Decrement Operators in Belief Change
Abstract
While research on iterated revision is predominant in the field of iterated belief change, the class of iterated contraction operators received more attention in recent years. In this article, we examine a nonprioritized generalisation of iterated contraction. In particular, the class of weak decrement operators is introduced, which are operators that by multiple steps achieve the same as a contraction. Inspired by Darwiche and Pearl’s work on iterated revision the subclass of decrement operators is defined. For both, decrement and weak decrement operators, postulates are presented and for each of them a representation theorem in the framework of total preorders is given. Furthermore, we present two types of decrement operators which have a unique representative.
Keywords:
belief revision, belief contraction, nonprioritized change, gradual change, forgetting, decrement operator1 Introduction
Changing beliefs in a rational way in the light of new information is one of the core abilities of an agent  and thus one of the main concerns of artificial intelligence. The established AGM theory [1] deals with desirable properties of rational belief change. The AGM approach provides properties for different types of belief changes. If new beliefs are incorporated into an agent’s beliefs while maintaining consistency, this is called a revision. Expansion adds a belief unquestioned to an agent’s beliefs, and contraction removes a belief from an agent’s beliefs. Building upon the characterisations of these kinds of changes and the underlying principle of minimal change, the theory fanned out in different directions and subfields.
The field of iterated belief revision examines the properties of belief revision operators which, due to their nature, can be applied iteratively. In this subfield, one of the most influential articles is the seminal paper [7] by Darwiche and Pearl (DP), establishing the insight that belief sets are not a sufficient representation for iterated belief revision. An agent has to encode more information about her belief change strategy into her epistemic state  where the revision strategy deeply corresponds with conditional beliefs. This requires additional postulates that guarantee intended behaviour in forthcoming changes. The common way of encoding, also established by Darwiche and Pearl [7], is an extension of Katsuno and Mendelzon’s characterisation of AGM revision in terms of plausibility orderings [10], where it is assumed that the epistemic states contain an order of the worlds (or interpretations).
Similar work has been done in recent years for iterated contraction. Chopra, Ghose, Meyer and Wong [6] contributed postulates for contraction on epistemic states. Caridroit, Konieczny and Marquis [4] provided postulates for iterated contraction on epistemic states and a characterisation with plausibility orders in the style of Katsuno and Mendelzon. By this characterisation, the main characteristic of a contraction with is that the worlds of the previous state remain plausible and that the most plausible countermodels of become plausible.
However, in the subfield of nonprioritised belief change, or more specifically, in the field of gradual belief change much work remains to be done on contraction. An important generalisation of iterated revision operators are the class of improvement operators by Konieczny and Pino Pérez [12], which achieve the state of an revision by multiple steps in a gradual way. These kind of changes where intensively studied by Konieczny, Pino Pérez, Booth, Fermé and Grespan [11, 3]. A counterpart of improvement operators for the case of contraction is missing. This article fills this gap. We investigate the contraction analogon to improvement operators, which we call decrement operators. The leading idea is to examine a class of operators which lead, after enough consecutive applications, to the same states as an (iterative) contraction would do.
The research presented in this paper is also motivated by the quest for a formalisation of forgetting operators within the field of knowledge representation and reasoning (KRR). In a recent survey article by Eiter and KernIsberner [8] the connection between contraction and forgetting of a belief is dealt with from a KRR point of view. Steps towards a general framework for kinds of forgetting in commonsense based belief management, revealing links to wellknown KRR methods, are taken in [2]. However, for the fading out of rarely used beliefs that takes places in humans gradually over time, or for the change of routines, e.g. in established workflows, often requiring many iterations and the intentional forgetting of the previous routines, counterparts in the formal methods of KRR are missing. With our work on decrement operators, we provide some basic building blocks that may prove useful for developing a formalisation of these psychologically inspired forgetting operations.
In summary, the main contributions of this paper are^{1}^{1}1The full proofs for all theorems given here can be found in the appendix.:

Postulates for operators which allow to perform contractions gradually.

Representation theorems for these classes in the framework or epistemic states and total preorders.

Two types of decrement operators with unique representatives.
The rest of the paper is organised as follows. Section 2 briefly presents the required background on belief change. Section 3 introduces the main idea and the postulates along with a representation theorem for weak decrement operators. In Section 4 the weak decrement operators are restricted by DPlike iteration postulates, leading to the class of decrement operators; we give also a representation theorem for the class of decrement operators. In Section 5 two special types of decrement operators are specified. We close the paper with a discussion and point out future work in Section 6.
2 Background
Let be a propositional signature (non empty finite set of propositional variables). The propositional language is the smallest set, such that for every and , if . We omit often and write instead of . We write formulas in with lower Greek letters , propositional variables with lower case letters , abbreviate by and sometimes write for . The set of the usual propositional interpretations is identified with the set of corresponding complete conjunctions, also called worlds, over . Propositional entailment is denoted by , and is the deductive closure of . This is lifted to a set by defining . For a set of worlds and a total preorder (reflexive and transitive relation) over , we denote with the set of all worlds in the lowest layer of that are elements in . For a total preorder , we denote with its strict variant, i.e. iff and ; with the direct successor variant, i.e. iff and there is no such that ; and we write iff and .
2.1 Epistemic States and Belief Changes
Every agent is equipped with an epistemic state, sometimes also called belief state, that maintains all necessary information for her belief apparatus. With we denote the set of all epistemic states. Without defining what a epistemic state is, we assume that for every epistemic state we can obtain the set of plausible sentences of , which is deductively closed. For the purpose of this article, the set can be represented by a single propositional formula and we will switch between these two representations as needed. We write iff and we define . A belief change operator over is a (leftassociative) function . We denote with the ntimes application of by to [12].
Darwiche and Pearl [7] propose that an epistemic state should be equipped with an ordering of the worlds (interpretations), where the compatibility with is ensured by the socalled faithfulness. A mapping is called faithful assignment if the following is satisfied [10]:
Konieczny and Pino Pérez give a stronger variant of faithful assignments for iterated belief change [12], which ensures that the mapping is compatible with the belief change operator with respect to syntax independence.
Definition 1 (Strong Faithful Assignment [12])
Let be a belief change operator. A function that maps each epistemic state to a total preorder on interpretations is said to be a strong faithful assigment with respect to if:
(SFA1)  
(SFA2)  
(SFA3) 
We will make use of strong faithful assignments for the characterisation of decrement operators.
2.2 Iterated Contraction
Postulates for AGM contraction in the framework of epistemic states where given by Chopra, Ghose, Meyer and Wong [6], and by Caridroit, Konieczny and Marquis [4] for propositional formula. We give here the formulation by Chropra et al. [6]:
(C1)  
(C2)  
(C3)  
(C4)  
(C5)  
(C6)  
(C7) 
For an explanation of these postulates we refer to the article of Caridroit et al. [4], where also a characterisation in terms of total preorders is given.
Proposition 1 (AGM Contraction for Epistemic State [4])
In addition to the postulates (C1) to (C7), Konieczny and Pino Pérez give DPlike postulates for intended iteration behaviour of contraction [13]. In the following, we call these class of operators iterated contraction operators, which are characterized by the following proposition.
Proposition 2 (Iterated Contraction[13])
A belief change operator is an iterated contraction operator if and only there exists a faithful assignment such that the following is satisfied:
2.3 Improvement Operators
The idea of (weak) improvements is to split the process of an AGM revision for epistemic states [7, p. 7ff] into multiple steps of an operator . For such a gradual operator define , where is smallest integer such that . In the initial paper about improvement operators [12], Konieczny and Pino Pérez gave postulates for , such that is an AGM revision for epistemic states. Due to space reasons, we refer the interested reader to the original paper for the postulates [12]. The following representation theorem gives an impression on weak improvement operators.
Proposition 3 (Weak Improvement Operator[12, Thm. 1])
A belief change operator is a weak improvement operator if and only if there exists a strong faithful assignment such that:
Furthermore, the class of weak improvement operators is restricted by DPlike iteration postulates [12] to an unique operator. Again, we refer to the work of Konieczny and Pino Pérez [12] for these postulates, and only present the characterisation in the framework of total preorders.
Proposition 4 (Improvement Operator[12, Thm. 2])
A weak improvement operator is an improvement operator if and only there exists a strong faithful assignment which fulfils
(S1)  
(S2)  
(S3)  
(S4)  
(S5) 
and satisfies:
In the following section we use the basic ideas of (weak) improvement operators as a starting point for developing the weak decrement operators.
3 Weak Decrement Operators
A property of a contraction operator is that the success condition of contraction is instantaneously achieved, i.e., if is believed in a state () then after the contraction with , it is not believed any more (). As a generalisation, we define hesitant contractions as operators who achieve the success condition of contraction after multiple consecutive applications.
Definition 2
A belief change operator is called a hesitant contraction operator if the following postulates are fulfilled:
(hesitant success) 
If is an hesitant contraction operator, then we define a corresponding operator by , where if , otherwise is the smallest integer such that .
The following Example 1 shows an modelling application for hesitant belief change operators.
Example 1
Addison bought a new mobile with much easier handling. She does no longer have to press a sequence of buttons to access her favourite application. However, it takes multiple changes of her epistemic state before she contracts the belief of having to press the sequence of buttons for her favourite application.
We now introduce weak decrement operators, which fulfil AGMlike contraction postulates, adapted for the decrement of beliefs.
Definition 3 (Weak Decrement Operator)
A belief change operator is called a weak decrement operator if the following postulates are fulfilled:
(D1)  
(D2)  
(D3)  
(D4)  
(D5)  
(D6)  
(D7) 
The postulates (D1) to (D7) correspond to the postulates (C1) to (C7). By (D1) a weak decrement does not add new beliefs, and together with (D2) the beliefs of an agent are not changed if is not believed priorly. (D3) ensures that after enough consecutive application a belief is removed. (D4) is the recovery postulate, stating that removing and then adding again recovers all initial beliefs. The postulate (D5) ensures syntax independence in the case of iteration. (D6) and (D7) state that a contraction of a conjunctive belief is constrained by the results of the contractions with each of the conjuncts alone.
For the class of weak decrement operators the following representation theorem holds:
Theorem 3.1 (Representation Theorem for Weak Decrement Operators)
Let be a belief change operator. Then the following items are equivalent:

is a weak decrement operator

there exists a strong faithful assignment with respect to such that:
(decrement success)
From Theorem 3.1 we easily get the following corollary:
Corollary 1
4 Decrement Operators
We now introduce an ordering on the formulas in order to shorten our notion in the following postulates.
Definition 4
Let be a hesitant change operator, then we define for every epistemic state and every two formula :
With we denote the strict variant of and define if and there is no such that .
Intuitively means that in the state the agent is more willing to remove the belief than the belief .
For the iteration of decrement operators we give the following postulates:
(D8) 
(D8) states that a prior decrement with does not influence the beliefs of an decrement with if .
(D9) 
(D9) states that a prior decrement with does not influence the beliefs of an decrement with if .
(D10) 
The postulate (D10) states that if a belief in is believed after a decrement of and the removal of , then only a removal of does not influence the belief in if implies .
(D11) 
By (D11), if and do not share anything, then a decrease of does not influence this belief.
(D12) 
By (D12), if in the state the agent prefers removing a consequence of minimally more than removing a consequence of , then after a decrement of , she is more willing to remove the consequence of .
(D13) 
(D13) axiomatically enforces that a single step does not add any beliefs.
We call operators that fulfil these postulates decrement operators.
Definition 5 (Decrement Operator)
On the semantic side, we define a specific form of strong faithful assignment which implements decrementing on total preorders.
Definition 6 (Decreasing Assignment)
Let be a hesitant belief change operator. A strong faithful assignment with respect to is said to be a decreasing assignment (with respect to ) if the following postulates are satisfied:
(DR8)  
(DR9)  
(DR10)  
(DR11)  
(DR12)  
(DR13) 
The postulates (DR8) to (DR11) are the same as given by Konieczny and Pino Pérez [13] for iterated contraction (cf. Proposition 2). The postulate (DR12) states that a world of which is minimally less plausible than a world of should be made at least as plausible as this world of . (DR13) ensures that (together with the other postulates) that world in stays plausible after a decrement.
The main result is that decrement operators are exactly those which are compatible with a decreasing assignment.
Theorem 4.1 (Representation Theorem for Decrement Operators)
Let be a belief change operator. Then the following items are equivalent:

is a decrement operator

there exists a decreasing assignment with respect to that satisfies (decrement success), i.e.:
The following proposition presents a nice property of decrement operators: Like AGM contraction for epistemic sates (cf. Proposition 1) a decrement operators keeps plausible worlds; and only the least unplausible counterworlds may become plausible.
Proposition 5
Let be a hesitant belief change operator. If there exists a decreasing assignment with respect to , then we have:
(partial success) 
5 Specific Decrement Operators
Unlike improvement operators [12], there is no unique decrement operator. The reason for this is, that if for and , and it is not required otherwise by (DR12), then the relative plausibility of and might not be changed by a decrement operator , i.e. Example 2 demonstrates this.
Layer 2  

\hdashlineLayer 1  
\hdashlineLayer 0 
Example 2
We capture this observation by two types of decrement operators. In the first case, the decrement operator improves the plausibility of a countermodel whenever it is possible.
Definition 7 (Type1 Decrement Operator)
A decrement operator is a type1 decrement operator if there exists a decreasing assignment with:
(DR14) 
The second type of decrement operators keeps the order whenever possible. We capture the cases when this is possible by the following notion. If is a total preorder on worlds, we say is frontal with respect to , if (1.) there is no such that , and (2.) there is no such that . We define the second type of decrement operators as follows.
Definition 8 (Type2 Decrement Operator)
A decrement operator is a type2 decrement operator if there exists a decreasing assignment with:
(DR15) 
Indeed, one can show that by the additional constraints the type1 decrement operators and the type2 decrement operators are both unique.
6 Discussion and Future Work
We provide postulates and representation theorems for gradual variants of AGM contractions in the DarwichPearl framework of epistemic states. These socalled weak decrement operators are a generalisation of AGM contraction for epistemic states. Additionally, we give postulates for intended iterative behaviour of these operators, forming the class of decrement operators. For both classes of operators we presented a representation theorem in the framework of total preorders. This fills a gap in the area of nonprioritized belief change.
For future work, we want to investigate the interrelation between (weak) decrement operators and (weak) improvement operators. One approach is to generalize the Levi identity [14] and Haper identity [9] to these operators. Another approach could be the direct definition of a contraction operator from improvement operators, as suggested by Konieczny and Pino Pérez [12]. For such operators, after achieving success, a next improvement may make certain models unplausible, while a decrement operator keeps the plausibility. While this already indicated a difference between the operators, the study of their specific interrelationship is part of future work. Another goal for future work is to generalize (weak) decrement operators to a more general class of gradual change operators [15]. Such operators are candidates for a formalisation of psychologically inspired forgetting operations. An immediate target towards this goal is to take a closer look at subclasses and interrelate them with the taxonomy of improvement operators [11].
Acknowledgements: We would like to thank Gabriele KernIsberner for fruitful discussions and her encouragement to follow the line of research leading to this paper. This work was supported by DFG Grant BE 1700/91 given to Christoph Beierle as part of the priority program "Intentional Forgetting in Organizations" (SPP 1921). Kai Sauerwald is supported by this Grant.
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Appendix A Proofs
This appendix contains full proofs for the two representation theorems and for Proposition 5. These proofs rely on three lemmata which are also proven here.
Proof
The proof is analogue to a proof by Caridroit et. al [5, Lem 13.].∎
Theorem A.1 (Representation Theorem for Weak Decrement Operators)
Let be a belief change operator. Then the following items are equivalent:

is a weak decrement operator

there exists a strong faithful assignment with respect to such that:
(decrement success)
Proof
For the (a) to (b)direction, is an hesitant contraction operator, and the corresponding operator is defined. We define the total preorder as follows:
We show that is a total preorder:
 Totality

Let . By definition , and therefore has at least one model and . By (hesitant success) there is an (and we choose here the smallest) such that . Therefore, either or .
 Reflexivity

Follows from totality.
 Transitivity

Let such that and . We differentiate by case:

If are not pairwise distinct, then transitivity is easily fulfilled (since is reflexive).

Assume that are pairwise distinct and for at least one we have . Then in each case it is easy to see that and thus, by (D1), for all it follows .

We show that is a strong faithful assignment with respect to .
 (SFA1)

Let . Then by (D1) we have . Therefore by definition of we have .
 (SFA2)
 (SFA3)

Follows directly from (D5).
We show that (decrement success) is fulfilled. We differentiate by case:

Case with . Then and by definition of we have , especially .

Case with . Then by (D1) we have . We show that every is an element of the set .
First, by (D4) we have . Then every which is an element of leads to a violation of (D4). Thus, we observe that every is an element of .
Second, towards a contradiction suppose such that . Let , and therefore . By definition we have and . By (D5) we have . Then by (D7) and by Lemma 1 we conclude . This shows .
Suppose is an element of such that . Without loss of generality we can assume ; thus, there exists at least one such that . By definition of we have . Clearly , and thus, . Since , we have . Therefore from (D7) we conclude and thus the contradiction . This completes the proof of .
For the (b) to (a)direction let be a belief change operator and a strong faithful assignment with respect to such that (decrement success) is fulfilled.
 (D3)

For and let be the smallest integer such that . By (decrement success) the existence of guaranteed. For , then and therefore is a hesitant contraction operator.
Since satisfies (D3) the corresponding operator is defined.
 (D1)

Follows directly by (decrement success).
 (D2)
 (D4)

Let and therefore . Then if and only if . By (decrement success) we conclude . Clearly, .
 (D5)

Follows by (SFA3).
 (D6)

By (decrement success) we have and we have . Furthermore, it holds that and therefore, we have:
 (D7)

Assume . Then by (decrement success) and (SFA3) we have . This implies that . By basic set theory we get . By (decrement success) this is equivalent to
In summary, the operator is an weak decrement operator. ∎
Lemma 2
Let be a belief change operator. If there exists a strong faithful assignment with respect to which satisfies (DR8), (DR9) and (DR11), then for every and we have:
(partial success) 
Proof
Let . If we are done, so it remains to show that in the case of .
We first show that if , then . Towards a contradiction suppose this is not the case, i.e. and . Then there a two cases: 1. There exists such that . We easy conclude that and thus, by (DR8), we have . Due to the faithfulness of the assignment , which is a contradiction. 2. For all we have . Then, by using , for all we must have . Thus, and from (DR11) we get . Again, due to the faithfulness of the assignment, we have , which is a contradiction. So every is an element of .
Now we show that every is an element of . Towards a contradiction suppose . Then there exists such that . By (DR9) we can conclude that , which is a contradiction to the assumed faithfulness of the assignment. ∎
Proposition 6
Let be a hesitant belief change operator. If there exists an decreasing assignment with respect to , then we have:
(partial success) 
Lemma 3
Let be a belief change operator, a strong faithful assignment with respect to and . Then if and only if for each and we have either or .
Proof
The "only if" direction. By definition of we have
which implies . Clearly, it follows that . In the case of we are done.
For the remaining case of suppose there exists such that . This implies that and . Thus, by definition we have . Similarly, we have , since and . Note that this implies . From the previous observations we conclude , and therefore . This leads to , which is a contradiction to . In summary it must be the case that either or .
For the "if" direction suppose that . This implies that and . Thus we have for every and some . Additionally, we have and . Thus we have for every and some . Note that is a total preorder, and thus, we have , a contradiction to the assumptions of or . ∎
Theorem A.2 (Representation Theorem for Decrement Operators)
Let be a belief change operator. Then the following items are equivalent:

is a decrement operator

there exists a decreasing assignment