -,¡3). Therefore, by (K-6), we have 
K--,a = K--,¡3, which according to Lemma 5.1 is equivalent to arr~8 if and 
only if ¡3rr~8, and we are done. 
(Irr4) We have to prove that if f- ¡3 then arr~¡3 for all a E'c. From f- ¡3, by (K-l), we 
have ¡3 E K-8 for aH 8 E'c. This implies, by Lemma 5.1, -,8rr~¡3 for aH 8 E ,c, 
which is equivalent to alI~¡3 for all a E ,c, and we are done. 
(Irrs) This postulate states that if a rJ K then -'arr~¡3 for all ¡3 E K. From a rJ K it 
follows by (K-3) that K-a = K. Let ¡3 E K. Then ¡3 E K-a. According to 
Lemma 5.1 this results in -'alI~¡3. 
(Irr6) Suppose ¡3 E K. We have to prove arr~(a V 11) for all a E'c. If ¡3 E K, it follO\vs 
by (K-5) that for any 8 E ,c we have ¡3 E (I( -8)+8. Then ¡3 E Cn(I( -8 U {8}) if 
and only if (8 --+ 11) E Cn(K-8) By (K-l) this is equivalent to (8 --+ ¡3) E K-b. 
Again, by (K-l), (-,8vl1) E K-8. Hence, accordingto Lemma5.1, -,81I~(-,8 V 11) 
for all 8 E ,c. In particular for 8 = -,a we llave arr~ (a V /3) for aH a E ,c. 
(1 rr7) Suppose that arr~8 and ¡31I~8. vVe have to prove (a V ¡3)1I~8. It fo11ows from alI~8 
and ¡31I~8, by Lemma 5.1, that 8 E K--,a and 8 E K--,¡3. Then, by (K-7), 
8 E K-(-,a 1\ -,¡3). Since f- (-,a 1\ -,11) f-t -,(a V 11), by (K-6), 8 E K--,(a V 13). 
But according to Lemma 5.1 this is equivalent to (aV ¡3)1I~8, which is the desired 
resulto 
(1 rra) We have to show that from (a V I1)rr~8 we can conclude alI~b or ¡3][~b. It follows 
from (a V ¡3)rr~b, by Lemma 5.1, that 8 E K--,(a V ¡3) = K-(-,a 1\ -,¡3). It 
follows, by (K-F) that at least one of the following cases holds: 
• Case 1: 8 E K--,a, or 
• Case 2: 8 E K--,¡3, or 
• Case 3: 8 E K--,a n K--,/3. 
7 
According to Lemma 5.1, if case 1 holds we can conclude that an~b. ATlalogously, 
if case 2 holds we conclude ¡3n~b. Finally, if case 3 holds both an~b and ,Bn~b, 
hold. This finishes our proof. 
• 
LEMMA 6.1. If the postulates for irrelevance relations are satisfied then the postu-
lates for contraction are satisfiecl. 
Proof. Suppose that the set of postulates for irrelevance relations are satisfied. Vve 
have to show that the set of postulates for contraction are satisfied. 
(K-l) \Ve have to prove that for every belief set K and every sentence a, K-a is a 
belief set, i. e., we have to prove 
1. if f- f3 then f3 E K-a, and 
2. K -a is closed under implications. 
For part 1, if f- f3 \ye have by (Irr4) and Lemma 5.1, that f3 E K----,b for all b E 1:..,. 
In particular, for a = ---,b we have f3 E K-a, whieh is the desired resulto 
For part 2 we need to show that if,B E K-a and,B --+ b E K-a then b E K-a. 
By Lemma 5.1, from,B E K-a we have ---'an~¡3, and from f3 --+ b E K-a we 
have ---'an~,B --+ b. Then , by postulate (Irr2) we can conclude ---'an~b. Hence, by 
Lemma 5.1, b E K-a, and we are done. 
(K-2) Suppose that ,B E K-a. We want to prove ,B E K. But from f3 E K-a, by 
Lemma 5.1, we have that ---'an~f3, and by (Def n~), we have f3 E K. 
(K-3) Suppose that a fj K. Then, by (Irrs), we have ---'an~f3 for all f3 E K, and by 
Lemma 5.1, we can conclude K-a = K. 
(K-4) Suppose that J.L a. It follows by (Irrl) that it is not the case that ---'an~a holds. 
Then, according to Lemma 5.1, we can condude a fj K-a, which is the desired 
resulto 
(K-5) Suppose that b E K. Then, by (Irr6), we have f3n~(,B V b) for all ,B E 1:..,. 
In particular, for f3 = ---,a we have ---'an~(---'a V b). Then, by Lemma 5.1, we 
have that K-a f- -:oa V b, which is equivalent to K-a f- a --+ b. Therefore, 
(K-a) U {a} f- b, and then, by (Def K+a), we can condude ,B E (K-a)+a. 
(K-6) We have to show that from f- a +-+ ,B we can conclude K-a = K-,B. From 
f- a +-+ f3 we have, f- ---,a +-+ ---,f3. Then, by (Irr3), we have ---'an~b if and only if 
---'f3n~b for all bE 1:.." which by Lemma 5.1 is equivalent to K-a = K-¡3. 
(K-7) We have to show that if b E K-a and b E K-,B then b E K-(a 1\ /3). By 
Lemma 5.1, we have from b E K-a and 8 E K-¡3, ---'an~8 and ---'an~8, respec-
tively. Then, by (Irr7), we have that (---'aV---'f3)n~b, Since f- (---,av---,f3) +-+ ---'(al\¡3), 
by (Irr3), ---,(a 1\ (3)n~8. Hence, by Lemma 5.1, we obtain 8 E K-(a 1\ (3), which 
is the desired resulto 
8 
(K-S) \Ve know by Lemma 3.2 that ifpostulates (K-1) to (K-6) are valid and if (K-F) 
is also valid, we can conclude (K-S) is valido Therefore, it is enough to show that 
(K-F) is valido Postulate (K-F) states that at least one of the following cases 
holds: 
• Case 1: K-(Q!\,B) = K-Q, or 
• Case 2: K-(Q!\,B) = K-,B, or 
• Case 3: K-(Q!\,B) = K-Q n K-,B. 
According to Lemma 5.1 to prove that at least one of thA aboye cases holds is 
equivalent to prove that at least one of the following cases holds: 
• Case 1: (,Q V ,,B)n~ó if and only if ,Qn~Ó, or 
• Case 2: (,Q V ,,B)JI~ó if and only if ,,BJI~ó, or 
• Case 3: (,Q V ,j3)JI~ó if and only if ,Qn~Ó and ,,BJI~ó. 
After sorne algebraic manipulations we can conclude that proving that at least 
one of the above cases holds is equivalent to prove that the following two eases 
simultaneously hold: 
• Case a: ,Qn~Ó and ,,8JI~ó implies (,Q V ,,B)JI~ó. 
• Case b: (,Q V ,,8)JI~ó implies ,QJI~ó or ,,8JI~ó. 
It is easy to see that case a is equivalent to (Irr7), while case b is equivalent to 
(Irrs). This concludes our proof. 
• 
THEOREM 7.1. The postulates for irrelevance relations (lrrl) .. (lrrs) are satisfied if 
and only if the postulates for contraction (K-1) .. (K-S) are satisfied. 
Proof. Straightforward from Lemma 5.2 and Lemma 6.1. 
• 
References 
[AGM85] Carlos Alchourrón, Peter Gardenfors, and David Makinson. On the Logic oE 
Theory Change: Partial Meet Contraction and Revision Functions. The Journal 
of Symbolic Logic, 50:510-530, 1985. 
[Ga88] Peter Gardenfors. Knowledge in Flux: Modeling the Dynamics oE Epistemic 
States. The MIT Press, Brndford Books, Cambridge, Massachusetts, 1988. 
[Ga92] Peter Gardenfors and Hans Rott. BelieE Revision. Lund University Congnitive 
Studies, 1992. 
[Ma97] Ana G. Maguitman. Formalización y uso del concepto de relevancia en sistemas 
lógicos. Tesis de Magister en Ciencias de la Computación. Universidad Nacional 
del Sur, 1997. 
9 


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