Conditional Syntax and Semantic Splittings of Strongly and Weakly Consistent Belief Bases

Abstract

Conditional syntax splitting for inductive inference from conditional belief bases has been proposed as a generalization of syntax splitting, which also covers cases where the conditionals in the subbases may share some atoms. While p-entailment and system Z fail to satisfy conditional syntax splitting, two other inductive inference operators, lexicographic inference and system W, have been shown to satisfy this property for reasoning from strongly consistent belief bases. In this article, we introduce the concept of conditional semantic splitting of a belief base. For both syntax splitting and semantic splitting, we take not only strongly consistent belief bases into account, but also belief bases that are only weakly consistent, enforcing some worlds to be fully infeasible. We show that c-representations satisfy a core postulate relating conditional splittings on the syntax and the semantic level. Based on these findings, we investigate conditional syntax splitting for nonmonotonic inference with c-representations. Regarding single c-representations, we utilize the concept of selection strategies and show that a straightforward property of the selection strategy leads to inference operators satisfying conditional syntax splittings. Furthermore, we show that c-inference, taking all c-representations of a belief base into account fully complies with conditional syntax splitting, and we prove that credulous and weakly skeptical inference based on c-representations also satisfies conditional syntax splitting. In particular, we show that these syntax splitting properties hold for strongly consistent belief bases and also in the case of only weakly consistent belief bases.

Keywords

belief base strong consistency weak consistency syntax splitting semantic splitting conditional splitting c-representation c-inference selection strategy

1. Introduction

In both human and formal reasoning it is often essential not to take into account the full amount of knowledge that is available to an intelligent agent, but to split the available knowledge into independent parts so that local reasoning on a part of the knowledge can be done. Toward this goal, the concept of syntax splitting was developed by Parikh (1999) for belief sets in order to formulate postulates for belief revision, and was later transferred to other structures and applications (e.g., Kern-Isberner & Brewka, 2017; Peppas et al., 2015), a related concept was introduced by Weydert (1998) as minimum irrelevance. Syntax splitting for nonmonotonic reasoning from conditional belief bases (Kern-Isberner et al., 2020) is a combination of the postulates relevance and independence, stating that only conditionals from the considered part of the syntax splitting of a belief base are relevant for corresponding inferences, and that inferences using only atoms from one part of the syntax splitting should be independent of additional information on the other parts. Furthermore, the concept of semantic splitting has been introduced with a focus on the models of a belief base (Beierle et al., 2021b).

From a theoretical point of view, these splittings are interesting because they implement a notion of (ir)relevance in inferences. But splitting techniques have also consequences for applications: they allow for breaking down conditional reasoning to the subbases relevant for a query, hence usually reducing the relevant subsignature significantly. From a cognitive point of view, splitting techniques bring in the concept of local reasoning, which accounts for the limited resources of humans. Moreover, local reasoning is also fundamental to all works on probabilistic networks (Pearl, 1988).

Full splittings in the sense of Kern-Isberner et al. (2020), however, are quite rare in real-world applications. The concept of conditional syntax splitting for inference from conditional belief bases (Heyninck et al., 2023) is a generalization of syntax splitting which also allows the subbases to overlap syntactically, providing much more realistic application scenarios. The relevance of conditional syntax splitting is further underpinned by the fact that the so called drowning-effect (Benferhat et al., 1993; Pearl, 1990) was formalized as a violation of conditional syntax splitting (Heyninck et al., 2023). The drowning effect is a phenomenon where, for some inductive inference relations, special subclasses will not properly inherit properties of their superclass. For example, given the knowledge that birds usually fly, penguins are usually birds, penguins usually do not fly, and birds usually have wings, an inductive inference operator suffering from the drowning effect would not be able to conclude that penguins usually have wings. This means that if an inference operator satisfies conditional syntax splitting, it is also guaranteed to avoid the drowning effect in general, and not just for the canonical example. p-Entailment, characterized by the axioms of system P (Adams, 1965; Kraus et al., 1990), and system Z (Goldszmidt & Pearl, 1996) are known to suffer from the drowning problem, and in the paper introducing the concept of conditional syntax splitting (Heyninck et al., 2023), it is demonstrated that p-entailment and system Z do not satisfy this property, while it is shown that the two inductive inference operators lexicographic inference (Lehmann, 1995) and system W (Komo & Beierle, 2020, 2022) fully comply with conditional syntax splitting.

In this article, we extend the study of conditional splittings of belief bases in several directions, in particular by introducing conditional semantic splittings, by extending the application of conditional syntax splitting also to belief bases exhibiting only a weak notion of consistency. As a basis for our investigations, we use ranking functions (ordinal conditional functions [OCFs]; Spohn, 1988) as a well-established and popular semantics for conditional belief bases. Besides their popularity, another reason for using OCFs is that the work on conditional syntax splitting is inspired by probabilistic techniques. Since OCFs can be understood as qualitative abstractions of logarithmic probabilities, they provide a perfect mediating framework to realize such probabilistic ideas for qualitative nonmonotonic reasoning. We answer the question to what extent c-representations (Kern-Isberner, 2001), which are a specific subset of ranking models of a belief base, and inference with them satisfy conditional semantic splitting and conditional syntax splitting by investigating skeptical, weakly skeptical, and credulous reasoning with c-representations over three different subsets of models.

It should be noted that so far the notions of conditional syntax and conditional semantic splittings mentioned above have only been investigated in the context of belief bases that are consistent in the sense of the well-known consistency test by Goldszmidt and Pearl (1996). This notion of consistency, called strong consistency in the following, requires models where every possible world is at least somewhat plausible. In contrast, a weakly consistent belief base may only have models where some worlds are fully infeasible and thus completely implausible (Haldimann et al., 2023). With a focus on taking also weakly consistent belief bases into account, this article provides the following main contributions:

(1)
We introduce the concept of conditional semantic splitting for semantics based on Spohn’s ranking functions (Spohn, 1988), and a postulate (CSemSplit) relating conditional splittings on the syntax and on the semantic level, covering both strongly and weakly consistent belief bases.
(2)
We show that c-representations, which are special ranking functions obtained by summing up impacts assigned to falsified conditionals (Kern-Isberner, 2001, 2004), satisfy conditional semantic splitting. This also holds for extended c-representations (Haldimann et al., 2024) for weakly consistent belief bases.
(3)
We study the relationship between conservative c-representations, which minimize the number of implausible worlds, and relaxations thereof, leading to a postulate ensuring classic preservation (Casini et al., 2019).
(4)
Regarding reasoning with respect to a single c-representation or a single extended c-representation, we utilize the concept of selection strategies (Beierle & Kern-Isberner, 2021; Kern-Isberner et al., 2020) and show that the property of preserving the impacts chosen for certain subbases for inferences on the full belief base leads to inference operators satisfying conditional syntax splitting.
(5)
Without any requirement regarding a selection strategy, we prove that c-inference, extended c-inference, and also conservative extended c-inference, which are the skeptical inference relations taking all c-representations, extended c-representations, or conservative extended c-representations, respectively, of a belief base into account (Beierle et al., 2018, 2021a; Haldimann et al., 2024) fully comply with conditional syntax splitting.
(6)
We show that also credulous and weakly skeptical inference with c-representations, extended c-representations, or conservative extended c-representations (Beierle et al., 2021a; Haldimann et al., 2025) satisfy conditional syntax splitting.
This article is a revised and largely extended version of our conference paper presented at the 21st International Conference on Principles of Knowledge Representation and Reasoning (KR 2024; Beierle et al., 2024b). The additions include, in particular, all aspects related to weakly consistent belief bases. We apply the notions of conditional syntax splitting and conditional semantic splitting accordingly. While c-representations extended to weakly consistent belief bases and the correspondingly extended c-inference operator have been introduced only recently (Haldimann et al., 2024), we show here that extended c-representations satisfy conditional semantic splitting and that extended c-inference and also inference with respect to extended c-representations obtained via an appropriate selection strategy satisfy conditional syntax splitting. We study relaxations of conservative c-representations for weakly consistent belief bases and formulate a postulate ensuring classic preservation for induced inductive inference operators based on them. Furthermore, we broaden our investigations of splitting properties of c-representation-based inference systems by addressing also credulous and weakly skeptical c-inference.

The rest of this article is organized as follows. In Section 2, we present the basics of conditional logic needed here, and in Section 3, we recall the concept of conditional syntax splitting. In Section 4, we introduce the notion of conditional semantic splitting, show that it is satisfied by c-representations and by extended c-representations, and we study the relationship between conservative c-representations and relaxations thereof. In Section 5, we present postulates for selection strategies and prove that these postulates ensure that the corresponding inference operators for single c-representations and for extended c-representations satisfy classic preservation and conditional syntax splitting, respectively. In Section 6, we show that c-inference taking all c-representations of strongly consistent belief bases, and also extended c-inference taking all extended c-representations, or all conservative extended c-representations, respectively, of weakly consistent belief bases into account, fully comply with conditional syntax splitting, without reference to any selection strategies. In Section 7, we expand these results to credulous and weakly skeptical c-inference, and in Section 8, we conclude and point out further work.
2. Formal Basics

Let $L$ be the propositional language over a finite signature $Σ$ with atoms $a, b, c, \dots$ , with the usual connectives $\neg$ (not), $\land$ (and), $\lor$ (or), and $\to$ (material implication), and with formulas $A, B, C, \dots .$ For conciseness of notation, we may omit the logical and-connector, writing $A B$ instead of $A \land B$ , and overlining formulas will indicate negation, that is, $\bar{A}$ means $\neg A$ . Let $Ω$ denote the set of possible worlds over $L$ ; $Ω$ will be taken here simply as the set of all propositional interpretations over $L$ . $ω ⊨ A$ means that the propositional formula $A \in L$ holds in the possible world $ω \in Ω$ ; then $ω$ is called a model of $A$ , and the set of all models of $A$ is denoted by $Mod (A)$ . For propositions $A, B \in L$ , $A ⊨ B$ holds iff $Mod (A) \subseteq Mod (B)$ , as usual. By slight abuse of notation, we will use $ω$ both for the model and the corresponding conjunction of all positive or negated atoms. This will allow us to use $ω$ both as an interpretation and a proposition, which will ease notation a lot. Since $ω ⊨ A$ means the same for both readings of $ω$ , no confusion will arise.

For subsets $Θ$ of $Σ$ , let $L (Θ)$ or short $L_{Θ}$ denote the propositional language defined by $Θ$ , with an associated set of interpretations $Ω (Θ)$ or short $Ω_{Θ}$ . Note that while each sentence of $L (Θ)$ can also be considered as a sentence of $L$ , the interpretations $ω^{Θ} \in Ω (Θ)$ are not elements of $Ω (Σ)$ if $Θ \neq Σ$ . But each interpretation $ω \in Ω$ can be written uniquely in the form $ω = ω^{Θ} ω^{\bar{Θ}}$ with concatenated $ω^{Θ} \in Ω (Θ)$ and $ω^{\bar{Θ}} \in Ω (\bar{Θ})$ , where $\bar{Θ} = Σ ∖ Θ$ is the complement of $Θ$ in $Σ$ . Note that the syntactical reading of interpretations as conjunctions makes perfect sense here. According to this reading, $ω$ is a conjunction of $ω^{Θ}$ and $ω^{\bar{Θ}}$ (with omitted $\land$ symbol). $ω^{Θ}$ is called the reduct of $ω$ to $Θ$ (Delgrande, 2017). If $Ω^{'} \subseteq Ω$ is a subset of models, then ${Ω^{'} |}_{Θ} = {ω^{Θ} | ω \in Ω^{'}} \subseteq Ω (Θ)$ restricts $Ω^{'}$ to a subset of $Ω (Θ)$ . In the following, we will often consider the case that $Σ_{1}, Σ_{2}$ are disjoint subsignatures of $Σ$ , then we write $ω^{i}$ instead of $ω^{Σ_{i}}$ for the reducts to ease notation.

By making use of a conditional operator $|$ , we introduce the language $(L | L)$ of conditionals over $L$ :

(L | L) = {(B | A) ∣ A, B \in L} .

A conditional

(B | A)

expresses a plausible, defeasible rule “If

A

then plausibly (usually, possibly, probably, typically)

B

” and is a trivalent logical entity with the evaluation (with

u

for unknown or indeterminate; de Finetti, 1937):

⟦ (B | A) ⟧_{ω} = {\begin{cases} 1 & iff ω ⊨ A B & (verification), \\ 0 & iff ω ⊨ A \bar{B} & (falsification), \\ u & iff ω ⊨ \bar{A} & (not applicable) . \end{cases}

(1)

A conditional $(F | E)$ is called self-fulfilling if $E ⊨ F$ , that is, there is no world that can falsify it.

A popular semantic framework that is often used for interpreting conditionals is provided by OCFs. OCFs $κ : Ω \to N \cup {\infty}$ with $κ^{- 1} (0) \neq \emptyset$ , also called ranking functions, were introduced (in a more general form) first by Spohn (1988). They express degrees of belief of propositional formulas $A$ by specifying degrees of disbeliefs of their negations $\bar{A}$ . More formally, we have $κ (A) := min {κ (ω) ∣ ω ⊨ A}$ , so that $κ (A \lor B) = min {κ (A), κ (B)}$ . A proposition $A$ is believed if $κ (\bar{A}) > 0$ (which implies $κ (A) = 0$ ). The uniform OCF $κ_{u}$ is defined by $κ_{u} (ω) = 0$ for all $ω \in Ω$ , representing an epistemic state of complete ignorance.

Degrees of plausibility can also be assigned to conditionals by setting $κ (B | A) = κ (A B) - κ (A)$ . A conditional $(B | A)$ is accepted in the epistemic state represented by $κ$ , written as $κ ⊨ (B | A)$ , iff $κ (A B) < κ (A \bar{B})$ or $κ (A) = \infty$ , that is, iff $A B$ is more plausible than $A \bar{B}$ or $A$ is deemed infeasible. Conditional belief bases $Δ$ (over $L$ ) consist of finitely many conditionals from $(L ∣ L)$ . An OCF $κ$ accepts $Δ$ , denoted by $κ ⊨ (B | A)$ , if $κ ⊨ (B | A)$ for all $(B | A) \in Δ$ . Consistency of a conditional belief base $Δ$ can be defined in terms of OCFs (Pearl, 1990). According to the well-known consistency test by Goldszmidt and Pearl (1996), $Δ$ is consistent iff there is an OCF $κ$ such that $κ ⊨ Δ$ and $κ^{- 1} (\infty) = \emptyset$ . In the following, we will call this notion strong consistency because it requires models $κ$ that view every possible world to be at least somewhat plausible. A more liberal notion of consistency used in other approaches (e.g., Casini et al., 2019; Giordano et al., 2015; Haldimann et al., 2023), allows for models where some worlds may be fully infeasible. A belief base $Δ$ is weakly consistent iff there is an OCF $κ$ such that $κ ⊨ Δ$ (Haldimann et al., 2023), thus, such a $κ$ represents that all worlds in $κ^{- 1} (\infty)$ are fully infeasible.

The nonmonotonic inference relation induced by an OCF $κ$ is given by Spohn (1988) (2)

The marginal of $κ$ on $Θ \subseteq Σ$ , denoted by ${κ |}_{Θ}$ , is defined by ${κ |}_{Θ} (ω^{Θ}) = κ (ω^{Θ})$ for any $ω^{Θ} \in Ω (Θ)$ . Note that this marginalization is a special case of the general forgetful functor $M o d (σ)$ from $Σ$ -models to $Θ$ -models (Beierle & Kern-Isberner, 2012) where $σ$ is the inclusion from $Θ$ to $Σ$ .

To formalize inductive inference from conditional belief bases, the notion of inductive inference operators has been introduced (Kern-Isberner et al., 2020). An inductive inference operator (on $L$ ) is a mapping $C$ that assigns to each conditional belief base $Δ \subseteq (L ∣ L)$ an inference relation on $L$ , that is, such that the following properties hold:

Direct Inference (DI) if $(B | A) \in Δ$ then , and

Trivial Vacuity (TV) implies $A ⊨ B$ .

An inductive inference operator for OCFs $C^{o c f}$ maps each belief base $Δ$ to an OCF $κ$ over $Ω (Σ)$ accepting $Δ$ ; the inference relation assigned to a belief base $Δ$ is then the inference relation induced by $κ$ according to equation (2).

An example for an inductive inference operator capable of handling also weakly consistent belief bases is extended system Z, based on the notion of tolerance and the extended Z-partition of a conditional belief base (Goldszmidt & Pearl, 1996).

Definition 1 ((Extended) Z-partition, (Goldszmidt & Pearl, 1990; Lehmann, 1995))

A conditional $(B | A)$ is tolerated by $Δ = {(B_{i} | A_{i}) ∣ i = 1, \dots, n}$ if there is a world $ω \in Ω$ such that $ω$ verifies $(B | A)$ and $ω$ does not falsify any conditional in $Δ$ , that is, $ω ⊨ A B$ and $ω ⊨ ⋀_{i = 1}^{n} (\bar{A_{i}} \lor B_{i})$ .

The (extended) Z-partition $E P (Δ) = (Δ^{0}, \dots, Δ^{k}, Δ^{\infty})$ of a belief base $Δ$ is the ordered partition of $Δ$ constructed by letting $Δ^{i}$ be the inclusion-maximal subset of $Δ ∖ (⋃_{j = 0}^{i - 1} Δ^{j})$ that is tolerated by $Δ ∖ (⋃_{j = 0}^{i - 1} Δ^{j})$ until $Δ^{k + 1} = \emptyset$ . The set $Δ^{\infty}$ is the remaining set of conditionals $Δ ∖ (⋃_{j = 0}^{k} Δ^{j})$ .

The (extended) Z-ranking function $κ_{Δ}^{z}$ is defined as follows: For $ω \in Ω$ , if a conditional in $Δ^{\infty}$ is applicable to $ω$ define $κ_{Δ}^{z} (ω) = \infty$ . If not, let $Δ^{j}$ be the last partition in $E P (Δ)$ that contains a conditional falsified by $ω$ , and define $κ_{Δ}^{z} (ω) = j + 1$ . If $ω$ does not falsify any conditional in $Δ$ , then let $κ_{Δ}^{z} (ω) = 0$ .

Because the sets $Δ^{i}$ are chosen inclusion-maximal, the Z-partition and thus the $κ^{z}$ ranking function is unique (Pearl, 1990). System Z yields the inductive inference operator

C^{z} : Δ \mapsto κ_{Δ}^{z},

which maps belief bases

Δ

to the OCF

κ_{Δ}^{z}

, yielding the inference relation induced by

κ_{Δ}^{z}

via equation (2).

3. Conditional Syntax Splitting

Syntax splittings describe that a belief base contains completely independent information about different parts of the signature. Let us first recall the notion of syntax splitting as introduced in Kern-Isberner et al. (2020). A conditional belief base $Δ$ splits into subbases $Δ_{1}, Δ_{2}$ if there are disjoint subsignatures $Σ_{1}, Σ_{2} \subseteq Σ$ such that $Δ = Δ_{1} \cup Δ_{2}$ , $Δ_{i} \subset (L_{i} | L_{i}), L_{i} = L (Σ_{i})$ for $i = 1, 2$ , $Σ_{1} \cap Σ_{2} = \emptyset$ , and $Σ_{1} \cup Σ_{2} = Σ$ . This is denoted as

Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} .

Syntax splittings were generalized in Heyninck et al. (2023) to conditional syntax splittings, which allow subbases to share the atoms in a given subsignature

Σ_{3}

Definition 2 (Conditional syntax splitting, (Heyninck et al., 2023))

We say a conditional belief base $Δ$ can be split into subbases $Δ_{1}$ , $Δ_{2}$ conditional on a subsignature $Σ_{3}$ , if there are $Σ_{1}, Σ_{2} \subseteq Σ$ such that $Δ_{i} = Δ \cap (L (Σ_{i} \cup Σ_{3}) ∣ L (Σ_{i} \cup Σ_{3}))$ for $i = 1, 2$ , the signatures $Σ_{1}$ , $Σ_{2}$ , and $Σ_{3}$ are pairwise disjoint, and $Σ = Σ_{1} \cup Σ_{2} \cup Σ_{3}$ . This is denoted as

Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} ∣ Σ_{3} .

(3)

However, conditional syntax splittings in general do not ensure complete independence of $Δ_{1}$ and $Δ_{2}$ in the sense that inferences involving only symbols from one of the subbases can safely be done taking into account just that subbase because they are independent of the other subbase, as it is the case in (unconditional) syntax splitting. To fix this, safe conditional syntax splittings were introduced.

Definition 3 (Safe conditional syntax splitting, (Heyninck et al., 2023))

A conditional belief base $Δ$ with $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} ∣ Σ_{3}$ can be safely split into subbases $Δ_{1}$ , $Δ_{2}$ conditional on a subsignature $Σ_{3}$ , writing:

Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3},

(4)

if the following safety property holds $for i, i^{'} \in {1, 2}, i \neq i^{'}$ :

for every ω^{i} ω^{3} \in Ω (Σ_{i} \cup Σ_{3}), there is an ω^{i^{'}} \in Ω (Σ_{i^{'}}) such that ω^{i} ω^{i^{'}} ω^{3} ⊭ \underset{(F | E) \in Δ_{i^{'}}}{⋁} E \land \neg F .

(5)

Safe conditional syntax splittings guarantee (conditional) independence of conditionals in $Δ_{1}$ and $Δ_{2}$ . In essence, the safety property ensures that any complete conjunction over $Σ_{3}$ may not require the falsification of a conditional in $Δ_{1}$ or $Δ_{2}$ . For a more detailed explanation on why this is necessary, see Heyninck et al. (2023).

Note that, unlike syntax splitting, conditional syntax splitting does not require the subbases $Δ_{1}$ and $Δ_{2}$ to be disjoint. For the remainder of this paper, we will use the notation introduced in the following straightforward proposition.

Proposition 4

If $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ then

\begin{aligned} Δ & = Δ_{1 ∖ 3} \cup Δ_{2 ∖ 3} \cup Δ_{3}, \end{aligned}

(6)

where

Δ_{1 ∖ 3}

Δ_{2 ∖ 3}

, and

Δ_{3}

are pairwise disjoint and are given by:

\begin{aligned} Δ_{3} & = Δ_{1} \cap Δ_{2}, \end{aligned}

(7)

\begin{aligned} Δ_{1 ∖ 3} & = Δ_{1} ∖ Δ_{3}, \end{aligned}

(8)

\begin{aligned} Δ_{2 ∖ 3} & = Δ_{2} ∖ Δ_{3} . \end{aligned}

(9)

Note that $Δ_{3} \subseteq (L (Σ_{3}) | L (Σ_{3}))$ , more precisely $Δ_{3} = Δ \cap (L (Σ_{3}) | L (Σ_{3}))$ , and $Δ_{i ∖ 3} \subseteq (L (Σ_{i} \cup Σ_{3}) | L (Σ_{i} \cup Σ_{3}))$ for $i \in {1, 2}$ . Furthermore, for $ω \in Ω$ and $A \in L (Σ_{i} \cup Σ_{3})$ we have that

ω ⊨ A iff ω^{i} ω^{3} ⊨ A .

(10)

We illustrate the notion of safe conditional syntax splitting with an example.

Example 5 (

Δ^{b}

)

Consider $Σ = {b, p, f, w}$ representing (b)irds, (p)enguins, (f)lying entities, and (w)inged entities. Let $Δ^{b} = {(f | b), (\bar{f} | p), (b | p), (w | b)}$ be a belief base describing the well-known penguin triangle together with the expression that birds usually have wings. Then

Δ^{b} = {(f | b), (\bar{f} | p), (b | p)} ⋃_{{p, f}, {w}}^{s} {(w | b)} ∣ {b}

is a conditional syntax splitting with

Σ_{1} = {p, f}, Σ_{2} = {w}

, and

Σ_{3} = {b}

. According to the notation in Proposition 4, we have

Δ_{3}^{b} = \emptyset

Δ_{1}^{b} =

Δ_{1 ∖ 3}^{b} = {(f | b), (\bar{f} | p), (b | p)}

, and

Δ_{2}^{b} =

Δ_{2 ∖ 3}^{b} = {(w | b)}

We can extend any $ω^{1} \in Ω (Σ_{1} \cup Σ_{3})$ by any $ω^{'} \in Ω (Σ_{2})$ with $ω^{'} ⊨ w$ without falsifying a conditional in $Δ_{2}^{b}$ . Similarly we can extend any $ω^{2} \in Ω (Σ_{2} \cup Σ_{3})$ by any $ω^{″} \in Ω (Σ_{1})$ with $ω^{″} ⊨ \bar{p} f$ without falsifying a conditional in $Δ_{1}^{b}$ . Thus, the splitting is safe.

The following is an example for a weakly consistent belief base with a safe conditional syntax splitting.

Example 6 ( $Δ^{b +}$ )

We extend $Δ^{b}$ from Example 5 with an additional conditional $δ_{5} = (⊥ | \bar{w})$ , modeling the constraint that we only consider worlds with winged entities to be feasible. In this way, we obtain the weakly consistent belief base $Δ^{b +} = {(f | b), (\bar{f} | p), (b | p), (w | b), (⊥ | \bar{w})}$ . Then

Δ^{b +} = {(f | b), (\bar{f} | p), (b | p)} ⋃_{{p, f}, {w}}^{s} {(w | b), (⊥ | \bar{w})} ∣ {b}

is a safe conditional syntax splitting of

Δ^{b +}

Every syntax splitting is also a safe conditional syntax splitting (Haldimann, 2024). In fact, syntax splittings coincide with conditional syntax splittings conditional on $Σ_{3} = \emptyset$ in a straightforward manner. Note also that every conditional syntax splitting with $Σ_{3} = \emptyset$ is trivially safe, leading to the following observation.

Proposition 7
Let $Δ$ be a consistent belief base. Then we have
$Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} iff Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ \emptyset .$

The postulates conditional relevance (CRel) and conditional independence (CInd) for inference from belief bases with conditional syntax splitting (Heyninck et al., 2023) are inspired by the postulates (Rel) and (Ind) (Kern-Isberner et al., 2020). (CRel) and (CInd) describe that inference over $Δ_{1}$ and $Δ_{2}$ should be independent if we have full information, that is, a full conjunction, on the “conditional pivot” $Σ_{3}$ . While for the original definition of (CRel) and (CInd) it was implicitly assumed that $Δ$ is strongly consistent (Heyninck et al., 2023), here we specifically also consider weakly consistent belief bases.

(CRel) (Heyninck et al., 2023). An inductive inference operator satisfies (CRel) if for $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , for $i \in {1, 2}$ , $A, B \in L_{Σ_{i}}$ , and a complete conjunction $E \in L_{Σ_{3}}$ we have that

Thus, an inductive inference operator satisfies conditional relevance if, for every safe conditional syntax splitting, inference in the language of $Σ_{i} \cup Σ_{3}$ depends only on the conditionals in $Δ_{i}$ , that is, only on those conditionals in that same language.

(CInd) (Heyninck et al., 2023). An inductive inference operator satisfies (CInd) if for $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , for $i, j \in {1, 2}, i \neq j$ , for any $A, B \in L_{Σ_{i}}$ , $D \in L_{Σ_{j}}$ , and a complete conjunction $E \in L_{Σ_{3}}$ such that we have

The requirement that was added here. Otherwise, (CInd) would require for every formula $A \in L_{Σ_{1}} \cup L_{Σ_{2}}$ . Conditional independence requires that, given complete knowledge of $Σ_{3}$ , inferences in the language of $Σ_{i} \cup Σ_{3}$ are independent of every formula over the language of $Σ_{j}$ .

The postulate (CSynSplit) is the combination of (CRel) and (CInd):

(CSynSplit) (Heyninck et al., 2023). An inductive inference operator satisfies (CSynSplit) if it satisfies (CRel) and (CInd).

While the conditional syntax splitting postulates can be applied to all belief bases, they are trivially satisfied for belief bases that are not weakly consistent. This is because belief bases that are not weakly consistent cannot have a safe conditional syntax splitting, as the following proposition states.
Proposition 8
Let $Δ$ be a belief base that is not weakly consistent. Then there is no $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ .
Proof.
First, note that weak consistency of a belief base $Δ$ can be characterized by the existence of a world $ω$ falsifying no conditional in $Δ$ . Let $Δ$ be a belief base that is not weakly consistent and let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} ∣ Σ_{3}$ . We show that $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} ∣ Σ_{3}$ cannot be safe. We discern two cases. (1)
Either $Δ_{1}$ or $Δ_{2}$ is not weakly consistent or both: Let $Δ_{i}$ be the subbase that is not weakly consistent. Then there is no world such that no conditional in $Δ_{i}$ is falsified. Thus, the safety condition cannot be satisfied.
(2)
Both $Δ_{1}$ and $Δ_{2}$ are weakly consistent: Then there is some $ω \in Ω$ that falsifies no conditionals in $Δ_{1}$ . With (10) this means that $ω^{1} ω^{3}$ falsifies no conditional in $Δ_{1}$ . Toward a contradiction, assume that $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ . Then the safety property would demand that there is $ω^{2} \in Ω (Σ_{2})$ such that $ω^{1} ω^{2} ω^{3}$ falsifies no conditional in $Δ_{2}$ . But then $ω^{1} ω^{2} ω^{3}$ falsifies no conditional in $Δ$ as a whole and therefore $Δ$ would be weakly consistent, contradiction.

For the rest of this article, we will therefore focus on the case of weakly consistent belief bases when investigating compliance with conditional syntax splitting postulates. Conditional syntax splitting is closely related to the notion of conditional $κ$ -independence for OCFs.
Definition 9 (Conditional $κ$ -independence, (Spohn, 2012))

Let $Σ_{1}, Σ_{2}, Σ_{3} \subseteq Σ,$ where $Σ_{1}, Σ_{2},$ and $Σ_{3}$ are pairwise disjoint and let $κ$ be an OCF. $Σ_{1}, Σ_{2}$ are conditionally $κ$ -independent given $Σ_{3}$ , in symbols $Σ_{1} ⊥ ⊥_{κ} Σ_{2} | Σ_{3}$ , if for all $ω^{1} \in Ω (Σ_{1}), ω^{2} \in Ω (Σ_{2})$ , and $ω^{3} \in Ω (Σ_{3})$ with $κ (ω^{3}) < \infty$ , it holds that $κ (ω^{1} | ω^{2} ω^{3}) = κ (ω^{1} | ω^{3})$ .

Note that in the expression $κ (ω^{1} | ω^{2} ω^{3}) = κ (ω^{1} | ω^{3})$ the $ω^{i}$ are treated not as worlds but as formulas over $Σ$ . The following lemma provides a useful alternative characterization of conditional $κ$ -independence.

Lemma 10
Let $Σ_{1}, Σ_{2}, Σ_{3} \subseteq Σ,$ where $Σ_{1}, Σ_{2},$ and $Σ_{3}$ are pairwise disjoint and let $κ$ be an OCF. $Σ_{1}, Σ_{2}$ are conditionally $κ$ -independent given $Σ_{3}$ iff for all $A \in L (Σ_{1}), B \in L (Σ_{2})$ and all complete conjunctions $C \in L (Σ_{3})$ with $κ (C) < \infty$ it holds that
$κ (A B C) = κ (A C) + κ (B C) - κ (C) .$

Proof.
Note that the equation $κ (ω^{1} | ω^{2} ω^{3}) = κ (ω^{1} | ω^{3})$ from Definition 9 is equivalent to
$κ (ω^{1} ω^{2} ω^{3}) = κ (ω^{1} ω^{3}) + κ (ω^{2} ω^{3}) - κ (ω^{3})$
(11)
by applying the definition of ranks of conditionals (cf. Section 2). Note that while $κ$ is built over $Σ$ , the $ω^{i}$ are treated here as conjunctions over their respective signatures. We show both directions of the “iff” separately.

Direction $\Rightarrow$ : Let $Σ_{1}, Σ_{2}$ be conditionally $κ$ -independent given $Σ_{3}$ . Then (11) holds for all $ω^{1} \in Ω (Σ_{1})$ , $ω^{2} \in Ω (Σ_{2})$ , and $ω^{3} \in Ω (Σ_{3})$ . Now let $ω^{1} ω^{3}$ be the world with minimal rank in the models of $A C$ . Assume the same for $ω^{2} ω^{3}$ and $B C$ . Note that since $C$ is a complete conjunction over $Σ_{3}$ , $ω^{3}$ must be a world with minimal rank in the models of $C$ . Thus we can rewrite (11) to
$κ (ω^{1} ω^{2} ω^{3}) = κ (A C) + κ (B C) - κ (C) .$
(12)
Clearly $ω^{1} ω^{2} ω^{3} ⊨ A B C$ . We now show that $ω^{1} ω^{2} ω^{3}$ also has minimal rank with this property. Toward a contradiction assume $ω^{1} ω^{2} ω^{3}$ did not have minimal rank with this property. Then there is some $ω^{'}$ with $ω^{'} ⊨ A B C$ and $κ (ω^{'}) < κ (ω^{1} ω^{2} ω^{3})$ . Since $Σ_{1}, Σ_{2}$ and $Σ_{3}$ are disjoint, $ω^{'}$ can be split into $ω^{' 1} \in Ω (Σ_{1})$ , $ω^{' 2} \in Ω (Σ_{2})$ , $ω^{' 3} \in Ω (Σ_{3})$ . Thus, it must hold that
$κ (ω^{' 1} ω^{' 2} ω^{' 3}) = κ (ω^{' 1} ω^{' 3}) + κ (ω^{' 2} ω^{' 3}) - κ (ω^{' 3}) .$
(13)
Since $κ (ω^{'}) < κ (ω^{1} ω^{2} ω^{3})$ it must hold that $κ (ω^{' 1} ω^{' 3}) < κ (ω^{1} ω^{3})$ or $κ (ω^{' 2} ω^{' 3}) < κ (ω^{2} ω^{3})$ or $κ (ω^{' 3}) > κ (ω^{3})$ . The first inequality cannot hold, as $ω^{' 1} ω^{' 3} ⊨ A C$ , but as per our assumption $ω^{1} ω^{3}$ is minimal with this property. Analogously the second inequality cannot hold. The third inequality does not hold either, as $C$ is a full conjunction and therefore $ω^{3} = ω^{' 3}$ . Thus $κ (ω^{'}) < κ (ω^{1} ω^{2} ω^{3})$ cannot hold and $ω^{1} ω^{2} ω^{3}$ must have minimal rank with $ω^{1} ω^{2} ω^{3} ⊨ A B C$ . Then we can rewrite (12) to
$κ (A B C) = κ (A C) + κ (B C) - κ (C) .$
(14)
Because $κ (C) < \infty$ , the right-hand side of (14) is always well-defined, completing the proof for this direction.

Direction $\Leftarrow$ : For the other direction, assume (14) holds for all $A \in L (Σ_{1}), B \in L (Σ_{2})$ , and all complete conjunctions $C \in L (Σ_{3})$ with $κ (C) < \infty$ . Let $ω^{1} \in Ω (Σ_{1}), ω^{2} \in Ω (Σ_{2})$ , and $ω^{3} \in Ω (Σ_{3})$ . As we have stated previously, all worlds can be represented by a complete conjunction of all literals of their signature. Let $A$ be such a conjunction for $ω^{1}$ , $B$ for $ω^{2}$ and $C$ for $ω^{3}$ . Then (14) is equivalent to (11), completing the proof.

Conditional independence for OCFs expresses that information on $Σ_{2}$ is redundant for $Σ_{1}$ if full information on $Σ_{3}$ is available and used. We can now state the following result regarding (CRel) for OCFs as follows:
Proposition 11 (Adapted from Heyninck et al., 2023)

An inductive inference operator for OCFs $C^{o c f} : Δ \mapsto κ_{Δ}$ satisfies (CRel) if for any $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , we have $κ_{Δ_{i}} = κ_{Δ} |_{Σ_{i} \cup Σ_{3}}$ for $i \in {1, 2}$ .

Originally, the other direction of Proposition 11 was stated to also hold (Heyninck et al., 2023, Proposition 7). We show now that this is not the case.

Proposition 12
There is an inductive inference operator for OCFs $C^{o c f} : Δ \mapsto κ_{Δ}$ that satisfies (CRel) but there is a safe conditional syntax splitting $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ such that $κ_{Δ_{i}} = κ_{Δ} |_{Σ_{i} \cup Σ_{3}}$ for $i \in {1, 2}$ does not hold.
Proof.
Let $Δ = {(a | c), (b | c)}$ with $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3} = {(a | c)} ⋃_{{a}, {b}}^{s} {(b | c)} ∣ {c}$ . Consider the OCFs given in Figure 1, and let $C^{o c f}$ be the inductive inference operator defined by:
$C^{o c f} (Δ) = {\begin{cases} κ_{Δ_{1}} & if Δ = {(a | c)}, \\ κ_{Δ}^{z} & otherwise . \end{cases}$
Clearly $C^{o c f}$ is OCF-based. The ranking function $κ_{Δ}^{z}$ obtained via $C^{o c f}$ from $Δ$ and its marginalization $κ_{Δ}^{z} |_{Σ_{1} \cup Σ_{3}}$ are shown in Figure 1(a). The ranking function $κ_{Δ_{1}}$ obtained via $C^{o c f}$ from $Δ_{1}$ is shown in Figure 1(b). As can be seen, (CRel) is satisfied, but $κ_{Δ_{1}} \neq κ_{Δ}^{z} |_{Σ_{1} \cup Σ_{3}}$ . Note, however, that $κ_{Δ_{1}}$ and $κ_{Δ}^{z} |_{Σ_{1} \cup Σ_{3}}$ are inferentially equivalent (Beierle & Kutsch, 2019) because they induce the same inference relation, that is, .
Figure 1.
Ranking functions for the proof of Proposition 12. (a) Ranking function $κ_{Δ}^{z}$ and its marginalization to $Σ_{1} \cup Σ_{3}$ . (b) Ranking function $κ_{Δ_{1}}$ .

However, this does not influence any of the other results in Heyninck et al. (2023), as this direction is never used. Next we also state a result connecting (CInd) to $κ$ -independence for OCFs.
Proposition 13 (Adapted from Heyninck et al., 2023)

An inductive inference operator for OCFs $C^{o c f} : Δ \mapsto κ_{Δ}$ satisfies (CInd) if for any $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ we have $Σ_{1} ⊥ ⊥_{κ_{Δ}} Σ_{2} | Σ_{3}$ .

Proposition 13 lets us use conditional $κ$ -independence to derive (CInd), while Proposition 11 allows us to derive (CRel) via marginalization of OCFs, both of which are useful for showing that an inference operator for OCFs satisfies (CInd) or (CRel), respectively.

4. Conditional Semantic Splitting

For the rest of this article, we focus on inference with OCF-based semantics. While syntax splitting notions aim at splitting a belief base based on its syntactical representation, a splitting notion at the semantical level was introduced in the form of semantic splitting (Beierle et al., 2021b). In analogy to syntax splitting, these semantic splittings split a belief base into disjoint subbases. We now extend this notion by introducing the new concept of conditional semantic splitting, allowing for the subbases to share conditionals as in the case of conditional syntax splitting. We show that extended c-representations and conservative c-representations (Haldimann et al., 2024) satisfy conditional semantic splitting.

4.1. Model Combinations and Semantic Splittings

First, we introduce the notion of model combinations for ranking models.

Definition 14 (Model combination)

Let $M_{1}, M_{2}$ be sets of OCFs over $Σ$ . Model combinations of $M_{1}$ and $M_{2}$ , denoted by $M_{1} \oplus M_{2}$ and by $M_{1} ⊖ M_{2}$ , respectively, are given by:

\begin{aligned} M_{1} \oplus M_{2} & = {κ ∣ κ (ω) = κ_{1} (ω) + κ_{2} (ω), κ_{1} \in M_{1}, κ_{2} \in M_{2}}, \\ M_{1} ⊖ M_{2} & = {κ ∣ κ (ω) = κ_{1} (ω) - κ_{2} (ω), κ_{1} \in M_{1}, κ_{2} \in M_{2}} . \end{aligned}

Note that in general, $M_{1} \oplus M_{2}$ or $M_{1} ⊖ M_{2}$ may contain functions that are not ranking functions because, for example, no $ω$ is mapped to $0$ . We consider different subclasses of ranking models for conditional belief bases in this paper, for example, system Z ranking functions or c-representations. The following definition provides a joint formal concept for focusing on such subclasses.

Definition 15
An (OCF-based) semantics $S e m$ for conditional belief bases is a function mapping a belief base $Δ$ over $Σ$ to a set of models $Mod_{Σ}^{S e m} (Δ) \subseteq Mod_{Σ} (Δ)$ where $Mod_{Σ} (Δ) = {κ ∣ κ ⊨ Δ}$ .

A conditional semantic splitting of $Δ$ builds on the combination of models given by an OCF-based semantics Sem and generalizes the notion of semantic splitting (Beierle et al., 2021b). Semantic splittings and conditional semantic splittings apply the notion of splittings to the model level, yielding a desirable splitting property to evaluate OCF-based semantics.
Definition 16 (Conditional semantic splitting)

$Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2} ∣ Σ_{3}$ is a conditional semantic splitting of $Δ$ for a semantic $S e m$ if

Mod_{Σ}^{Sem} (Δ) = Mod_{Σ}^{Sem} (Δ_{1}) \oplus Mod_{Σ}^{Sem} (Δ_{2}) ⊖ Mod_{Σ}^{Sem} (Δ_{3}) .

Intuitively, if a belief base $Δ$ has a conditional semantic splitting, then the models of the subbases $Δ_{1}$ and $Δ_{2}$ can be computed independently and then combined to obtain models of $Δ$ . Because it might be the case that $Δ_{3} = Δ_{1} \cap Δ_{2} \neq \emptyset$ , the conditionals in $Δ_{3}$ are counted twice and thus their influence must be subtracted again to obtain models of $Δ$ . This yields the base for the following postulate.

(CSemSplit). An OCF-based semantic $S e m$ satisfies (CSemSplit) if every safe splitting $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ is also a conditional semantic splitting of $Δ$ .

We first give two examples of popular inference operators that do not satisfy conditional semantic splitting.

Example 17
System P is an axiom system stating desirable properties for nonmonotonic reasoning with conditionals (Adams, 1965; Kraus et al., 1990). It also characterizes a semantic that maps a belief base $Δ$ to all its models, that is, $Mod_{Σ}^{System P} (Δ) = Mod_{Σ} (Δ)$ . This semantics does not satisfy (CSemSplit) which can be illustrated with $Δ = Δ_{1} \cup Δ_{2}$ and $Δ_{1} = {(a | ⊤)}$ and $Δ_{2} = {(b | ⊤)}$ . Obviously, ${Δ_{1}, Δ_{2}}$ is a syntax splitting and thus a safe conditional syntax splitting of $Δ$ (Proposition 7). Furthermore,
$\begin{aligned} κ_{1} & = {a b \mapsto 1, a \bar{b} \mapsto 0, \bar{a} b \mapsto 1, \bar{a} \bar{b} \mapsto 1} accepts Δ_{1}, and \\ κ_{2} & = {a b \mapsto 1, a \bar{b} \mapsto 1, \bar{a} b \mapsto 0, \bar{a} \bar{b} \mapsto 1} accepts Δ_{2}, \end{aligned}$
but $κ_{1} + κ_{2} = {a b \mapsto 2, a \bar{b} \mapsto 1, \bar{a} b \mapsto 1, \bar{a} \bar{b} \mapsto 2}$ is not even a ranking function and would also not model $Δ$ if it was normalized by reducing all ranks by 1.
Example 18
(Extended) system Z (cf. Definition 1) yields a model semantics given by $Mod_{Σ}^{Sem} (Δ) = {κ_{Δ}^{z}}$ . This semantics also does not satisfy (CSemSplit). Consider $Δ^{b}$ and the conditional syntax splitting in Example 5. We have $E P (Δ^{b}) = {{(f | b), (w | b)}, {(b | p), (\bar{f} | p)}}$ , $E P (Δ_{1}^{b}) = {{(f | b)}, {(b | p), (\bar{f} | p)}}$ and $E P (Δ_{2}^{b}) = {{(w | b)}}$ . Then we get $κ_{Δ}^{z} (\bar{p} b \bar{f} \bar{w}) = 1 \neq 2 = κ_{Δ_{1}}^{z} (\bar{p} b \bar{f}) + κ_{Δ_{2}}^{z} (b \bar{w}) - κ_{Δ_{3}}^{z} (b)$ . Thus (CSemSplit) is not satisfied.

While system P and system Z do not satisfy conditional semantic splitting we will consider semantics satisfying this property in the next section.
4.2. Extended c-Representations and Relaxations of Extended c-Representations

Among the OCF models of $Δ$ , c-representations are special ranking models obtained by assigning individual integer impacts to the conditionals in $Δ$ and generating the world ranks as the sum of impacts of falsified conditionals.

Definition 19 (c-Representation; Kern-Isberner, 2001, 2004)

A c-representation of a strongly consistent $Δ = {(B_{1} | A_{1}), \dots, (B_{n} | A_{n})}$ is an OCF $κ$ constructed from nonnegative integer impacts $η_{i} \in N_{0}$ assigned to each $(B_{i} | A_{i})$ such that $κ$ accepts $Δ$ and is given by:

\begin{aligned} κ (ω) = \sum_{\begin{matrix} 1 ⩽ i ⩽ n \\ ω ⊨ A_{i} {\bar{B}}_{i} \end{matrix}} η_{i} . \end{aligned}

(15)

c-Representations can conveniently be specified using a constraint satisfaction problem (for detailed explanations, see Kern-Isberner, 2001, 2004):

Definition 20 (

C R (Δ)

, (Beierle et al., 2018; Kern-Isberner, 2001))

Let $Δ = {(B_{1} | A_{1}), \dots, (B_{n} | A_{n})}$ . The constraint satisfaction problem for c-representations of $Δ$ , denoted by $C R (Δ)$ , is given by the conjunction of the constraints, for all $i \in {1, \dots, n}$ :

\begin{aligned} η_{i} & ⩾ 0, \end{aligned}

(16)

\begin{aligned} η_{i} & > min_{ω ⊨ A_{j} B_{j}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} \bar{B_{k}} \end{matrix}} η_{k} - min_{ω ⊨ A_{j} {\bar{B}}_{j}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} \bar{B_{k}} \end{matrix}} η_{k} . \end{aligned}

(17)

Note that (16) expresses that falsification of conditionals should make worlds not more plausible, and (17) ensures that $κ$ as specified by (15) accepts $Δ$ . A solution of $C R (Δ)$ is a vector $\vec{η} = (η_{1}, \dots, η_{n})$ of natural numbers. $S o l (C R (Δ))$ denotes the set of all solutions of $C R (Δ)$ . For $\vec{η} \in S o l (C R (Δ))$ and $κ$ as in equation (15), $κ$ is the OCF induced by $\vec{η}$ and is denoted by $κ_{\vec{η}}$ . $C R (Δ)$ is sound and complete (Beierle et al., 2018; Kern-Isberner, 2001): for every $\vec{η} \in S o l (C R (Δ))$ , $κ_{\vec{η}}$ is a c-representation with $κ_{\vec{η}} ⊨ Δ$ , and for every c-representation $κ$ with $κ ⊨ Δ$ , there is $\vec{η} \in S o l (C R (Δ))$ such that $κ = κ_{\vec{η}}$ . Thus, c-representations yield an OCF-based model semantics

Mod_{Σ}^{c-rep} (Δ) = {κ_{\vec{η}} ∣ \vec{η} \in S o l (C R (Δ))} .

For an impact vector

\vec{η}

, we will simply write

{\vec{η}}^{1}

and

{\vec{η}}^{2}

for the corresponding projections

{\vec{η} |}_{Δ_{1}}

and

{\vec{η} |}_{Δ_{2}}

, and

({\vec{η}}^{1}, {\vec{η}}^{2})

for their composition. Similarly, we will write

({\vec{η}}^{1}, η_{j})

for the composition of a vector with a singular natural number

η_{j}

. We illustrate c-representations and

C R (Δ)

with an example.

Example 21 (

Δ^{b}

continued)

Consider the belief base $Δ^{b}$ from Example 5.

$C R (Δ^{b})$ contains $η_{i} ⩾ 0$ for $i \in {1, 2, 3, 4}$ and the following constraints:

\begin{aligned} η_{1} > & min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ b f \end{matrix}} \sum_{\begin{matrix} j \neq 1 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} - min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ b \bar{f} \end{matrix}} \sum_{\begin{matrix} j \neq 1 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j}, \\ η_{2} > & min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ p \bar{f} \end{matrix}} \sum_{\begin{matrix} j \neq 2 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} - min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ p f \end{matrix}} \sum_{\begin{matrix} j \neq 2 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j}, \\ η_{3} > & min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ p b \end{matrix}} \sum_{\begin{matrix} j \neq 3 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} - min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ p \bar{b} \end{matrix}} \sum_{\begin{matrix} j \neq 3 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} . \\ η_{4} > & min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ b w \end{matrix}} \sum_{\begin{matrix} j \neq 4 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} - min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ b \bar{w} \end{matrix}} \sum_{\begin{matrix} j \neq 4 \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} . \end{aligned}

Figure 2(a) shows some solutions for

Δ^{b}

as well as their corresponding induced c-representations. For example,

{\vec{η}}_{1} = (1, 2, 2, 1) \in S o l (C R (Δ^{b}))

{\vec{η}}_{1}^{1} = (1, 2, 2) \in S o l (C R (Δ_{1 ∖ 3}^{b})),

and

{\vec{η}}_{1}^{2} = (1) \in S o l (C R (Δ_{2 ∖ 3}^{b}))

Figure 2.

(a) Verification (v) and falsification (f) on worlds, impact vectors ${\vec{η}}_{1}, {\vec{η}}_{2}, {\vec{η}}_{3}$ for $Δ^{b}$ and their induced c-representations $κ_{{\vec{η}}_{1}}, κ_{{\vec{η}}_{2}}, κ_{{\vec{η}}_{3}}$ . (b) Verification and falsification of the additional conditional $δ_{5} \in Δ^{b +}$ , impact vectors ${\vec{η}}_{4}, {\vec{η}}_{5}$ for $Δ^{b +}$ and their induced extended c-representations $κ_{{\vec{η}}_{4}}, κ_{{\vec{η}}_{5}}$ . (c) The system Z ranking functions $κ_{Δ^{b}}^{z}$ and $κ_{Δ^{b +}}^{z}$ . These tables are used in Examples 21, 24, and 30.

Recently, c-representations have been generalized to also cover weakly consistent belief bases, yielding extended c-representations.

Definition 22 (Extended c-representation, (Haldimann et al., 2024))

An extended c-representation of a belief base $Δ = {(B_{1} | A_{1}), \dots, (B_{n} | A_{n})}$ over $Σ$ is a ranking function $κ_{\vec{η}}$ constructed from impacts $\vec{η} = (η_{1}, \dots, η_{n})$ with $η_{i} \in N_{0} \cup {\infty}, i \in {1, \dots, n}$ assigned to each conditional $(B_{i} | A_{i})$ such that $κ_{\vec{η}}$ accepts $Δ$ and is given by:

\begin{aligned} κ_{\vec{η}} (ω) = & \sum_{\begin{matrix} 1 ⩽ i ⩽ n \\ ω ⊨ A_{i} {\bar{B}}_{i} \end{matrix}} η_{i} . \end{aligned}

(18)

We will denote the set of all extended c-representations of

Δ

Mod_{Σ}^{e c} (Δ)

Every c-representation is also an extended c-representation (Haldimann et al., 2024). Just like c-representations, extended c-representations can also be specified using a constraint satisfaction problem, based on the constraint satisfaction problem $C R (Δ)$ for c-representations.

Definition 23 (

{C R}_{Σ}^{e x} (Δ)

, (Haldimann et al., 2024))

Let $Δ = {(B_{1} | A_{1}), \dots, (B_{n} | A_{n})}$ be a belief base over $Σ$ . The constraint satisfaction problem for extended c-representations of $Δ$ , denoted by ${C R}_{Σ}^{e x} (Δ)$ , on the constraint variables ${η_{1}, \dots, η_{n}}$ ranging over $N_{0} \cup {\infty}$ is given by the conjunction of the constraints, for all $i \in {1, \dots, n}$ :

η_{i} ⩾ 0.

(19)

The constraint system ${C R}_{Σ}^{e x} (Δ)$ is sound and complete (Haldimann et al., 2024), thus extended c-representations yield an OCF-based model semantics

Mod_{Σ}^{e c} (Δ) = {κ_{\vec{η}} ∣ \vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))} .

We illustrate the notion of extended c-representations and their constraint system with an example.

Example 24 (

Δ^{b}, Δ^{b +}

continued)

Let $Δ^{b}$ , $Δ^{b +}$ as in Examples 5, 6, and 21. The constraint for $(⊥ | \bar{w})$ in ${C R}_{Σ}^{e x} (Δ^{b +})$ is given by:

min_{\begin{matrix} ω \in Ω (Σ) \\ ω ⊨ \bar{w} \end{matrix}} \sum_{\begin{matrix} 1 ⩽ j ⩽ n \\ ω ⊨ A_{k} \bar{B_{k}} \\ (B_{k} | A_{k}) \in Δ^{b +} \end{matrix}} η_{k} = \infty or η_{5} > min_{\begin{matrix} ω ⊨ ⊥ \end{matrix}} \sum_{\begin{matrix} j \neq 5 \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ^{b +} \end{matrix}} η_{k} - min_{\begin{matrix} ω ⊨ \bar{w} \end{matrix}} \sum_{\begin{matrix} j \neq 5 \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ^{b +} \end{matrix}} η_{k} .

Every possible solution of this constraint requires choosing

η_{5} = \infty

, yielding, for example, the impact vector

{\vec{η}}_{4}^{e x} = (1, 2, 2, 1, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ^{b +}))

{\vec{η}}_{5}^{e x} = (1, \infty, 2, 1, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ^{b +}))

as possible solutions of

{C R}_{Σ}^{e x} (Δ^{b +})

The corresponding extended c-representations can be seen in Figure 2(b).

Among the extended c-representations, special attention should be given to the so-called conservative c-representations, assigning $\infty$ to a world only if it is strictly necessary for accepting all conditionals in $Δ$ .

Definition 25 (Conservative c-representation, (Haldimann et al., 2024), adapted)

Let $Δ$ be a belief base. Then $C Mod_{Σ}^{e c} (Δ)$ is the set of conservative c-representations $κ_{\vec{η}}$ of $Δ$ such that for all worlds $ω$ , we have $κ_{\vec{η}} (ω) = \infty$ only if for every OCF $κ$ , $κ ⊨ Δ$ implies $κ (ω) = \infty$ .

Note that every conservative c-representation is also an extended c-representation (Haldimann et al., 2024). Because $κ_{Δ}^{z}$ is the unique minimal ranking function accepting a belief base $Δ$ in the sense that $κ_{Δ}^{z} (ω) ⩽ κ (ω)$ for every $ω$ and every OCF $κ$ accepting $Δ$ (Goldszmidt & Pearl, 1990), we immediately obtain the following lemma.

Lemma 26
Let $Δ$ be a weakly consistent belief base and let $κ$ be a conservative c-representation of $Δ$ . Then $κ_{Δ}^{z} (ω) < \infty$ iff $κ_{\vec{η}} (ω) < \infty$ .

The set of conservative c-representations has been shown to admit exactly the same inferences as the set of extended c-representations if all ranking functions are taken into account.
Proposition 27 (Haldimann et al., 2024)

Let $Δ$ be a belief base and let $A, B \in L (Σ)$ . Then

Conservative c-representations can be specified using a simplified constraint satisfaction problem, taking into account only those conditionals whose impacts are not assigned $\infty$ .

Definition 28 ( ${C R S}_{Σ}^{e x} (Δ)$ , (Haldimann et al., 2024))

Let $Δ = {(B_{1} | A_{1}), \dots, (B_{n} | A_{n})}$ be a belief base over $Σ$ with the extended Z-partition $E P (Δ) = {Δ^{0}, \dots, Δ^{m}, Δ^{\infty}}$ . Let

J_{Δ} = {j ∣ (B_{j} | A_{j}) \in Δ ∖ Δ^{\infty} s.t. A_{j} \bar{B_{j}} \land (\underset{(D | C) \in Δ^{\infty}}{⋀} (\bar{C} \lor D)) ≢ ⊥} .

The simplified constraint satisfaction problem for extended c-inference of

Δ

, denoted by

{C R S}_{Σ}^{e x} (Δ)

, on the constraint variables

{η_{j_{1}}, \dots, η_{j_{l}}}, j_{k} \in J_{Δ}

ranging over

N_{0}

is given by the constraints

{c r s}_{j}^{e x Δ}

, for all

j \in J_{Δ}

\begin{aligned} ({c r s}_{j}^{e x Δ}) η_{i} > min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ A_{i} B_{i} \end{matrix}} \sum_{\begin{matrix} j \in J_{Δ} \\ j \neq i \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} - min_{\begin{matrix} ω \in Ω_{Σ} \\ ω ⊨ A_{i} {\bar{B}}_{i} \end{matrix}} \sum_{\begin{matrix} j \in J_{Δ} \\ j \neq i \\ ω ⊨ A_{j} \bar{B_{j}} \end{matrix}} η_{j} . \end{aligned}

(21)

By assigning the value $\infty$ to all constraint variables not mentioned in ${C R S}_{Σ}^{e x} (Δ)$ , we obtain the set of impact vectors ${S o l}_{Δ}^{J + \infty}$ characterizing conservative c-representations.

Definition 29 (

{S o l}_{Δ}^{J + \infty}

, (Haldimann et al., 2024))

Let $Δ$ be a belief base, $n = | Δ |$ , and let $J_{Δ}$ be defined as above. For ${\vec{η}}^{J} \in S o l ({C R S}_{Σ}^{e x} (Δ))$ let ${\vec{η}}^{J + \infty} \in (N_{0} \cup {\infty})^{n}$ be the impact vector with

η_{i}^{J + \infty} = {\begin{cases} η_{i} & for i \in J_{Δ}, \\ \infty & otherwise. \end{cases}

Then

{S o l}_{Δ}^{J + \infty} := {{\vec{η}}^{J + \infty} ∣ {\vec{η}}^{J} \in S o l ({C R S}_{Σ}^{e x} (Δ))}

The set of impact vectors ${S o l}_{Δ}^{J + \infty}$ is sound and complete, that is, it characterizes exactly the set of conservative c-representations (Haldimann et al., 2024), yielding an OCF-based model semantics

C Mod_{Σ}^{e c} (Δ) = {κ_{\vec{η}} ∣ \vec{η} \in {S o l}_{Δ}^{J + \infty}} .

We illustrate the differences between conservative c-representations and extended c-representations by giving an example.

Example 30 ( $Δ^{b +}$ continued)

Let $Δ^{b +}$ as in Example 6. This belief base is weakly consistent with $E P (Δ^{b +}) = (Δ^{0}, Δ^{1}, Δ^{\infty}) = ({(f | b), (w | b)}, {(\bar{f} | p), (b | p)}, {(⊥ | \bar{w})})$ . Let ${\vec{η}}_{4} = (1, 2, 2, 1, \infty)$ and ${\vec{η}}_{5} = (1, \infty, 2, 1, \infty)$ be as in Example 24 with $κ = κ_{{\vec{η}}_{4}}$ and $κ^{'} = κ_{{\vec{η}}_{5}}$ . Then $κ$ is a conservative c-representation, that is, ${\vec{η}}_{4} \in {S o l}_{Δ^{b +}}^{J + \infty}$ but $κ^{'}$ is not conservative, as $κ^{'} (b p f w) = \infty$ , but $κ_{Δ^{b +}}^{z} (b p f w) = 2$ . The ranking functions $κ = κ_{{\vec{η}}_{4}}$ and $κ^{'} = κ_{{\vec{η}}_{5}}$ can be seen in Figure 2(b). The system Z ranking functions $κ_{Δ^{b}}^{z}$ and $κ_{Δ^{b +}}^{z}$ can be seen in Figure 2(c).

Consider the inference relations and derived from $κ$ and $κ^{'}$ , respectively. First, we have that if for all . However, additionally allows for further, potentially unintended inferences to be drawn. For example, we have both, and , admitting the inferences that flying penguins are usually birds and not birds at the same time, due to the fact that $κ_{Δ}^{'} (p f) = \infty$ , that is, according to $κ^{'}$ , flying penguins are deemed infeasible. On the other hand, regarding $κ$ , we have and , that is, according to $κ$ , flying penguins are usually birds, which seems more intuitive.

We now look at the differences between conservative and extended c-representations more formally. For this, we introduce the notion of relaxation of a conservative ranking function and of an impact vector.

Definition 31 (Relaxation of conservative c-representations)

Let $κ$ be a conservative c-representation and let $κ^{'}$ be an extended c-representation. Then $κ^{'}$ is a relaxation of $κ$ , if $κ^{'} (ω) = κ (ω)$ or $κ^{'} (ω) = \infty$ . Similarly, ${\vec{η}}^{'} = ({\vec{η}}_{1}^{'}, \dots, {\vec{η}}_{n}^{'}) \in S o l ({C R}_{Σ}^{e x} (Δ))$ is a relaxation of $\vec{η} = ({\vec{η}}_{1}, \dots, {\vec{η}}_{n}) \in {S o l}_{Δ}^{J + \infty}$ if, for $i \in {1, \dots, n}$ , $η_{i}^{'} = η_{i}$ or $η_{i}^{'} = \infty$ . We call a relaxation proper if additionally $κ \neq κ^{'}$ or $\vec{η} \neq {\vec{η}}^{'}$ , respectively.

The intuition here is that every conservative c-representation, induced by an impact vector $\vec{η}$ , has a corresponding set of extended c-representations, whose impact vectors share the same values as $\vec{η}$ , except that they may unnecessarily assign the value $\infty$ to some constraint variable(s). For example, in Example 24, the impact vector ${\vec{η}}_{5}$ is a relaxation of ${\vec{η}}_{4}$ , and furthermore $κ_{{\vec{η}}_{5}}$ is a relaxation of $κ_{{\vec{η}}_{4}}$ . We now show that the inferences of a conservative c-representation and its relaxations are, in general, incomparable.

Proposition 32
There is a belief base $Δ$ such that there are a conservative c-representation $κ$ of $Δ$ and a relaxation $κ^{'}$ of $κ$ such that both and hold.
Proof.
Let $Δ = {(a | ⊤), (b | ⊤)}$ . Let $\vec{η} = (2, 1) \in {S o l}_{Δ}^{J + \infty}$ and $κ = κ_{\vec{η}}$ . Let ${\vec{η}}^{'} = (2, \infty) \in {C R}_{Σ}^{e x} (Δ)$ and $κ^{'} = κ_{{\vec{η}}^{'}}$ . Clearly $κ^{'}$ is a relaxation of $κ$ . The ranking functions can be seen in Figure 3. then and , because but , and but .
Figure 3.
Ranking functions for the proof of Proposition 32.

While we have studied the differences between conservative and extended c-representations in this section, in the next section, we will show that both semantics satisfy conditional semantic splitting.
4.3. Extended and Conservative c-Representations Satisfy Semantic Splitting

A fundamental property of c-representations is that for any syntax splitting $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}} Δ_{2}$ the composition of any impact vectors for the subbases yields an impact vector for $Δ$ , and vice versa (Kern-Isberner et al., 2020). Before proving a generalization of this observation to conditional syntax splitting, we state two useful lemmata.

Lemma 33
Let $Δ$ be a conditional belief base, $\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))$ , and $(F | E)$ a self-fulfilling conditional, where $E ⊨ F$ . Then $(\vec{η}, η) \in S o l ({C R}_{Σ}^{e x} (Δ \cup {(F | E)}))$ for any $η \in N \cup {\infty}$ , and accordingly $κ_{\vec{η}} (ω) = κ_{(\vec{η}, η)} (ω)$ for all $ω \in Ω$ .
Proof.
Since $ω ⊭ E \bar{F}$ for all worlds $ω$ , the impact $η$ assigned to $(F | E)$ only has to satisfy $η ⩾ 0$ , and it does not appear in the sum-expression (15) defining an extended c-representation.
Lemma 34
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ . Then all conditionals in $Δ_{3} = Δ_{1} \cap Δ_{2}$ are self-fulfilling.
Proof.
Let $i, i^{'} \in {1, 2}$ where $i \neq i^{'}$ and $(B | A) \in Δ_{3}, A, B \in L (Σ_{3})$ .

Toward a contradiction, assume there was some $ω$ with $ω ⊨ A \bar{B}$ . Then for $ω^{3} = ω |_{Σ_{3}}$ it must also hold that $ω^{3} ⊨ A \bar{B}$ . Due to the safety property (5), $ω^{3}$ must have extensions $ω^{1} \in Ω (Σ_{1})$ and $ω^{2} \in Ω (Σ_{2})$ such that no conditional in $Δ_{1}$ , respectively, $Δ_{2}$ is falsified. Since $(B | A) \in Δ_{3}$ and thus $(B | A) \in Δ_{1}$ and $(B | A) \in Δ_{2}$ , we get $ω^{3} ⊭ A \bar{B}$ , contradicting our assumption.

Note that in our example base $Δ^{b}$ , $Δ_{3}$ is empty while $Σ_{3}$ is not. The crucial (conditional) link between $Δ_{1}$ and $Δ_{2}$ is given semantically by $Σ_{3}$ .

The following proposition provides the key for showing that c-representations satisfy conditional semantic splitting.
Proposition 35
For any $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , where $Δ_{3} = Δ_{1} \cap Δ_{2}$ , we have $S o l ({C R}_{Σ}^{e x} (Δ)) = {({\vec{η}}^{1}, {\vec{η}}^{2}, {\vec{η}}^{3}) ∣ {\vec{η}}^{i} \in S o l ({C R}_{Σ}^{e x} (Δ_{i ∖ 3})), i \in {1, 2}; {\vec{η}}^{3} \in (N \cup {\infty})^{| Δ_{3} |}}$ , i.e.,
$S o l ({C R}_{Σ}^{e x} (Δ)) = S o l ({C R}_{Σ}^{e x} (Δ_{1 ∖ 3})) \times S o l ({C R}_{Σ}^{e x} (Δ_{2 ∖ 3})) \times (N \cup \infty)^{| Δ_{3} |} .$

Proof.
We will first consider the case that $Δ_{3} = \emptyset$ . Then, we have that $Δ = Δ_{1 ∖ 3} \cup Δ_{2 ∖ 3}$ .

Let $i, i^{'} \in {1, 2}$ and $i \neq i^{'}$ . Let $Δ_{1 ∖ 3} = {(B_{1} | A_{1}), \dots, (B_{n_{1}} | A_{n_{1}})}$ , $Δ_{2 ∖ 3} = {(B_{n_{1} + 1} | A_{n_{1} + 1}), \dots, (B_{n_{1} + n_{2}} | A_{n_{1} + n_{2}})}$ , $Δ = {(B_{1} | A_{1}), \dots, (B_{n} | A_{n})}$ , thus $n = n_{1} + n_{2}$ . Then, for $(B_{j} | A_{j}) \in Δ$ , we have $(B_{j} | A_{j}) \in Δ_{i ∖ 3}$ iff $(B_{j} | A_{j}) \notin$ $Δ_{i^{'} ∖ 3}$ . We start with the following assumption:

(S1) ${\vec{η}}^{1} \in S o l ({C R}_{Σ}^{e x} (Δ_{1 ∖ 3}))$ , ${\vec{η}}^{2} \in S o l ({C R}_{Σ}^{e x} (Δ_{2 ∖ 3})) .$

Let us denote the constraint variables in ${C R}_{Σ}^{e x} (Δ_{1 ∖ 3})$ with $η_{1}^{1}, \dots, η_{n_{1}}^{1}$ and in ${C R}_{Σ}^{e x} (Δ_{2 ∖ 3})$ with $η_{n_{1} + 1}^{2}, \dots, η_{n}^{2}$ . Hence, we can write the constraints in ${C R}_{Σ}^{e x} (Δ_{i ∖ 3})$ as:
$\begin{aligned} min_{\begin{matrix} ω ⊨ A_{j} \end{matrix}} & \sum_{\begin{matrix} 1 ⩽ j ⩽ n \\ ω ⊨ A_{k} \bar{B_{k}} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i} = \infty or η_{j}^{i} > min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i} - min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i} . \end{aligned}$
(22)
Due to the safety property (5), ${C R}_{Σ}^{e x} (Δ_{1 ∖ 3})$ does not mention any constraint variable from ${C R}_{Σ}^{e x} (Δ_{2 ∖ 3})$ and vice versa, thus (S1) is equivalent to:

(S2) $({\vec{η}}^{1}, {\vec{η}}^{2}) \in S o l (Γ 1)$ , $Γ 1 = {C R}_{Σ}^{e x} (Δ_{1 ∖ 3}) \cup {C R}_{Σ}^{e x} (Δ_{2 ∖ 3})$ .

We will now consider the two constraints (20a) and (20b) separately. First, we consider (20b). We can write the constraints of form (20b) in ${C R}_{Σ}^{e x} (Δ_{i ∖ 3})$ as:
$\begin{aligned} η_{j}^{i} > \underset{V_{\min} (j, i)}{\underset{⏟}{min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i}}} - \underset{F_{\min} (j, i)}{\underset{⏟}{min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i}}} . \end{aligned}$
(23)

For $(B_{j} | A_{j}) \in Δ_{i ∖ 3}$ , let $V_{\min} (j, i^{'})$ and $F_{\min} (j, i^{'})$ be:
$\begin{aligned} V_{\min} (j, i^{'}) = min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i^{'} ∖ 3} \end{matrix}} η_{k}^{i^{'}}, \end{aligned}$
(24)

$\begin{aligned} F_{\min} (j, i^{'}) = min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i^{'} ∖ 3} \end{matrix}} η_{k}^{i^{'}} . \end{aligned}$
(25)
Note that both $V_{\min} (j, i^{'})$ and $F_{\min} (j, i^{'})$ only involve impacts corresponding to falsified conditionals from the subbase the conditional $(B_{j} | A_{j})$ does not belong to.

Due to the safety property (5), any world $ω \in Ω$ that minimizes the sum in $V_{\min} (j, i^{'})$ , falsifies no conditionals in $Δ_{i^{'} ∖ 3}$ . Analogously, this holds for $F_{\min} (j, i^{'})$ and therefore we have $V_{\min} (j, i^{'}) = F_{\min} (j, i^{'}) = 0$ .

Thus, adding $V_{\min} (j, i^{'}) - F_{\min} (j, i^{'})$ to the right-hand side of (23) yields the following constraint having the same set of solutions as (23):
$\begin{aligned} η_{j}^{i} > \underset{V_{\min} (j, i)}{\underset{⏟}{min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i}}} + \underset{V_{\min} (j, i^{'})}{\underset{⏟}{min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i^{'} ∖ 3} \end{matrix}} η_{k}^{i^{'}}}} - \underset{F_{\min} (j, i)}{\underset{⏟}{min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i}}} - \underset{F_{\min} (j, i^{'})}{\underset{⏟}{min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ_{i^{'} ∖ 3} \end{matrix}} η_{k}^{i^{'}}}} . \end{aligned}$
(26)

Because of the safety property, for every world $ω$ minimizing $V_{\min} (j, i)$ (or $F_{\min} (j, i)$ , respectively) there must also be a world $ω^{'}$ minimizing both $V_{\min} (j, i)$ and $V_{\min} (j, i^{'})$ (or $F_{\min} (j, i)$ and $F_{\min} (j, i^{'})$ , respectively), thus the $V_{\min}$ -minimizations and the $F_{\min}$ -minimizations in (26) can be combined without changing the set of solutions. Together with the fact that $Δ_{3} = \emptyset$ and $Δ = Δ_{i ∖ 3} \cup Δ_{i^{'} ∖ 3}$ , this yields the constraint:
$\begin{aligned} η_{j}^{i} > min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ \end{matrix}} η_{k}^{i} - min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ \end{matrix}} η_{k}^{i} . \end{aligned}$
(27)

Notice that the constraints (23), (26), and (27) all have the same set of solutions. In the case of non-extended c-representations specified by the constraint system $C R (Δ)$ , this constraint system only contains constraints of the form (20b). Then, we can replace each constraint (23) in $Γ 1$ by constraint (27), and, using $η_{1}^{1}, \dots, η_{n_{1}}^{1}, \dots, η_{n_{1} + 1}^{2}, \dots, η_{n}^{2}$ as constraint variables expressing ${C R}_{Σ}^{e x} (Δ)$ , we have that (S2) and thus (S1) is equivalent to the condition $({\vec{η}}^{1}, {\vec{η}}^{2}) \in S o l ({C R}_{Σ}^{e x} (Δ))$ and we are done.

Next we consider the case of the inequality (20a). We can write the constraints of form (20a) in ${C R}_{Σ}^{e x} (Δ_{i ∖ 3})$ as:
$min_{\begin{matrix} ω ⊨ A_{j} \end{matrix}} \sum_{\begin{matrix} 1 ⩽ j ⩽ n \\ ω ⊨ A_{k} \bar{B_{k}} \\ (B_{k} | A_{k}) \in Δ_{i ∖ 3} \end{matrix}} η_{k}^{i} = \infty .$
(28)
Due to the safety property (5) there is an extension $ω^{i^{'}} \in Ω (Σ_{2})$ of $ω^{i} ω^{3}$ such that no conditional in $Δ_{i^{'} ∖ 3}$ is falsified. Since we are looking for a world that minimizes the sum expression, such a world will not falsify any conditionals in $Δ_{i^{'} ∖ 3}$ . Thus, we can rewrite (28) to
$min_{\begin{matrix} ω ⊨ A_{j} \end{matrix}} \sum_{\begin{matrix} 1 ⩽ j ⩽ n \\ ω ⊨ A_{k} \bar{B_{k}} \\ (B_{k} | A_{k}) \in Δ \end{matrix}} η_{k}^{i} = \infty,$
(29)
where (28) and (29) have the same set of solutions.

Because the constraints (23), (26), and (27) all have the same set of solutions, and the same holds for the constraints (28) and (29), we have that (22) has the same set of solutions as:
$\begin{aligned} min_{\begin{matrix} ω ⊨ A_{j} \end{matrix}} \sum_{\begin{matrix} 1 ⩽ j ⩽ n \\ ω ⊨ A_{k} \bar{B_{k}} \\ (B_{k} | A_{k}) \in Δ \end{matrix}} η_{k}^{i} = \infty or η_{j}^{i} > min_{\begin{matrix} ω ⊨ A_{j} B_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ \end{matrix}} η_{k}^{i} - min_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \end{matrix}} \sum_{\begin{matrix} k \neq j \\ ω ⊨ A_{k} {\bar{B}}_{k} \\ (B_{k} | A_{k}) \in Δ \end{matrix}} η_{k}^{i} . \end{aligned}$
(30)
Therefore, (S2) and thus also (S1) is equivalent to:

(S3) $({\vec{η}}^{1}, {\vec{η}}^{2}) \in S o l (Γ 2)$ , where $Γ 2$ is obtained from $Γ 1$ by replacing each constraint (22) by (30).

Using $η_{1}^{1}, \dots, η_{n_{1}}^{1}, \dots, η_{n_{1} + 1}^{2}, \dots, η_{n}^{2}$ as constraint variables for expressing ${C R}_{Σ}^{e x} (Δ)$ , we observe that $Γ 2 = {C R}_{Σ}^{e x} (Δ)$ .

Next, we consider the case that $Δ_{3} \neq \emptyset$ . Due to Lemmas 34 and 33, the impact $η$ assigned to $(F | E)$ in $S o l ({C R}_{Σ}^{e x} (Δ_{3}))$ has no influence on $S o l ({C R}_{Σ}^{e x} (Δ_{1 ∖ 3}))$ or on $S o l ({C R}_{Σ}^{e x} (Δ_{2 ∖ 3}))$ , and, furthermore, $S o l ({C R}_{Σ}^{e x} (Δ_{3})) = (N \cup \infty)^{| Δ_{3} |}$ , completing the proof.

Next we will work toward showing an analogous result for conservative c-representations.

We first state the following result, stating that arbitrarily shrinking the belief bases $Δ_{1}$ and $Δ_{2}$ of a safe conditional syntax splitting $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ yields again a safe conditional syntax splitting.
Lemma 36
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ . Then $Δ^{'} = Δ_{1}^{'} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2}^{'} ∣ Σ_{3}$ for all $Δ_{1}^{'} \subseteq Δ_{1}$ , $Δ_{2}^{'} \subseteq Δ_{2}$ , $Δ^{'} = Δ_{1}^{'} \cup Δ_{2}^{'}$ .
Proof.
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ and let $i, i^{'} \in {1, 2}, i \neq i^{'}$ . Let $Δ_{1}^{'} \subseteq Δ_{1}$ , $Δ_{2}^{'} \subseteq Δ_{2}$ , $Δ^{'} = Δ_{1}^{'} \cup Δ_{2}^{'}$ . Then clearly $Δ^{'} = Δ_{1}^{'} ⋃_{Σ_{1}, Σ_{2}} Δ_{2}^{'} ∣ Σ_{3}$ . Toward a contradiction assume that the splitting is not safe. Then there is a world $ω^{i} ω^{3}$ such that there is no extension $ω^{i^{'}}$ such that $ω^{i} ω^{3} ω^{i^{'}} ⊨ A \bar{B}$ for some $(B | A) \in Δ_{i^{'}}^{'}$ . However, because $Δ_{i^{'}}^{'} \subseteq Δ_{i^{'}}$ , we have $(B | A) \in Δ_{i^{'}}$ leading to a contradiction.

Conservative c-representations are linked to the system Z ranking function $κ^{z}$ , cf. Definition 25. The following proposition relates the notion of tolerance underlying the definition of $κ^{z}$ to conditional syntax splitting.
Proposition 37 (Heyninck et al., 2023)

Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ and, for $i \in {1, 2}$ , $δ_{i} \in Δ_{i}$ . Then $δ_{i}$ is tolerated by $Δ$ iff $δ_{i}$ is tolerated by $Δ_{i}$ .

Regarding the Z-partition, we adapt a lemma for strongly consistent belief bases (Haldimann, 2024) and extend it to weakly consistent belief bases by adding statement (4) in the following lemma.

Lemma 38 (Based on Haldimann, 2024)

Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , $E P (Δ) = (Δ^{1}, \dots, Δ^{n}, Δ^{\infty})$ , and, for $i \in {1, 2}$ , $E P (Δ_{i}) = (Δ_{i}^{1}, \dots, Δ_{i}^{l_{i}}, Δ_{i}^{\infty})$ .

For $i \in {1, 2}$ and $j \in {0, \dots, l_{i}}$ it holds that $Δ_{i}^{j} = Δ^{j} \cap Δ^{i}$ .

$max {l_{1}, l_{2}} = n$ .

If $l_{1} ⩽ l_{2}$ we have $Δ^{j} = {\begin{cases} Δ_{1}^{j} \cup Δ_{2}^{j} & for j = 1, \dots, l_{2}, \\ Δ_{2}^{j} & for j = l_{1} + 1, \dots, n . \end{cases}$

$Δ^{\infty} = Δ_{1}^{\infty} \cup Δ_{2}^{\infty}$ .

Proof.
With the exception of $Δ^{\infty}$ , the tolerance partitions for weakly consistent belief bases are constructed in the same way as for strongly consistent belief bases. Therefore, the proofs of the first three statements are the same way as in Haldimann (2024). For proving statement (4), we show both directions of the subset relationships expressed by the equation in that statement. First, let $δ \in Δ^{\infty}$ . Then there is $i \in {1, 2}$ such that $δ \in Δ_{i}$ . Toward a contradiction, assume $δ \notin Δ_{i}^{\infty}$ . Then, there is some $k < \infty$ such that $δ \in Δ_{i}^{k}$ . With (3) this implies $δ \in Δ^{k}$ , contradicting our assumption.

For the other direction, let $δ \in Δ_{1}^{\infty} \cup Δ_{2}^{\infty}$ . Again, toward a contradiction assume $δ \notin Δ^{\infty}$ . Then there is $k < \infty$ such that $δ \in Δ^{k}$ . With (3) this implies $δ \in Δ_{1}^{k} \cup Δ_{2}^{k}$ , contradicting our assumption.

We can now show a result analogous to Proposition 35 for conservative c-representations.
Proposition 39
For any $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , where $Δ_{3} = Δ_{1} \cap Δ_{2}$ , we have ${S o l}_{Δ}^{J + \infty} = {({\vec{η}}^{1}, {\vec{η}}^{2}, {\vec{η}}^{3}) ∣ {\vec{η}}^{i} \in {S o l}_{Δ_{i ∖ 3}}^{J + \infty}, i \in {1, 2}; {\vec{η}}^{3} \in N^{| Δ_{3} |}}$ , that is,
${S o l}_{Δ}^{J + \infty} = {S o l}_{Δ_{1 ∖ 3}}^{J + \infty} \times {S o l}_{Δ_{2 ∖ 3}}^{J + \infty} \times N^{| Δ_{3} |} .$

Proof.
Again, we first consider the case where $Δ_{3} = \emptyset$ . Let $E P (Δ) = (Δ^{0}, \dots, Δ^{n}, Δ^{\infty})$ be the extended Z-partition of $Δ$ , and let $E P (Δ_{i}) = (Δ_{i}^{0}, \dots, Δ_{i}^{m}, Δ_{i}^{\infty})$ be the extended Z-partitions of $Δ_{i}$ for $i \in {1, 2}$ . Let $i^{'} \in {1, 2}$ such that $i \neq i^{'}$ and let $(B_{j} | A_{j}) \in Δ_{i}$ . We consider two cases:

Case $η_{j}^{i} = \infty$ : Then either (a) $(B_{j} | A_{j}) \in Δ_{i}^{\infty}$ or (b) there is no $ω^{i} ω^{3} \in Ω (Σ_{i} \cup Σ_{3})$ with $ω^{i} ω^{3} ⊨ A_{j} \bar{B_{j}} \land (\underset{(D | C) \in Δ_{i}^{\infty}}{⋀} (\bar{C} \lor D))$ . In the case of (a) we have that $(B_{j} | A_{j}) \in Δ^{\infty}$ iff $(B_{j} | A_{j}) \in Δ_{i}^{\infty}$ due to Lemma 38 and thus $η_{j} = \infty$ iff $η_{j}^{i} = \infty$ . In the case of (b), due to the safety property, $ω^{i} ω^{3}$ has an extension $ω^{i^{'}} \in Ω (Σ_{i^{'}})$ such that $ω^{i} ω^{3} ⊨ A_{j} \bar{B_{j}} \land (\underset{(D | C) \in Δ_{i}^{\infty}}{⋀} (\bar{C} \lor D))$ iff $ω^{i} ω^{i^{'}} ω^{3} ⊨ A_{j} \bar{B_{j}} \land (\underset{(D | C) \in Δ^{\infty}}{⋀} (\bar{C} \lor D))$ and therefore $η_{j} = \infty$ iff $η_{j}^{i} = \infty$ .

Case $η_{j}^{i} \neq \infty$ : Then $j \in J_{Δ}$ , thus the constraint system ${C R S}_{Σ}^{e x} (Δ_{i})$ applies to $η_{j}^{i} \neq \infty$ and, utilizing Lemma 36, we can apply the same arguments that we used for inequality (20b) in the case of ${C R}_{Σ}^{e x} (Δ_{i})$ in the proof of Proposition 35, yielding that ${\vec{η}}^{1} \in S o l ({C R S}_{Σ}^{e x} (Δ_{1 ∖ 3})), {\vec{η}}^{2} \in S o l ({C R S}_{Σ}^{e x} (Δ_{2 ∖ 3}))$ is equivalent to $({\vec{η}}^{1}, {\vec{η}}^{2}) \in S o l ({C R S}_{Σ}^{e x} (Δ))$ . Together with the first case, we obtain that ${\vec{η}}^{1} \in {S o l}_{Δ_{1}}^{J + \infty}, {\vec{η}}^{2} \in {S o l}_{Δ_{2}}^{J + \infty}$ is equivalent to $({\vec{η}}^{1}, {\vec{η}}^{2}) \in {S o l}_{Δ}^{J + \infty}$ . This can then be extended to the case where $Δ_{3} \neq \emptyset$ , analogous to the proof of Proposition 35.

Propositions 35 and 39 show that, just like for syntax splittings, for safe conditional syntax splittings, the impact vectors for the subbases can be determined independently, yielding an impact vector for $Δ$ through composition. We illustrate this with an example.
Example 40 ( $Δ^{b +}$ continued)

Recall $Δ^{b +}$ and its safe conditional syntax splitting from Example 6. Because $Δ_{3}^{b +} = \emptyset$ we have that $Δ_{1}^{b} = Δ_{1 ∖ 3}^{b +}$ and $Δ_{2}^{b +} = Δ_{2 ∖ 3}^{b +}$ . According to Proposition 35, in order to obtain a solution for ${C R}_{Σ}^{e x} (Δ^{b})$ it suffices to determine solutions for ${C R}_{Σ}^{e x} (Δ_{1}^{b +})$ and ${C R}_{Σ}^{e x} (Δ_{2}^{b +})$ separately, where ${C R}_{Σ}^{e x} (Δ_{1}^{b +})$ consists of the first three constraints from Example 21 and ${C R}_{Σ}^{e x} (Δ_{2}^{b +})$ consists of the fourth one and the constraint from Example 24. Thus, we have that, for example, ${\vec{η}}_{4} = (1, 2, 2, 1, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ^{b +}))$ can be obtained by composing ${\vec{η}}_{4}^{1} = (1, 2, 2) \in S o l ({C R}_{Σ}^{e x} (Δ_{1}^{b +}))$ and ${\vec{η}}_{4}^{2} = (1, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ_{2}^{b +}))$ , see also Figure 2, that is, for the projections of ${\vec{η}}_{4}$ we have ${\vec{η}}_{4} |_{Δ_{1}^{b +}} = {\vec{η}}_{4}^{1}$ , and ${\vec{η}}_{4} |_{Δ_{2}^{b +}} = {\vec{η}}_{4}^{2}$ . We can also compose ${\vec{η}}_{2}^{1} = (3, 4, 4) \in S o l ({C R}_{Σ}^{e x} (Δ_{1}^{b +}))$ and ${\vec{η}}_{4}^{2} = (1, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ_{2}^{b +}))$ to obtain ${\vec{η}}_{6} = (3, 4, 4, 1, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ^{b +}))$ .

With Proposition 35 we can now show that extended c-representations satisfy conditional semantic splitting.

Proposition 41
Extended c-representations satisfy (CSemSplit).
Proof.
Let $κ$ be an extended c-representation for a belief base $Δ$ and let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , where $Δ_{3} = Δ_{1} \cap Δ_{2}$ . First, in lieu of Lemma 34, it is clear that $Δ_{3}$ must be strongly consistent. We have to show that this is also a conditional semantic splitting, that is, that
$Mod_{Σ}^{ec} (Δ) = Mod_{Σ}^{ec} (Δ_{1}) \oplus Mod_{Σ}^{ec} (Δ_{2}) ⊖ Mod_{Σ}^{ec} (Δ_{3})$
(31)
holds. With Proposition 35 we have that every $\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))$ can be split into $({\vec{η}}^{1}, {\vec{η}}^{2}, {\vec{η}}^{3})$ such that ${\vec{η}}^{1} \in S o l ({C R}_{Σ}^{e x} (Δ_{1 ∖ 3}))$ , ${\vec{η}}^{2} \in S o l ({C R}_{Σ}^{e x} (Δ_{2 ∖ 3}))$ , and ${\vec{η}}^{3} \in S o l ({C R}_{Σ}^{e x} (Δ_{3}))$ . Vice versa, for every ${\vec{η}}^{1} \in S o l ({C R}_{Σ}^{e x} (Δ_{1 ∖ 3}))$ , ${\vec{η}}^{2} \in S o l ({C R}_{Σ}^{e x} (Δ_{2 ∖ 3}))$ , and ${\vec{η}}^{3} \in S o l ({C R}_{Σ}^{e x} (Δ_{3}))$ we have $({\vec{η}}^{1}, {\vec{η}}^{2}, {\vec{η}}^{3}) \in S o l ({C R}_{Σ}^{e x} (Δ))$ . Therefore, we have that
$Mod_{Σ}^{ec} (Δ) = Mod_{Σ}^{ec} (Δ_{1 ∖ 3}) \oplus Mod_{Σ}^{ec} (Δ_{2 ∖ 3}) \oplus Mod_{Σ}^{ec} (Δ_{3}) .$
(32)
With Lemmas 34 and 33, we have that $Mod_{Σ}^{ec} (Δ_{1}) = Mod_{Σ}^{ec} (Δ_{1 ∖ 3})$ , $Mod_{Σ}^{ec} (Δ_{2}) = Mod_{Σ}^{ec} (Δ_{2 ∖ 3}),$ and $Mod_{Σ}^{ec} (Δ_{3}) = {κ_{u}}$ , where $κ_{u}$ is the uniform OCF mapping every world $ω$ to 0 (cf. Section 2). Thus, (32) is equivalent to (31), finishing the proof.

Analogously, we can use Proposition 39 to show that also conservative c-representations satisfy conditional semantic splitting.
Proposition 42
Conservative c-representations satisfy (CSemSplit).
Proof.
Analogous to the proof of Proposition 41 utilizing Proposition 39.

Because conservative c-representations coincide with (nonextended) c-representations for strongly consistent belief bases, Proposition 42 also applies to c-representations for strongly consistent belief bases. The splitting properties of c-representations on the semantic level and on the level of impact vectors provide an important base for showing conditional syntax splitting properties with respect to inference with c-representations.
5. Conditional Syntax Splitting and Inference with Respect to Single c-Representations

Choosing a c-representation $κ$ for any given belief base $Δ$ and assigning to $Δ$ , thereby extending the beliefs in $Δ$ , yields an inductive inference operator . Principally, for every $Δ$ , the c-representation $κ$ may be chosen arbitrarily. The concept of selection strategies for c-representations (Beierle & Kern-Isberner, 2021) can govern the choice of $κ$ via appropriate postulates. Selection strategies have been used successfully for investigating and proving properties, for example, for inference with c-representations and (unconditional) syntax splitting for strongly consistent belief bases (Kern-Isberner et al., 2020), or for a kinematics principle in iterated revision (Sezgin et al., 2021). The adaptation of selection strategies to weakly consistent belief bases and extended c-representations is straightforward.

Definition 43 (Selection strategy $σ$ , adapted from Beierle & Kern-Isberner, 2021)

A selection strategy (for extended c-representations) is a function

σ : Δ \mapsto \vec{η}

assigning to each weakly consistent conditional belief base

Δ

an impact vector

\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))

Each such selection strategy determines an inductive inference operator for weakly consistent belief bases and extended c-representations.

Definition 44 (Inductive inference operator $C_{σ}^{e c}$ , adapted from Kern-Isberner et al., 2020)

An inductive inference operator for extended c-representations with selection strategy $σ$ is a function

C_{σ}^{e c} : Δ \mapsto κ_{σ (Δ)},

where

σ

is a selection strategy and, as before,

is obtained via equation (2) from

κ_{σ (Δ)}

Because each satisfies both (DI) and (TV), $C_{σ}^{e c}$ is an inductive inference operator for every selection strategy $σ$ . A recent example for a specific selection strategy are minimal core c-representations (Wilhelm et al., 2024).

Without any further restriction, a selection strategy $σ$ may assign any impact vector $\vec{η} \in {C R}_{Σ}^{e x} (Δ)$ to a belief base $Δ$ . For instance, $σ$ may always choose the impact vector $\vec{η} = (\infty, \dots, \infty)$ , thus assigning the impact $\infty$ to each conditional, because $(\infty, \dots, \infty) \in S o l ({C R}_{Σ}^{e x} (Δ))$ (Haldimann et al., 2024). This choice maximizes the set $κ_{\vec{η}}^{- 1} (\infty)$ , deeming as many worlds as possible to be considered infeasible, and it maximizes the set of formulas $A \in L$ with , rendering many conclusions trivial and unhelpful, and leading to an inductive inference operator that coincides with classical deduction with respect to the material counterparts of all involved conditionals.

Proposition 45 (infinity selection strategy $τ$ yields material counterpart inference )

For any belief base $Δ$ , let $\tilde{Δ} = {A_{i} \to B_{i} ∣ (B_{i} | A_{i}) \in Δ}$ be the material counterpart of $Δ$ . For formulas $A, B$ , we call $B$ a material counterpart inference from $A$ in the context of $Δ$ , denoted by , if: Let $τ$ be the infinity selection strategy assigning to each weakly consistent belief base $Δ$ the impact vector $\vec{η} = (\infty, \dots, \infty)$ . Then for all weakly consistent belief bases $Δ$ , we have , that is, for all formulas $A, B$ , the following holds:

(33)
Proof.
First, observe the following two facts. (a) The models of $\tilde{Δ}$ are exactly those worlds that do not falsify any conditional in $Δ$ , and (b) for every world $ω \in Ω$ either $κ_{τ (Δ)} (ω) = 0$ or $κ_{τ (Δ)} (ω) = \infty$ . We show both directions of the iff in (33) separately.

Direction “ $\Rightarrow$ ”: Assume . Then either $κ_{τ (Δ)} (A) = \infty$ or $κ_{τ (Δ)} (A B) < κ_{τ (Δ)} (A \bar{B})$ . We make a case distinction. In the first case $κ_{τ (Δ)} (A) = \infty$ , which implies that all worlds $ω \in Ω$ with $ω ⊨ A$ must falsify some conditional in $Δ$ . Together with (1) this implies $\tilde{Δ} ⊨ \bar{A}$ and thus $\tilde{Δ} ⊨ A \to B$ . In the second case $κ_{τ (Δ)} (A B) < κ_{τ (Δ)} (A \bar{B})$ implies with (b) $κ_{τ (Δ)} (A \bar{B}) = \infty$ . Then $\tilde{Δ} ⊨ \neg (A \bar{B})$ and thus $\tilde{Δ} ⊨ A \to B$ .

Direction “ $\Leftarrow$ ”: Assume $\tilde{Δ} ⊨ A \to B$ , that is, $\tilde{Δ} ⊨ \bar{A} \lor B$ . We discern two cases: $\tilde{Δ} ⊨ \bar{A}$ and $\tilde{Δ} ⊭ \bar{A}$ . In the case that $\tilde{Δ} ⊨ \bar{A}$ we have $\tilde{Δ} ⊭ A$ and thus with (a) and (b) that $κ_{τ (Δ)} (A) = \infty$ and therefore . In the case that $\tilde{Δ} ⊭ \bar{A}$ we have $κ_{τ (Δ)} (\bar{A}) = \infty$ and thus $κ_{τ (Δ)} (A) = 0$ . We also have $\tilde{Δ} ⊨ B$ due to $\tilde{Δ} ⊨ \bar{A} \lor B$ and thus with (a) also $κ_{τ (Δ)} (B) = 0$ . Since $\tilde{Δ} ⊨ B$ we have $\tilde{Δ} ⊭ \bar{B}$ and thus $κ_{τ (Δ)} (\bar{B}) = \infty$ . Altogether, we have $κ_{τ (Δ)} (\bar{A} \lor \bar{B}) = \infty$ , implying $κ_{τ (Δ)} (A B) = 0$ and $κ_{τ (Δ)} (A \bar{B}) = \infty$ , meaning .

Thus, while adding more inferences to an inference relation may be desirable, there should be a restriction for inferences leading to a contradiction, that is, inferences of the form . Such an inference expresses that considers $A$ to be inconsistent, thereby preventing the entailment of any other informative beliefs from $A$ . A corresponding restriction is obtained by the (Classic Preservation) postulate (Casini et al., 2019), stating that should only hold if $A$ entails $⊥$ according to system P, that is, if where is the system P inductive inference operator (Adams, 1965; Kraus et al., 1990; cf. Example 17).

(Classic Preservation) adapted from Casini et al. (2019) . An inductive inference operator satisfies (Classic Preservation) if for all belief bases $Δ$ and $A, B \in L$ it holds that iff .

Because system Z satisfies (Classic Preservation), this postulate can also be characterized using system Z as stated by the following lemma.
Lemma 46 (Haldimann et al., 2023)

For a weakly consistent belief base $Δ$ and a formula $A$ we have $κ_{Δ}^{z} (A) = \infty$ iff .

As illustrated above, there are selection strategies $σ$ such that $C_{σ}^{e c}$ does not satisfy (Classic Preservation), for example, due to selecting $σ (Δ) = (\infty, \dots, \infty)$ , thereby assigning worlds rank $\infty$ while this may be not enforced by the conditionals in $Δ$ . Toward avoiding this phenomenon, the following postulate restricts selection strategies to only yield conservative c-representations.

(CI) A selection strategy $σ$ satisfies conservative impact if, for every belief base $Δ$ , $σ (Δ) \in {S o l}_{Δ}^{J + \infty}$ .

For selection strategies satisfying (CI) we show that (Classic Preservation) is satisfied.

Proposition 47
Let $σ$ be a selection strategy. If $σ$ satisfies (CI) then $C_{σ}^{e c}$ satisfies (Classic Preservation).
Proof.
Let $Δ$ be a belief base and $σ$ be a selection strategy satisfying (CI). Then $κ = κ_{σ (Δ)}$ is a conservative c-representation. With Lemma 46 we need to show that iff $κ_{Δ}^{z} (A) = \infty$ . Because $σ$ satisfies (CI), according to Lemma 26, we have for all $ω \in Ω$ that $κ_{Δ}^{z} (ω) < \infty$ iff $κ (ω) < \infty$ , or, equivalently, $κ_{Δ}^{z} (ω) = \infty$ iff $κ (ω) = \infty$ . Thus, $κ (A) = \infty \, iff \, κ_{Δ}^{z} (A) = \infty$ , completing the proof.

For a belief base $Δ$ and a selection strategy $σ$ satisfying (CI), no proper relaxation $κ^{'}$ of $κ_{σ (Δ)}$ satisfies (Classic Preservation). This is because for any relaxation $κ^{'}$ of $κ$ with $κ^{'} \neq κ$ , there must be some world $ω \in Ω$ such that $κ (ω) \neq κ^{'} (ω)$ . According to Definition 31, this means $κ^{'} (ω) = \infty$ and $κ (ω) < \infty$ . According to Definition 25, because $κ (ω) < \infty$ , we have $κ_{Δ}^{z} (ω) < \infty$ , leading to a violation of (Classic Preservation) by $κ^{'}$ according to Lemma 46.

In principle, for every $Δ$ , a selection strategy $σ$ may choose some impact vector independently from the choices for all other belief bases. The following postulate characterizes selection strategies that preserve the impacts chosen for subbases that are part of a safe conditional syntax splitting of $Δ$ .

(IP-CSP) A selection strategy $σ$ is impact preserving w.r.t. conditional belief base splitting if, for $i \in {1, 2}$ , we have $σ (Δ_{i}) = {σ (Δ) |}_{Δ_{i}}$ for every safe conditional syntax splitting $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ .

We illustrate (IP-CSP) with an example.
Example 48 ( $Δ^{b}$ continued)

Recall Example 21. Let $σ$ be a selection strategy with $σ (Δ^{b}) = (1, 2, 2, 1)$ . Then (IP-CSP) requires that $σ (Δ_{1 ∖ 3}^{b}) = (1, 2, 2)$ and $σ (Δ_{2 ∖ 3}^{b}) = (1)$ .

An algorithm for generating selection strategies satisfying a corresponding impact preserving postulate for (unconditional) syntax splittings (Kern-Isberner et al., 2020) is introduced in Beierle and Kern-Isberner (2021), providing a basis for an algorithm for generating selection strategies satisfying (IP-CSP).

As a first proposition with respect to (IP-CSP), we show that inference with extended c-representations based on an impact preserving selection strategy satisfies (CRel).

Proposition 49
Let $σ$ be a selection strategy satisfying (IP-CSP). Then $C_{σ}^{e c}$ satisfies (CRel).
Proof.
Let $ω^{i} ω^{3} \in Ω (Σ_{i} \cup Σ_{3})$ . Note that here both $κ_{Δ_{i}}$ and $κ |_{Σ_{i} \cup Σ_{3}}$ are defined on worlds in $Ω (Σ_{i} \cup Σ_{3})$ . According to the marginalization of ranking functions (cf. Section 2) we have:
$\begin{aligned} κ |_{Σ_{i} \cup Σ_{3}} (ω^{i} ω^{3}) = κ (ω^{i} ω^{3}) & = min {\sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ \end{matrix}} η_{j} ∣ ω ⊨ ω^{i} ω^{3}} . \end{aligned}$
(34)
Due to the safety property (5), there is an extension $ω^{i^{'}}$ of $ω^{i} ω^{3}$ such that $ω^{i} ω^{3} ω^{i^{'}}$ falsifies no conditional in $Δ_{i^{'}}$ . Therefore, we can simplify (34) by only considering $Δ_{i}$ as follows:
$\begin{aligned} κ |_{Σ_{i} \cup Σ_{3}} (ω^{i} ω^{3}) & = \sum_{\begin{matrix} ω^{i} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{i} \end{matrix}} η_{j} . \end{aligned}$
(35)
Note that we no longer need to consider a minimum over worlds, since $Δ_{i} \subseteq (L (Σ_{i} \cup Σ_{3}) | L (Σ_{i} \cup Σ_{3}))$ and $ω^{i} ω^{3} \in Ω (Σ_{i} \cup Σ_{3})$ is a full conjunction; thus, every world that is minimal with respect to $κ$ and that is a model of $ω^{i} ω^{3}$ falsifies the same conditionals in $Δ_{i}$ as $ω^{i} ω^{3}$ . Because $σ$ satisfies (IP-CSP) we have $η_{j}^{i} = η_{j} |_{Δ_{j}}$ and thus (35) is equivalent to
$\begin{aligned} κ |_{Σ_{i} \cup Σ_{3}} (ω^{i} ω^{3}) & = \sum_{\begin{matrix} ω^{i} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{i} \end{matrix}} η_{j}^{i}, \end{aligned}$
(36)
which is the definition of $κ_{Δ_{i}} (ω^{i} ω^{3})$ . This holds for all $ω^{i} ω^{3}$ . Accordingly $κ_{Δ_{i}} = κ |_{Σ_{i} \cup Σ_{3}}$ which, together with Proposition 11, implies (CRel).

Before addressing (CInd) for inference with single extended c-representations, we first relate extended c-representations to conditional independence via Proposition 13 by stating the following proposition.
Proposition 50
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , and let $κ$ be an extended c-representation with $κ ⊨ Δ$ . Then $Σ_{1} ⊥ ⊥_{κ} Σ_{2} | Σ_{3}$ .
Proof.
Let $ω = ω^{1} ω^{2} ω^{3}$ and let $\vec{η} \in S o l (Δ)$ such that $κ = κ_{\vec{η}}$ . Recall the definition of c-representations (15). We can rewrite (15) to:
$\begin{aligned} κ_{\vec{η}} (ω) & = \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{1 ∖ 3} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{2 ∖ 3} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{3} \end{matrix}} η_{j} . \end{aligned}$
(37)
Note that, due to Lemma 34, the sums over the conditionals of $Δ_{3}$ cannot evaluate to $\infty$ . By simply adding and subtracting the last sum of (37) we obtain the following equation:
$\begin{aligned} κ_{\vec{η}} (ω) & = \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{1 ∖ 3} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{2 ∖ 3} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{3} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{3} \end{matrix}} η_{j} - \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{3} \end{matrix}} η_{j} . \end{aligned}$
(38)
Then we can combine the sums for $Δ_{1 ∖ 3}$ and $Δ_{2 ∖ 3}$ with the sum for $Δ_{3}$ to obtain sums for $Δ_{1}$ and $Δ_{2}$ , respectively.
$\begin{aligned} κ_{\vec{η}} (ω) & = \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{1} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{2} \end{matrix}} η_{j} - \sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{3} \end{matrix}} η_{j} . \end{aligned}$
(39)

Since $Δ_{1}$ is in $L (Σ_{1} \cup Σ_{3})$ , $Δ_{2}$ is in $L (Σ_{2} \cup Σ_{3})$ , and $Δ_{3}$ is in $L (Σ_{3})$ , we can use (10) to simplify (39):
$\begin{aligned} κ_{\vec{η}} (ω) & = \sum_{\begin{matrix} ω^{1} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{1} \end{matrix}} η_{j} + \sum_{\begin{matrix} ω^{2} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{2} \end{matrix}} η_{j} - \sum_{\begin{matrix} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{3} \end{matrix}} η_{j} . \end{aligned}$
(40)

Due to the safety property, $ω^{i} ω^{3}$ cannot falsify any conditional in $Δ_{j}$ for $i, \in 1, 2$ . Thus (40) can be rewritten to:
$\begin{aligned} κ_{\vec{η}} (ω) & = \sum_{\begin{matrix} ω^{1} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ \end{matrix}} η_{j} + \sum_{\begin{matrix} ω^{2} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ \end{matrix}} η_{j} - \sum_{\begin{matrix} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ \end{matrix}} η_{j} . \end{aligned}$
(41)
By applying the definition of c-representations again, we obtain
$κ_{\vec{η}} (ω^{1} ω^{2} ω^{3}) = κ_{\vec{η}} (ω^{1} ω^{3}) + κ_{\vec{η}} (ω^{2} ω^{3}) - κ_{\vec{η}} (ω^{3}),$
(42)
which is equivalent to $κ_{\vec{η}} (ω^{1} | ω^{2} ω^{3}) = κ_{\vec{η}} (ω^{1} | ω^{3})$ . The proof is completed by the fact that $κ_{\vec{η}} (ω^{3}) < \infty$ due to Lemma 34.

Proposition 50 yields a straightforward way to prove that inference with respect to single extended c-representations satisfies (CInd).
Proposition 51
Let $σ$ be a selection strategy. Then $C_{σ}^{e c}$ satisfies (CInd).
Proof.
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ and let $σ$ be a selection strategy. Let $i, i^{'} \in {1, 2}$ with $i \neq i^{'}$ . Let $η_{j}^{i}$ be the impact of $(B_{j} | A_{j}) \in Δ_{i}$ . Due to Proposition 50, we know that $Σ_{1} ⊥ ⊥_{κ_{Δ}} Σ_{2} | Σ_{3}$ holds. Thus, with Proposition 13, $C_{σ}^{c-rep}$ satisfies (CInd).

Note that Proposition 51 holds for every selection strategy $σ$ ; in particular, it is not necessary for $σ$ to satisfy (IP-CSP). By combining Propositions 49 and 51, we reach the following result.
Proposition 52
Let $σ$ be a selection strategy satisfying (IP-CSP). Then $C_{σ}^{e c}$ satisfies (CSynSplit).
Proof.
Follows directly from the satisfaction of (CRel) and (CInd) according to Propositions 49 and 51.

Thus inference with single extended c-representations satisfies (CSynSplit) if the underlying selection strategy satisfies (IP-CSP). Note that (IP-CSP) is required only for (CRel) because (CInd) holds for inference with single extended c-representations in general.

With the help of Proposition 52, we can easily derive that deductive inference with respect to the material counterpart of a belief base satisfies conditional syntax splitting.
Proposition 53
Material counterpart inference satisfies (CSynSplit).
Proof.
Because the infinity selection strategy $τ$ satisfies (IP-CSP) and material counterpart inference coincides with $C_{τ}^{e c}$ according to Proposition 45, the claim follows due to Proposition 52.

The results in this section also apply to (non-extended) c-representations for strongly consistent belief bases, because they are a subset of extended c-representations in the context of strongly consistent belief bases.
6. c-Inference and Extended c-Inference Satisfy Conditional Syntax Splitting

The inductive inference operator c-inference is the skeptical inference operator obtained by taking all c-representations of a belief base $Δ$ into account (Beierle et al., 2016a, 2018), that is, holds iff for all c-representations $κ$ of $Δ$ . Correspondingly, extended c-inference is defined with respect to all extended c-representations.

Definition 54 (Extended c-inference, , (Beierle et al., 2016a; Haldimann et al., 2024))

Let $Δ$ be a belief base and let $A, B \in L$ . Then $B$ is an extended c-inference from $A$ in the context of $Δ$ , denoted by , iff holds for all extended c-representations $κ$ of $Δ$ , yielding the inductive inference operator

For strongly consistent belief bases c-inference and extended c-inference coincide (Haldimann et al., 2024). Before proving that extended c-inference and thus also c-inference satisfy conditional syntax splitting, we show a proposition stating the following observations.

Consider a safe conditional syntax splitting of $Δ$ into $Δ_{1}$ and $Δ_{2}$ , and an extended c-representation $κ_{\vec{η}}$ determined by a solution vector $\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))$ together with its projections $κ_{{\vec{η}}^{1}}$ and $κ_{{\vec{η}}^{2}}$ to $Δ_{1}$ and $Δ_{2}$ , respectively. Then the rank of any consistent formula $F_{i}$ over the language $L (Σ_{i} \cup Σ_{3})$ of $Δ_{i}$ under the projection $κ_{{\vec{η}}^{i}}$ coincides with the rank of the formula rank determined by $κ_{\vec{η}}$ , while its rank under the other projection $κ_{{\vec{η}}^{i^{'}}}$ evaluates to zero.

Proposition 55
For any $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ , for all $\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))$ and $F_{i} \in L (Σ_{i} \cup Σ_{3})$ with $F_{i} ≢ ⊥$ , $i \in {1, 2}$ , we have $κ_{{\vec{η}}^{1}} (F_{2}) = κ_{{\vec{η}}^{2}} (F_{1}) = 0$ and $κ_{\vec{η}} (F_{i}) = κ_{{\vec{η}}^{i}} (F_{i})$ .
Proof.
Let $i, i^{'} \in {1, 2}, i \neq i^{'}$ . We show $κ_{{\vec{η}}^{i}} (F_{i^{'}}) = κ_{{\vec{η}}^{i^{'}}} (F_{i}) = 0$ first. Consider some world $ω^{i} ω^{3} \in Ω (Σ_{i} \cup Σ_{3})$ with $ω^{i} ω^{3} ⊨ F_{i}$ . Then due to the safety property (5) there is some $ω^{i^{'}} \in Ω (Σ_{i^{'}})$ such that $ω^{i} ω^{3} ω^{i^{'}}$ does not falsify any conditional in $Δ_{i^{'}}$ . Then we have $κ_{{\vec{η}}^{i^{'}}} (ω^{i} ω^{3} ω^{i^{'}}) = 0$ and thus $κ_{{\vec{η}}^{i^{'}}} (F_{i}) = 0$ .

Next we show $κ_{\vec{η}} (F_{i}) = κ_{{\vec{η}}^{i}} (F_{i})$ . We have:
$\begin{aligned} κ_{\vec{η}} (F_{i}) & = min {\sum_{\begin{matrix} ω ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ \end{matrix}} η_{j} ∣ ω ⊨ F_{i}} . \end{aligned}$
(43)
Let $ω^{i} ω^{3} = ω |_{Σ_{1} \cup Σ_{3}}$ then $ω^{i} ω^{3} ⊨ F_{i}$ . Furthermore, $ω$ and $ω^{i} ω^{3}$ falsify the same conditionals in $Δ_{i ∖ 3}$ , since $Δ_{i ∖ 3} \in (L (Σ_{i} \cup Σ_{3}) | L (Σ_{i} \cup Σ_{3}))$ . Due to the safety property (5) there is some extension $ω^{i^{'}} \in Ω (Σ_{i^{'}})$ such that $ω^{i} ω^{3} ω^{i^{'}}$ does not falsify any conditional in $Δ_{i^{'} ∖ 3}$ . Clearly $ω^{i} ω^{3} ω^{i^{'}}$ is a minimal world in the sense of (43) if $ω^{i} ω^{3}$ is a minimal world in the sense of (43). Since $ω^{i} ω^{3} ω^{i^{'}}$ does not falsify any conditional in $Δ_{i^{'}}$ we can omit $Δ_{i^{'}}$ from (43) in the following way:
$\begin{aligned} κ_{\vec{η}} (F_{i}) & = min {\sum_{\begin{matrix} ω^{i} ω^{3} ⊨ A_{j} {\bar{B}}_{j} \\ (B_{j} | A_{j}) \in Δ_{i ∖ 3} \end{matrix}} η_{j}^{i} ∣ ω^{i} ω^{3} ⊨ F_{i}} . \end{aligned}$
(44)
Thus, we have $κ_{\vec{η}} (F_{i}) = κ_{{\vec{η}}^{i}} (F_{i})$ .

The next proposition shows that extended c-inference satisfies conditional syntax splitting. Note that since in general the inference relation cannot be represented by an OCF, no corresponding characterization of syntax splitting based on OCFs is applicable to it. Thus, the techniques used in the proofs of the propositions for here are different from those used in previous proofs referring to OCF-based characterizations of syntax splittings.
Proposition 56
Extended c-inference $C^{e c}$ satisfies (CSynSplit).
Proof.
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ . W.l.o.g. assume $A, B \in L (Σ_{1}), D \in L (Σ_{2}), \dot{B} \in {B, \bar{B}}$ and assume $E \in L (Σ_{3})$ is a complete conjunction with .

To prove that $C^{e c}$ satisfies (CRel) we need to show that iff . By applying the definition of we obtain:
$\begin{aligned} \forall \vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ)) : κ_{\vec{η}} (A E B) < κ_{\vec{η}} (A E \bar{B}) or κ_{\vec{η}} (A E) = \infty, \\ iff \forall {\vec{η}}^{1} \in S o l ({C R}_{Σ}^{e x} (Δ_{1})) : κ_{{\vec{η}}^{1}} (A E B) < κ_{{\vec{η}}^{1}} (A E \bar{B}) or κ_{{\vec{η}}^{1}} (A E) = \infty . \end{aligned}$
(45)
For the direction $\Rightarrow$ of (45) observe that with Proposition 35, every $\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))$ can be split into $\vec{η} = ({\vec{η}}^{1 ∖ 3}, {\vec{η}}^{2 ∖ 3}, {\vec{η}}^{3})$ where, with Lemma 33, ${\vec{η}}^{1} = ({\vec{η}}^{1 ∖ 3}, {\vec{η}}^{3}) \in S o l ({C R}_{Σ}^{e x} (Δ_{1}))$ .

For the other direction $\Leftarrow$ of (45), with Proposition 35 and Lemma 33, we have that every ${\vec{η}}^{1} = ({\vec{η}}^{1 ∖ 3}, {\vec{η}}^{3}) \in S o l ({C R}_{Σ}^{e x} (Δ_{1}))$ with ${\vec{η}}^{1 ∖ 3} \in S o l ({C R}_{Σ}^{e x} (Δ_{1 ∖ 3}))$ and ${\vec{η}}^{3} \in (N \cup \infty)^{| Δ_{3} |}$ has an extension ${\vec{η}}^{2 ∖ 3} \in S o l ({C R}_{Σ}^{e x} (Δ_{2 ∖ 3}))$ such that $({\vec{η}}^{1 ∖ 3}, {\vec{η}}^{2 ∖ 3}, {\vec{η}}^{3}) \in S o l ({C R}_{Σ}^{e x} (Δ))$ . Therefore, to show (45) it suffices to show that
$κ_{\vec{η}} (A E B) < κ_{\vec{η}} (A E \bar{B}) or κ_{\vec{η}} (A E) = \infty iff κ_{{\vec{η}}^{1}} (A E B) < κ_{{\vec{η}}^{1}} (A E \bar{B}) or κ_{{\vec{η}}^{1}} (A E) = \infty$
(46)
for all $\vec{η} = ({\vec{η}}^{1 ∖ 3}, {\vec{η}}^{2 ∖ 3}, {\vec{η}}^{3}) \in S o l ({C R}_{Σ}^{e x} (Δ))$ with ${\vec{η}}^{1} = ({\vec{η}}^{1 ∖ 3}, {\vec{η}}^{3})$ . With Proposition 55 this follows directly because $κ_{\vec{η}} (A E \dot{B}) = κ_{{\vec{η}}^{1}} (A E \dot{B})$ since $A, B \in L (Σ_{1})$ , $E \in L (Σ_{3})$ and $Δ_{1} \subseteq (L (Σ_{1} \cup Σ_{3}) | L (Σ_{1} \cup Σ_{3}))$ .

Next, we prove that $C^{e c}$ satisfies (CInd). We need to show iff . Again, due to Proposition 35 it suffices to show that
$κ_{\vec{η}} (A E B) < κ_{\vec{η}} (A E \bar{B}) or κ_{\vec{η}} (A E) = \infty iff κ_{\vec{η}} (A D E B) < κ_{\vec{η}} (A D E \bar{B}) or κ_{\vec{η}} (A D E) = \infty$
(47)
for all $\vec{η} \in S o l ({C R}_{Σ}^{e x} (Δ))$ . First we consider the case that $κ_{\vec{η}} (A E) < \infty$ and $κ_{\vec{η}} (A D E) < \infty$ . Since Proposition 50 states that $Σ_{1}$ and $Σ_{2}$ are conditionally $κ_{\vec{η}}$ -independent given $Σ_{3}$ and $κ_{\vec{η}} (D E) < \infty$ due to we have with Lemma 10 that $κ_{\vec{η}} (A \dot{B} D E) = κ_{\vec{η}} (A \dot{B} E) + κ_{\vec{η}} (D E) - κ_{\vec{η}} (E)$ and therefore (47) holds. In the case that $κ_{\vec{η}} (A E) = \infty$ and/or $κ_{\vec{η}} (A D E) = \infty$ , again, with Proposition 50 and Lemma 10 we have that $κ_{\vec{η}} (A D E) = κ_{\vec{η}} (A E) + κ_{\vec{η}} (D E) - κ_{\vec{η}} (E)$ . Because , we have $κ_{\vec{η}} (D E) < \infty$ and $κ_{\vec{η}} (E) < \infty$ . Thus $κ_{\vec{η}} (A D E) = \infty$ iff $κ_{\vec{η}} (A E) = \infty$ .
Proposition 57
c-Inference $C^{c}$ satisfies (CSynSplit).
Proof.
Because extended c-inference $C^{e c}$ limited to strongly consistent belief bases coincides with c-inference $C^{c}$ , the claim follows from Proposition 56.

We give an example illustrating Propositions 55, 56, and 57.
Example 58 ( $Δ^{b}$ continued)

Recall Example 21. According to Proposition 55, for $p \land f \in L (Σ_{1})$ , we get $κ_{{\vec{η}}_{1}^{1}} (p \land \bar{f}) = 1$ and $κ_{{\vec{η}}_{1}^{2}} (p \land \bar{f}) = 0$ from $κ_{{\vec{η}}_{1}} (p \land \bar{f}) = 1$ without having to compute $κ_{{\vec{η}}_{1}^{1}}$ or $κ_{{\vec{η}}_{1}^{2}}$ . This also works in the other direction, where we do not have to compute $κ_{{\vec{η}}_{1}}$ if we have knowledge about $κ_{{\vec{η}}_{1}^{1}}$ .

Next, consider again the safe conditional syntax splitting given in Example 5. Taking the constraints in Example 21 into account, it follows that holds. With Proposition 57 we know that $C^{e c}$ satisfies (CRel) and (CInd). From (CRel) we conclude . Furthermore, according to (CInd), we know that and .

A skeptical inference relation based on extended c-representations is obtained by taking all conservative extended c-representations of $Δ$ into account. The resulting inference relation, called conservative extended c-inference, coincides with extended c-inference (cf. Proposition 27) and thus also satisfies (CSynSplit).

Proposition 59
Conservative extended c-inference satisfies (CSynSplit).
Proof.
Follows directly from Propositions 27 and 56.

We have shown that c-inference, extended c-inference, and conservative extended c-inference for weakly and strongly consistent belief bases, respectively, fully comply with conditional syntax splitting. Note that just as the proof that c-inference satisfies (unconditional) syntax splitting (Kern-Isberner et al., 2020), also proving satisfaction of (CSynSplit) by $C^{c}$ and by $C^{e c}$ does not require the use of selection strategies.
7. Credulous and Weakly Skeptical c-Inference and Their Extended Versions Satisfy Conditional Syntax Splitting

Besides reasoning with respect to all models of a belief base, two other modes of reasoning have been investigated and instantiated for c-representations (Beierle et al., 2016b): credulous and weakly skeptical reasoning.

Definition 60 (Credulous/weakly skeptical c-inference, (Beierle et al., 2021a))

Let $Δ$ be a strongly consistent belief base and let $A$ , $B$ be formulas.

$B$ is a credulous c-inference from $A$ in the context of $Δ$ , denoted by , iff there is a c-representation $κ$ of $Δ$ with .

$B$ is a weakly skeptical c-inference from $A$ in the context of $Δ$ , denoted by , iff $A \equiv ⊥$ or there is a c-representation $κ$ of $Δ$ with and there is no c-representation $κ^{'}$ of $Δ$ with .

These inference modes have been adapted to belief bases that are not strongly consistent by taking extended and conservative extended c-representations, respectively, into account.

Definition 61 ((Conservative) extended credulous/weakly skeptical c-inference, (Haldimann et al., 2025))

Let $Δ$ be a belief base and let $A$ , $B$ be formulas.

$B$ is an extended credulous c-inference from $A$ in the context of $Δ$ , denoted by , iff (i) $Δ$ is not weakly consistent or (ii) there is an extended c-representation $κ$ of $Δ$ with .

$B$ is a conservative extended credulous c-inference from $A$ in the context of $Δ$ , denoted by , iff (i) $Δ$ is not weakly consistent or (ii) there is a conservative extended c-representation $κ \in C Mod_{Σ}^{e c} (Δ)$ of $Δ$ with .

$B$ is an extended weakly skeptical c-inference from $A$ in the context of $Δ$ , denoted by , iff (i) $Δ$ is not weakly consistent, (ii) there is an extended c-representation $κ^{\circ}$ of $Δ$ with $κ \circ (A) = \infty$ , or (iii) there is an extended c-representation $κ$ of $Δ$ with and there is no extended c-representation $κ^{'}$ of $Δ$ with .

$B$ is a conservative extended weakly skeptical c-inference from $A$ in the context of $Δ$ , denoted by , iff (i) $Δ$ is not weakly consistent, (ii) there is a conservative extended c-representation $κ^{\circ}$ of $Δ$ with $κ^{\circ} (A) = \infty$ , or (iii) there is a conservative extended c-representation $κ$ of $Δ$ with and there is no conservative extended c-representation $κ^{'}$ of $Δ$ with .

The inference methods given in Definitions 60 and 61 all satisfy (DI) and (TV) and are thus inductive inference operators. However, so far, none of them has been investigated with respect to their syntax splitting properties. In the following, we will show that all six operators fully comply with conditional syntax splitting and thus also with (unconditional) syntax splitting. We start with the inference operators defined with respect to all extended c-representations.

Proposition 62
Extended credulous c-inference and extended weakly skeptical c-inference satisfy (CSynSplit).
Proof.
Both extended credulous c-inference and extended weakly skeptical c-inference have been shown to coincide with material counterpart inference (Haldimann et al., 2025, Proposition 64 and 74). Thus, the claim follows due to Proposition 53.

In the following two propositions, we show that also the inference operators defined with respect to all conservative extended c-representations satisfy conditional syntax splitting.
Proposition 63
Conservative extended credulous c-inference satisfies (CSynSplit).
Proof.
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ and let $i, i^{'} \in {1, 2}, i \neq i^{'}$ . We show (CRel) and (CInd) separately.

(CRel): Let $A, B \in L (Σ_{i})$ and let $E$ be a full conjunction over $Σ_{3}$ . We need to show iff , which is equivalent to showing that there is a conservative extended c-representation $κ$ of $Δ$ with $κ (A E B) < κ (A E \bar{B})$ iff there is a conservative extended c-representation $κ_{i}$ of $Δ_{i}$ with $κ_{i} (A E B) < κ_{i} (A E \bar{B})$ . Let $\vec{η}$ be an impact vector such that $κ = κ_{\vec{η}}$ is a conservative extended c-representation of $Δ$ with $κ (A E B) < κ (A E \bar{B})$ . With Proposition 39 there is an impact vector ${\vec{η}}_{i}$ such that $κ_{i} = κ_{{\vec{η}}_{i}}$ is a conservative extended c-representation of $Δ_{i}$ and ${\vec{η}}_{i} = \vec{η} |_{Δ_{i}}$ . Vice versa, every conservative extended c-representation $κ_{i}$ of $Δ_{i}$ is based on an impact vector ${\vec{η}}_{i}$ that can be extended to an impact vector $\vec{η}$ such that $κ = κ_{\vec{η}}$ is a conservative extended c-representation of $Δ$ and $\vec{η} |_{Δ_{i}} = {\vec{η}}_{i}$ . With Proposition 55, we get $κ (F) = κ_{i} (F)$ for all $F \in L (Σ_{i} \cup Σ_{3})$ , implying $κ (A E B) < κ (A E \bar{B})$ iff $κ_{i} (A E B) < κ_{i} (A E \bar{B})$ and thus iff .

(CInd): Let $A, B \in L (Σ_{i})$ , let $D \in L (Σ_{i^{'}})$ , and let $E$ be a full conjunction over $Σ_{3}$ with . We need to show iff . Assume . Further, there is a conservative extended c-representation $κ$ such that . Then there is a selection strategy $σ$ such that the inductive inference operator $C_{σ}^{e c}$ maps $Δ$ to $κ$ , that is, $C_{σ}^{e c} (Δ) = κ$ . According to Proposition 51, the inference operator $C_{σ}^{e c}$ satisfies (CInd). Thus implies and thus . The other direction is analogous.
Proposition 64
Conservative extended weakly skeptical c-inference satisfies (CSynSplit).
Proof.
Let $Δ = Δ_{1} ⋃_{Σ_{1}, Σ_{2}}^{s} Δ_{2} ∣ Σ_{3}$ and let $i, i^{'} \in {1, 2}, i \neq i^{'}$ . We show (CRel) and (CInd) separately.

(CRel): Let $A, B \in L (Σ_{i})$ and let $E$ be a full conjunction over $Σ_{3}$ . We need to show iff . In the proof of Proposition 63, we have already shown that (1) there is a conservative extended c-representation $κ$ of $Δ$ with $κ (A E B) < κ (A E \bar{B})$ iff there is a conservative extended c-representation $κ_{i}$ of $Δ_{i}$ with $κ_{i} (A E B) < κ_{i} (A E \bar{B})$ . In particular, this proof also implies that $κ (A) = \infty$ iff $κ_{i} (A) = \infty$ , covering case (ii) from Definition 61. We now show that (2) there is no conservative extended c-representation $κ$ of $Δ$ with $κ (A E \bar{B}) < κ (A E B)$ iff there is no conservative extended c-representation $κ_{i}$ of $Δ_{i}$ with $κ_{i} (A E \bar{B}) < κ_{i} (A E B)$ . We show this via contraposition. Assume there were some conservative extended c-representation $κ$ with $κ (A E \bar{B}) < κ (A E B)$ . Then, utilizing the same arguments as in the proof of Proposition 63, we obtain that there must also be a conservative extended c-representation $κ_{i}$ with $κ_{i} (A E \bar{B}) < κ_{i} (A E B)$ . The other direction is analogous. With (1) and (2), we get iff .

(CInd): Let $A, B \in L (Σ_{i})$ , let $D \in L (Σ_{i^{'}})$ , and let $E$ be a full conjunction over $Σ_{3}$ with . We need to show iff . In the proof of Proposition 63, we have already shown that (1) there is a conservative extended c-representation $κ$ of $Δ$ with $κ (A E B) < κ (A E \bar{B})$ and $κ (A D E B) < κ_{i} (A D E \bar{B})$ . In particular, this covers (ii) from Definition 61. We now show that (2) there is no conservative extended c-representation $κ$ of $Δ$ with $κ (A E \bar{B}) < κ (A E B)$ iff there is no conservative extended c-representation $κ^{'}$ with $κ^{'} (A D E \bar{B}) < κ^{'} (A D E B)$ . We show this via contraposition. Assume there were some conservative extended c-representation $κ$ with $κ (A E \bar{B}) < κ (A E B)$ . Then, utilizing the same arguments as in the proof of Proposition 63, we obtain that $κ (A E D \bar{B}) < κ (A E D B)$ . The other direction is analogous. With (1) and (2), we obtain iff .

Finally, we show that conditional syntax splitting is also satisfied by the two inference operators defined only for strongly consistent belief bases.
Proposition 65
Credulous c-inference and weakly skeptical c-inference satisfy (CSynSplit).
Proof.
For all strongly consistent belief bases, conservative extended credulous c-inference coincides with credulous c-inference (Haldimann et al., 2025, Lemma 68), and conservative extended weakly skeptical c-inference coincides with weakly skeptical c-inference (Haldimann et al., 2025, Lemma 78). Thus, the claim follows due to Propositions 63 and 64.

Thus, all inference operators obtained by reasoning with c-representations across three different reasoning modes—credulous, weakly skeptical, and skeptical—and with respect to three different sets of models—c-representations, extended c-representations, and conservative extended c-representations—fully comply with syntax splitting and conditional syntax splitting.
8. Conclusions and Future Work

For inductive reasoning from conditional belief bases, the concept of conditional syntax splitting has been introduced in the literature as a generalization of syntax splitting. It is applicable also to cases where the conditionals in the subbases share some atoms, and it leads to a formalization of the drowning effect which had been described previously only by means of examples. However, previous work on splitting often implicitly or explicitly assumes that the considered belief bases are (strongly) consistent in the sense of the well-known consistency test by Goldszmidt and Pearl (1996).

In this article, we extended the study of conditional splitting. by taking also weakly consistent belief bases into account and thus allowing for some worlds to be completely infeasible. We introduced the concept of conditional semantic splitting for OCF-based semantics of conditional belief bases. We showed that c-representations, which exhibit notable properties desirable for nonmonotonic reasoning, satisfy the conditional semantic splitting postulate (CSemSplit). For inference based on single c-representations, we showed that the concept of selection strategies leads to inductive inference operators satisfying the conditional syntax splitting postulate (CSynSplit). Furthermore, we proved that c-inference, which is obtained by taking all c-representations of a belief base into account, also satisfies (CSynSplit) and thus fully complies with conditional syntax splitting, and we have shown that also credulous and weakly skeptical reasoning with c-representations satisfies conditional syntax splitting. All results regarding c-representations and inference based on them are shown for strongly consistent belief bases and also for their extended and conservative extended versions covering weakly consistent belief bases. This results in different inference operators based on c-representations that all fully comply with conditional syntax splitting, showing that c-representations behave very well with respect to splitting properties. We also elaborated on the relationship between conservative extended c-representations and relaxations thereof, leading to a postulate for selection strategies ensuring classic preservation.

Our current and future work includes exploiting the benefits of conditional splitting in implementations of inductive inference, for example, in the reasoning platform InfOCF and its underlying library (Beierle et al., 2017, 2024a, 2025; Kutsch & Beierle, 2021), and to extend the study of splitting to also cover case splitting and the kinematics principle (Kern-Isberner et al., 2023).

Footnotes

ORCID iDs

Christoph Beierle

Lars-Phillip Spiegel

Jonas Haldimann

Marco Wilhelm

Jesse Heyninck

Gabriele Kern-Isberner

Acknowledgments

We would like to thank all reviewers for their valuable comments and hints, which helped us to improve this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—512363537, grant BE 1700/12-1 awarded to Christoph Beierle. Lars-Phillip Spiegel was supported by this grant.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Conditional Syntax and Semantic Splittings of Strongly and Weakly Consistent Belief Bases

Abstract

Keywords

1. Introduction

3. Conditional Syntax Splitting

Definition 2 (Conditional syntax splitting, (Heyninck et al., 2023))

Example 6 ( Δ b + )

4. Conditional Semantic Splitting

4.1. Model Combinations and Semantic Splittings

Definition 14 (Model combination)

Definition 19 (c-Representation; Kern-Isberner, 2001, 2004)

Definition 25 (Conservative c-representation, (Haldimann et al., 2024), adapted)

Definition 28 ( C R S Σ e x ( Δ ) , (Haldimann et al., 2024))

Example 30 ( Δ b + continued)

Definition 31 (Relaxation of conservative c-representations)

Lemma 38 (Based on Haldimann, 2024)

Definition 43 (Selection strategy σ , adapted from Beierle & Kern-Isberner, 2021)

Definition 44 (Inductive inference operator C σ e c , adapted from Kern-Isberner et al., 2020)

Proposition 45 (infinity selection strategy τ yields material counterpart inference )

Definition 54 (Extended c-inference, , (Beierle et al., 2016a; Haldimann et al., 2024))

Definition 60 (Credulous/weakly skeptical c-inference, (Beierle et al., 2021a))

Definition 61 ((Conservative) extended credulous/weakly skeptical c-inference, (Haldimann et al., 2025))

Footnotes

ORCID iDs

Acknowledgments

Funding

Declaration of Conflicting Interests

References

Example 6 ( $Δ^{b +}$ )

Definition 28 ( ${C R S}_{Σ}^{e x} (Δ)$ , (Haldimann et al., 2024))

Example 30 ( $Δ^{b +}$ continued)

Definition 43 (Selection strategy $σ$ , adapted from Beierle & Kern-Isberner, 2021)

Definition 44 (Inductive inference operator $C_{σ}^{e c}$ , adapted from Kern-Isberner et al., 2020)

Proposition 45 (infinity selection strategy $τ$ yields material counterpart inference )