On Definite Iterated Belief Revision with Belief Algebras^††thanks: This is an extended version of the corresponding IJCAI 2025 paper with the same title.

Updating or revising the beliefs of an agent in light of new evidence is a fundamental process in both everyday life and scientific activities. For instance, Newton’s laws of motion were widely accepted for centuries until discoveries at very small scales or very high speeds revealed their limitations. Similarly, our knowledge is continuously updated and enriched through learning and communication. To formalize this process, researchers in artificial intelligence have developed the subfield of belief change (see, e.g., Doyle (1979); Harper (1976)). Among the most influential contributions to this field is the AGM framework Alchourron et al. (1985), which has inspired numerous extensions and applications in areas such as game theory and argumentation Williams (1996); van Harmelen et al. (2008); Fermé and Hansson (2011); Diller et al. (2015); Zhang (2010).

Various research efforts have focused on representing belief information in iterated revision. Spohn Spohn (1988) introduced the concept of ordinal conditional function (OCF) to encode preference information over worlds and developed a process called conditionalization to revise OCFs. Williams Williams (1994) proposed a formula-based counterpart to OCF, mapping formulas to ordinals based on their resistance to change. Darwiche and Pearl Darwiche and Pearl (1997) advanced the field by representing belief information as total preorders on worlds and extending the AGM framework with four postulates to characterize iterated revision.

Several researchers have improved the DP (Darwiche and Pearl) framework’s settings. Benferhat et al. Benferhat et al. (2005) used partial preorders, while Peppas and Williams Peppas and Williams (2014) employed semiorders. Ma et al. Ma et al. (2015) revised epistemic states with partial epistemic states. Andrikopoulou et al. Andrikopoulou et al. (2025) discussed belief revision under filters that are subsets of partially ordered sets. Benferhat et al. Benferhat et al. (2000) enhanced the representation of new evidence by using an epistemic state, proposing postulates for minimal-model preserving operators and proving the uniqueness of the revision result given a total preorder and a new total preorder as evidence. This aligns with our discussion in Section 5, as their operator satisfies (RE1)–(RE3). Meanwhile, new frameworks and semantic structures continue to be proposed Liberatore (2024); Bonanno (2025).

Many works aim to define the most “reasonable” revision rules, yet achieving consensus on rationality remains challenging. These debates often resemble philosophical discussions, focusing on abstract principles rather than practical implementations. The rationality of the basic AGM rules has been questioned Aravanis (2023). Sauerwald and Thimm Sauerwald and Thimm (2024) considered the realizability of AGM revision and contraction operators in Epistemic Spaces. Some researchers, such as Booth and Meyer (2006); Jin and Thielscher (2007); Nayak et al. (2003), have observed that the DP framework can produce counter-intuitive revision results. To address that issue, they proposed modifications or additions to the DP postulates. For instance, Nayak, Pagnucco, and Peppas Nayak et al. (2003) introduced conjunction postulates, treating consistent evidences as order-independent, while Jin and Thielscher Jin and Thielscher (2007) proposed a weaker independence postulate. Notably, these works primarily focus on rule construction, with limited attention to operator selection.

While the exploration of revision rules continues, the potential and application prospects of belief revision have also received significant attention Hunter and Boyarinov (2022); Baroni et al. (2022). This shift reflects a growing interest in leveraging belief revision for practical, real-world problems. In applications such as constrained differential privacy Liu et al. (2023) and text generation by large language models Wilie et al. (2024), the focus has shifted towards applications of revision. In industrial agent design, selecting a specific update algorithm is crucial to enable agents to iteratively refine their beliefs. To address this, our work focuses on representing belief information through belief algebra building on Meng et al. (2015), and proposes a unique revision operator by strengthening revision rules in a natural way, aiming to provide a deterministic and practical solution for real-world applications.

We recall some necessary knowledge for what will follow.

Given two belief algebras $G_{1}=(2^{W},\gg_{1})$ and $G_{2}=(2^{W},\gg_{2})$ on $W$, we denote by $G_{1}\subseteq G_{2}$ iff $\gg_{1}\subseteq\gg_{2}$. Also, we will not distinguish between $G_{1}\cup G_{2}$ and $\gg_{1}\cup\gg_{2}$, and $G_{1}\cap G_{2}$ and $\gg_{1}\cap\gg_{2}$, whenever it is self-explanatory based on context. For each belief algebra $G=(2^{W},\gg)$, $Tr(W)=\{(U,\emptyset)\mid U\subseteq W,U\neq\emptyset\}$ is always contained in $\gg$ by $(A1)$. It is not difficult to verify that $(2^{W},Tr(W))$ is a belief algebra. Then for each belief algebra $G$ on $W$, we always have $(2^{W},Tr(W))\subseteq G$.

A subset of $R_{W}$ (defined in Definition 1) can generate a belief algebra.

If $\operatorname{Gen}(\Omega)$ also satisfies (A2), then $(2^{W},\operatorname{Gen}(\Omega))$ is a belief algebra by definition, and we will also use $\operatorname{Gen}(\Omega)$ to denote this belief algebra. Here we always assume that the agent is rational and $\operatorname{Gen}(\Omega)$ is always a belief algebra when $\Omega$ represents some belief information. Given $\Omega$, $\operatorname{Gen}(\Omega)$ can be obtained by closing $\Omega$ under (A1), (A3), and (A4).

With the operator $\operatorname{Gen}(\cdot)$, one can see that each CBA is entirely determined by preferences on the sets consisting of a single world.

There is a 1-1 correspondence between CBAs and total preorders, which is given as follows.

Each CBA is totally decided by preferences on the sets consisting of a single world, which is implied straightforwardly by Lemma 4.

The following proposition shows how to construct new belief algebras from existing belief algebras.

The following result unveils the structure of $\gg$, which consists of levels of subsets of worlds.

The above lemma shows that the backbone is the core structure of a belief algebra. (Ch2) shows that for each $U_{i}$ in a backbone, any subsets of $U_{i}$ can not be compared with one another. In terms of belief preferences, this means that an agent has no preference on subsets of $U_{i}$. Another important concept is the support of a subset of $W$ w.r.t. some backbone.

Figure 1 gives an illustration of the above definitions. Moreover, the preferences of an agent are “consistent” with the backbone in the sense of property (1) in Lemma 6 that follows.

The above lemma shows that there are different belief algebras that have the same backbone $\Delta$, and there is a unique complete belief algebra with $\Delta$ as its backbone, viz., the largest one containing all the pairs $(U,V)$ s.t. $I(U)\gg I(V)$. Roughly speaking, the backbone of a belief algebra is the “core” belief information that reflects the main preferences of the agent. Taking Figure 1 as an example, a belief algebra $G$ with $\{U_{1}\gg U_{2}\gg U_{3}\gg\cdots\gg U_{n}\}$ as its backbone may or may not contain $V_{1}\gg V_{2}$, but if it is CBA, then it must contain $V_{1}\gg V_{2}$, and it should never contain $V_{2}\gg V_{1}$ because $I(V_{1})=U_{1}\gg U_{2}=I(V_{2})$.

Note that $G_{1}\leq G_{2}$ is different from $G_{1}\subseteq G_{2}$. It actually means that $G_{1}$ and $G_{2}$ contain the same “core” belief information, but $G_{1}$ is “less informational” than $G_{2}$, in the sense that any revision result of $G_{1}$ should be contained in that of $G_{2}$, as we will see later in the postulate (RA5). Let $BA_{L}$ be the set of belief algebras over the worlds of $L$. It is easy to check that $\leq$ as defined above is a partial order on $BA_{L}$. We also use $BA_{L}$ to denote the partial order set $(BA_{L},\leq)$.

Before discussing the general case of revision with belief algebras, we first consider a simplified scenario where the agent’s current belief, the new evidence, and the revision result are all represented as complete belief algebras (CBAs; see Definition 2). This process is equivalent to revising an existing total preorder $\preceq_{1}$ by a new total preorder $\preceq_{2}$ to obtain a revised total preorder $\preceq_{3}$.

Inspired by existing research (e.g., Alchourron et al. (1985); Darwiche and Pearl (1997); Ma et al. (2015)), we propose the following revision rules for this setting. The revision of a total preorder by another total preorder, denoted as $\preceq_{1}\circ\preceq_{2}=\preceq_{3}$, has been studied in Benferhat et al. (2000) and is known as a minimal-model preserving operator. This operator can be equivalently characterized by the following postulates:

1. (RE1) $\preceq_{3}$ is a total preorder.

2. (RE2) $\prec_{2}\subseteq\prec_{3}$, where $\prec_{i}=\{(x,y)\mid x\preceq_{i}y\wedge y\not\preceq_{i}x\}$.

3. (RE3) If $\omega\sim_{2}\omega^{\prime}$, then: $\omega\prec_{3}\omega^{\prime}$ if and only if $\omega\prec_{1}\omega^{\prime}$.

The postulate (RE1) is natural by underlying setting. The postulate (RE2) requires that the new belief information (corresponding to the strict part of the new total preorder) must be fully incorporated into the revision result. Finally, (RE3) ensures that for worlds with equal (non)preference under $\preceq_{2}$, the preference relation from $\preceq_{1}$ is preserved in $\preceq_{3}$. This allows the revision process to retain more of the original belief information while incorporating the new evidence.

Interestingly, it can be shown that revision results satisfying the above postulates are unique.

In this section, we extend our discussion to a more general framework of iterated belief revision based on a belief algebra, rather than restricting ourselves to CBAs. This framework allows for a richer representation of belief states and supports more flexible revision processes. For simplicity, we assume that the agent is rational (i.e. both her belief and the new evidence do not contain “conflicting information”). If not specified otherwise, we always suppose that the agent’s current belief state is a belief algebra $G_{1}$, the new evidence is a belief algebra $G_{2}$, and the revision result is also a belief algebra denoted by $G_{1}\bullet G_{2}$, where $\bullet$ is a revision operator from $BA_{L}\times BA_{L}$ to $BA_{L}$.

To give postulates to characterize rational revisions, following literature, e.g., Alchourron et al. (1985); Darwiche and Pearl (1997); Jin and Thielscher (2007), we first assume that new evidence has a higher preference, that is, belief information of $G_{2}$ is more believable than $G_{1}$. Then $G_{2}$ should be kept in $G_{1}\bullet G_{2}$, and we have:

1. (RA1) $G_{2}\subseteq G_{1}\bullet G_{2}$

Since $G_{1}$ and $G_{2}$ collectively cover all the agent’s belief information, we assume that $G_{1}\bullet G_{2}$ is entirely determined by $G_{1}$ and $G_{2}$ and is generated by some subset of $G_{1}\cup G_{2}$. This leads to the following postulate:

1. (RA2) There is an $\Omega\subseteq G_{1}\cup G_{2}$ s.t. $G_{1}\bullet G_{2}=\operatorname{Gen}(\Omega)$.

(RA2) requires that the revision result cannot be generated with information outside $G_{1}\cup G_{2}$, and if $G_{1}$ and $G_{2}$ have conflicting information, one should choose a consistent subset in order to generate a belief algebra.

Notice that a complete belief algebra is equivalent to a total preorder on worlds. Revising a complete belief algebra $G_{1}$ by another complete belief algebra $G_{2}$ is equivalent to revising a total preorder by another total preorder. The following postulates (RA3) and (RA4) are thus inspired by (RE1) and (RE3) for revising total preorders in Section 5, respectively.

1. (RA3) Suppose $G_{1}$ and $G_{2}$ are complete belief algebras. Then $G_{1}\bullet G_{2}$ is also a complete belief algebra.

2. (RA4) Suppose $G_{1}=(2^{W},\gg_{1})$ and $G_{2}=(2^{W},\gg_{2})$ are complete belief algebras, and $I_{2}(\{\omega\})=I_{2}(\{\omega^{\prime}\})$ in $G_{2}$. Then $(\{\omega\},\{\omega^{\prime}\})\in G_{1}\bullet G_{2}$ iff $\{\omega\}\gg_{1}\{\omega^{\prime}\}$.

This above theorem generalizes Theorem 3 and shows that, under the postulates (RA1)-(RA4), the revision operator $\bullet$ is deterministic when applied to complete belief algebras. Specifically, the revision result $G_{1}\bullet G_{2}$ is uniquely determined by combining the preference relations from $G_{1}$ and $G_{2}$ in a principled way. The set $\Lambda(G_{1},G_{2})$ captures the preferences from $G_{1}$ that are consistent with the structure of $G_{2}$, ensuring that the revision process preserves as much of the original belief information as possible while fully incorporating the new evidence. The following corollary establishes a direct correspondence between the revision operator $\bullet$ for complete belief algebras and the revision operator $\circ$ for their equivalent total preorders.

For the case where $G_{1}$ and $G_{2}$ are possibly incomplete belief algebras, we include the following postulate:

1. (RA5) If $G_{1}\leq G_{1}^{\prime}$, $G_{2}\leq G_{2}^{\prime}$ then $G_{1}\bullet G_{2}\subseteq G_{1}^{\prime}\bullet G_{2}^{\prime}$.

Recall that $G_{1}\leq G_{1}^{\prime}$ means that $G_{1}\subseteq G_{1}^{\prime}$ and $G_{1}$ and $G_{1}^{\prime}$ have the same backbone. (RA5) means that if $G_{1}$ and $G_{2}$ contain less information than $G^{\prime}_{1}$ and $G^{\prime}_{2}$, respectively, and they have the same core belief information (i.e., same backbones), then $G_{1}\bullet G_{2}$ also contains less belief information than $G^{\prime}_{1}\bullet G^{\prime}_{2}$. From (RA5), the following proposition is easy to verify.

The above proposition shows that $\operatorname{Com}(G_{1})\bullet\operatorname{Com}(G_{2})$ is an upper bound of $G_{1}\bullet G_{2}$ (under $\subseteq$).