Probabilistic Belief Contraction Using Argumentation

1 Introduction

The epistemic or belief state of an intelligent agent is generally in a state of flux with new beliefs being added and old beliefs being removed. The field of belief change [Alchourr´on et al., 1985; G¨ardenfors, 1988; Peppas, 2008] deals with modelling the different types of changes an agent's belief state may undergo. The impetus for change is usually a received piece of information (sentence). Conventionally belief change is studied under a propositional logic framework. In its most rudimentary form, the belief state of an agent is represented by a belief set, that is a set of sentences that is possibly closed under logical consequence. 1

If an agent is less certain about one sentence than another, additional mechanisms are needed to capture information about the level of uncertainties. A number of approaches exist for this purpose, including epistemic entrenchment of beliefs [G¨ardenfors, 1988; Nayak, 1994], ranking functions [Spohn, 1988], probability measures [G¨ardenfors, 1988], possibility theory [Dubois et al., 1994], and DempsterShafer belief functions [Shafer and others, 1976]. In this paper we deal with belief states represented as probability functions which effectively assign probability 1 to beliefs, 0 to disbeliefs, and different intermediate values to non-beliefs. When a belief state is represented as a probability function, a (probabilistic) belief change operation (under a given sentential input) becomes a mapping from probability functions to probability functions. Belief contraction is the process by which a sentence h that is a belief becomes a non-belief. It is well known that in a deterministic framework belief contraction can be defined through belief revision employing the Harper Identity. 2 The counterpart of the Harper Identity in the probabilistic setup is presented in [G¨ardenfors, 1988] where probabilistic belief contraction is taken to be a mixture of two probability functions P and P ∗ ¬h where P represents the original belief state, and P ∗ ¬h the result of revising P by the negation of the sentence being contracted (h). This mixture takes as ingredients a proportion of P and a proportion of P ∗ ¬h . However, it is not clear exactly what these proportions should be and to the best of our knowledge, there is no work in literature addressing this. The primary aim of this paper is to propose a plausible solution to this problem. The main idea is to view belief contraction as an argumentation process involving two agents X and Y with belief states P and P ∗ ¬h . X argues for h whereas Y argues for ¬h and the goal is to reach a compromise which is represented by the contracted belief state P - h . Thus, the input for a belief change process is not only a naked piece of information - it should be accompanied by arguments to support it. Different probability functions will generate arguments of different strengths for X and Y and we exploit this feature to make an informed choice as to what the proportions of the mixture should be. Section 2 introduces probabilistic belief states and Section 3 briefly reviews probabilistic belief contraction and its problems. In Section 4, we give an overview of argumentation theory and our argumentation framework for determining the mixture proportion is presented in Section 5 followed by the discussion and conclusion in Section 6.

∗This research is partially funded by ARC grant DP150104133. 1 When the set of beliefs is not closed, it is customary to call it a

belief base.

2 Background

We assume a propositional language L consisting of a finite, nonempty set of propositional variables (atoms) PS along with the standard logical connectives ∧, ∨, ¬ and →. We take the background logic represented by the provability relation ⊢ to be the classical logic. Lower case Roman letters like a and b represent sentences of L and a set of sentences is represented by upper case Roman letters like A and B. Given the finitary nature of L, the information content of an agent's belief state can be represented as a single sentence. We reserve the sentence k for this purpose. Alternatively, K is reserved to represent the information content of an agent's belief state when it is represented as a set of sentences. An interpretation or a possible world of L is a function from the set of propositional atoms to {0, 1}. A model of a sentence x is an interpretation that satisfies x, by [x] we represent the set of models of x and by |[x]| the number of models of x. We also say two sentences x and y are equivalent, denoted as x ≡ y, if they have exactly the same set of models, i.e [x] = [y]. By extension, A ≡ x iff ∧ A ≡ x. Lower case Greek letters like α and β denote worlds, upper case Greek letters denote sets of worlds and Ω is the set of all worlds. ω⊥ ∈ Ω, called the absurd world, is a special world where every sentence holds. ω⊥ allows an agent to have inconsistent beliefs, i.e. it believes in contradictions. The letter P is reserved for representing probability (distribution) functions that assign a non-negative probability mass to each member of Ω such that the total mass of Ω is 1, with the proviso that P(ω⊥) = 1 or P(ω⊥) = 0. Note that if P(ω⊥) = 1, then all other worlds will have zero probability mass. 3 The probability of any sentence x ∈ L is the sum of the probability mass assigned by P to its models.

2Kh = K ∩K∗¬h, for any belief set K and sentence h.

Let P be a probability function over Ω repesenting the belief state. The belief set K (mutatis mutandis, k) is the top of P, that is, the set of all sentences that have a probability of 1. Given a probability function P and its associated belief set K, a sentence h is said to be consistent with K (and, by extension, consistent with P) if P(h) > 0. If P(h) is 1, then the sentence h is a belief, if P(h) is 0, it is a disbelief, and otherwise it is a non-belief. The converse issue of generating a belief state from a belief set is problematic since there can be several probability functions that can be associated with K. This is an important foundational question addressed by [Lindstr¨om and Rabinowicz, 1989] [Boutilier, 1995] and [Voorbraak, 1999] among others in the context of probabilistic belief revision. We assume that a unique probability function P has been chosen through some process as the one that best represents K, subject to the condition that it assigns to each model of K except ω⊥ some non-zero probability mass whenever P is stable or maximally stable, which are defined below:

Definition 1. Given a belief state P , we say:

• P is stable iff P(ω⊥) = 0,
• P is unstable iff P(ω⊥) = 1 and,
• P is maximally stable iff P is stable and for all ω, P (ω) > 0, where ω ̸= ω⊥.

Note that all maximally stable belief states are also stable but not all stable belief states are maximally stable. When we refer to a belief state as stable, we mean stable but not maximally stable unless otherwise noted. If P is unstable, [K] = {ω⊥}. Thus contradictions form part of the belief set K and we will say K or P is inconsistent. If P is stable, then [K] ⊂ Ω and contingent sentences form part of K. If P is maximally stable, then [K] = Ω and K only consists of tautologies. In this case, P assigns non-zero probability mass to all worlds in Ω except ω⊥. We will assume that the epistemic agent is not opinionated, that is, there is at least one sentence in L that she neither believes nor disbelieves. 4 This is captured by the assumption that the agent's beliefs allow at least three models, i.e. |[K]| ≥ 3.5

3 To respect ∑ ω∈Ω P (ω) = 1 (Kolmogorov's Second Axiom).

3 Probability Contraction

A probabilistic contraction function maps a probability function, under a sentential input, to another probability function. Contraction is the process by which a sentence h that is initially a belief becomes a non-belief. Semantically, if the agent's belief state prior to the contraction is P, then P assigns non-zero probability mass to the h-worlds but not to the ¬h-worlds whereas after contraction both h-worlds and ¬h-worlds will have non-zero probability mass. We let P - h represent the belief state obtained as a result of contracting sentence h from P. [G¨ardenfors, 1988] proposed five basic postulates (P -1)-(P -5) that a rational agent should adhere to when contracting a belief. Motivation behind and justification for these postulates are provided in [G¨ardenfors, 1988]. While it is possible to independently define contraction functions, we take revision as a more natural belief change process and define contraction in terms of it. For this work, it suffices to know that the revision of a belief state P by a sentence ¬h results in a new belief state P ∗ ¬h where ¬h is a belief and h is a disbelief, i.e. P ∗ ¬h (h) = 0 and P ∗ ¬h (¬h) = 1. Given a probability distribution P and a sentence h, the probability contraction function is defined in terms of revision as follows [G¨ardenfors, 1988]:

Definition 2. Given belief state P, belief h and the revised belief state P ∗ ¬h , the contracted belief state P - h is obtained as follows:

for all sentence x and where 0 ≤ ϵ ≤ 1.

Often it is simply written: P - h = PϵP ∗ ¬h , with P ϵP ∗ ¬h called the ϵ-mixture of P and P ∗ ¬h . By K∗¬h and Kh , we denote the belief sets associated with P ∗ ¬h and P - h respectively.

It is clear from Definition 2 that one needs to only specify ϵ and a probability function P ∗ ¬h in order to construct a probability contraction function. From a functional point of view, the role of ϵ is clear. It determines how much of the original belief state P to retain. However, in using this definition, three issues immediately surface: 1) What exactly is ϵ? 2) What should the value of ϵ be and how does one justify that value? and, 3) How should P ∗ ¬h be constructed? As to the first issue, it has been argued [G¨ardenfors, 1988] that ϵ represents a measure of the degree of closeness to the beliefs in P . This view is in line with the extreme cases. When ϵ = 1, contraction fails, indicating the agent totally closes up her mind and keeps its belief state unchanged. On the other hand, when ϵ = 0, contraction is reduced to revision by ¬h and the agent is totally open to accepting P ∗ ¬h as its new belief state. In the non-extreme cases 0 < ϵ < 1, the agent opens up its mind partially and is willing to part only with a certain proportion of her belief state. What makes an agent close or open up its mind is the second issue and its answer lies in how ϵ is measured. This will depend on one's interpretation of the notion of closeness. One natural candidate for this potential measure of closeness is epistemic entrenchment and its variants (such as possibility measure or ranking functions). In this paper we consider an alternative approach towards the interpretation and implementation of this measure ϵ. We imagine a dialogic process going on in between the agent and an adversary, the agent defending its belief h and the adversary pushing for ¬h. A typical case would be when there is some support for h, but the agent is prepared to open up its mind and consider the adversary's proposal. In this case the agent suspends the belief h without disbelieving it altogether, retains a measure of probability depending on how firmly it was defended in the first place. Hence what we need in addition to the probability measure is an argumentation framework. Regarding P ∗ ¬h , as it is altogether a separate issue, in the rest of the paper we will assume that it is obtained via some suitable mechanism.

4 An agent with a unstable belief state is opinionated since it believes in every sentence.

5 It is 3 (not 2) to account for ω⊥ which is the model of any sentence.

4 Argumentation Framework

While Pollock is widely considered to be the founder of formal argumentation systems [Pollock, 1987], most current work on argumentation is based on the abstract argumentation framework of [Dung, 1995] which inspired the framework in [Amgoud and Cayrol, 1998; Amgoud and Prade, 2004] that employs possibility theory. Our work below is based on the latter and on [Hunter, 2013] where probabilistic arguments are considered.

Definition 3. An argument is a pair (H, h) where h is a sentence of L, H ⊆ L and, a) H is consistent, b) H ⊢ h, c) there is no proper subset H ′ of H such that H′ ⊢ h and, d) H ̸= {h′ }, where h′ ≡ h.

H is called the support of h. First, we need to make the notion of consistency clear as in our framework we have the absurd world ω⊥. We will say H is consistent iff [H] ̸= {ω⊥}. Thus, an argument that has a contradiction as support is considered invalid. Definition 3 is essentially the same as those in [Amgoud and Prade, 2004; Hunter, 2013] except that it imposes an extra condition (d). Thus, in our framework, we refrain from arguments whose conclusion and support are logically equivalent. Traditionally arguments of this sort are considered suffering from the fallacy of petitio principii [Iacona and Marconi, 2005] - we accuse one of this when we claim that they are begging the question, so to speak.

Let A denote the set of all arguments that can be generated from L. Defeasibility of arguments is captured by the following definition:

Definition 4. Let (H1, h1) and (H2, h2) be two arguments in A. Argument (H1, h1) rebuts (H2, h2) iff h1 ≡ ¬h2, and (H1, h1) undercuts (H2, h2) iff h ≡ ¬h1 for some h ∈ H2.

Now we consider the strength of an argument. We assume that the strength of an argument will be sensitive to the joint probability of its premises. This is in line with the approachs taken in [Hunter, 2013; Skyrms, 1986]. In [Pollock, 1987; Prakken and Horty, 2012], the weakest link principle, that an argument's strength is the minimum of the strengths of the argument's premises, is endorsed as a way to determine an argument's strength. Unfortunately, in using probabilities to quantify uncertainty, one cannot assume that the probability of the argument is equal to the minimum of the probability of its premises. However, it is known that for two events A and B, P(A&B) ≤ P(A) and P(A&B) ≤ P(B), and in this sense it conforms to the weakest link principle.

Definition 5. Given a probability distribution P and an argument (H, h) where H = {h1, . . . , hn}, the strength of the argument is denoted as level(H, h) and is given by the probability of the conjunction of its support P(h1 ∧ . . . ∧ hn).

Given a belief state P , we denote the set of arguments that can be generated for a conclusion (belief) h as APh . The strength of each argument in APh can be determined from P as in Definition 5. Next we define how one argument may defend itself from another:

Definition 6. Let (H1, h1) and (H2, h2) be two arguments in A. If (H2, h2) rebuts or undercuts (H1, h1), then (H1, h1) defends itself against (H2, h2) iff level(H1, h1) ≥ level(H2, h2).

5 Contraction via Argumentation

Assume there is an agent X whose belief state is P and amongst other sentences has h as one of its beliefs. Recall that the new belief state P - h of X after h has been removed from its set of beliefs can be computed by the mixture P ϵP ∗ ¬h as given in Definition 2. Our primary goal is to propose a plausible way of determining ϵ. Though P ∗ ¬h represents the revision of X's belief state P by ¬h, it is convenient for illustrative purposes to think of P ∗ ¬h as the belief state of another agent Y. We now give the basic idea of the argumentation process and its role in determining ϵ. We imagine that X has been approached by Y to give up its belief in h. However, X is reluctant to do so and an argumentation process ensues. Since h is the sentence of interest, X equips itself with the set of arguments APh for h and similarly Y with AP ∗ ¬h for ¬h. 6 X and Y begin by presenting to each other their best possible arguments. If X defends itself against Y's argument and Y can't do the same, then the situation favors X and P - h should be closer to X 's belief state P . The value assigned to ϵ should then reflect this. Should the situation favor Y, we must again take this into account in determining ϵ. If however, there is a tie, i.e. both X and Y defend against each other's arguments, then the argumentation process continues with the agents resorting to comparing the number of arguments that they possess in order to break the tie. The details of the argumentation process are presented later.

6 We have abused notation here to improve readability and represented AP ∗¬h ¬h as AP ∗ ¬h .

Any sentence considered as a belief by an agent could be one of three types: a tautology, a contradiction or a contingent sentence. The combination of different types of belief states and beliefs give rise to what we call contraction scenarios as shown in Table 1. The first step in our work is to identify which scenarios are possible and which are not. This will help us to focus on what kind of belief state and beliefs we should be looking out for. It is worth keeping in mind that the semantic interpretation of revision of P by ¬h means none of the h worlds will have non-zero probability mass assigned by P ∗ ¬h or in other words, the total probability mass is completely contained within the ¬h worlds.

In Table 1, 7 h represents a current belief that we desire to remove, and hence ¬h is the sentence by which we must revise the belief state in the process. The cases (S2, M1, M2) and (U3) need some discussion. (S2) is not "possible" since the agent in a stable belief state does not have any contradictory beliefs to remove. Similarly, in (M1) and (M2), an agent in a maximal belief states has only tautological beliefs, and there are no contingent or contradictory beliefs to remove. As to (U3), we stipulate that removal of a tautology from an unstable belief state leaves it unchanged, that is contraction is not successful. The last column describes the intermediate state P ∗ ¬h that the process is assumed to go through. The five contraction scenarios that interest us are (S1), (S3), (M3), (U1) and (U2). As we will show later, the latter four are limiting cases where ϵ takes on extremal values.

Any belief state and belief in (S1), (S3), (M3), (U1) and (U2) belong to one of five types: (T1)-(T5) that are shown in Table 2. This is useful as it will allows us to learn about the existence and level of arguments in the various contraction scenarios as shown in Propositon 1 below and summarized in Table 2 as well. For instance for (S3), we need to consider only argument sets that can be generated from (T 2) and (T 4). Furthermore, arguments in T2 are of level 0, level 1 and level between 0 and 1 as seen in Table 2.

7 We recall that by stable, we mean stable but not maximally stable belief states.

Proposition 1. Given a belief state P and a belief h:

• if P is stable and h is contingent or a tautology, then there are arguments of level 0, level 1 and level between 0 and 1 in APh ,
• if P is maximal and h is a tautology, then there are only arguments of level between 0 and 1 in APh ,
• if P is unstable and h is a contradiction, then there are no arguments in APh ,
• If P is unstable and h is contingent or a tautology, then there are only arguments of level 1 in APh .

Given an agent's argument set APh , ideally we would not want an argument (H, h) ∈ APh to be attacked by other arguments in APh . An argument that is attacked should at least be able to defend itself. The next propositions offer us this reassurance.

Proposition 2. Given a belief state P and a belief h, if:

• P is stable and h is contingent (T1) or,
• P is stable and h is a tautology (T2, T3) or, 8
• P is unstable and h is a contradiction (T4) or,
• P is unstable and h is contingent (T5), then there are no rebuts nor undercuts in APh .

We are interested in contraction scenarios (S1), (S3), (M3), (U1) and (U2). In each scenario, there are two argument sets we are interested in: argument set APh of X and AP ∗ ¬h of Y. An argumentation process will generally involve both rebuttals and undercuts. However, as the following simple theorem shows APh and AP ∗ ¬h do not have any undercutting arguments:

Theorem 1. For any (S1), (S3), (M3), (U1) or (U2), if (H, h) and (H′ , ¬h) are arguments in APh and AP ∗ ¬h respectively, then (H, h) and (H′ , ¬h) only rebut but do not undercut each other.

Theorem 1 greatly simplifies the argumentation process as X and Y no longer have to worry about their arguments being undercut by the other and we use this feature to define when one argument set is preferred over the other simply based on the strength of the arguments that they possess.

Definition 7. Two arguments are equivalent iff h ≡ h′ and H ≡ H′ .

8 Here, stable includes maximally stable belief states as well.

Definition 8. Given APh , by APh /∼, we denote the set of equivalence classes of APh where each class consists of arguments of the same level.

Definition 9. Given APh /∼ and a ∈ APh /∼,

• level(a) is the level of a and,
• uniq(a) is the number of arguments unique upto logical equivalence in a,
• cP is the equivalence class of the highest level not equal to 1,
• if cP exists then best(APh /∼) = level(cP ) else best(APh /∼) = 0.

Example 1. Let PS = {a, b, c} and k ≡ a ∨ b → a ∧ b ∧ ¯ c. Thus, [k] = {¯a¯b¯ c, ¯a¯bc, ab¯ c}. Let P(¯a¯b¯ c) = 0.35, P(¯a¯bc) = 0.4, P (ab¯ c) = 0.25. 9 Let h = a ↔ b. Since [h] = [k] ∪ {abc} and P(h) = 1, h is a belief. Any argument (H, h) such that [H] ⊂ [h] is a valid argument. However, only if [H] ∩ [k] ̸= ∅ will level(H, h) > 0. It is clear that for any argument (H, h) with level 1, it must be that [k] ⊂ [H]. In this example, all arguments with level 1 must be equivalent to k, since [k] is the largest proper subset of [h]. Thus, if a is the equivalence class in APh /∼ s.t. level(a) = 1, then uniq(a) = 1. The next highest level possible for arguments is 0.75. For any argument (H, h) of level 0.75, it must be that either [H] = {¯a¯b¯ c, ¯a¯bc} or [H] = {¯a¯b¯ c, ¯a¯bc, abc} as these are the only two subsets of [h] whose probabilities sum to 0.75. Let a′ ∈ APh /∼ s.t. level(a) = 0.75, then cP = a′ , uniq(cP ) = 2 and best(APh /∼) = 0.75.

Definition 10. Given APh /∼ and AP ′ h′ /∼:

Case I: APh andAP ′ h′ are equally preferred ( APh ≈ AP ′ h′ ) iff best(APh /∼) = best(AP ′ h′/∼) and uniq(cP ) = uniq(cP ′ ),

In Definition 9, best(APh /∼) returns the highest level amongst all equivalence classes in APh /∼ that are not of level 1, and 0 if there is no such class. We have adopted the view in Definition 10 that where possible the winner of an argument should be the one who possesses stronger (higher level) arguments (II-a). However, if two agents have arguments of the same strength, then the winner is decided by counting the number of arguments (II-b). Thus, level(·) has precedence over uniq(·). In [Amgoud and Prade, 2004], arguments with a small number of formulas in the support are preferred in order to reduce the chances of being undercut. However, as we do not have undercutting arguments (Theorem 1), we do not differentiate between the arguments in cP or cP ′. The reader will note that in the definition of best(·), level 1 classes are ignored. Consider scenario (S1) where both argument sets for X and Y are of type (T1) and thus always have level 1 arguments. This means preference over the arguments are determined using Case II-b. It is easily seen then that there is no need to use a probabilistic belief state in the first place as ties may be broken simply based on the number of models of an argument. Thus to exploit probabilities, we refrain from classes of level one. Finally, for the sake of simplicity, we focussed only on one equivalences class in Definition 10 but it is certainly possible to take other classes into consideration.

9 Recall that only k-worlds have non-zero probabilities.

Determining ϵ

Recall that the contracted belief state P - h will be constructed according to the formula P - h = ϵ · P + (1 - ϵ) · P ∗ ¬h . For each case in Definition 10, we justify and show how ϵ is determined. If the argument sets of X and Y are equally preferred, then the proportion of X and Y's belief state to be used for computing P - h should be equal as shown below.

Now if X's argument set is preferred to Y's, because its arguments are at a higher level compared to Y's, then a greater proportion of X's belief state should be "retained" in P - h . This proportion should depend on how much stronger X 's argument is and this is captured in Case II-a below. If X wins then ϵ > 0.5 and if X loses, then ϵ < 0.5.

Case II-a: If APh ̸≈ AP ∗ ¬h , then:

The final case is when X's argument set is preferred to Y's because it has more arguments than Y even though their arguments are of the same level. It is tempting to normalize the number of arguments but this is not very desirable. Let uniq(cP ) = 1 for X and uniq(cP ∗ ) = 2 for Y. By normalizing uniq(cP ), we get ϵ ≈ 0.33. ϵ becomes disproportionately skewed and a large proportion Y's belief state will be used for P - h even though Y has only one more argument than X. Instead, in such a case, we imagine that a losing agent (one with fewer arguments) commits to retaining at most a certain proportion of its belief state, represented by αmax which must be less than 0.5. At the same time, no matter how many more arguments the winning agent has, the losing agent is determined to retain a certain proportion of its belief state and this is represented by αmin, where 0 < αmin < αmax. The actual proportion of the losing agent's belief state to be used in the mixture, α, will be between αmin and αmax, and depend on the difference, diff , between the number of arguments of the two agents. The minimum value of diff will be 1 at which point we need α = αmax and for all other values of diff , α < αmax. A desirable feature for α is for it to not jump too much between small changes of diff , i.e. we would like it to decrease slowly towards αmin as diff becomes bigger. A function for α that has these properties is the slowly decreasing log function shown below:

Case II-b: If APh ̸≈ AP ∗ ¬h , let:

where δ = e 1 αmax-αmin - 1 and diff = |uniq(cP ) - uniq(cP ∗ )|. 10

• (i) If APh ≺ AP ∗ ¬h , then ϵ = α, else

Finally, remember that ϵ is the proportion of X's belief state to keep. Therefore, if X is the losing agent (IIb-i) then ϵ = α, whereas if X wins (IIb-ii) , then ϵ = 1 - α.

Argumentation Scenarios

Table 3 shows the ϵ values of scenarios (S3), (M3), (U1) and (U2). We discuss here only (S3), but the reasoning is similar for the others. (S1) which is not included in the table is discussed separately. In (S3), X's argument set APh is of type (T2) consisting of a stable belief state P and a tautologicalbelief h. X is challenged by Y, whose belief state P ∗ ¬h is unstable and pushes for X to instead accept a contradiction ¬h as a belief. From Table 2, we see that best(APh /∼) is some non-zero number less than 1, where as best(AP∗ ¬h/∼) = 0 (see Definition 10). Thus, AP∗ ¬h ≺ APh (Case II-a) and we get ϵ = 1. If X's belief state is stable or maximally stable, and is challenged by Y whose belief state is unstable (as in cases S3 and M3) then X wins outright and refuses to give up any of its beliefs and P - h is the same as P. On the other hand, if X's belief state is unstable and Y with a stable or maximally stable belief state challenges it (as in cases U1 and U2), then Y wins outright and X is forced to give up all its beliefs. In the latter case, P - h is simply P ∗ ¬h . The value of ϵ for (S1) is best illustrated by the Example 2 below. In (S1), P and P ∗ ¬h are both stable, and beliefs h and ¬h are both contingent.

Example 2 (continued from Example 1). Recall that h = a ↔ b, uniq(cP ) = 2 and best(APh /∼) = 0.75. Let P ∗ ¬h be such that P ∗ ¬h (¯ab¯ c) = 0.5, P ∗ ¬h (¯abc) = 0.3, P ∗ ¬h (a¯b¯ c) = 0.2. Note that P ∗ ¬h (h) = 0. By similar reasoning to Example 1, we get uniq(cP ∗ ¬h) = 2 and best(AP ∗ ¬h/ ∼) = 0.8. Since best(APh /∼) < best(AP ∗ ¬h/∼), APh ≺ AP ∗ ¬h (Case II-a), and ϵ ≈ 0.75 0.75+0.8 ≈ 0.48. Thus, P - h = 0.48P + 0.52P ∗ ¬h . Consider now P ∗ ¬h (¯ab¯ c) = 0.5, P ∗ ¬h (¯abc) = 0.25, P ∗ ¬h (a¯b¯ c) = 0.25, then best(APh /∼) = best(AP ∗ ¬h/∼) = 0.75. However, uniq(cP ∗ ¬h) = 4 > uniq(cP ). Thus APh ≺ AP ∗ ¬h (Case IIb). Let αmax = 0.49 and αmin = 0.1. We get diff = 2 , δ ≈ 11.99 and α ≈ 0.48. Since APh ≺ AP ∗ ¬h , we set ϵ = α ≈ 0.48 and P - h = 0.48P +0.52P ∗ ¬h .

As we can see, (S1) is the only scenario where 0 < ϵ < 1 and neither X nor Y are willing to give up all of their beliefs.

10 e is Euler's number. αmin is the horizontal asymptote.

In X's original belief state P, the maximum level of support for ¬h is 0, where as in P - h , it has a maximum level of 1-ϵ.

6 Discussion and Conclusion

The work in [Hunter, 2013] is closely related to ours and it considers how two agents that have arguments over two different probability distributions, Pi and Pj , may combine their probability distributions to reflect their possibly conflicting views. The combined probability distribution is defined as: Pi⊕j(A) = max(Pi(A), Pj(A)), where A is any argument in A and Pi(A) is the level of the argument (Definition 5). However, it is also shown that Pi⊕j(A) will only be consistent iff Pi and Pj do not diverge, which only occurs when Pi(ω) = Pj(ω), for all ω ∈ Ω. In all the contractions scenarios that we considered in our work, P and P ∗ ¬h are fully divergent since they do not agree on any models and using Pi⊕j(A) will result in an inconsistent probability distribution which violates the first postulate (P -1) for probabilistic belief contraction. In [Amgoud and Prade, 2004], the main focus is on two agents that use argumentation to negotiate over goals under a possibilistic logic setting. In relation to belief revision, they define the revision of a belief set by an argument (H, h) as a new belief where the argument is forced to hold, i.e. with a possibility degree of 1. Since our work deals with contraction, we do not fully accept arguments from AP ∗ ¬h except in scenarios (U1) and (U2) where X's initial belief state is unstable. In [Ramachandran et al., 2012; Olsson, 1995] probabilistic belief contraction is also addressed but the rationale behind their approach is based on information theory. They assume that the entropy of the contracted belief state should be higher than original belief state since information is lost and attempt to construct the contracted belief state directly.

When the contraction of a probabilistic belief state is viewed as the mixture of the original belief set and the revised belief state, the question of what mixture proportion should be has been largely ignored. By exploiting the quantitative nature of probabilities and using ideas from argumentation theory, we presented a framework that can determine a credible value for the proportions of the mixture. It is worth noting that a belief state is not simply a probability distribution P, but rather a pair 〈P, σ〉 where σ represents an appropriate similarity measure such as a ranking or a distance function. Although we don't have any explicit similarity measure, we suspect that the argumentation support structure for the input sentence h somehow captures the relevant aspects of a similarity measure and further exploration of this idea will be taken up in future. We plan to look at refining the procedure to determine preferred argument sets by examining more than one equivalence class as done here and to consider argument sets with beliefs other than the belief to be removed as conclusion. Argumentation theory offers an alternative way of accounting for and thinking about probabilistic belief change in general, and this work is a first step in that direction.