Can We Improve Without Worsening? Belief Change in Rational Rankings

Abstract

With the aim of providing a natural and expressive domain for defining change operators, we introduce the epistemic space of rational rankings. We show that this epistemic space is highly useful for understanding various aspects of belief dynamics, particularly those related to the improvement of new information. It enables us to define certain classes of improvement operators, a generalization of iterated revision operators, in a natural and intuitive way. A key feature of rational rankings is the possibility of defining improvement operators in such a way that the negation of the new information does not worsen. This is impossible within the frameworks of total preorders or ordinal conditional functions, two well-known epistemic spaces. Another notable aspect of this space is that the behavior of these operators can be characterized by a few simple equations and inequalities, whose meaning remains transparent. Additionally, there are no stationary states: The epistemic state resulting from applying an operator to a prior epistemic state and new information is always distinct from the prior state. Finally, we prove that this class of operators is indeed a subclass of improvement operators. Consequently, we show that these operators exhibit desirable behavior when iterated sufficiently, ultimately converging to Darwiche and Pearl revision operators.

No man ever steps in the same river twice, for it’s not the same river and he’s not the same man.

Heraclitus

1. Introduction

Unlike Heraclitus’ quote, in many models of belief revision, the new information might not affect the epistemic state (Alchourrón et al., 1985; Darwiche & Pearl, 1997). Thus, the epistemic state remains the same even if one piece of information arrives multiple times. However, this is not always true, and we would like to model situations in which new information always produces a change. Another behavior observed in widely used models of belief revision (total preorders (TPOs) or ranking functions as an instantiation of epistemic states) is that, in general, after revision, there is a loss of plausibility of the negation of the input. However, one might want the plausibility of the negation of the input to remain the same before and after revision, following the minimal change principle.

Let us show some concrete situations in which the phenomena pointed out in the previous paragraphs are realized.

Example 1

The situation depicted in Figure 1, shows a example in which we have an epistemic state $\mathrm{\Psi }$ represented as a TPO and a formula $\alpha$ , the new piece of information, such that for any revision operator $\circ$ that complies with Darwiche and Pearl’s postulates, necessarily $\mathrm{\Psi }\circ \alpha =\mathrm{\Psi }$ . As usual, TPOs over worlds are outlined graphically putting the worlds on layers and the lower the level, the more plausible it is. In our example we take the worlds over the propositional language $\{p,q,r\}$ the formula $\alpha$ has exactly as models those appearing in the the three lowest levels, drawn in orange, for instance $\alpha =p\leftrightarrow \mathrm{\neg }q$ . The reason why any revision operator satisfying the Darwiche and Pearl postulates does not produce change in the situation depicted in Figure 1 is that the plausibility order of models of $\alpha$ does not change (postulate DP1), the plausibility order of models of $\mathrm{\neg }\alpha$ does not change (postulate DP2), and finally the plausibility order of models of $\alpha$ does nor worsen with respect the models of $\mathrm{\neg }\alpha$ (postulates DP3 and DP4). And since in $\mathrm{\Psi }$ all models of $\alpha$ are strictlty more plausible than the models of $\mathrm{\neg }\alpha$ the only possible output for $\mathrm{\Psi }\circ \alpha$ is $\mathrm{\Psi }$ .

Figure 1.

No movements: A situation in which no Darwiche and Pearl (DP) operator produces change.

The other situation which is unavoidable in representations of epistemic states as TPOs or ranking functions, the loss of plausibility of the models of the negation of the new piece of information, is illustrated in next example.

Example 2

Suppose we have the epistemic state represented by the TPO $\mathrm{\Phi }$ of Figure 2 and the new piece of information is the formula $\beta$ having as set of models $\{101,100\}$ . Then any revision operator $\circ$ satisfying the postulates of Darwiche and Pearl, produces as output the TPO $\mathrm{\Phi }\circ \beta$ of Figure 2 in which the minimal models are those of $\beta$ and the models of $\mathrm{\neg }\beta$ conserve the same relations as in $\mathrm{\Phi }$ . What is important to stress is that the models of $\mathrm{\neg }\beta$ are less plausible in $\mathrm{\Phi }\circ \beta$ than in $\mathrm{\Phi }$ .

Figure 2.

Loss of plausibility of counter-models of the new information.

In the following example, we present a situation in which permanent change is desirable and, at the same time, it illustrates why we don’t want the plausibility of the negation of the new piece of information to decrease after the change:

Example 3

Dr. Roberts, MD, has been practicing medicine for 20 years. Since he began his practice, he has applied treatment A to the common cold. Treatment A always works, so he has great confidence in this treatment, and this confidence increases with every successful case even when its effectiveness against the common cold was already part of his beliefs. One day he observes a case of common cold where treatment A did not work. Backed by all his experience, his confidence in treatment A does not decrease, but he gives some credit to the idea that treatment A might not work in some cases.

The epistemic states and operators we introduce in this work allow modeling this kind of situation where the degree of confidence in treatment A does not change in numerical terms after receiving the information that the treatment does not work. However, after receiving the information that treatment A does not work, the degree of confidence in this new information increases. Of course, if this new information is repeated a sufficient number of times, the doctor must change his opinion about the benefits of treatment A.

Another feature that one would like to model is that the changes are produced little by little (and sometimes very little by little!) as in the following example:

Example 4

Carlos is learning English. He has only rudimentary knowledge. He takes an initial test and doesn’t even manage to obtain the A1 certificate. Due to work requirements, he needs to obtain the C2 certificate, the highest level. He knows that to achieve this, he will have to keep working hard every single day. Every day, he checks how his English level is progressing. He knows he is improving—but very slowly. So he thinks that one way to measure his improvement is as a small daily epsilon that, after many days, will allow him to reach the C2 level.

We claim that the structures we are going to introduce, rational rankings, allow us to perform revisions and improvements without worsening in an absolute way. However, even if the plausibility values remain unchanged, some models that were previously believed may fall out of the beliefs after an update. This can be interpreted as their relative plausibility has worsened from the perspective of the more entrenched beliefs. That said, the richness of our numerical epistemic states could be used beyond belief change, namely, to other logic-based AI applications where absolute or conditional plausibility values are directly exploited. For example, it could make sense to reuse those rational rankings as such in other contexts as in Bauters et al. (2017), Casini et al. (2023), Ma and Liu (2011) or van Eijck and Kai Li (2017) where epistemic states are used not to model change, but as a foundation for inference/prediction/interaction. In such settings, plausibility information is valuable in its own right independent of belief revision.

Having given these previous examples to illustrate our motivations, we will now establish the context of our work in detail and where it fits within the current lines of research.

Our starting point is the logical model for belief change proposed by Alchourrón, Gärdenfors, and Makinson (Alchourrón et al., 1985), known as the AGM framework. This framework adheres to three main principles: Coherence, success, and minimal change. Specifically, when dealing with the incorporation of new information into a corpus of beliefs (belief revision), these principles form the foundation of the revision process (revision operators). These operators are functions that map the epistemic state of an agent and the new information into a new epistemic state that is coherent (i.e., free of contradictions), incorporates the new information, and remains as close as possible to the original epistemic state. In the logical AGM framework, epistemic states are represented as logical theories (sets of sentences closed under logical deduction), and the new information is represented by a logical formula. Typically, propositional logic serves as the underlying logic. When the logic is finite propositional logic, Katsuno and Mendelzon (Katsuno & Mendelzon, 1991) proposed a simplified model (the KM framework) where the epistemic state corresponds to a propositional formula, and the new information is also a propositional formula. The KM framework provides an easily understandable set of rationality principles and includes a useful representation theorem: Revision operators are characterized as functions mapping a formula (an epistemic state) into a TPO over the set of interpretations. ¹

Both the AGM and KM models are effective for explaining a single step of revision. However, the KM representation theorem led Darwiche and Pearl (Darwiche & Pearl, 1997) to demonstrate that these frameworks are inadequate for iterated revision processes. Consequently, they proposed a new model for iterated revision, referred to as the Darwiche and Pearl (DP) framework. The core concept in the DP framework is the notion of a (complex) epistemic state. These epistemic states differ from formulas and theories; they are objects associated with certain beliefs. For instance, the simplest examples of epistemic states are TPOs over interpretations, where the beliefs associated with a TPO are represented by a formula whose models are precisely the minimal models of the TPO.

While TPOs provide a natural and straightforward setting for studying DP revision operators, more general settings exist where DP revision operators can be defined. For example, ordinal conditional functions (OCFs) proposed by Spohn (Spohn, 1988) offer such a general setting. Indeed, there are DP revision operators defined on OCFs that cannot be described in terms of TPOs (see Aravanis et al., 2019 and Schwind et al., 2022). It is crucial to recognize that a belief change operator, such as a DP revision operator, must be defined within a specific “epistemic space,” and numerous epistemic spaces, including some quite uncommon ones, exist (Schwind et al., 2022).

Another important area of research in belief change, particularly in belief revision, involves non-prioritized change operators (Hansson, 1997, 1999). In these operators, the success postulate is relaxed. This category includes credibility-limited revision operators (Booth et al., 2012; Hansson et al., 2001), where only information from a credible set is incorporated, and information not in this set is rejected with no impact on the current belief state. Another family of non-prioritized revision operators aims to capture the idea that the plausibility of new information increases. These improvement operators (Konieczny et al., 2010b; Konieczny & Pino Pérez, 2008; Medina Grespan & Pino Pérez, 2013) uniquely achieve success after several iterations of the same information, contrasting with credibility-limited revision operators, where some pieces of information are never incorporated even after multiple iterations. However, these two approaches have been successfully merged in credibility-limited improvement operators (Booth et al., 2014). Notably, research on these operator families has been primarily conducted within the framework of TPOs as the epistemic space, though some improvement operators have also defined in the space of OCFs (Konieczny & Pino Pérez, 2008).

In this paper, we introduce a novel complex epistemic space called rational rankings, where rankings are represented by rational numbers. This exploration is motivated by the idea that studying structures beyond TPOs and OCFs can offer deeper insights into the properties that improvement operators should possess in broader contexts. The framework of rational rankings is both rich and intuitive, allowing us to define operators on this space that capture the increase in plausibility of new information. This behavior is characterized by a set of simple equations and inequalities with clear, intuitive meanings. We show that these operators fit within the class of improvement operators proposed in the literature and demonstrate that, after sufficient iterations, they function as Darwiche and Pearl’s revision operators.

2. Preliminaries

We consider a finite set of propositional variables $P=\{x_1,x_2,\dots ,x_n\}$ and the set of propositional formulas, ${\mathcal{L}}_P$ built up from $P$ and the usual connectives. ${{\mathcal{L}}_P}^{*}$ denotes the set of consistent formulas from ${\mathcal{L}}_P$ . The symbols $\mathrm{\top }$ and $\perp$ denote respectively the tautology and the contradiction. The set $\mathrm{\Omega }=\{w_1,w_2,\dots ,w_{2^n}\}$ is the set of all interpretations (or worlds) that is the classical truth functions on $P$ . Propositional formulas will be denoted in general by lower case Greek letters like $\alpha$ and their set of models is denoted by $\text{⟦}\alpha \text{⟧}$ .

Let $⪯$ be a a total pre-order, that is a reflexive ( $x⪯x$ ), transitive ( $(x⪯y\wedge y⪯z)\to x⪯z$ ) and total ( $x⪯y\vee y⪯x$ ) relation over $\mathrm{\Omega }$ . The corresponding strict relation $\prec$ is defined as $x\prec y$ iff $x⪯y$ and $y⪯̸x$ , and the corresponding equivalence relation $\simeq$ is defined as $x\simeq y$ iff $x⪯y$ and $y⪯x$ . We denote $\omega \prec \prec {\omega }^{\prime }$ when $\omega \prec {\omega }^{\prime }$ and there is no ${\omega }^{″}$ such that $\omega \prec {\omega }^{″}\prec {\omega }^{\prime }$ .

Operators of change adequate to iteration (Darwiche & Pearl, 1997) have to be defined in epistemic spaces (Schwind et al., 2022). Let us recall their definition more precisely.

Definition 1

An epistemic space is an ordered pair $\mathcal{E}=⟨U,\mathcal{B}⟩$ where $U$ is a set whose elements are called epistemic states and $\mathcal{B}$ is a projection function that maps each $\mathrm{\Psi }\in U$ to a consistent propositional formula that represents the belief set associated with $\mathrm{\Psi }$ .

Thus an epistemic state is just an element of an epistemic space, very much as a vector is just an element of a vector space.

The two more typical examples of epistemic spaces are the tpo epistemic space, ${\mathcal{E}}_{\mathrm{tpo}}$ (tpo is the abbreviation of TPO) and the ocf epistemic space, ${\mathcal{E}}_{\mathrm{ocf}}$ (ocf is the abbreviation OCF). ² More precisely, ${\mathcal{E}}_{\mathrm{tpo}}=⟨U_{\mathrm{tpo}},B_{\mathrm{tpo}}⟩$ where $U_{\mathrm{tpo}}$ is the set of all TPOs over $\mathrm{\Omega }$ and $B_{\mathrm{tpo}}$ maps each TPO $⪯$ in $U_{\mathrm{tpo}}$ to a consistent formula $\psi \in {{\mathcal{L}}_P}^{*}$ such that $\text{⟦}\psi \text{⟧}=min(\mathrm{\Omega },⪯)$ . And ${\mathcal{E}}_{\mathrm{ocf}}=⟨U_{\mathrm{ocf}},B_{\mathrm{ocf}}⟩$ where $U_{\mathrm{ocf}}$ is the set of all ocfs over $\mathrm{\Omega }$ and $B_{\mathrm{ocf}}$ maps an ocf $\kappa$ to a consistent formula $\psi$ such that $\text{⟦}\psi \text{⟧}=\{\omega :\kappa (\omega )=0\}$ .

Given an epistemic space $\mathcal{E}=⟨U,\mathcal{B}⟩$ , a change operator $\circ$ is a function of the form $\circ :U\times {\mathcal{L}}_P⟶U$ . General epistemic states are noted by upper case Greek letters like $\mathrm{\Psi }$ . The output of an operator is noted in infix notation. Thus, $\mathrm{\Psi }\circ \alpha$ is the epistemic state obtained from $\mathrm{\Psi }$ with the change produced by the information $\alpha$ .

Given a change operator $\circ$ we define for every formula $\alpha$ :

$$\begin{array}{rl}\mathrm{\Psi }{\circ }^0\alpha & :=\mathrm{\Psi }\\ \mathrm{\Psi }{\circ }^{k+1}\alpha & :=(\mathrm{\Psi }{\circ }^k\alpha )\circ \alpha k\ge 0\end{array}$$

The improvement operators proposed in Konieczny and Pino Pérez (2008) have an important property, the success at iteration (I1), which allows stating postulates in terms of a corresponding revision operator. More precisely, given a change operator $\circ$ of the form $\circ :U\times {\mathcal{L}}_P⟶U$ defined in an epistemic space $\mathcal{E}=⟨U,\mathcal{B}⟩$ , which satisfies (I1): For all $\mathrm{\Psi }$ and $\alpha$ , there exists $n$ such that $\mathcal{B}(\mathrm{\Psi }{\circ }^n\alpha )⊢\alpha$ , we can define the operator $\star :U\times \mathcal{L}⟶U$ associated to the operator $\circ$ in the following way:

$$\mathrm{\Psi }\star \alpha =\mathrm{\Psi }{\circ }^k\alpha \text{ with }k=min\{n\in \mathbb{N}:\mathrm{\Psi }{\circ }^n\alpha ⊢\alpha \}$$

(1)

Then, improvement operators are defined as follows:

Definition 2 [Konieczny and Pino Pérez (2008)]

The operator $\circ :U\times {\mathcal{L}}_P⟶U$ is an improvement operator if it satisfies the following postulates:

(I1)

There exists $n$ such that $\mathcal{B}(\mathrm{\Psi }{\circ }^n\alpha )⊢\alpha$

(I2)

If $\mathcal{B}(\mathrm{\Psi })\wedge \alpha ⊬\mathrm{\perp }$ , then $\mathcal{B}(\mathrm{\Psi }\star \alpha )\equiv \mathcal{B}(\mathrm{\Psi })\wedge \alpha$

(I3)

If $\alpha ⊬\mathrm{\perp }$ , then $\mathcal{B}(\mathrm{\Psi }\circ \alpha )⊬\mathrm{\perp }$

(I4)

For any positive integer $n$ if ${\alpha }_i\equiv {\beta }_i$ for all $i\le n$ then $\mathcal{B}(\mathrm{\Psi }\circ {\alpha }_1\circ \cdots \circ {\alpha }_n)\equiv \mathcal{B}(\mathrm{\Psi }\circ {\beta }_1\circ \cdots \circ {\beta }_n)$

(I5)

$\mathcal{B}(\mathrm{\Psi }\star \alpha )\wedge \beta ⊢\mathcal{B}(\mathrm{\Psi }\star (\alpha \wedge \beta ))$

(I6)

If $\mathcal{B}(\mathrm{\Psi }\star \alpha )\wedge \beta ⊬\mathrm{\perp }$ , then $\mathcal{B}(\mathrm{\Psi }\star (\alpha \wedge \beta ))⊢\mathcal{B}(\mathrm{\Psi }\star \alpha )\wedge \beta$

(I7)

If $\alpha ⊢\mu$ then $\mathcal{B}((\mathrm{\Psi }\circ \mu )\star \alpha )\equiv \mathcal{B}(\mathrm{\Psi }\star \alpha )$

(I8)

If $\alpha ⊢\mathrm{\neg }\mu$ then $\mathcal{B}((\mathrm{\Psi }\circ \mu )\star \alpha )\equiv \mathcal{B}(\mathrm{\Psi }\star \alpha )$

(I9)

If $\mathcal{B}(\mathrm{\Psi }\star \alpha )⊬\mathrm{\neg }\mu$ then $\mathcal{B}((\mathrm{\Psi }\circ \mu )\star \alpha )⊢\mu$

In Konieczny and Pino Pérez (2008), a representation theorem of improvement operators in terms of TPOs over worlds is provided.

It is worth remarking that postulate (I4) is stronger than the DP postulate (R4), that is, $\mathcal{B}(\mathrm{\Psi }\circ \alpha )\equiv \mathcal{B}(\mathrm{\Psi }\circ \beta )$ whenever $\alpha \equiv \beta$ ; for a discussion on this, see Konieczny and Pino Pérez (2008). Actually, (I4) aims to code at the level of beliefs the fact that if $\alpha \equiv \beta$ , then $\mathrm{\Psi }\circ \alpha$ and $\mathrm{\Psi }\circ \beta$ are exactly the same epistemic state, which is clearly stronger than (R4) (Darwiche & Pearl, 1997).

Definition 3 (Gradual assignment)

Given an epistemic space $\mathcal{E}=⟨E,\mathcal{B}⟩$ and an operator over this space, a mapping $\mathrm{\Psi }\mapsto {⪯}_{\mathrm{\Psi }}$ associating each epistemic state $\mathrm{\Psi }\in E$ with a TPO over worlds ${⪯}_{\mathrm{\Psi }}$ is called a gradual assignment for $\circ$ if the following conditions are satisfied:

(1)

If $\omega ,{\omega }^{\prime }\in \text{⟦}\mathcal{B}(\mathrm{\Psi })\text{⟧}$ , then $\omega {\simeq }_{\mathrm{\Psi }}{\omega }^{\prime }$

(2)

If $\omega \in \text{⟦}\mathcal{B}(\mathrm{\Psi })\text{⟧}$ and ${\omega }^{\prime }\notin \text{⟦}\mathcal{B}(\mathrm{\Psi })\text{⟧}$ , then $\omega {\prec }_{\mathrm{\Psi }}{\omega }^{\prime }$

(3)

For any positive integer $n$ , if ${\alpha }_i\equiv {\beta }_i$ for any $i\le n$ ,

then ${⪯}_{\mathrm{\Psi }\circ {\alpha }_1\circ \dots \circ {\alpha }_n}={⪯}_{\mathrm{\Psi }\circ {\beta }_1\circ \dots \circ {\beta }_n}$

(4)

If $\omega ,{\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ then $\omega {⪯}_{\mathrm{\Psi }}{\omega }^{\prime }\iff \omega {⪯}_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$

(5)

If $\omega ,{\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ then $\omega {⪯}_{\mathrm{\Psi }}{\omega }^{\prime }\iff \omega {⪯}_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$

(6)

If $\omega \in \text{⟦}\alpha \text{⟧},{\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ then $\omega {⪯}_{\mathrm{\Psi }}{\omega }^{\prime }\Rightarrow \omega {\prec }_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$

We have the following representation theorem for the class of improvement operators:

Theorem 1

A change operator $\circ$ is an improvement operator if and only if there exists a gradual assignment such that for each epistemic state and each formula $\alpha$ , $\text{⟦}B(\mathrm{\Psi }\star \alpha )\text{⟧}=min(\text{⟦}\alpha \text{⟧},{\le }_{\mathrm{\Psi }})$ .

Note that conditions 4 and 5 align with Darwiche and Pearl’s conditions (CR1) and (CR2) respectively (Darwiche & Pearl, 1997). Condition 6 corresponds to condition (P) introduced in Booth and Meyer (2006), Jin and Thielscher (2007) for defining admissible DP revision operators. Consequently, the operator $\star$ associated with an improvement operator $\circ$ is an admissible DP revision operator.

Let us give some examples of operators defined on ${\mathcal{E}}_{\mathrm{tpo}}$ , the space of TPOs: The natural revision operator (Boutilier, 1996), the lexicographic revision operator (Nayak, 1994) and the one-improvement operator (Konieczny et al., 2010b; Konieczny & Pino Pérez, 2008).

Example 5

The natural revision operator ${\circ }_{\mathrm{N}}$ is defined on the epistemic space ${\mathcal{E}}_{tpo}$ mapping each tpo $\mathrm{\Psi }\in {\mathcal{E}}_{tpo}$ and each formula $\alpha$ into the tpo $\mathrm{\Psi }{\circ }_{\mathrm{N}}\alpha$ that satisfies $min(\mathrm{\Psi }{\circ }_{\mathrm{N}}\alpha )=min(\text{⟦}\alpha \text{⟧},{⪯}_{\mathrm{\Psi }})$ along with the following condition:

• (N)

If $\omega ,{\omega }^{\prime }\notin min(\text{⟦}\alpha \text{⟧},{⪯}_{\mathrm{\Psi }})$ , then $\omega {⪯}_{\mathrm{\Psi }}{\omega }^{\prime }\iff \omega {⪯}_{\mathrm{\Psi }{\circ }_{\mathrm{N}}\alpha }{\omega }^{\prime }$ ,

where ${⪯}_{\mathrm{\Psi }}$ denotes $\mathrm{\Psi }$ and ${⪯}_{\mathrm{\Psi }{\circ }_{\mathrm{N}}\alpha }$ denotes $\mathrm{\Psi }{\circ }_{\mathrm{N}}\alpha$ .

That is, Boutilier’s natural revision operator on ${\mathcal{E}}_{\mathrm{tpo}}$ selects the set of all minimal models of $\mu$ according to an input tpo and defines this set as the first level of the revised tpo, leaving the rest of the tpo unchanged.

Example 6

Nayak’s lexicographic operator ${\circ }_{\mathrm{L}}$ is defined also in the epistemic space ${\mathcal{E}}_{tpo}$ by putting $min(\mathrm{\Psi }{\circ }_{\mathrm{L}}\alpha )=min(\text{⟦}\alpha \text{⟧},\mathrm{\Psi })$ , plus conditions 4 and 5 of a gradual assignment, and:

• (L)

If $\omega \models \alpha$ and ${\omega }^{\prime }⊭\alpha$ , then $\omega {\prec }_{\mathrm{\Psi }{\circ }_{\mathrm{L}}\alpha }{\omega }^{\prime }$

Lexicographic revision reorders all models of $\alpha$ to be higher than all models of $\mathrm{\neg }\alpha$ , while preserving the internal relationships within the worlds of $\alpha$ and within the worlds of $\mathrm{\neg }\alpha$ .

Example 7

The one-improvement operator $\odot$ is defined also in the epistemic space ${\mathcal{E}}_{tpo}$ by mapping each tpo ${⪯}_{\mathrm{\Psi }}\in {\mathcal{E}}_{tpo}$ and each formula $\mu$ into the tpo ( ${⪯}_{\mathrm{\Psi }}\odot \mu$ ) that satisfies conditions 4, 5 and 6 of a gradual assignment, and the following additional conditions:

If $\omega \in \text{⟦}\alpha \text{⟧},{\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ then ${\omega }^{\prime }{\prec }_{\mathrm{\Psi }}\omega \Rightarrow {\omega }^{\prime }{⪯}_{\mathrm{\Psi }\odot \alpha }\omega$

If $\omega \in \text{⟦}\alpha \text{⟧},{\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ then ${\omega }^{\prime }{\prec \prec }_{\mathrm{\Psi }}\omega \Rightarrow \omega {⪯}_{\mathrm{\Psi }\odot \alpha }{\omega }^{\prime }$

These additional conditions characterize the behavior of $\odot$ . They state that the models of $\alpha$ at a level just above a level with models of $\mathrm{\neg }\alpha$ move down to this level.

Let us now illustrate how the behavior of these operators differs from each other.

Example 8 Adapted From Schwind et al. (2022)

Let $P=\{p,q,r\}$ . Figure 3 shows a tpo $\mathrm{\Psi }$ over worlds ³ , and the revised tpos $\mathrm{\Psi }{\circ }_{\mathrm{N}}(p\iff \mathrm{\neg }q)$ , $\mathrm{\Psi }{\circ }_{\mathrm{L}}(p\iff \mathrm{\neg }q)$ and $\mathrm{\Psi }\odot (p\iff \mathrm{\neg }q)$ . We have $Bel(\mathrm{\Psi })\equiv p\wedge q$ , $Bel(\mathrm{\Psi }{\circ }_{\mathrm{N}}(p\iff \mathrm{\neg }q))\equiv Bel(\mathrm{\Psi }{\circ }_{\mathrm{L}}(p\iff \mathrm{\neg }q))\equiv p\wedge \mathrm{\neg }q$ , but $Bel(\mathrm{\Psi }\odot (p\iff \mathrm{\neg }q))\equiv p$ and the three associated tpos are very different. Therefore, it is easy to identify scenarios (sequences of input formulas) that result in different beliefs for each tpo.

Figure 3.

The natural revision operator ${\circ }_{\mathrm{N}}$ , the lexicographic revision operator ${\circ }_{\mathrm{L}}$ and the one-improvement operator $\odot$ at work.

Note that there are other improvement operators defined in ${\mathcal{E}}_{\mathrm{ocf}}$ , the space of OCFs, which can not be represented in the space of tpos (see Schwind et al., 2022).

3. Generalized Improvement Operators

We would like to consider operators that improve only a portion of the new information (for instance, only the most plausible models of the new information). This approach could lead to a “natural improvement” operator reminiscent of Boutilier’s natural revision. To clarify this idea, we will define the operator ${\circ }_{\mathrm{N}\mathrm{I}}$ on the space ${\mathcal{E}}_{\mathrm{tpo}}$ , which emulates this behavior. First, given a tpo ${⪯}_{\mathrm{\Psi }}$ and a formula $\mu$ , let ${\mu }_{\mathrm{\Psi }}$ be a formula such that $\text{⟦}{\mu }_{\mathrm{\Psi }}\text{⟧}=min(\text{⟦}\mu \text{⟧},{⪯}_{\mathrm{\Psi }})$ . Then define ${⪯}_{\mathrm{\Psi }}{\circ }_{\mathrm{N}\mathrm{I}}\mu ={⪯}_{\mathrm{\Psi }}\odot {\mu }_{\mathrm{\Psi }}$ . Figure 4 illustrates the behavior of this operator at work.

Figure 4.

The natural improvement operator ${\circ }_{\mathrm{N}\mathrm{I}}$ at work.

It is easy to see that the operator ${\circ }_{\mathrm{N}\mathrm{I}}$ does not satisfy (I9), which corresponds to condition 6 of a gradual assignment. This is evident from Figure 4 where the world 010 remains at the same level as the world 000 after the change. However, the operator ${\circ }_{\mathrm{N}\mathrm{I}}$ satisfies postulate (I1), which is central to the improvement principle. In fact, it is clear that the operator $\star$ associated with ${\circ }_{\mathrm{N}\mathrm{I}}$ is indeed the natural revision operator ${\circ }_{\mathrm{N}}$ . Thus, it is reasonable to replace postulate (I9) with “non-worsening postulates”, those corresponding to postulates (C3) and (C4) of Darwiche and Pearl. This leads to the following definition, which extends the class of improvement operators:

Definition 4 (Generalized Improvement Operator)

The operator $\circ :U\times {\mathcal{L}}_P⟶U$ is a generalized improvement operator if it satisfies the postulates (I1-I8) plus the following ones:

(I9a)

If $\mathcal{B}(\mathrm{\Psi }\star \alpha )⊢\mu$ then $\mathcal{B}((\mathrm{\Psi }\circ \mu )\star \alpha )⊢\mu$

(I9b)

If $\mathcal{B}(\mathrm{\Psi }\star \alpha )⊬\mathrm{\neg }\mu$ , then $\mathcal{B}((\mathrm{\Psi }\circ \mu )\star \alpha )⊬\mathrm{\neg }\mu$

Obviously enough, if an operator sastisfies (I9), then it satisfies (I9a) and (I9b). As a consequence, every improvement operator is a generalized improvement operator.

From the semantic perspective, we have the following:

Definition 5

Given an epistemic space $\mathcal{E}=⟨U,\mathcal{B}⟩$ and an operator over this space, a mapping $\mathrm{\Psi }\mapsto {⪯}_{\mathrm{\Psi }}$ associating each epistemic state $\mathrm{\Psi }\in E$ with a TPO over worlds ${⪯}_{\mathrm{\Psi }}$ is called a generalized gradual assignment if it satisfies conditions 1–5 of a gradual assignment and the following conditions:

• 6a.

If $\omega \in \text{⟦}\alpha \text{⟧}$ and ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ then $\omega {\prec }_{\mathrm{\Psi }}{\omega }^{\prime }\Rightarrow \omega {\prec }_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$

6b.

If $\omega \in \text{⟦}\alpha \text{⟧}$ and ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ then $\omega {⪯}_{\mathrm{\Psi }}{\omega }^{\prime }\Rightarrow \omega {⪯}_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$

Conditions 6a and 6b above correspond to Darwiche and Pearl’s conditions (CR3) and (CR4) (Darwiche & Pearl, 1997). Then, we can prove with a technique similar to the proof of Theorem 1 the following result for the class of operators satisfying (I1):

Theorem 2

A change operator $\circ$ is a generalized improvement operator if and only if there exists a generalized gradual assignment such that $\text{⟦}B(\mathrm{\Psi }\star \alpha )\text{⟧}=min(\text{⟦}\alpha \text{⟧},{⪯}_{\mathrm{\Psi }})$ .

It is easy to see that if $\circ$ is a generalized improvement operator, then its corresponding operator $\star$ is a DP revision operator. More interestingly, we obtain the following correspondence:

Theorem 3

An operator is a generalized improvement operator satisfying (R1) if and only if it is a DP revision operator satisfying (I4).

4. Rational Improvement Operators

The goal of this section is to better understand the behavior of various improvement operators within a specific epistemic space: The space of rational rankings. Given that the order on $\mathbb{Q}$ is dense and unbounded, it more naturally captures the dynamics of improvement operators. The density of $\mathbb{Q}$ is particularly advantageous because it enables the insertion of an intermediate layer between two existing layers without altering their ranks. This capability is not possible with OCFs when the layers are consecutive. This approach aligns with the principle of improving without changing the rank of models that contradict the new information.

Although rankings into real numbers have been studied in Spohn (2009), there are key differences, which we will explore in Section 7.

A rational ranking is a function $r:\mathrm{\Omega }\to \mathbb{Q}$ . The set of all rational rankings is denoted by $\mathcal{R}(\mathbb{Q})$ . For a given rational ranking $r$ , we denote $r_k$ as the preimage $r^{-1}(k)$ for $k\in \mathbb{Q}$ , that is, $r_k=r^{-1}(k)=\{\omega \in \mathrm{\Omega }:r(\omega )=k\}$ . The (possibly empty) set $r_k$ represents the $k$ th level of the ranking $r$ .

Given any consistent formula $\alpha$ and a ranking $r$ , we define the rank of $\alpha$ in $r$ , denoted by $r(\alpha )$ , as:

$$r(\alpha )=max\{k\in \mathbb{Q}:r_k\cap \text{⟦}\alpha \text{⟧}\ne \mathrm{\varnothing }\}$$

(2)

The following notation, $top(\alpha )$ , which denotes the set of most plausible models of $\alpha$ in $r$ , will be useful:

$$top(\alpha ):=r_{r(\alpha )}\cap \text{⟦}\alpha \text{⟧}$$

Definition 6

The epistemic space of rational rankings, ${\mathcal{E}}_{\mathbb{Q}}=⟨U,\mathcal{B}⟩$ is defined by taking $U$ as the set of rational rankings and $\mathcal{B}$ as the function from $U$ to ${{\mathcal{L}}_P}^{*}$ defined as:

$$\omega \in \text{⟦}\mathcal{B}(r)\text{⟧}⟺r(\omega )=r(\mathrm{\top })$$

(3)

The intended meaning of a rational ranking $r$ as an epistemic state is that $r({\omega }^{\prime })<r(\omega )$ indicates that the world $\omega$ is more plausible than the world ${\omega }^{\prime }$ . The beliefs associated with $r$ correspond to a formula whose models are precisely those with the highest rank.

In this work, we adopt the perspective that the value assigned to a world in a rational ranking directly represents its plausibility, in contrast to OCFs. We find it natural to associate higher ranks with greater plausibility.

A significant difference from OCFs is that rational rankings are not normalized, meaning that a rational ranking can have beliefs with arbitrarily high ranks. This characteristic allows for the plausibility of models representing new information to be increased without altering the plausibility of models representing the negation of that information.

Definition 7

Given a rational ranking $r$ , its underlying preorder ${⪯}_r$ is defined by:

$$\omega {⪯}_r{\omega }^{\prime }⟺r({\omega }^{\prime })\le r(\omega )$$

(4)

Definition 8

The canonical assignment $\mathfrak{c}\mathfrak{a}$ on rankings is the function $r\stackrel{\mathfrak{c}\mathfrak{a}}{⟼}{⪯}_r$ mapping each ranking $r$ to its underlying preorder ${⪯}_r$ .

Observation 1

Since $\mathcal{B}(r)=r(\mathrm{\top })$ , we have that

$$\mathcal{B}(r)=max(\text{⟦}\mathrm{\top }\text{⟧},{⪯}_r)$$

Definition 9 (Rational Improvement Operators)

$\circ$ is a rational improvement operator if there is a rational number $c>0$ such that the following properties are satisfied for all rational ranking $r$ , all consistent formulae $\alpha$ and all worlds $\omega$ , ${\omega }^{\prime }$ :

(RI1)

If $\omega \in \text{⟦}\alpha \text{⟧}$ then $r(\omega )\le (r\circ \alpha )(\omega )$

(RI2)

there is a ${\omega }^{″}\in \text{⟦}\alpha \text{⟧}$ such that $r({\omega }^{″})+c<(r\circ \alpha )({\omega }^{″})$

(RI3)

if $\omega \notin \text{⟦}\alpha \text{⟧}$ , then $r(\omega )=(r\circ \alpha )(\omega )$

(RI4)

if $\omega ,{\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ then $r(\omega )\le r({\omega }^{\prime })⟺(r\circ \alpha )(\omega )\le (r\circ \alpha )({\omega }^{\prime })$

(RI5)

If $\alpha \equiv \beta$ and $\omega \in \text{⟦}\alpha \text{⟧}$ then $(r\circ \alpha )(\omega )=(r\circ \beta )(\omega )$

Postulate (RI1) states that for any ranking function $r$ , the plausibility of the models of $\alpha$ does not decrease after a rational improvement by $\alpha$ . (RI2) asserts that at least one model of $\alpha$ sees its plausibility strictly increased after a rational improvement by $\alpha$ . Moreover, this increase has a lower bound for any formula $\alpha$ in any rational ranking $r$ . (RI3) requires that a rational improvement by $\alpha$ does not alter the plausibility of the models of $\mathrm{\neg }\alpha$ . This can be seen as a strengthening of condition 5 of a gradual assignment, which requires that the relative plausibility between the models of $\mathrm{\neg }\alpha$ is not altered after revision by $\alpha$ . This is also a somewhat extreme account of minimal change principle: While we improve the plausibility of new information $\alpha$ , we are specifically told not to move any possible world unrelated to $\alpha$ . (RI4) ensures that the relative order of the models of $\alpha$ should remain unchanged after a rational improvement by $\alpha$ . Lastly, (RI5), together with (RI3), ensures the independence of syntax: A rational improvement by a formula or an equivalent one should result in the same ranking function.

It is important to note that postulates (RI1-RI5) are both natural and flexible, allowing for the definition of a wide variety of rational improvement operators, including those with a uniform behavior. But, one could also define an operator $\circ$ such that, for some models of $\alpha$ , the plausibility increases only from the second time $\alpha$ is encountered in a series of consecutive inputs of $\alpha$ : $r(\omega )=(r\circ \alpha )(\omega )<((r\circ \alpha )\circ \alpha )(\omega )$ .

The next result is crucial in proving that rational improvement operators are indeed generalized improvement operators.

Theorem 4

If $\circ$ is a rational improvement operator, then it satisfies (I1).

Proof.

Let $\mathrm{\Delta }(\alpha )=\lceil r(\mathrm{\top })-min\{r(\omega )\in \mathbb{Q}:\omega \in \text{⟦}\alpha \text{⟧}\}\rceil$ . Since each element in $\text{⟦}\alpha \text{⟧}$ is a binary sequence, we can well-order this set with lexicographic order. Now define the function:

$$\begin{array}{rl}f:\mathbb{N}& ⟶\text{⟦}\alpha \text{⟧}\\ k& \mapsto \underset{{<}_{\text{lex}}}{min}\{\omega \in \text{⟦}\alpha \text{⟧}:(r{\circ }^{k+1}\alpha )(\omega )>(r{\circ }^k\alpha )(\omega )+c\}\end{array}$$

This function is well defined since (RI2) implies that the set $\{\omega \in \text{⟦}\alpha \text{⟧}:(r{\circ }^{k+1}\alpha )(\omega )>(r{\circ }^k\alpha )(\omega )+c\}$ is non-empty. Since $\text{⟦}\alpha \text{⟧}$ is finite, the pigeonhole principle implies there is some ${\omega }_0\in \text{⟦}\alpha \text{⟧}$ with $|f^{-1}({\omega }_0)|={\mathrm{\aleph }}_0$ . Hence

$$f^{-1}({\omega }_0)=\{k_1,k_2,k_3,\dots \}$$

Now, using (RI1), it is easy to that

$$(r{\circ }^{k_n}\alpha )({\omega }_0)>r({\omega }_0)+nc$$

By the Archimedean property, there is a natural number $N$ such that $Nc>\mathrm{\Delta }(\alpha )$ , hence

$$(r{\circ }^{k_N}\alpha )({\omega }_0)>r({\omega }_0)+Nc>r({\omega }_0)+\mathrm{\Delta }(\alpha )>r(\mathrm{\top })$$

As $(r{\circ }^{k_N}\alpha )({\omega }_0)\ge (r{\circ }^{k_N}\alpha )({\omega }_0)$ , we have that $(r{\circ }^{k_N}\alpha )({\omega }_0)>r(\mathrm{\top })$ . Since $(r{\circ }^n\alpha )(\omega )=r(\omega )$ for every $\omega \notin \text{⟦}\alpha \text{⟧}$ and every $n\in \mathbb{N}$ , we have that $\omega \notin \text{⟦}\alpha \text{⟧}⟹(r{\circ }^n\alpha )(\omega )\le r(\mathrm{\top })$ . Thus, the models in $(r{\circ }^{k_N}\alpha )(\mathrm{\top })$ are a subset of $\text{⟦}\alpha \text{⟧}$ hence $\mathcal{B}(r{\circ }^{k_N}\alpha )⊢\alpha$ . That is, $\circ$ satisfies (I1).

We can now state that:

Theorem 5

Every rational improvement operator is a generalized improvement operator.

The idea of the proof is to show that the canonical assignment $r\mapsto {⪯}_r$ is a generalized gradual assignment. This, along with Theorem 4, allows us to apply Theorem 2 to reach our conclusion.

Proof.

Let $\circ$ be a rational improvement operator. Theorem 4 implies that the associated revision operator $\star$ is well-defined. Hence $\mathcal{B}(r\star \alpha )⊢\alpha$ , thus $\text{⟦}\mathcal{B}(r\star \alpha )\text{⟧}\subseteq \text{⟦}\alpha \text{⟧}$ . Since $\text{⟦}\mathcal{B}(r\star \alpha )\text{⟧}=r_{r(\mathrm{\top })}$ , it follows from the definition of ${⪯}_r$ that

$$\text{⟦}\mathcal{B}(r\star \alpha )\text{⟧}=min(\text{⟦}\mathrm{\top }\text{⟧},{⪯}_r)$$

We now want to prove that $r\mapsto {⪯}_r$ is a generalized gradual assignment for $\circ$ . Properties (1.) and (2.) are immediate from the definition of ${⪯}_r$ and the definition of $\mathcal{B}(r)$ . To prove Property (3.), it is enough to prove it for $n=1$ and then apply an inductive argument. To do so, suppose $\alpha \equiv \beta$ . Then, (RI3) along with (RI5) implies that $r\circ \alpha =r\circ \beta$ , thus ${⪯}_{(r\circ \alpha )}={⪯}_{(r\circ \beta )}$ . Property (4.) follows directly from (RI4). Property (5.) is an immediate consequence of (RI3). Properties (6a.) and (6b.) follow from (RI1) and (RI3). Therefore, $r\mapsto {⪯}_r$ is a generalized gradual assignment for $\circ$ and Theorem 2 implies that $\circ$ is a generalized improvement operator.

Observation 2

Since every rational improvement operator $\circ$ is a generalized improvement operator, it follows that its associated operator $\star$ is a DP revision operator.

5. Examples and Properties

In this section, we introduce several families of rational improvement operators, discuss some of their properties, and provide characterizations for some of them.

Example 9

Let $t\in {\mathbb{Q}}^{+}$ . The $t$ -translation is the operator

$${\oplus }_t:U\times \mathcal{L}⟶U$$

such that, if $r$ is a ranking function and $\alpha$ is a sentence, the operator ${\oplus }_t$ is defined as

$$(r{\oplus }_t\alpha )(\omega )=\{\begin{array}{ll}r(\omega )& \text{ if }\omega \notin \text{⟦}\alpha \text{⟧}\\ r(\omega )+t& \text{ if }\omega \in \text{⟦}\alpha \text{⟧}\end{array}$$

Satisfaction of (RI1), (RI3) and (RI4) is a direct consequence of the definition of ${\oplus }_t$ . Taking any $\omega \in \text{⟦}\alpha \text{⟧}$ and $c=t/2$ , we get that $r(\omega )+c<r(\omega )+t=(r{\oplus }_t\alpha )(\omega )$ , hence ${\oplus }_t$ satisfies (RI2). Lastly, if $\alpha \equiv \beta$ we have that $(r{\oplus }_t\alpha )(\omega )=(r{\oplus }_t\beta )(\omega )=r(\omega )+t$ for every $\omega \in \text{⟦}\alpha \text{⟧}$ since $\text{⟦}\alpha \text{⟧}=\text{⟦}\beta \text{⟧}$ , and we also have $(r{\oplus }_t\alpha )(\omega )=(r{\oplus }_t\beta )(\omega )=r(\omega )$ for every $\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ since $\text{⟦}\mathrm{\neg }\alpha \text{⟧}=\text{⟦}\mathrm{\neg }\beta \text{⟧}$ , so (RI5)) is satisfied. Therefore, ${\oplus }_t$ is a rational improvement operator.

These operators are named $t$ -translations because they uniformly move the entire set of models of $\alpha$ “upwards”. Translation operators are reminiscent of $(A\to x)$ -conditionalization (Spohn, 2009). However, while conditionalizations are revision operators, translations are not.

Example 10

We can model the case presented in Example 3 using $t$ -translations. First, fix a positive rational number $t_0$ . Let $\alpha \in {\mathcal{L}}_P$ be a formula such that $\text{⟦}\alpha \text{⟧}$ is exactly the set of all possible worlds where treatment A cures common cold. Suppose $r$ is a ranking representing the epistemic state of Dr. Roberts when he began his practice. Assume $r$ is a state of “complete ignorance” that is all possible worlds are in the same level. Moreover, we can assume they are at level $0$ . Suppose $r^{\prime }$ is the ranking representing the epistemic state of Dr. Roberts right before he observes some scenario where treatment A did not cure common cold. Then $r^{\prime }=r{\oplus }_{t_0}^n\alpha$ for some (large) $n$ . We also know that $\alpha$ is a part of Dr. Roberts beliefs, thus $r_{r^{\prime }(\mathrm{\top })}^{\prime }\subseteq \text{⟦}\alpha \text{⟧}$ with $r^{\prime }(\mathrm{\top })=n{t}_0$ . Moreover, $r^{\prime }(\omega )=r(\omega )=0$ for every $\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ . Hence $(r^{\prime }{\oplus }_{t_0}\mathrm{\neg }\alpha )(\omega )=t_0$ for every $\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ and $(r^{\prime }{\oplus }_{t_0}\mathrm{\neg }\alpha )(\omega )=n{t}_0$ for every $\omega \in \text{⟦}\alpha \text{⟧}$ .

We can also model the situation showed in Example 4 using $t$ -translations as follows:

Example 11

First, fix a positive rational number $\epsilon$ such that $\epsilon =1/n$ where $n$ is big enough. Let $\alpha \in {\mathcal{L}}_P$ be a formula such that $\text{⟦}\alpha \text{⟧}$ is exactly the set of all possible worlds where the English is fluent. As before, suppose $r$ is a ranking representing the epistemic state where Carlos begin his training. We can assume that $r$ is a state of “neutral” where all possible worlds are at level $0$ . After $k$ days of training, we can expect that the level of Carlos has improved to $k\epsilon$ . This can be modeled by $r{\oplus }_{\epsilon }^k\alpha$ . Thus, in that state the English level of Carlos is $k\epsilon$ . Then, if we suppose that the level of the C2 certificate is $M$ , a very large number, we know (by the Archimedean property) that there will be a number $k$ of days such that $k\epsilon >M$ . Therefore the beliefs in the state $r{\oplus }_{\epsilon }^k\alpha$ is $\alpha$ with a degree bigger that $M$ , that is in such state Carlos is assured to obtain the C2 certificate.

It is interesting to note that the $t$ -translations satisfy the following interesting commutativity property:

• (Com)

$(r\circ \alpha )\circ \beta =(r\circ \beta )\circ \alpha$

This property a two-step adaptation to rational rankings of the commutativity property introduced in Schwind and Konieczny (2020).

When an operator satisfies (Com) for every ranking $r$ and formulas $\alpha$ and $\beta$ , it is called a commutative operator.

Proposition 1

The operator ${\oplus }_t$ is a commutative operator.

Proof.

Let $\omega \in \mathrm{\Omega }$ . The proof is direct by (RI3) in the case when $\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}\cup \text{⟦}\mathrm{\neg }\beta \text{⟧}$ .

Now suppose that $\omega \in \text{⟦}\alpha \text{⟧}\setminus \text{⟦}\beta \text{⟧}$ . Then

$$\begin{array}{rlr}((r{\oplus }_t\alpha ){\oplus }_t\beta )(\omega )& =(r{\oplus }_t\alpha )(\omega )& \mathbf{\text{(RI3)}}\\ & =r(\omega )+t\end{array}$$

and

$$\begin{array}{rl}((r{\oplus }_t\beta ){\oplus }_t\alpha )(\omega )& =(r{\oplus }_t\beta )(\omega )+t\\ & =r(\omega )+t& \mathbf{\text{(RI3)}}\end{array}$$

Hence,

$$((r{\oplus }_t\alpha ){\oplus }_t\beta )(\omega )=((r{\oplus }_t\beta ){\oplus }_t\alpha )(\omega )=r(\omega )+t$$

The proof is similar for the case when $\omega \in \text{⟦}\beta \text{⟧}\setminus \text{⟦}\alpha \text{⟧}$ . The remaining case is when $\omega \in \text{⟦}\alpha \text{⟧}\cap \text{⟦}\beta \text{⟧}$ :

$$\begin{array}{rlr}((r{\oplus }_t\alpha ){\oplus }_t\beta )(\omega )& =(r{\oplus }_t\alpha )(\omega )+t& \omega \in \text{⟦}\beta \text{⟧}\\ & =(r(\omega )+t)+t& \omega \in \text{⟦}\alpha \text{⟧}\\ & =r(\omega )+2t\\ ((r{\oplus }_t\beta ){\oplus }_t\alpha )(\omega )& =(r{\oplus }_t\beta )(\omega )+t& \omega \in \text{⟦}\alpha \text{⟧}\\ & =(r(\omega )+t)+t& \omega \in \text{⟦}\beta \text{⟧}\\ & =r(\omega )+2t\end{array}$$

Thus for every $\omega \in \mathrm{\Omega }$ :

$$((r{\oplus }_t\alpha ){\oplus }_t\beta )(\omega )=((r{\oplus }_t\beta ){\oplus }_t\alpha )(\omega )$$

This shows that ${\oplus }_t$ satisfies (Com).

Observation 3

By the previous proposition and the results in Schwind and Konieczny (2020) (Proposition 9 therein) the t-translation operators can not be representable as improvement operators in the space of tpos. And the impossibility also remains for the revision operators corresponding to the translations.

An interesting behavior of this family of operators is captured by the following definition:

Definition 10

A rational improvement operator $\circ$ is a Nayak operator iff for each formula $\alpha$ , there is some $N\in \mathbb{N}$ such that:

$$\omega \in \text{⟦}\alpha \text{⟧},{\omega }^{\prime }\notin \text{⟦}\alpha \text{⟧}⟹(r{\circ }^{\mathrm{N}}\alpha )({\omega }^{\prime })<(r{\circ }^{\mathrm{N}}\alpha )(\omega )$$

These operators are named as such because, with a sufficient number of iterations, the underlying preorders behave similarly to the lexicographical revision proposed by Nayak (Nayak, 1994).

Observation 4

Notice that $\{{\oplus }_t:t\in {\mathbb{Q}}^{+}\}$ is a family of Nayak operators.

However, not all Nayak operators belong to this family. For instance, in Example 15, one can see operators that fall outside this classification.

More generally, we have the following definition:

Definition 11

An operator $*$ defined in the space of rational rankings represents an operator $\circ$ defined in the space of TPOs when for any ranking $r$ and any formula $\alpha$ we have ${⪯}_{r*\alpha }={⪯}_r\circ \alpha$ , that is, if the underlying TPO of $r*\alpha$ coincides with the TPO of changing by $\alpha$ the underlying TPO of $r$ with the operator $\circ$ . In other words, if the following diagram commutes:

$$\begin{array}{ccccccc}& r& & \stackrel{*,\alpha }{\to }& & r*\alpha & \\ \mathfrak{c}\mathfrak{a}& \downarrow & & & & \downarrow & \mathfrak{c}\mathfrak{a}\\ & {⪯}_r& & \stackrel{\circ ,\alpha }{\to }& & {⪯}_{r*\alpha }& \end{array}$$

(5)

where $\mathfrak{c}\mathfrak{a}$ is the canonical assignment of Definition 8.

An interesting observation is that when a revision operator $*$ defined on rational rankings represents a revision operator $\circ$ defined on TPOs in the sense of the previous definition then $*$ is an instantiation of $\circ$ in the sense of Schwind et al. (2022). However, previous definition not only concerns revision operators, it concerns also improvement operators (Figures 5 to 7).

Figure 5.

Iterative behavior of ${\oplus }_t$ with $t=2$ .

Figure 6.

Iterative behavior of ${*}_{\text{top}-t}$ with $t=2$ .

Figure 7.

Iterative behavior of ${\circ }_Z$ .

Now, given a translation operator ${\oplus }_t$ , define the operator ${\overline{\star }}_t$ associated to it, in the following way: First, for a ranking $r$ and a formula $\alpha$ define $\underset{\_}{r}(\alpha )$ as $min\{r(\omega ):\omega \in \text{⟦}\alpha \text{⟧}\}$ and second put $r{\overline{\star }}_t\alpha =r{\oplus }_t^k\alpha$ where $k=min\{n:\underset{\_}{r{\oplus }_t^n\alpha }(\alpha )\}>r{\oplus }_t^n\alpha (\mathrm{\neg }\alpha )$ . Note that ${\overline{\star }}_t$ is well defined because ${\oplus }_t$ is a translation and moreover it is easy to see that ${\overline{\star }}_t$ represents the lexicographical revision operator ${\circ }_{\mathrm{L}}$ of Nayak within the meaning of Definition 11.

Another interesting family of rational operators exhibits behavior inspired by Boutilier’s natural revision operator (Boutilier, 1996). First let us define what is a Boutilier operator in our framework.

Definition 12

A rational improvement operator $\circ$ is a Boutillier operator if the operator $\star$ associated to it, represents the operator ${\circ }_{\mathrm{N}}$ , that is, the diagram 5 holds for $\star$ and ${\circ }_{\mathrm{N}}$ .

Example 12

Given $t>0$ with $t\in \mathbb{Q}$ , the top- $t$ -translation is the operator

$${*}_{\text{top}-t}:U\times \mathcal{L}⟶U$$

assigning to each ranking $r$ and each sentence $\alpha$ the ranking ${*}_{\text{top}-t}$ given by

$$(r{*}_{\text{top}-t}\alpha )(\omega )=\{\begin{array}{ll}r(\omega )+t& \text{ if }\omega \in r_{r(\alpha )}\cap \text{⟦}\alpha \text{⟧}\\ r(\omega )& \text{ otherwise }\end{array}$$

It is easy to see from this definition that ${*}_{\text{top}-t}$ is a rational improvement operator. Moreover, the $\star$ operator associated to ${*}_{\text{top}-t}$ represents the operator ${\circ }_{\mathrm{N}}$ of revision natural in the sense of Definition 11.

Summarizing we have:

Observation 5

$\{{*}_{\text{top}-t}:t\in {\mathbb{Q}}^{+}\}$ is a family of Boutillier operators.

However, as we will see in Example 14, not all Boutilier operators belong to this family.

Proposition 2

The operator ${*}_{\text{top}-t}$ is not commutative.

Proof.

Consider a ranking $r$ and formulas $\alpha ,\beta$ such that $r(\alpha )=r(\beta )$ . Let $n=r(\alpha )$ , and let $\omega \in \text{⟦}\alpha \text{⟧}\setminus \text{⟦}\beta \text{⟧}$ , ${\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}\cap \text{⟦}\beta \text{⟧}$ such that $r(\omega )=r({\omega }^{\prime })=n$ . Now

$$\begin{array}{rl}(r{*}_{\text{top}-t}\alpha )(\omega )& =(r{*}_{\text{top}-t}\alpha )({\omega }^{\prime })\\ & =n+t\end{array}$$

Now since ${\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}\cap \text{⟦}\beta \text{⟧}$ we have

$$(r{*}_{\text{top}-t}\alpha )(\alpha )=(r{*}_{\text{top}-t}\alpha )(\beta )=n+t$$

and $((r{*}_{\text{top}-t}\alpha ){*}_{\text{top}-t}\beta )(\omega )=n+t$ since $\omega \notin \text{⟦}\beta \text{⟧}$ . However, $(r{*}_{\text{top}-t}\beta )(\omega )=n$ and $(r{*}_{\text{top}-t}\beta )({\omega }^{\prime })=n+t$ . As ${\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ , this means that $(r{*}_{\text{top}-t}\beta )(\alpha )=n+t$ , hence $(r{*}_{\text{top}-t}\beta )(\omega )\ne (r{*}_{\text{top}-t}\beta )(\alpha )$ and $((r{*}_{\text{top}-t}\beta ){*}_{\text{top}-t}(\alpha ))(\omega )=n$ . Therefore

$$((r{*}_{\text{top}-t}\alpha ){*}_{\text{top}-t}(\beta ))(\omega )\ne ((r{*}_{\text{top}-t}\beta ){*}_{\text{top}-t}(\alpha ))(\omega )$$

It is important to note that simply increasing the plausibility of models of new information does not necessarily result in a rational improvement operator. The manner in which this increase is implemented is crucial. The following example illustrates this point.

Example 13

Consider the operator ${\circ }_Z$ that associates each pair $(r,\alpha )$ with a new ranking $r{\circ }_Z\alpha$ defined as follows: If $r(\alpha )\ne r(\mathrm{\top })$ , then

$$(r{\circ }_Z\alpha )(\omega )=\{\begin{array}{ll}r(\omega )& \text{if }\omega \notin \text{⟦}\alpha \text{⟧}\\ r(\omega )+{\displaystyle \frac{r(\mathrm{\top })-r(\alpha )}{2}}& \text{if }\omega \in \text{⟦}\alpha \text{⟧}\end{array}$$

and if $r(\alpha )=r(\mathrm{\top })$ :

$$(r{\circ }_Z\alpha )(\omega )=\{\begin{array}{ll}r(\omega )& \text{if }\omega \notin \text{⟦}\alpha \text{⟧}\\ r(\omega )+1& \text{if }\omega \in \text{⟦}\alpha \text{⟧}\end{array}$$

Notice there is no $c\in {\mathbb{Q}}^{+}$ such that this operator satisfies (RI2), so it is not a rational improvement operator. Clearly, when $r(\alpha )<r(\mathrm{\top })$ , the models of $\alpha$ are “improving” with repeated applications of ${\circ }_Z$ . However, this operator does not generally satisfy (I1), meaning there is no associated $\star$ revision operator. Therefore, ${\circ }_Z$ is not an improvement operator in the usual sense. This demonstrates that simply increasing the ranks of models of new information is insufficient to define an improvement operator.

Operators exhibiting this behavior can be referred to as Zeno operators, as their behavior resemble the phenomena of Zeno Paradox. There is a goal that is approached but never fully achieved: If $r(\alpha )<r(\mathrm{\top })$ and $\omega \in \text{⟦}\alpha \text{⟧}$ then for every $n\in \mathbb{N}$ , $(r{\circ }^n\alpha )(\omega )<r(\mathrm{\top })$ .

We can use the concept from Example 13 to define a Boutilier operator that is not part of the family described in Example 12. In Example 14, the operator is designed so that the highest-ranked models of $\alpha$ always improve by $t$ , while some lower-ranked models of $\alpha$ move in a Zeno-like manner.

Example 14

The idea of the operator ${*}_t$ , which we are going to define, is very simple: The top models of $\alpha$ improve by $t$ . The “isolated” models of $\alpha$ (those in a layer with only models of $\alpha$ ) improve by half of the distance to the consecutive upward rank, while the rest of the models don’t move. More precisely, let $q_1,q_2,\dots ,q_m$ be the image of the ranking $r$ where $q_i>qi+1$ . Define $t_i=(q_i-q_{i+1})/2$ . Then, fix $t$ as a positive rational number and define ${*}_t$ as follows:

1. (1)

$(r{*}_t\alpha )(\omega )=r(\omega )+t$ if $\omega \in top(\alpha )$ ,

(2)

$(r{*}_t\alpha )(\omega )=r(\omega )+t_i$ if $r(\omega )=q_{i+1},\omega \in \text{⟦}\alpha \text{⟧}$ and there is no ${\omega }^{\prime }\notin \text{⟦}\alpha \text{⟧}$ with $r({\omega }^{\prime })=q_{i+1}$ ,

(3)

$(r{*}_t\alpha )(\omega )=r(\omega )$ in any other case.

It is easy to see that ${*}_t$ is a rational improvement operator. Moreover, it is clear from the definition of ${*}_t$ that it is a Boutillier operator because its associated $\star$ operator represents ${\circ }_{\mathrm{N}}$ .

With a similar idea we can define a Nayak operator which is not a $t$ -translation for any $t\in {\mathbb{Q}}^{+}$ .

Example 15

The idea of the operator ${\oplus }_{tt}$ we are about to define is very simple: The top models of $\alpha$ improve by $t$ . The other models of $\alpha$ improve by $t/2$ . More precisely,

$$(r{\oplus }_{tt}\alpha )(\omega )=\{\begin{array}{ll}r(\omega )+t& \text{ if }\omega \in top(\alpha )\\ r(\omega )+t/2& \text{ if }\omega \in \text{⟦}\alpha \text{⟧}\setminus top(\alpha )\\ r(\omega )& \text{ if }\omega \notin \text{⟦}\alpha \text{⟧}\end{array}$$

It is easy to see that ${\oplus }_{tt}$ is a rational improvement operator. And it is quite simple to verify that it is a Nayak operator different from ${\oplus }_t$ for any $t\in {\mathbb{Q}}^{+}$ .

Although epistemic states always change with applications of rational improvement operators, the underlying preorders have a fixed point.

Theorem 6

Let $\circ$ be a rational improvement operator, $r$ a rational ranking and $\alpha$ a propositional formula. Then there is some $m\in \mathbb{N}$ such that ${⪯}_{r{\circ }^n\alpha }={⪯}_{r{\circ }^m\alpha }$ for every $n\ge m$ .

Proof.

Let $U_{\mathrm{tpo}}$ be the set of all TPOs on $\mathrm{\Omega }$ . Since $\mathrm{\Omega }$ is a finite set, $U_{\mathrm{tpo}}$ is also a finite set. Now, define a function

$$\begin{array}{rl}f:\mathbb{N}& ⟶U_{\mathrm{tpo}}\\ k& \mapsto {⪯}_{r{\circ }^k\alpha }\end{array}$$

Hence, there is some preorder $⪯$ on $\mathrm{\Omega }$ that has an infinite preimage $S\subseteq \mathbb{N}$ . Suppose $m$ is the least element of $S$ . We are going to prove, using induction, that $S=\{n\in \mathbb{N}:m\le n\}$ , that is, $S$ is a final segment of $\mathbb{N}$ thus ${⪯}_{r{\circ }^m\alpha }$ is the desired fixed point.

Suppose $k\in S$ , then $⪯={⪯}_{r{\circ }^k\alpha }$ . Given any two $\omega ,{\omega }^{\prime }$ we have to prove that $\omega {⪯}_{r{\circ }^k\alpha }{\omega }^{\prime }⟺\omega {⪯}_{r{\circ }^{k+1}\alpha }{\omega }^{\prime }$ . This is immediate if $\omega ,{\omega }^{\prime }\notin \text{⟦}\alpha \text{⟧}$ by (RI3) and when $\omega ,{\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ by (RI4).

If $\omega \in \text{⟦}\alpha \text{⟧}$ and ${\omega }^{\prime }\notin \text{⟦}\alpha \text{⟧}$ :

$$\omega {⪯}_{r{\circ }^k\alpha }{\omega }^{\prime }⟺(r{\circ }^k\alpha )({\omega }^{\prime })\le (r{\circ }^k\alpha )(\omega )$$

(6)

Since ${\omega }^{\prime }\notin \text{⟦}\alpha \text{⟧}$ we have that $(r{\circ }^k\alpha )({\omega }^{\prime })=(r{\circ }^{k+1}\alpha )({\omega }^{\prime })$ and since $\omega \in \text{⟦}\alpha \text{⟧}$ we have that $(r{\circ }^k\alpha )(\omega )\le (r{\circ }^{k+1}\alpha )(\omega )$ , thus $(r{\circ }^k\alpha )({\omega }^{\prime })\le (r{\circ }^k\alpha )(\omega )$ implies that $(r{\circ }^{k+1}\alpha )({\omega }^{\prime })\le (r{\circ }^{k+1}\alpha )(\omega )$ hence $\omega {⪯}_{r{\circ }^k\alpha }{\omega }^{\prime }⟹\omega {⪯}_{r{\circ }^{k+1}\alpha }{\omega }^{\prime }$ .

If $\omega \notin \text{⟦}\alpha \text{⟧}$ , ${\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ and $\omega {⪯}_{r{\circ }^k\alpha }{\omega }^{\prime }$ , suppose $\omega {⋠}_{r{\circ }^{k+1}\alpha }{\omega }^{\prime }$ . Then $(r{\circ }^{k+1}\alpha )(\omega )<(r{\circ }^{k+1}\alpha )({\omega }^{\prime })$ . Since $S$ is infinite, there is some $p>k+1$ such that ${⪯}_{r{\circ }^p\alpha }={⪯}_{r{\circ }^k\alpha }$ that is such that $(r{\circ }^p\alpha )({\omega }^{\prime })\le (r{\circ }^p\alpha )(\omega )$ .

Since $\omega \notin \text{⟦}\alpha \text{⟧}$ , we have that $(r{\circ }^k\alpha )(\omega )=(r{\circ }^n\alpha )(\omega )$ for every $n\in \mathbb{N}$ thus

$$\begin{array}{rl}(r{\circ }^p\alpha )({\omega }^{\prime })& \le (r{\circ }^p\alpha )(\omega )\\ & =(r{\circ }^{k+1}\alpha )(\omega )\\ & <(r{\circ }^{k+1}\alpha )({\omega }^{\prime })\end{array}$$

thus $(r{\circ }^p\alpha )({\omega }^{\prime })<(r{\circ }^{k+1}\alpha )({\omega }^{\prime })$ . Since $p>k+1$ , there is some $\ell \in \mathbb{N}$ , $n>1$ , such that $p=k+\ell$ , therefore

$$(r{\circ }^{k+\ell }\alpha )({\omega }^{\prime })<(r{\circ }^{k+1}\alpha )({\omega }^{\prime })$$

which is impossible because of (RI1).

Hence, for every pair $\omega ,{\omega }^{\prime }\in \mathrm{\Omega }$ and every $k\in S$ :

$$\omega {⪯}_{r{\circ }^k\alpha }{\omega }^{\prime }⟹\omega {⪯}_{r{\circ }^{k+1}\alpha }{\omega }^{\prime }$$

In order to prove the reciprocal statement, suppose $\omega {⋠}_{r{\circ }^k\alpha }{\omega }^{\prime }$ . Since ${⪯}_{r{\circ }^k\alpha }$ is a TPO, this implies that ${\omega }^{\prime }{\prec }_{r{\circ }^k\alpha }\omega$ , but by the previous discussion this implies ${\omega }^{\prime }{\prec }_{r{\circ }^{k+1}\alpha }\omega$ , hence $\omega {⋠}_{r{\circ }^{k+1}\alpha }{\omega }^{\prime }$ . Thus we have proved $\omega {⪯}_{r{\circ }^{k+1}\alpha }{\omega }^{\prime }⟹\omega {⪯}_{r{\circ }^k\alpha }{\omega }^{\prime }$ by contraposition.

Therefore, for every $k>m$ we have that ${⪯}_{r{\circ }^k\alpha }={⪯}_{r{\circ }^{k+1}\alpha }$ , that is, $S$ is a final segment of $\mathbb{N}$ as we wanted to prove.

Let us call $m_0$ the minimum $m$ satisfying Theorem 6. Then the following observation is easy to see:

Observation 6

$m_0\ge k$ where

$$k=min\{n\in \mathbb{N}:r{\circ }^n\alpha ⊢\alpha \}$$

When $\circ$ is a Boutillier operator $m_0=k$ . If $\circ$ is Nayak, then generally $k<m_0$ .

Example 16

Consider the relation ‘ $\equiv$ ’ of logical equivalence between formulas in ${\mathcal{L}}_P$ . Since $\mathrm{\wp }(\mathrm{\Omega })$ is a finite set and each formula in ${\mathcal{L}}_P$ can be identified with its set of models, the logical equivalence relation partitions ${\mathcal{L}}_P$ on a finite set of equivalence classes.

Let $w$ be a function mapping each equivalence class in ${\mathcal{L}}_P{/}_{\equiv }$ to a positive rational number. Denote by $t_{\alpha }$ the image of the equivalence class of $\alpha$ via $w$ . Now we define the operator ${\oplus }_w$ :

$$(r{\oplus }_w\alpha )(\omega )=\{\begin{array}{ll}r(\omega )+t_{\alpha }& \text{if }\omega \in \text{⟦}\alpha \text{⟧}\\ r(\omega )& \text{else}\end{array}$$

(7)

Suppose $t_0=min\{t\in \mathbb{Q}:t=t_{\alpha }\text{ for some }\alpha \in {\mathcal{L}}_P\}$ . Take $c=t_0/2$ . Then it is straight forward to prove that ${\oplus }_w$ is a rational improvement operator. We can call ${\oplus }_w$ a weighted translation and $w$ a weight function.

Weighted translations generalize translations. Consider a fixed positive rational number $t$ , then translation ${\oplus }_t$ is ${\oplus }_w$ where $w$ is the weight function such that $t_{\alpha }=t$ for every $\alpha \in {\mathcal{L}}_P$ .

5.1. A Characterization of Translations

The translation operators are the operators introduced in Example 9. More formally we have the following definition:

Definition 13

A rational improvement operator is a translation if it is ${\oplus }_t$ for some $t\in {\mathbb{Q}}^{+}$ .

There are two additional postulates we can include in order to capture all the homogeneity behind the translations: For any rankings $r$ , $r^{\prime }$ , any formulas $\alpha$ , $\beta$ and any worlds $\omega$ and ${\omega }^{\prime }$

• (T1)

If $\omega \in \text{⟦}\alpha \text{⟧}$ and ${\omega }^{\prime }\in \text{⟦}\beta \text{⟧}$ then $(r\circ \alpha )(\omega )-r(\omega )=(r\circ \beta )({\omega }^{\prime })-r({\omega }^{\prime })$ .

(T2)

$\omega \in \text{⟦}\alpha \text{⟧}⟹(r\circ \alpha )(\omega )-r(\omega )=(r^{\prime }\circ \alpha )(\omega )-r^{\prime }(\omega )$ .

(T1) says that the increment of the rank of a given model does not depend on the formula and (T1) says that said increment does not depend of the ranking.

Lemma 1

Postulate (T1) implies

$$\begin{array}{}\text{(T1W)}& \omega ,{\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}⟹r(\omega )-r({\omega }^{\prime })=(r\circ \alpha )(\omega )-(r\circ \alpha )({\omega }^{\prime })\end{array}$$

Proof.

First, notice that replacing $\beta$ by $\alpha$ in (T1) we obtain precisely (T1W), thus $\text{(T1)}⟹\text{(T1W)}$ .

Theorem 7

Let $\circ$ be a rational improvement operator. Then $\circ$ is a translation iff it satisfies properties (T1) and (T2).

Proof.

It is easy to check that every translation satisfies (T1) and (T2). On the other hand, suppose $\circ$ is a rational improvement operator satisfying (T1) and (T2). We prove it is a translation. First, take any ranking $r$ , any consistent formula $\alpha$ , and take $\omega$ a model of $\alpha$ with $(r\circ \alpha )(\omega )-r(\omega )>0$ (which exists by (RI2)). Define

$$t=(r\circ \alpha )(\omega )-r(\omega )$$

Postulate (T2) implies that $t$ is independent of $r$ . (T1W) implies that $t$ is also independent of $\omega$ and (T1) implies that $t$ does not depend on $\alpha$ . Hence $t$ depends only on $\circ$ . (RI3) entails that $(r\circ \alpha )(\omega )=r(\omega )$ for every $\omega \notin \text{⟦}\alpha \text{⟧}$ . By the very same definition of $t$ , we have that $(r\circ \alpha )(\omega )=r(\omega )+t$ for every $\omega \in \text{⟦}\alpha \text{⟧}$ . Therefore $\circ ={\oplus }_t$ for some $t>0$ and we can conclude that postulates (T1) and (T2), together with the postulates for rational improvement, characterize translations.

Note that Postulate (T1W) is strictly weaker than (T1). To prove this, enumerate the (finite) set of equivalence classes of ${\mathcal{L}}_P$ under the relation $\equiv$ given by $\alpha \equiv \beta ⟺\text{⟦}\alpha \text{⟧}=\text{⟦}\beta \text{⟧}$ . We have classes $C_1,\dots ,C_N$ with $N=2^{2^n}$ . Define ${\circ }_{\equiv }$ :

$$r{\circ }_{\equiv }\alpha =r{\oplus }_k\alpha ⟺\alpha \in C_k$$

It is easy to see that this is a rational improvement operator that satisfies (T1W) but not (T1), hence $\text{(T1W)}\nRightarrow \text{(T1)}$ .

One can wonder whether postulates (T1) and (T1) are independent of each other or not. The answer is positive:

Proposition 3

Postulates (T1) and (T1) are independent.

Proof.

Notice that ${\circ }_{\equiv }$ satisfies (T2) thus $\text{(T2)}\nRightarrow \text{(T1)}$ .

Finally, given an enumeration $r^1,r^2,\dots$ of rankings, consider the function ${*}_k$ defined as $r{*}_k\alpha =r{\oplus }_k\alpha$ iff $r=r^k$ . This is a rational improvement operator satisfying (T1) but not (T2), therefore $\text{(T1)}\nRightarrow \text{(T1))}$ .

Let us finish this section about translations with one observation about them.

Note that $r{\oplus }_t\mathrm{\top }$ is the the ranking $r^{\prime }$ obtained from $r$ by shifting the rank of all the models by $t$ . Thus, clearly, $r^{\prime }\ne r$ but the underlying preorder of both rankings is the same. Moreover, we have the following:

Observation 7

Given $r$ and $\alpha$ , if $r^{\prime }=(r{\oplus }_t\alpha ){\oplus }_t\mathrm{\neg }\alpha$ we have that $r^{\prime }=r{\oplus }_t\mathrm{\top }$ . Thus, $r$ and $r^{\prime }$ have the same underlying preorder but $r\ne r^{\prime }$ . In other words, the operator ${\oplus }_t$ has a cancelation property at the level of underlying orders.

Observation 8

Weighted translations, defined in Example 16, are not translations because their very definition violates postulate (T1) of translations when there are two formulas with different weights.

6. One Improvement and Half Improvement Operators in the Setting of Rational Rankings

An interesting feature of the epistemic space of rational rankings is its richness and flexibility. This allows us to define some operators appearing in the literature in an easy and natural way. As examples of our claim, we will define the one-improvement and half-improvement operators ⁴ Konieczny et al. (2010b); Konieczny and Pino Pérez (2008) on the epistemic space ${\mathcal{E}}_{\mathbb{Q}}$ . Nevertheless, these operators will not be rational improvement operators, but they are going to be improvement operators.

Definition 14

Given an epistemic space $\mathcal{E}$ , consider a gradual assignment $\mathrm{\Psi }\mapsto {⪯}_{\mathrm{\Psi }}$ for some change operator $\circ$ .

1. (1)

The assignment is a systematic enhancement if it satisfies:

• (S4)

If $\omega \in \text{⟦}\alpha \text{⟧}$ , ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ , then ${\omega }^{\prime }{\prec }_{\mathrm{\Psi }}\omega ⟹{\omega }^{\prime }{⪯}_{\mathrm{\Psi }\circ \alpha }\omega$ .

(S5)

If $\omega \in \text{⟦}\alpha \text{⟧}$ , ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ , then ${\omega }^{\prime }{\prec \prec }_{\mathrm{\Psi }}\omega ⟹\omega {⪯}_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$ . ⁵

(2)

The assignment is half gradual if it satisfies (S4) and

• (SH1)

If $\omega \in \text{⟦}\alpha \text{⟧}$ , ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ , ${\omega }^{\prime }{\prec \prec }_{\mathrm{\Psi }}\omega$ and there is no ${\omega }^{″}\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ such that ${\omega }^{″}{\simeq }_{\mathrm{\Psi }}\omega$ , then $\omega {⪯}_{\mathrm{\Psi }\circ \alpha }{\omega }^{\prime }$ .

(SH2)

If $\omega \in \text{⟦}\alpha \text{⟧}$ , ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ , ${\omega }^{\prime }{\prec \prec }_{\mathrm{\Psi }}\omega$ and there is a ${\omega }^{″}\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ such that ${\omega }^{″}{\simeq }_{\mathrm{\Psi }}\omega$ , then ${\omega }^{\prime }{\prec }_{\mathrm{\Psi }\circ \alpha }\omega$ .

Our setting is more general than the setting of Konieczny et al. (2010b), Medina Grespan and Pino Pérez (2013) but the following result is still valid.

Theorem 8 Konieczny et al. (2010b), Medina Grespan and Pino Pérez (2013)

A change operator $\circ$ in some epistemic space $\mathcal{E}$ is a one-improvement (resp. a half-improvement) operator iff there is a systematic enhancement (resp. a half-gradual assignment) $\mathrm{\Psi }\mapsto {⪯}_{\mathrm{\Psi }}$ for $\circ$ such that

$$\text{⟦}\mathcal{B}(\mathrm{\Psi }\star \alpha )\text{⟧}=min(\text{⟦}\alpha \text{⟧},{⪯}_{\mathrm{\Psi }})$$

Now we define operators $\odot$ and $\oslash$ and prove that they are, respectively, the one-improvement and the half-improvement operators in our epistemic space.

Observation 9

Since systematic enhancements and every half gradual assignments are gradual assignments, we have that one-improvement and half-improvement are indeed improvement operators.

Given a ranking $r$ , its range is the set:

$$\text{range}(r)=\{k\in \mathbb{Q}:r_k\ne \mathrm{\varnothing }\}$$

Since $\mathrm{\Omega }$ is a finite set, $\text{range}(r)$ is also finite for every ranking $r$ that is there is some $m\in \mathbb{N}$ and some finite sequence of rational numbers $i_1<i_2<\cdots <i_m$ such that $\text{range}(r)=\{i_1,\dots ,i_m\}$ .

For each $j\in \{1,\dots ,m\}$ , define

$${\mathrm{\Delta }}_j^r=\{\begin{array}{ll}{r}_{j+1}-r_j& \text{if }j=1,\dots ,m-1\\ 1& \text{if }j=m\end{array}$$

(8)

Definition 15

We define the operator $\odot$ on the epistemic space ${\mathcal{E}}_{\mathbb{Q}}$ as the function $\odot :U\times {\mathcal{L}}_P⟶U$ mapping each pair $(r,\alpha )$ to the ranking $r\odot \alpha$ defined by

$$(r\odot \alpha )(\omega )=\{\begin{array}{ll}r(\omega )& \text{if }\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}\\ r(\omega )+{\mathrm{\Delta }}_j^r& \text{if }\omega \in \text{⟦}\alpha \text{⟧}\cap r_{i_j}\end{array}$$

This function is well defined since each element of $\mathrm{\Omega }$ is in exactly one non-empty level of the ranking $r$ .

Equivalently, if $\omega \in \text{⟦}\alpha \text{⟧}$ and $r(\omega )=i_j$ with $j<m$ then $(r\odot \alpha )(\omega )=i_{j+1}$ . In the other hand, if $j=m$ then $(r\odot \alpha )(\omega )=i_m+1$ .

The one-improvement operator is not a rational improvement operator since it is impossible to find a constant $c$ as in Definition 9 that suits every ranking.

Lemma 2

For every ranking $r$ and every consistent formula $\alpha$ , there exists a natural number $n$ such that $\mathcal{B}(r{\odot }^n\alpha )⊢\alpha$ that is operator $\star$ is defined for $\odot$ .

Proof.

Let $r$ be a rational ranking and $\alpha$ any consistent formula. Let $\{i_1,\dots ,i_m\}$ be the range of $r$ with $i_1<\cdots <i_m$ . Suppose $\omega \in \text{⟦}\alpha \text{⟧}\cap r_{i_j}$ for some $1\le j\le m$ . By the definition of $\odot$ and an inductive argument it is easy to see that, given a natural number $k$ :

$$(r{\odot }^k\alpha )(\omega )=\{\begin{array}{ll}{i}_{j+k}& \text{if }0\le k\le m-j\\ r(\mathrm{\top })+(k-(m-j))& \text{if }m-j<k\end{array}$$

If $i_j=r(\alpha )$ , then

$$(r{\odot }^n\alpha )(\omega )=r(\mathrm{\top })+1$$

for $n=m-j+1$ . Since $(r{\odot }^n\alpha )({\omega }^{\prime })=r({\omega }^{\prime })$ for every ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ we have that

$$\mathcal{B}(r{\odot }^n\alpha )⊢\alpha$$

that is the revision operator $\star$ associated to $\odot$ is defined.

Lemma 3

The canonical assignment is gradual for $\odot$ .

Proof.

The canonical assignment satisfies properties (1) to (6) in Definition 3:

Properties (1) and (2) are immediate consequences of the definitions of canonical assignment and $\mathcal{B}(r)$ . Property (3) follows easily by induction on $n$ . Properties (4), (5) and (6) follow directly from the definitions of the operator $\odot$ and the canonical assignment.

Lemma 4

The canonical assignment is a systematic enhancement for operator $\odot$ .

Proof.

By Lemma 3, we know the canonical assignment is gradual for $\odot$ . It remains to prove that it satisfies (S4) and (S5). These is immediate from the Definition of $\odot$ . Let $\omega \in \text{⟦}\alpha \text{⟧}$ and ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ , then:

If ${\omega }^{\prime }{\prec }_r\omega$ , we have that $r(\omega )<r({\omega }^{\prime })$ . Suppose $\text{range}(r)=\{i_1,\dots ,i_m\}$ , then $r(\omega )=i_j,r({\omega }^{\prime })=i_k$ for some pair $i_j,i_k\in \text{range}(r)$ such that $i_1\le i_j<i_k\le i_m$ . If $k=j+1$ , then $(r\odot \alpha )(\omega )=i_k$ and $(r\odot \alpha )({\omega }^{\prime })=r({\omega }^{\prime })$ hence $(r\odot \alpha )(\omega )=(r\odot \alpha )({\omega }^{\prime })$ . If $k>j+1$ then $(r\odot \alpha )(\omega )=i_{j+1}$ and $(r\odot \alpha )(\omega )=r(\omega )=i_k$ , then $(r\odot \alpha )(\omega )<(r\odot \alpha )({\omega }^{\prime })$ . Hence ${\omega }^{\prime }{\prec }_r\omega$ implies $(r\odot \alpha )(\omega )\le (r\odot \alpha )({\omega }^{\prime })$ thus ${\omega }^{\prime }{⪯}_{r\odot \alpha }\omega$ and the cannonical assignment satisfies (S4).

Now, if ${\omega }^{\prime }{\prec \prec }_r\omega$ , we have that $r(\omega )=i_j$ and $r({\omega }^{\prime })=i_{j+1}$ . Thus $(r\odot \alpha )(\omega )=(r\odot \alpha )({\omega }^{\prime })=i_{j+1}$ , hence $\omega {\simeq }_{r\odot \alpha }{\omega }^{\prime }$ therefore $\omega {⪯}_{r\odot \alpha }{\omega }^{\prime }$ and (S5) is satisfied.

Lemma 5

If $r\mapsto {⪯}_r$ is the canonical assignment, then

$$\text{⟦}\mathcal{B}(r\star \alpha )\text{⟧}=min(\text{⟦}\alpha \text{⟧},{⪯}_r)$$

for every ranking $r$ and every formula $\alpha$ , if $\star$ is the revision operator associated to $\odot$ .

Proof.

By Lemma 2, the revision operator $\star$ is defined for the operator $\odot$ . Thus, given any ranking $r$ and a formula $\alpha$ , there is some $n\in \mathbb{N}$ such that

$$\text{⟦}\mathcal{B}(r{\odot }^n\alpha )\text{⟧}\subseteq \text{⟦}\alpha \text{⟧}$$

and

$$\text{⟦}\mathcal{B}(r{\odot }^m\alpha )\text{⟧}⊈\text{⟦}\alpha \text{⟧}$$

for every $m<n$ .

Since the canonical assignment is gradual, it is easy to check, for every $k\in \mathbb{N}$ and every pair of worlds $\omega ,{\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ , that

$$(r{\odot }^k\alpha )({\omega }^{\prime })\le (r{\odot }^k\alpha )(\omega )⟺r({\omega }^{\prime })\le r(\omega )$$

Therefore

$$\omega \in (r{\odot }^n\alpha )(\mathrm{\top })⟺r({\omega }^{\prime })\le r(\omega )$$

for every ${\omega }^{\prime }\in \text{⟦}\alpha \text{⟧}$ . As $(r{\odot }^n\alpha )(\mathrm{\top })=\mathcal{B}(r{\odot }^n\alpha )$ , we have

$$\omega \in \text{⟦}\mathcal{B}(r{\odot }^n\alpha )\text{⟧}⟺\omega \in r(\alpha )$$

Theorem 9

Operator $\odot$ on rational rankings is a one-improvement operator.

Proof.

Direct from Lemmas 2, 3, 4 and 5.

It is worth to note that Lemmas 2, 3, 4 and 5 prove formally that the operator $\odot$ defined on rational rankings represents the one improvement operator ⁶ defined on TPOs within the meaning of Definition 11.

We can also define a half-improvement operator in this setting but, as in the one-improvement case, it can not be a rational improvement operator but a (generalized) improvement operator. This is not the only way to define a half-improvement operator in this epistemic space but it is natural.

Definition 16

We define the half-improvement operator $\oslash$ on the epistemic space ${\mathcal{E}}_{\mathbb{Q}}$ as the function $\oslash :U\times {\mathcal{L}}_P⟶U$ mapping each pair $(r,\alpha )$ to the ranking $r\oslash \alpha$ defined by

$$(r\oslash \alpha )(\omega )=\{\begin{array}{ll}r(\omega )& \text{if }\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}\\ r(\omega )+{\mathrm{\Delta }}_j^r& \text{if }\omega \in \text{⟦}\alpha \text{⟧}\cap r_{i_j}\mathrm{\&}{r}_j\subseteq \text{⟦}\alpha \text{⟧}\\ r(\omega )+{\displaystyle \frac{{\mathrm{\Delta }}_j^r}{2}}& \text{if }\omega \in \text{⟦}\alpha \text{⟧}\cap r_{i_j}\mathrm{\&}{r}_j⊈\text{⟦}\alpha \text{⟧}\end{array}$$

Lemma 6

For every ranking $r$ and every consistent formula $\alpha$ , there exists a natural number $n$ such that $\mathcal{B}(r{\oslash }^n\alpha )⊢\alpha$ therefore the revision operator $\star$ is defined for $\oslash$ .

Proof.

Let $r$ be a ranking, and $\alpha$ a formula. It suffice to find a natural number $n$ such that $\mathcal{B}(r{\oslash }^n\alpha )⊢\alpha$ for every $\omega \in \text{⟦}\alpha \text{⟧}$ such that $r(\omega )=r(\alpha )$ . Then the set

$$\{n\in \mathbb{N}:\mathcal{B}(r{\oslash }^n\alpha )⊢\alpha \}$$

is non empty, so it has a least element and this means that the operator $\star$ is defined.

Let $\text{range}(r)=\{i_1,\dots ,i_m\}$ . It is enough to prove that there is some $n\in \mathbb{N}$ such that $(r{\oslash }^n\alpha )(\omega )=(r{\oslash }^n\alpha )(\mathrm{\top })$ for every $\omega \in \text{⟦}\alpha \text{⟧}$ with $r(\omega )=r(\alpha )$ and $(r{\oslash }^n\alpha )({\omega }^{\prime })<(r{\oslash }^n\alpha )(\mathrm{\top })$ for each ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ . Define the $r$ -distance between $i_k$ and $i_p$ as

$$d_r(i_k,i_p):=|k-p|$$

We proceed by induction on the $r$ -distance between $r(\mathrm{\top })$ and $r(\alpha )$ .

Suppose $d_r(r(\mathrm{\top }),r(\alpha ))=0$ . Then $r(\alpha )=i_m=r(\mathrm{\top })$ . There are two cases: Either $r_{i_m}\cap \text{⟦}\mathrm{\neg }\alpha \text{⟧}=\mathrm{\varnothing }$ or $r_{i_m}\cap \text{⟦}\mathrm{\neg }\alpha \text{⟧}\ne \mathrm{\varnothing }$ . In the first case, there is nothing to prove and $n=0$ . In the second case, given $\omega \in r_{i_m}\cap \text{⟦}\alpha \text{⟧}$ , we have that $(r\oslash \alpha )(\omega )=i_m+1/2$ while $(r\oslash \alpha )({\omega }^{\prime })=r({\omega }^{\prime })$ for every ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ . Thus in this case $n=1$ .

Now suppose $d_r(r(\mathrm{\top }),r(\alpha ))=k+1$ . For simplicity, say $r(\alpha )=i_{\ell }$ for some suitable $\ell$ , then $d_r(r(\mathrm{\top }),r(\alpha ))=m-\ell$ . Again, we have two cases. If $r_{i_{\ell }}\cap \text{⟦}\mathrm{\neg }\alpha \text{⟧}=\mathrm{\varnothing }$ , then $(r\oslash \alpha )(\omega )=i_{\ell +1}$ for every $\omega \in r_{i_{\ell }}$ . Now, since $r(\alpha )=i_{\ell }$ and $(r\oslash \alpha )({\omega }^{\prime })=r({\omega }^{\prime })$ for every ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ , we have $j_1,\dots ,j_t\in \mathbb{Q}$ such that $\text{range}(r\oslash \alpha )=\{j_1,\dots ,j_t,i_{\ell +1},\dots ,i_m\}$ and $(r\oslash \alpha )(\alpha )=i_{\ell +1}$ .

Notice that the $(r\oslash \alpha )$ -distance between $(r\oslash \alpha )(\mathrm{\top })$ and $(r\oslash \alpha )(\alpha )$ is $m-(\ell +1)=k$ , hence we can apply the inductive hypothesis to $r\oslash \alpha$ , thus there is some natural number $n$ such that $((r\oslash \alpha ){\oslash }^n\alpha )(\omega )=(r{\oslash }^{n+1}\alpha )(\mathrm{\top })$ and $(r{\oslash }^{n+1}\alpha )({\omega }^{\prime })=r({\omega }^{\prime })<(r{\oslash }^{n+1}\alpha )(\mathrm{\top })$ for every ${\omega }^{\prime }\in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ and we are done.

If $r_{i_{\ell }}\cap \text{⟦}\mathrm{\neg }\alpha \text{⟧}\ne \mathrm{\varnothing }$ , then $(r{\oslash }^2\alpha )(\omega )=i_{\ell +1}$ and we can prove the conclusion of the theorem similarly as we did in the first case.

We are going to prove that, indeed, $\oslash$ is the half-improvement operator in this setting by proving that the canonical assignment is a half gradual assignment such that $\text{⟦}\mathcal{B}(r\star \alpha )\text{⟧}=min(\text{⟦}\alpha \text{⟧},{⪯}_r)$ where $\star$ is the revision operator associated to $\oslash$ (Konieczny et al., 2010a).

Lemma 7

The canonical assignment is a gradual assignment for $\oslash$ . Moreover, it is a half gradual assignment.

Proof.

Similar to the proof of Lemma 3.

Lemma 8

If $r\mapsto {⪯}_r$ is the canonical assignment, then

$$\text{⟦}\mathcal{B}(r\star \alpha )\text{⟧}=min(\text{⟦}\alpha \text{⟧},{⪯}_r)$$

for every ranking $r$ and every formula $\alpha$ , if $\star$ is the revision operator associated to $\oslash$ .

Proof.

Similar to the proof of Lemma 5.

Theorem 10

The operator $\oslash$ on rational rankings is a half improvement operator on rational rankings.

Proof.

Immediate from Lemmas 6, 7 and 8.

It is worth to note that Lemmas 6, 7 and 8 prove formally that the operator $\oslash$ defined on rational rankings represents the half improvement operator ⁷ defined on TPOs within the meaning of Definition 11.

Something interesting here is that rational improvement operators are not the only improvement operators defined on the epistemic space ${\mathcal{E}}_Q$ .

Observation 10

When we restrict the domain of the operator $\odot$ to rankings with ${\mathrm{\Delta }}_j=1$ for each $j$ , then the behavior of $\odot$ is identical to the behavior of ${\oplus }_1$ .

6.1. One-improvement on OCF Versus One-improvement on Rational Rankings

One-improvement and half-improvement operators are defined this way, because they are indeed respectively one-improvement and half-improvement operators on the underlying preorders of ranking functions. An advantage of working on rational rankings is that we can define a one-improvement operator such that for every ranking $r$ and every formula $\alpha$ , we have that $(r\odot \alpha )(\omega )=r(\omega )$ whenever $\omega \in \text{⟦}\mathrm{\neg }\alpha \text{⟧}$ that is we only “move” the models of the new information, which is in accordance with the minimal change principle. To illustrate this point, consider the ocf $\kappa$ depicted in Figure 8. The domain of $\kappa$ is the set $\mathrm{\Omega }$ of all possible worlds on variables $p,q,r$ considered in this order. The one-improvement operator on OCFs is defined in Konieczny and Pino Pérez (2008). Figure 8 shows its behavior.

Figure 8.

The one-improvement operator $\odot$ at work on a ordinal conditional function (OCF).

Following Schwind et al. (2022), the TPO associated to $\kappa$ is depicted on Figure 9. Although there are infinite rational rankings associated to this TPO we choose one and apply one-improvement operator three times on a particular ranking in Figure 10.

Figure 9.

The total preorder associated to $\kappa$ .

Figure 10.

The one-improvement operator $\odot$ on rational rankings.

7. Related Work

Soft Improvement

The notion of soft improvement operators introduced in Konieczny et al. (2010b) can be translated to the context of rational rankings as follows:

(r-Soft) If $\omega \in \text{⟦}\alpha \text{⟧}$ , ${\omega }^{\prime }\notin \text{⟦}\alpha \text{⟧}$ and $r({\omega }^{\prime })>r(\omega )$ , then $r({\omega }^{\prime })\ge (r\circ \alpha )(\omega )$

This means that the increase in plausibility cannot result in the rank of a model being strictly greater than that of a counter-model which was more plausible in the initial ranking.

Unfortunately, our rational improvement operators cannot satisfy (r-Soft). The reason is that if an operator satisfies (RI2), it necessarily violates (r-Soft). Specifically, we can find a rational ranking $r$ where the underlying order is linear and the difference between $r(\mathrm{\top })$ and the minimum rank in the image of $r$ is less than the constant $c$ given by property (RI2). If we choose an $\alpha$ such that $top(\mathrm{\top })\cap \text{⟦}\alpha \text{⟧}=\mathrm{\varnothing }$ , then it is clear that if $\omega$ is the model of $\alpha$ given by (RI2), it violates (r-Soft) because $(r\circ \alpha )(\omega )>r(\mathrm{\top })$ .

Nevertheless, for certain inputs, rational improvement operators can exhibit soft behavior. For instance, if $r$ is a rational ranking such that the image of $r$ consists of the set $\{q_1,\dots ,q_k\}$ with $q_i-1=q_{i+1}$ , then ${\oplus }_{\frac{1}{2}}$ displays soft behavior. In such cases, its behavior resembles that of the half improvement operator (Konieczny et al., 2010b).

Decrement Operators

The decrement operators introduced by Sauerwald et al. (Sauerwald & Beierle, 2019) are related to soft changes aimed at achieving contraction. The idea is to decrease the plausibility of the input until it is no longer believed. Related to this, we can define in the space of rational rankings operators that we call regressions. They are defined similarly to translations but use a negative rational number $t$ .

These operators will generally not be soft, but for certain inputs, they behave like decrement operators. For example, if $r$ is a rational ranking where the ranks in the image have a difference of 1 between consecutive ranks, then the regression operator ${\oplus }_{-\frac{1}{2}}$ behaves as a decrement operator.

General Properties for Iteration

In Schwind et al. (2023), some generalized DP postulates were proposed for revision. These postulates aimed to ensure homogeneity in the behavior of operators. In the space of TPOs, these postulates characterized a family reduced to only three operators: Nayak lexicographic revision, Boutilier natural revision, and the restrained revision of Booth and Meyer (Booth & Meyer, 2006). The semantic aspect of these postulates is satisfied by the translations. A natural question is whether these (or similar) postulates can characterize the family of translations.

Expressive Power of Rankings

Rational rankings are not more expressive than OCFs, under standard hypotheses (postulate (G) in Schwind et al., 2022). However, in this paper we aim to work with a meaningful, unnormalized structure and consider a framework with rationality postulates directly applied to these structures, rather than just at the level of beliefs. Within this context, OCF-based operators are clearly not rational improvement operators in the sense of Definition 9. In particular, we have seen in Example 2 that in normalized structures as OCF’s, it is unavoidable to worsen the plausibility degree of countermodels of the new information. On the contrary, in structures such as rational rankings is possible to improve or even revise without worsening the countermodels of the new information.

Spohn’s Work on Rankings

There are some important differences between our work and Spohn’s work (Spohn, 2009): First, Spohn defines normalized rankings into the positive real numbers, while our rational rankings are not normalized. Additionally, he defines rankings into the set $\mathbb{R}\cup \{-\mathrm{\infty },\mathrm{\infty }\}$ . In Spohn’s framework, negative ranks model degrees of disbelief, which is not the intention in our work. Moreover, the density of $\mathbb{Q}$ is a key property for our development, which is overlooked in Spohn’s work.

JLZ System

Weydert proposes in Weydert (2003) the JLZ System for default reasoning with the aim of building a canonical ranking for any given default base. In Weydert (2012), he defines a formalism for iterated belief revision where belief states are modeled by ranking measures and inputs are sets of graded doxastic conditionals.

The general structures considered in Weydert (2012) are also normalized unlike our structures. Moreover, both Spohn -see for instant Hild and Spohn (2008)- and Weydert consider ranks taking the value $\mathrm{\infty }$ . We don’t need this value.

Another important difference between our work and the work of Spohn and the work of Weydert is that we study mechanisms of very gradual changes that we call improvement. We have indded characterized a class of this operators for space of rational rankings.

8. Conclusions and Future Work

A first contribution of this work is the introduction and characterization of the class of generalized improvement operators. We have defined the epistemic space of rational rankings and the associated rational improvement operators. These operators consistently produce changes, and notably, improving new information does not require decreasing the plausibility of models that do not satisfy this new information. This framework effectively models scenarios such as the evolution of Dr. Roberts’ beliefs in Example 3.

The epistemic space of rational rankings allows to define in a simple and natural manner operators that represent one-improvement and half-improvement operators (Konieczny et al., 2010b) defined on TPOs (see Definitions 15 and 16). The definition of the one-improvement operator on OCFs is very complex in comparison (Konieczny & Pino Pérez, 2008). Half-improvement operators on OCFs has not been defined.

By working within the space of rational rankings, we can define $t$ -translations with very small values of $t$ . This flexibility allows us to model Carlos’ evolving beliefs in Example 4, demonstrating the need for many iterations to achieve a goal.

Another notable feature of using rational rankings is the ability to define regressions simply with negative numbers.

Through a few equations and inequalities with clear meanings, we have established a class of operators that captures the notion of improving the plausibility of the input: The rational improvement operators. Additionally, we have provided two independent postulates that characterize the translation operators.

The space of rational rankings can be very useful as a field of experimentation in order to find the properties characterizing well behaved operators. For instance, we can define operators as the Zeno operators (Example 13) which show that it is not enough to increase the plausibility of certain models of the input, for being a real improvement operator (to achieve success iterating the input enough number of times).

Some concepts, such as the softness of operators, are intrinsically contextual. As demonstrated, this is incompatible with (RI2). Future work should explore more relaxed notions than (RI2) that accommodate contextual operators, particularly postulates compatible with (r-Soft).

An important direction for future research is to determine the syntactic characterization of the classes of operators defined in this work. Additionally, investigating whether rankings on real numbers would provide a richer framework, potentially incorporating properties such as completeness, could be a valuable avenue for exploration.