arXiv CSopen access

Broadening the Applicability of Conditional Syntax Splitting for Reasoning from Conditional Belief Bases

Unknown · 2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

artificialintelligence knowledgerepresentation reasoning

artificial intelligence, reasoning, knowledge representation

Highlights Broadening the Applicability of Conditional Syntax Splitting for Reasoning from Conditional Belief Bases Lars-Phillip Spiegel, Jonas Haldimann, Jesse Heyninck, Gabriele Kern-Isberner, Christoph Beierle • Generalization of safe conditional syntax splitting for belief bases; • Postulates (CRelg ), (CIndg ), and (CSynSplitg ) for inductive inference operators;

arXiv:2604.12660v1 [cs.AI] 14 Apr 2026

• Identification of genuine splittings as a necessary condition for splittings beneficial for inductive inference; • Evaluation of established inductive inference operators with respect to generalized conditional syntax splitting; • Proofs that lexicographic inference, c-inference, inference with a single c-representation determined by an appropriate selection strategy, c-core-closure inference, and System W satisfy (CSynSplitg ); • Showing that (CSynSplitg ) implies (CSynSplit), but not the other way around.

Broadening the Applicability of Conditional Syntax Splitting for Reasoning from Conditional Belief Bases Lars-Phillip Spiegela,∗ , Jonas Haldimannb,c , Jesse Heyninckd,b , Gabriele Kern-Isbernere and Christoph Beierlea a FernUniversität in Hagen, Hagen, Germany b University of Cape Town and CAIR, Cape Town, South Africa c TU Wien, Vienna, Austria d Open Universiteit, Heerlen, 6419 AT, the Netherlands e TU Dortmund University, Dortmund, Germany

ARTICLE INFO

ABSTRACT

Keywords: conditional belief base syntax splitting conditional syntax splitting generalized conditional syntax splitting inductive inference inductive inference operator system Z lexicographic inference system W c-inference c-representation

In nonmonotonic reasoning from conditional belief bases, an inference operator satisfying syntax splitting postulates allows for taking only the relevant parts of a belief base into account, provided that the belief base splits into subbases based on disjoint signatures. Because such disjointness is rare in practice, safe conditional syntax splitting has been proposed as a generalization of syntax splitting, allowing the conditionals in the subbases to share some atoms. Recently this overlap of conditionals has been shown to be limited to trivial, self-fulfilling conditionals. In this article, we propose a generalization of safe conditional syntax splittings that broadens the applicability of splitting postulates. In contrast to safe conditional syntax splitting, our generalized notion supports syntax splittings of a belief base Δ where the subbases of Δ may share atoms and nontrivial conditionals. We illustrate how this new notion overcomes limitations of previous splitting concepts, and we identify genuine splittings, separating them from simple splittings that do not provide benefits for inductive inference from Δ. We introduce adjusted inference postulates based on our generalization of conditional syntax splitting, and we evaluate several popular inductive inference operators with respect to these postulates. Furthermore, we show that, while every inductive inference operator satisfying generalized conditional syntax splitting also satisfies conditional syntax splitting, the reverse does not hold.

∗ Corresponding author

[email protected] (L. Spiegel); [email protected] (J. Haldimann); [email protected] (J. Heyninck); [email protected] (G. Kern-Isberner); [email protected] (C. Beierle) ORCID (s): 0009-0001-1962-752X (L. Spiegel); 0000-0002-2618-8721 (J. Haldimann); 0000-0002-3825-4052 (J. Heyninck); 0000-0001-8689-5391 (G. Kern-Isberner); 0000-0002-0736-8516 (C. Beierle)

L.Spiegel et al.: Preprint submitted to Elsevier

Page 1 of 34

Broadening the Applicability of Conditional Syntax Splitting

Contents 1

Introduction

Formal Basics

Safe Conditional Syntax Splitting and its Limitations

Generalized Safe Conditional Syntax Splitting

Evaluating Inductive Inference Operators with respect to (CSynSplitg ) 5.1 System Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lexicographic Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 System W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 10 11 14

Evaluating Inductive Inference Operators based on c-Representations 6.1 c-Representations and 𝜅-Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Inference with Single c-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 c-Core closure Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 c-Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 16 20 21 24

(CSynSplitg ) properly strengthens (CSynSplit)

Conclusions and Future Work

References

A List of all conditional syntax splitting of belief base Δ𝑟𝑎𝑖𝑛 from Example 14.

B List of all conditional syntax splittings of belief base Δ𝑘 from the proof of Proposition 55.

L.Spiegel et al.: Preprint submitted to Elsevier

Page 2 of 34

Broadening the Applicability of Conditional Syntax Splitting

1. Introduction Both human and formal reasoning methods often rely on restricting the amount of information taken into account for a given reasoning task, tuning out unrelated facts and knowledge. The concept of syntax splitting (Parikh, 1999; Peppas, Williams, Chopra and Foo, 2015; Kern-Isberner and Brewka, 2017), and of the related idea of minimum irrelevance (Weydert, 1998) have been introduced as steps to formalize this goal. This idea has been transferred to inductive inference from conditional belief bases under the motto “syntax splitting = relevance + independence” in the form of postulates (Rel) and (Ind) for inductive inference operators (Kern-Isberner, Beierle and Brewka, 2020), taking splittings over a belief base Δ into account where the subbases Δ1 , Δ2 are given over disjoint subsignatures of Δ. In practice, such splittings are rare because the disjointness of subsignatures imposes a harsh restriction. The concept of conditional syntax splitting (Heyninck, Kern-Isberner, Meyer, Haldimann and Beierle, 2023) is an approach to overcome this restriction by allowing Δ1 and Δ2 to share atoms in their respective subsignatures. To ensure semantic (conditional) independence given the joint atoms, a safety condition has been formulated, enabling local reasoning within the subbases. The postulate of conditional relevance (CRel) implements this idea of localized reasoning, formalizing the ability to focus only on syntactically relevant parts of a belief base, while the postulate of conditional independence (CInd) describes the ability to leave aside syntactically irrelevant information (Heyninck et al., 2023). Furthermore, the postulate of conditional independence (CInd) for safe conditional splittings characterizes avoiding the drowning effect (Pearl, 1990; Benferhat, Dubois and Prade, 1993), yielding the first formal definition of the notorious drowning problem that had been described before only by specific examples (Heyninck et al., 2023). These splittings are not only interesting from a theoretical point of view by formalizing notions of conditional relevance and independence, but also have consequences for applications by allowing the breaking down of conditional reasoning to the subbases relevant for a given query, usually reducing the relevant signature significantly. Hence, broadening the applicability of such splittings provides not only theoretical insights, but also benefits for applications. It has been shown recently that the safety condition in (Heyninck et al., 2023) has the undesirable consequence that every conditional in the intersection of Δ1 and Δ2 is a trivial self-fulfilling conditional, meaning that it cannot be falsified (Beierle, Spiegel, Haldimann, Wilhelm, Heyninck and Kern-Isberner, 2024b), thus imposing a strong restriction on possible splitting benefits for inference. We develop a generalization of this safety condition, allowing the intersection of Δ1 and Δ2 to contain more meaningful conditionals. This greatly broadens the application possibilities of syntax splitting by increasing both the amount of splittings and the amount of belief bases where splittings can be exploited for inductive reasoning. The main contributions of this article are: • Generalization of safe conditional syntax splitting for belief bases; • Postulates (CRelg ), (CIndg ), and (CSynSplitg ) for generalized safe conditional syntax splitting; • Identification of the subclass of genuine splittings, separating them from the large class of simple splittings that have no benefits for inductive inference because existing postulates cannot be meaningfully applied to them; • Evaluation of established inductive inference operators with respect to generalized conditional syntax splitting; • Proofs that lexicographic inference (Lehmann, 1995), c-inference (Beierle, Eichhorn, Kern-Isberner and Kutsch, 2018, 2021a), inference with a single c-representation determined by an appropriate selection strategy (Beierle and Kern-Isberner, 2021), c-core-closure inference (Wilhelm, Kern-Isberner and Beierle, 2024), and System W (Komo and Beierle, 2020, 2022) satisfy (CSynSplitg ); • Showing that (CSynSplitg ) implies (CSynSplit), but not the other way around. This article is a revised and largely extended version of the paper previously published at IJCAI 2025 (Spiegel, Haldimann, Heyninck, Kern-Isberner and Beierle, 2025). In particular, we added the evaluation of further inductive inference operators, showing that lexicographic inference (Lehmann, 1995) and c-core closure inference (Wilhelm et al., 2024) both satisfy (CSynSplitg ). Furthermore, this articles contains all proofs which were not present in the conference paper, and we added more explanations of the concepts introduced and additional examples illustrating them. After recalling the needed background in Sect. 2, we point out the limitations of safe conditional splittings in Sect. 3. Next, we introduce the concepts of generalized safe and genuine conditional syntax splitting and present adapted L.Spiegel et al.: Preprint submitted to Elsevier

Page 3 of 34

Broadening the Applicability of Conditional Syntax Splitting

postulates for inference in Sect. 4. We evaluate inductive inference operators with respect to these new postulates in Sect. 5 and Sect. 6. In Sect. 7, we show that (CSynSplitg ) implies (CSynSplit) but not the other way around, before concluding in Sect 8.

2. Formal Basics Let  be a finitely generated propositional language over a signature Σ with atoms 𝑎, 𝑏, 𝑐, … and with formulas 𝐴, 𝐵, 𝐶, … We may write 𝐴𝐵 instead of 𝐴 ∧ 𝐵, and overline formulas to indicate negation, i.e., 𝐴 means ¬𝐴. If a statement holds for both 𝐴 and 𝐴, we will sometimes use 𝐴̇ to denote both formulas at the same time. Let Ω denote the set of possible worlds over , taken here simply as the set of all propositional interpretations over . 𝜔 ⊧ 𝐴 means that the propositional formula 𝐴 ∈  holds in 𝜔 ∈ Ω; in this case 𝜔 is called a model of 𝐴, and the set of all models of 𝐴 is denoted by Mod (𝐴). For propositions 𝐴, 𝐵 ∈ , 𝐴 ⊧ 𝐵 holds iff Mod (𝐴) ⊆ Mod (𝐵), as usual. We will use 𝜔 both for the model and the corresponding complete conjunction of all positive or negated atoms, allowing us to use 𝜔 both as an interpretation and a proposition. For Θ ⊆ Σ, let (Θ) or short Θ denote the propositional language defined by Θ, with associated set of interpretations Ω(Θ) or short ΩΘ . Note that while each formula of (Θ) can also be considered as a formula of , the interpretations 𝜔Θ ∈ Ω(Θ) are not elements of Ω(Σ) if Θ ≠ Σ. But each interpretation 𝜔 ∈ Ω can be written uniquely in the form 𝜔 = 𝜔Θ 𝜔Θ with concatenated 𝜔Θ ∈ Ω(Θ) and 𝜔Θ ∈ Ω(Θ), where Θ = Σ∖Θ. The world 𝜔Θ is called the reduct of 𝜔 to Θ (Delgrande, 2017). If Ω′ ⊆ Ω is a subset of models, then Ω′ |Θ = {𝜔Θ |𝜔 ∈ Ω′ } ⊆ Ω(Θ) restricts Ω′ to a subset of Ω(Θ). In the following, we will often denote subsignatures of Σ by Σ1 , Σ2 , … and write 𝜔𝑖 instead of 𝜔Σ𝑖 to ease notation. By making use of a conditional operator |, we introduce the language (|) = {(𝐵|𝐴) ∣ 𝐴, 𝐵 ∈ } of conditionals over . Conditionals (𝐵|𝐴) are meant to express plausible, defeasible rules “If 𝐴 then plausibly (usually, possibly, probably, typically etc.) 𝐵”. For a world 𝜔 a conditional (𝐵|𝐴) is either verified by 𝜔 if 𝜔 ⊧ 𝐴𝐵, falsified by 𝜔 if 𝜔 ⊧ 𝐴𝐵, or not applicable to 𝜔 if 𝜔 ⊧ 𝐴. A conditional (𝐹 |𝐸) is called self-fulfilling, or trivial, if 𝐸 ⊧ 𝐹 , i.e., there is no world that can falsify it. For a conditional 𝛿𝑖 = (𝐵𝑖 |𝐴𝑖 ) let 𝑣𝑒𝑟(𝛿𝑖 ) = {𝜔 ∈ Ω(Σ) ∣ 𝜔 ⊧ 𝐴𝑖 𝐵𝑖 } 𝑓 𝑎𝑙(𝛿𝑖 ) = {𝜔 ∈ Ω(Σ) ∣ 𝜔 ⊧ 𝐴𝑖 𝐵𝑖 } denote the sets of verifying and falsifying worlds, respectively. A belief base Δ (over Σ) is a set of finitely many conditionals from ( ∣ ). A popular semantic framework for interpreting conditionals are ordinal conditional functions (OCFs) 𝜅 ∶ Ω → ℕ ∪ {∞} with 𝜅 −1 (0) ≠ ∅. OCFs, also called ranking functions, introduced, in a more general form, by (Spohn, 1988). Intuitively, less plausible worlds are assigned higher numbers. Formulas are assigned the rank of their most plausible models, i.e., 𝜅(𝐴) ∶= min{𝜅(𝜔) ∣ 𝜔 ⊧ 𝐴}. The rank of (𝐵|𝐴) is 𝜅(𝐵|𝐴) = 𝜅(𝐴𝐵) − 𝜅(𝐴). A conditional (𝐵|𝐴) is accepted by 𝜅, written as 𝜅 ⊧ (𝐵|𝐴), iff 𝜅(𝐴𝐵) < 𝜅(𝐴𝐵), i.e., iff 𝐴𝐵 is more plausible than 𝐴𝐵. This is lifted to belief bases via 𝜅 ⊧ Δ if 𝜅 ⊧ (𝐵|𝐴) for all (𝐵|𝐴) ∈ Δ. Consistency of a belief base Δ can be defined in terms of OCFs (Pearl, 1990): Δ is (strongly) consistent iff there is an OCF 𝜅 such that 𝜅 ⊧ Δ and 𝜅(𝜔) < ∞ for all 𝜔 ∈ Ω. We focus on (strongly) consistent belief bases in the sense of (Pearl, 1990; Goldszmidt and Pearl, 1996) in order to elaborate our approach without having to deal with distracting technical particularities. The nonmonotonic inference relation |∼ 𝜅 induced by an OCF 𝜅 is given by (Spohn, 1988) 𝐴 |∼ 𝜅 𝐵

iff 𝐴 ≡ ⊥ or 𝜅(𝐴𝐵) < 𝜅(𝐴𝐵).

(1)

The marginal of 𝜅 on Θ ⊆ Σ, denoted by 𝜅|Θ , is defined by 𝜅|Θ (𝜔Θ ) = 𝜅(𝜔Θ ) for any 𝜔Θ ∈ Ω(Θ). Here 𝜔Θ is treated as a world in 𝜅|Θ (𝜔Θ ) but as a formula in 𝜅(𝜔Θ ). Note that this marginalization is a special case of the general forgetful functor 𝑀𝑜𝑑(𝜎) from Σ-models to Θ-models (Beierle and Kern-Isberner, 2012) where 𝜎 is the inclusion from Θ to Σ. To formalize inductive inference from belief bases, the notion of inductive inference operators was introduced (Kern-Isberner et al., 2020). An inductive inference operator is a mapping 𝐂 that assigns to each belief base Δ ⊆ ( ∣ ) an inference relation |∼ Δ on , i.e., 𝐂 ∶ Δ ↦ |∼ Δ , such that the following two properties hold: Direct Inference (DI): If (𝐵|𝐴) ∈ Δ then 𝐴 |∼ Δ 𝐵, and L.Spiegel et al.: Preprint submitted to Elsevier

Page 4 of 34

Broadening the Applicability of Conditional Syntax Splitting

Trivial Vacuity (TV): 𝐴 |∼ ∅ 𝐵 implies 𝐴 ⊧ 𝐵. A special subclass of inductive inference operators are OCF-based inductive inference operators 𝐂𝑜𝑐𝑓 ∶ Δ ↦ 𝜅Δ , assigning to each belief base Δ an OCF 𝜅Δ (Kern-Isberner et al., 2020). The inference relation for OCF-based inductive inference operators is then obtained via Equation (1). The following are examples for inductive inference operators: p-Entailment |∼ 𝑝 (Adams, 1975; Goldszmidt and Pearl, 1996) considers all models of a belief base and is the most cautious preferential inductive inference operator. It can be characterized by system P because it licenses precisely the inferences that can be obtained by iteratively applying the rules of system P (Lehmann and Magidor, 1992; Dubois and Prade, 1994). System Z |∼ 𝑧 (Goldszmidt and Pearl, 1996) determines the uniquely defined minimal ranking model of Δ, and it coincides with rational closure (Lehmann and Magidor, 1992). System Z is an example of an OCF-based inductive inference operator. Lexicographic inference |∼ 𝑙𝑥 (Lehmann, 1995) employs a comparison of the number of conditionals falsified by a world, and it extends rational closure. c-Inference |∼ 𝑐 (Beierle et al., 2018, 2021a) considers all c-representations which are special ranking functions obtained by summing up natural number impacts assigned to falsified conditionals (Kern-Isberner, 2001, 2004). System W |∼ 𝗐 (Komo and Beierle, 2020, 2022) is based on a preferred structure on worlds, and it captures both c-inference and system Z and thus rational closure. p-Entailment is extended by the four other inductive inference operators in the sense that all five inductive inference operators are preferential, while lexicographic inference extends all other four inductive inference operators; for an overview of the interrelationships among them see (Haldimann and Beierle, 2024). In the rest of this article we will evaluate these and further inductive inference operators with respect to their syntax splitting behaviour.

3. Safe Conditional Syntax Splitting and its Limitations Syntax splittings describe that a belief base contains completely independent information about different parts of the signature. According to (Kern-Isberner et al., 2020), a belief base Δ splits into subbases Δ1 , Δ2 if {Σ1 , Σ2 } is a partition of Σ such that Δ = Δ1 ∪ Δ2 , Δ𝑖 ⊊ (𝑖 |𝑖 ), 𝑖 = (Σ𝑖 ) for 𝑖 ∈ {1, 2}, Σ1 ∩ Σ2 = ∅, and Σ1 ∪ Σ2 = Σ, denoted as ⋃ Δ2 . (2) Δ = Δ1 Σ1 ,Σ2

Syntax splittings are very useful for formalizing the idea that independent information about different topics should not affect each other in reasoning. Syntax splittings were generalized in (Heyninck et al., 2023) to conditional syntax splittings, which allow subbases to share some atoms in a given subsignature Σ3 . Definition 1 (conditional syntax splitting (Heyninck et al., 2023)). A belief base Δ splits into subbases Δ1 ,Δ2 conditional on Σ3 , if there are Σ1 , Σ2 ⊆ Σ such that Δ𝑖 = Δ ∩ ((Σ𝑖 ∪ Σ3 ) ∣ (Σ𝑖 ∪ Σ3 )) for 𝑖 = 1, 2, and {Σ1 , Σ2 , Σ3 } is a partition of Σ. This is denoted as ⋃ Δ = Δ1 Δ2 ∣ Σ3 . (3) Σ1 ,Σ2

Unlike syntax splitting, conditional syntax splitting does not require the subbases Δ1 and Δ2 to be disjoint. For the remainder of this paper, we will use the notation introduced in the following straightforward proposition. ⋃ Proposition 2. Let Δ = Δ1 Σ1 ,Σ2 Δ2 ∣ Σ3 and let Δ3 = Δ1 ∩ Δ2

(4)

Δ1⧵3 = Δ1 ⧵ Δ3

(5)

L.Spiegel et al.: Preprint submitted to Elsevier

Page 5 of 34

Broadening the Applicability of Conditional Syntax Splitting

(6)

Δ2⧵3 = Δ2 ⧵ Δ3 . Then Δ1⧵3 , Δ2⧵3 , Δ3 are pairwise disjoint and

(7)

Δ = Δ1⧵3 ∪Δ2⧵3 ∪Δ3 . Proof. Pairwise disjointness and (7) follow immediately from (4), (5), and (6).

Note that in Proposition 2, Δ3 = Δ ∩ ((Σ3 )|(Σ3 )), implying that Δ3 ⊆ ((Σ3 )|(Σ3 )), and, for 𝑖 ∈ {1, 2}, Δ𝑖′ ⧵3 ⊆ ((Σ𝑖 ∪ Σ3 )|(Σ𝑖 ∪ Σ3 )). For 𝜔 ∈ Ω and 𝐴 ∈ (Σ𝑖 ) we have 𝜔1 𝜔3 𝜔2 ⊧ 𝐴

iff

(8)

𝜔𝑖 𝜔3 ⊧ 𝐴.

Given a complete conjunction over Σ3 , i.e., a formula uniquely describing a world in Ω(Σ3 ), conditional syntax splittings in general do not ensure complete independence of Δ1 and Δ2 (for details see (Heyninck et al., 2023, Example 6)). To fix this, safe conditional syntax splittings were introduced. ⋃ Definition 3 (safe conditional syntax splitting (Heyninck et al., 2023)). A belief base Δ = Δ1 Σ1 ,Σ2 Δ2 ∣ Σ3 can be safely split into subbases Δ1 , Δ2 conditional on a subsignature Σ3 , writing Δ = Δ1

𝗌 ⋃

(9)

Δ2 ∣ Σ3

Σ1 ,Σ2

if the following safety property holds for 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ : ′

′

for every 𝜔𝑖 𝜔3 ∈ Ω(Σ𝑖 ∪ Σ3 ), there is 𝜔𝑖 ∈ Ω(Σ𝑖′ ) such that 𝜔𝑖 𝜔3 𝜔𝑖 ̸⊧

⋁

𝐸 ∧ ¬𝐹 .

(10)

(𝐹 |𝐸)∈Δ𝑖′

The safety condition demands, in essence, that no complete conjunction over Σ3 may force the falsification of a conditional in Δ when considering Σ as a whole. Example 4 (Δ𝑠𝑢𝑛 ). Consider the belief base Δ𝑠𝑢𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑔|𝑏), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑜𝑟)} describing the following: If it is (r)ainy, then usually it is not (s)unny and vice versa. If it is rainy and sunny at the same time, then we can usually observe a rain(b)ow. Maybe superstitiously, we believe that there is usually some (g)old to be found at the end of the rainbow. Unrelated to this, we usually spend some time (o)utside if it is sunny and not rainy. If it is rainy, then we usually do not spend time outside. If, despite our normal habits, we do spend time outside and it is rainy, then we usually have an (u)mbrella. Δ𝑠𝑢𝑛 has a safe conditional syntax splitting Δ𝑠𝑢𝑛 = Δ𝑠𝑢𝑛 1

𝗌 ⋃

Δ𝑠𝑢𝑛 ∣ {𝑏} 2

(11)

{𝑔},{𝑠,𝑟,𝑜,𝑢}

where Σ1 = {𝑔}, Σ2 = {𝑠, 𝑟, 𝑜, 𝑢}, Σ3 = {𝑏}, Δ𝑠𝑢𝑛 = {(𝑔|𝑏)}, Δ𝑠𝑢𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑜𝑟)}, and 1 2 1 ∈ Ω(Σ ∪ Σ ) by any 𝜔′ ∈ Ω(Σ ) with 𝜔′ ⊧ 𝑠 ∧ 𝑟 ∧ 𝑜 ∧ 𝑢 without Δ𝑠𝑢𝑛 = ∅. This splitting is safe: We can extend any 𝜔 1 3 2 3 2 ∈ Ω(Σ ∪ Σ ) by any 𝜔′′ ∈ Ω(Σ ) with 𝜔′′ ⊧ 𝑔 without falsifying a conditional in Δ𝑠𝑢𝑛 . Similarly we can extend any 𝜔 2 3 1 2 falsifying a conditional in Δ𝑠𝑢𝑛 . 1 Safe conditional syntax splitting provides similar benefits for inductive inference as syntax splitting. Reasoning in the language of Δ1 is independent of the conditionals in Δ2 , and vice versa, given we have full knowledge over the atoms in Σ3 . However, it has been shown recently (Beierle et al., 2024b) that the safety property (10) imposes a strong, undesired restriction on Δ3 . ⋃ Lemma 5 ((Beierle et al., 2024b)). Let Δ = Δ1 𝗌Σ ,Σ Δ2 ∣ Σ3 , then Δ3 = Δ1 ∩ Δ2 contains only self-fulfilling 1 2 conditionals.

L.Spiegel et al.: Preprint submitted to Elsevier

Page 6 of 34

Broadening the Applicability of Conditional Syntax Splitting

While it is true that Δ3 can not contain “meaningful” information, the elements in Σ3 are still relevant and can occur in conditionals of both Δ1 and Δ2 (as we see in Example 4). A generalization of the safety property to avoid the effect described in Lemma 5 would be advantageous. Recall that Σ3 represents a sort of global knowledge, that should be considered in both subbases. However it is not always possible to find a safe splitting, given some intuitive or in practice desirable allocation of signature elements to Σ3 . Example 6 (Δ𝑠𝑢𝑛 cont.). Assume we want to reason based on Δ𝑠𝑢𝑛 , under the assumption that we have full knowledge about 𝑠 and 𝑟. A conditional syntax splitting reflecting our knowledge about the weather is ⋃ ∣ {𝑠, 𝑟} (12) Δ𝑠𝑢𝑛 Δ𝑠𝑢𝑛 = Δ𝑠𝑢𝑛 2 1 {𝑏,𝑔},{𝑜,𝑢}

where Σ1 = {𝑏, 𝑔}, Σ2 = {𝑜, 𝑢}, Σ3 = {𝑠, 𝑟}, Δ𝑠𝑢𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑔|𝑏)}, Δ𝑠𝑢𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), 1 2 𝑠𝑢𝑛 (𝑢|𝑜𝑟)}, and Δ3 = {(𝑠|𝑟), (𝑟|𝑠)}. Assume that we know that it is sunny and rainy at the same time, and we would like to know if there will usually be a rainbow, i.e., whether 𝑠𝑟 |∼Δ𝑠𝑢𝑛 𝑏 holds. Employing the splitting (12), it suffices to consider Δ𝑠𝑢𝑛 to answer this query because we have full knowledge about {𝑠, 𝑟}. However, because the conditionals 1 in Δ𝑠𝑢𝑛 are not self-fulfilling the splitting (12) is not safe. 3 Comparing the splittings (11) and (12), we can see that there are situations where (12) provides benefits not provided by (11). For instance, as it will be shown formally in the following sections, answering the query 𝑠𝑟 |∼Δ𝑠𝑢𝑛 𝑏 can be done using the subbase Δ𝑠𝑢𝑛 from (12) while the splitting (11) does not provide any advantage for answering this query. 1 Another limitation of safe conditional syntax splittings is that there exist belief bases where all safe conditional syntax splittings involve a subset relationship between the subbases. Example 7 (Δ𝑟𝑎𝑖𝑛 ). Starting from Δ𝑠𝑢𝑛 , we get rid of our superstitious beliefs by removing the signature element 𝑔 and all associated conditionals, yielding Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑜𝑟)}. Δ𝑟𝑎𝑖𝑛 has a splitting conditional on {𝑠, 𝑟} ⋃ Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)} {(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑜𝑟)} ∣ {𝑠, 𝑟} (13) {𝑏},{𝑜,𝑢}

⋃𝗌 𝑟𝑎𝑖𝑛 ∣ Σ satisfies Δ𝑟𝑎𝑖𝑛 ⊆ Δ𝑟𝑎𝑖𝑛 or which, however, is not safe. In fact, every safe splitting of Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 3 Σ1 ,Σ2 Δ2 1 2 1 𝑟𝑎𝑖𝑛 𝑟𝑎𝑖𝑛 Δ2 ⊆ Δ1 . ⋃ Every conditional syntax splitting Δ = Δ1 Σ1 ,Σ2 Δ2 ∣ Σ3 with Δ1 ⊆ Δ2 or Δ2 ⊆ Δ1 is of little use for inductive inference. Suppose Δ1 ⊆ Δ2 . Then answering any query over Σ2 ∪ Σ3 requires considering Δ as a whole because Δ2 = Δ. Furthermore, any query over Σ1 ∪ Σ3 can also not benefit from the splitting. This is because atoms of Σ1 can not appear in Δ1 , since all conditionals of Δ1 are defined over Σ3 as Δ1 = Δ1 ∩ Δ2 = Δ3 and full knowledge of Σ3 is required to make use of the splitting. The observations above give rise to two points. First, we will extend the notion of safety to cover conditional splittings like (12) and (13). Second, we will identify splittings that are useful for inductive inference.

4. Generalized Safe Conditional Syntax Splitting We first introduce generalized safe splittings as a generalization of safe splittings to cover cases where the subbases may share non-trivial conditionals, and we introduce genuine splittings, which identify splittings that provide benefits for inductive inference. ⋃ Definition 8 (generalized safe conditional syntax splitting). A belief base Δ = Δ1 Σ1 ,Σ2 Δ2 ∣ Σ3 can be generalized safely split into subbases Δ1 , Δ2 conditional on a subsignature Σ3 , writing Δ = Δ1

𝗀𝗌 ⋃

(14)

Δ2 ∣ Σ3

Σ1 ,Σ2

if the following generalized safety property holds for 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ : ′

′

for every 𝜔𝑖 𝜔3 ∈ Ω(Σ𝑖 ∪ Σ3 ), there is 𝜔𝑖 ∈ Ω(Σ𝑖′ ) such that 𝜔𝑖 𝜔3 𝜔𝑖 ̸⊧

⋁

𝐸 ∧ ¬𝐹 .

(15)

(𝐹 |𝐸)∈Δ𝑖′ ⧵3

L.Spiegel et al.: Preprint submitted to Elsevier

Page 7 of 34

Broadening the Applicability of Conditional Syntax Splitting

The deciding difference in (15) compared to (10) is that only conditionals in Δ𝑖′ ⧵3 are considered for the generalized safety property as opposed to all conditionals in Δ𝑖′ for the safety property. Generalized safety extends the notion of safety in the sense that every safe conditional syntax splitting is also generalized safe. The notions coincide whenever Δ3 = ∅. ⋃ Proposition 9. Let Δ be a belief base over Σ with a conditional syntax splitting 𝑆 ∶ Δ = Δ1 Σ1 ,Σ2 Δ2 ∣ Σ3 . 1. If 𝑆 is safe, then 𝑆 is generalized safe. 2. If Δ3 = Δ1 ∩ Δ2 = ∅, then 𝑆 is safe iff 𝑆 is generalized safe. ⋃ Proof. Let Δ be a belief base with Δ = Δ1 𝗌Σ ,Σ Δ2 ∣ Σ3 . 1. For 𝑖, 𝑗 ∈ {1, 2}, 𝑖 ≠ 𝑗, for every 𝜔3 ∈ Ω(Σ𝑖 ∪ Σ3 ) there 1

is 𝜔𝑗 ∈ Ω(Σ𝑗 ) such that 𝜔3 𝜔𝑗 does not falsify any conditional in Δ𝑖′ . Then 𝜔3 𝜔𝑗 does not falsify any conditional in ⋃𝗀𝗌 Δ𝑖′ ⧵3 and thus Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . 2. For Δ3 = ∅ we have Δ𝑖 = Δ𝑖⧵3 , thus the two properties are equivalent. 1

Generalized safety allows for more splittings adhering to a notion of safety. In particular, generalized safe splittings allow for non-trivial conditionals in Δ3 . Example 10 (Δ𝑠𝑢𝑛 , Δ𝑟𝑎𝑖𝑛 cont.). While not safe, the conditional syntax splittings in Examples 6 and 7 are generalized safe. For instance, in both examples, the conditional (𝑟|𝑠) can be falsified by 𝑟𝑠 ∈ Ω(Σ𝑖 ∪Σ3 ), thus making the splittings not safe, but since (𝑟|𝑠) ∈ Δ𝑠𝑢𝑛 and (𝑟|𝑠) ∈ Δ𝑟𝑎𝑖𝑛 , this fact does not lead to a violation of generalized safety. 3 3 With Lemma 10 and Lemma 9 we can see that there exist more generalized safe conditional syntax splittings than safe conditional syntax splittings. Thus, generalized safety properly extends the amount of belief bases that can be conditionally split while adhering to a notion of safety. We will now introduce postulates to evaluate inductive inference operators with respect to generalized safe conditional syntax splitting. The idea of these postulates is that for a belief base with (generalized safe conditional) syntax splitting, inference over one subsignature should be independent from the information about the other subsignature (given that the valuation of the shared subsignature is fixed). These postulates are analogous to the postulates conditional relevance and conditional independence (Heyninck et al., 2023) but we adapt them to include also generalized safe splittings. ⋃𝗀𝗌 (CRelg ) An inductive inference operator 𝐂 satisfies generalized conditional relevance if for any Δ = Δ1 Σ ,Σ Δ2 ∣ 1 2 Σ3 , for 𝑖 ∈ {1, 2} and any 𝐴, 𝐵 ∈ (Σ𝑖 ), and a complete conjunction 𝐸 ∈ (Σ3 ), 𝐴𝐸 ∣∼ Δ 𝐵

iff

𝐴𝐸 ∣∼ Δ𝑖 𝐵.

Thus, (CRelg ) restricts the scope of inference by requiring that inferences in the sub-language Σ1 ∪ Σ3 can be made taking only Δ1 into account. (CIndg ) An inductive inference operator 𝐂 satisfies generalized conditional independence if for any Δ = ⋃𝗀𝗌 Δ1 Σ ,Σ Δ2 ∣ Σ3 , for 𝑖, 𝑗 ∈ {1, 2}, 𝑗 ≠ 𝑖, and any 𝐴, 𝐵 ∈ (Σ𝑖 ), 𝐷 ∈ (Σ𝑗 ), and a complete conjunction 1 2 𝐸 ∈ (Σ3 ), such that 𝐷𝐸 ̸|∼ Δ ⊥ we have 𝐴𝐸 ∣∼ Δ 𝐵

iff

𝐴𝐸𝐷 ∣∼ Δ 𝐵.

Thus, an inductive inference operator satisfies (CIndg ) if, for any Δ that safely splits into Δ1 and Δ2 conditional on Σ3 , whenever we have all the necessary information about Σ3 , inferences from one sub-language are independent from formulas over the other sub-language. Analogously to conditional syntax splitting (Heyninck et al., 2023), generalized conditional syntax splitting (CSynSplitg ) is the combination of the two properties (CIndg ) and (CRelg ). (CSynSplitg ) An inductive inference operator 𝐂 satisfies generalized conditional syntax splitting if it satisfies (CRelg ) and (CIndg ). The difference between (CSynSplit) and our new variant (CSynSplitg ) is that (CSynSplit) is defined regarding safe conditional syntax splittings only, while our adjusted variant (CSynSplitg ) takes into account all generalized safe conditional syntax splittings. Thus, an inductive inference operator satisfying (CSynSplitg ) respects an increased number of conditional splittings. L.Spiegel et al.: Preprint submitted to Elsevier

Page 8 of 34

Broadening the Applicability of Conditional Syntax Splitting

Example 11 (Δ𝑟𝑎𝑖𝑛 cont.). Recall the splitting (13) for Δ𝑟𝑎𝑖𝑛 from Example 7. Let 𝐂 ∶ Δ ↦ |∼𝑟𝑎𝑖𝑛 Δ be an inductive g g inference operator that satisfies (CSynSplit ). Applying (CRel ), we obtain that the inference 𝑠𝑟 |∼𝑟𝑎𝑖𝑛 Δ 𝑏 holds iff the inference 𝑠𝑟 |∼𝑟𝑎𝑖𝑛 𝑏 holds. Thus, if we want to know whether the inference holds in Δ, it is sufficient to consider only Δ1 , Δ1 reducing the number of conditionals we need to take into account from 6 to 3 and the number of signature elements from 𝑟𝑎𝑖𝑛 5 to 3. By applying (CIndg ), we additionally know that the inferences 𝑠𝑟𝑜 |∼𝑟𝑎𝑖𝑛 Δ 𝑏 and 𝑠𝑟𝑜 |∼Δ 𝑏 hold if the inference 𝑠𝑟 |∼𝑟𝑎𝑖𝑛 Δ 𝑏 holds. In this way, we can localize our reasoning tasks for the entire belief base to a smaller subbase, given that our reasoning mechanism satisfies (CSynSplitg ). Because (CSynSplitg ) takes into account strictly more splittings than (CSynSplit), (CSynSplitg ) poses a harder requirement for an inductive inference operator to satisfy than (CSynSplit). This means that there are inductive inference operators that satisfy (CSynSplit), but do not satisfy (CSynSplitg ). We will formally prove this observation later in Section 7. For governing inductive inference, we are only interested in generalized safe or safe conditional syntax splittings. Lemma 7 shows that there exist belief bases that have safe conditional syntax splittings, but Δ1 is a subset of Δ2 or vice versa and therefore, even though the splitting is safe, it does not provide any meaningful information for inductive inference. In fact, given some Σ3 ⊆ Σ, every belief base has at least one syntax splitting conditional on Σ3 . Proposition 12. Let Δ be a belief base over a signature Σ. For every Σ3 ⊆ Σ, there exists the conditional syntax splitting ⋃ Δ=Δ (Δ ∩ ((Σ3 )|(Σ3 ))) ∣ Σ3 . (16) Σ⧵Σ3 ,∅

⋃ Proof. Let Δ be a belief base. Consider, for any Σ3 , the splitting Δ = Δ Σ⧵Σ3 ,∅ (Δ∩((Σ3 )|(Σ3 ))) ∣ Σ3 . This splitting is a conditional syntax splitting of Δ, because Δ ∩ (((Σ ⧵ Σ3 ) ∪ Σ3 )|((Σ ⧵ Σ3 ) ∪ Σ3 )) = Δ and Δ ∩ ((Σ3 )|(Σ3 )) = Δ3 with Σ = (Σ ⧵ Σ3 ) ∪ Σ3 and additionally Σ ⧵ Σ3 and Σ3 being disjoint. Thus, for every Σ3 ⊆ Σ, there exists a syntax splitting conditional on Σ3 . However, as we have elaborated at the end of Sect. 3, the conditional syntax splitting postulates can not be meaningfully applied to splittings of form (16) even if they are generalized safe. We now identify splittings that are meaningful with respect to inductive inference as so-called genuine splittings. Definition 13 (genuine splitting). Let Δ be a belief base over a signature Σ. A conditional syntax splitting Δ = ⋃ Δ1 Σ1 ,Σ2 Δ2 ∣ Σ3 of Δ is called genuine, if Δ1 ⊈ Δ2 and Δ2 ⊈ Δ1 . Note that genuine splittings can be equivalently characterized by Δ1⧵3 ≠ ∅ and Δ2⧵3 ≠ ∅. Intuitively, we call a splitting genuine if each subbase contains information that can not be found in the other subbase. Thus, genuine splittings help us determine those splittings where conditional syntax splitting postulates actually yield advantages for inductive inference. We call a splitting that is not genuine a simple splitting. Especially, the following types of splittings are simple. ⋃ ⋃𝗀𝗌 ⟨trivial⟩ (i) Δ = Δ 𝗌Σ,∅ ∅ ∣ ∅ or (ii) Δ = Δ ∅,∅ Δ ∣ Σ. ⟨set-empty⟩ Δ1 = ∅ or Δ2 = ∅. ⟨sig-empty⟩ Σ1 = ∅ or Σ2 = ∅. While the ⟨trivial⟩ splitting (i) is safe, the splitting (ii) is not safe, unless Δ contains self-fulfilling conditionals only. However, splitting (ii) is generalized safe. Furthermore, in the case that Σ contains no elements that do not appear in Δ, the simple splittings are exactly the ⟨sig-empty⟩ splittings. If Δ3 = ∅, then the simple splittings are exactly the ⟨set-empty⟩ splittings. Genuine splittings usually make out only a small subset of all conditional syntax splittings (see Proposition 12) and thus greatly reduce the number of splittings that need to be considered for generalized safety. We give an example to illustrate the importance of identifying genuine splittings. Example 14 (Δ𝑟𝑎𝑖𝑛 cont.). We continue Lemma 7. The belief base Δ𝑟𝑎𝑖𝑛 has a total of 37 conditional syntax splittings, out of which 32 are generalized safe splittings, but only 16 are safe splittings. Only 5 of the 37 splittings are genuine. L.Spiegel et al.: Preprint submitted to Elsevier

Page 9 of 34

Broadening the Applicability of Conditional Syntax Splitting

For this belief base all genuine splittings are generalized safe, while no safe splitting is genuine. The five genuine generalized safe splittings of Δ𝑟𝑎𝑖𝑛 are listed here. For a full list of all conditional syntax splittings we refer to the appendix. 𝗀𝗌 ⋃

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)}

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟}

{𝑏},{𝑜,𝑢}

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)}

𝗀𝗌 ⋃

{(𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑟, 𝑜}

{𝑏,𝑠},{𝑢} 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)}

⋃

{(𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑏, 𝑟, 𝑜}

{𝑠},{𝑢} 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)}

⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟, 𝑜}

{𝑏},{𝑢}

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)}

𝗀𝗌 ⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟, 𝑢}

{𝑏},{𝑜}

While in Example 14 the set of genuine and generalized safe splittings coincide, this does not hold in general. Lemma 14 shows that there exist belief bases for which no genuine, safe conditional syntax splitting exists, but a genuine, generalized safe splitting exists. Indeed, conditional syntax splittings that are both genuine and generalized safe are those splittings, where the properties of conditional relevance and conditional independence for inductive inference may be meaningfully applied.

5. Evaluating Inductive Inference Operators with respect to (CSynSplitg ) In this section we consider several inductive inference operators and evaluate whether they satisfy (CSynSplitg ).

5.1. System Z

System Z is an OCF-based inductive inference operator based on the ranking function 𝜅 𝑧 (Pearl, 1990). The definition of 𝜅 𝑧 crucially relies on the notion of tolerance. A conditional (𝐵|𝐴) is tolerated by a set of conditionals ⋀ Δ = {(𝐵1 |𝐴1 ), … , (𝐵𝑛 |𝐴𝑛 )} if there is a world 𝜔 ∈ Ω such that 𝜔 ⊧ 𝐴𝐵 and 𝜔 ⊧ 𝑛𝑖=1 (𝐴𝑖 ∨ 𝐵𝑖 ), i.e., iff 𝜔 verifies (𝐵|𝐴) and does not falsify any conditional in Δ. For every consistent knowledge base, the notion of tolerance yields a unique inclusion-maximal ordered partition, in the following denoted by 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ), of Δ where each ⋃ ⋃ Δ𝑖 is the (with respect to set inclusion) maximal subset of 𝑘𝑗=𝑖 Δ𝑗 that is tolerated by 𝑘𝑗=𝑖 Δ𝑗 . Intuitively, general conditionals of Δ are placed in the first sets of 𝑂𝑃 (Δ) while more specific conditionals are placed in later parts of the partition. The system Z ranking function 𝜅 𝑧 (𝜔) is defined as follows. If 𝜔 does not falsify any conditional in Δ, then let 𝜅 𝑧 (𝜔) = 0. Otherwise, let Δ𝑗 be the latest part in the tolerance partition containing a conditional falsified by 𝜔, and let 𝜅 𝑧 (𝜔) = 𝑗 + 1 (Goldszmidt and Pearl, 1996). System Z yields the inference relation induced by 𝜅 𝑧 , i.e., 𝐂𝐳 ∶ Δ ↦ |∼ 𝜅 𝑧 . Just like safe splittings, generalized safe splittings are respected by the notion of tolerance. ⋃𝗀𝗌 Proposition 15. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . Then, for any 𝑖 ∈ {1, 2}, Δ𝑖 tolerates (𝐵|𝐴) ∈ Δ𝑖 iff Δ tolerates (𝐵|𝐴). 1

Proof. First, assume (𝐵|𝐴) is tolerated by Δ𝑖 . Then there must be some 𝜔𝑖 𝜔3 such that 𝜔𝑖 𝜔3 ⊧ 𝐴𝐵 and there is no (𝐷|𝐶) ∈ Δ𝑖 such that 𝜔𝑖 𝜔3 ⊧ 𝐶𝐷. In particular, there is no such conditional in Δ3 . Due to generalized safety there is ′ ′ an extension 𝜔𝑖 such that there is no conditional (𝐹 |𝐸) ∈ Δ𝑖′ ⧵3 with 𝜔𝑖 𝜔3 𝜔𝑖 ⊧ 𝐸𝐹 . Since Δ = Δ𝑖 ∪ Δ𝑖′ ⧵3 , (𝐵|𝐴) is tolerated by Δ. The other direction is immediate. We also state the following proposition which extends Proposition 15 to the conditionals in Δ3 specifically. ⋃𝗀𝗌 Proposition 16. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . Then Δ3 tolerates (𝐵|𝐴) ∈ Δ3 iff Δ1 and Δ2 tolerate (𝐵|𝐴). 1

L.Spiegel et al.: Preprint submitted to Elsevier

Page 10 of 34

Broadening the Applicability of Conditional Syntax Splitting

Proof. Assume (𝐵|𝐴) is tolerated by Δ3 . Then there must be some 𝜔3 such that 𝜔3 ⊧ 𝐴𝐵 and there is no (𝐷|𝐶) ∈ Δ3 such that 𝜔3 ⊧ 𝐶𝐷. Due to the generalized safety property there are then extensions 𝜔1 , 𝜔2 of 𝜔3 such that 𝜔3 𝜔1 𝜔2 does not falsify any conditional in Δ ⧵ Δ3 . Thus, (𝐵|𝐴) is also tolerated in both Δ1 and Δ2 and also in Δ. Because Δ3 = Δ1 ∩ Δ2 the other direction is immediate. From Proposition 15 and Proposition 16 we can conclude the following lemma regarding the tolerance partition. ⋃𝗀𝗌 Proposition 17. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 and 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ). Let 𝑂𝑃 (Δ1 ) = (Δ01 , … , Δ𝑛1 ), 𝑂𝑃 (Δ2 ) = 1

(Δ02 , … , Δ𝑚 ), and 𝑂𝑃 (Δ3 ) = (Δ03 , … , Δ𝑝3 ). Let 𝑞 = min{𝑛, 𝑚}; furthermore, if 𝑞 = 𝑛 let 𝑖 = 1 and otherwise let 𝑖 = 2. 2 Then, for 𝑙 ∈ {0, … , 𝑞}, it holds that 1. (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙 implies (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙1 or (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙2 or both, 2. (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙𝑖 implies (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙 , and 3. (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙3 iff (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙1 and (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙2 .

For 𝑙 ∈ {𝑞 + 1, … , 𝑘} it holds that (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙 iff (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙𝑖 . Proof. For 𝑙 = 0 the lemma holds due to Proposition 15 and Proposition 16. For 𝑙 = 1, observe that Δ ⧵ Δ0 = ⋃𝗀𝗌 0 Δ1 ⧵ Δ1 Σ ,Σ Δ2 ⧵ Δ02 ∣ Σ3 and the lemma follows from Proposition 15 and Proposition 16 again. Note that 1

Proposition 15 also implies 0 < 𝑙 ⩽ 𝑘 we can derive that Δ ⧵ (Δ0 ∪ ⋯ ∪ Δ𝑙−1 ) = ⋃𝗀𝗌 max{𝑛, 𝑚}0 = 𝑘. For 𝑙−1 𝑙−1 0 Δ1 ⧵ (Δ1 ∪ ⋯ ∪ Δ1 ) Σ ,Σ Δ2 ⧵ (Δ2 ∪ ⋯ ∪ Δ2 ) ∣ Σ3 and again the lemma follows from Proposition 15 and 1 2 Proposition 16. For System Z we can then show the following result. Proposition 18. System Z satisfies (CRelg ), but does not satisfy (CIndg ) and thus does not satisfy (CSynSplitg ). ⋃𝗀𝗌 Proof. Let Δ be a belief base with 𝑂𝑃 (Δ) = (Δ0 , Δ1 , … , Δ𝑛 ). Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 be some generalized safe 1

conditional syntax splitting for Δ and 𝑂𝑃 (Δ𝑖 ) = (Δ0𝑖 , Δ1𝑖 , … , Δ𝑛𝑖 ) for 𝑖 ∈ {1, 2} the tolerance partition of subbase Δ𝑖 . 𝑚 With Lemma 17 we have that Δ𝑚 𝑖 ⊆ Δ for 𝑚 ∈ {0, 1, … , 𝑛}, because removing conditionals from Δ can not undo the generalized safety property. Due to the generalized safety property, every world in Ω(Σ𝑖 ∪ Σ3 ) has an extension in Ω(Σ𝑗 ) such that no conditional in Δ𝑖′ ⧵3 is falsified. Given that (CRelg ) only considers formulas over Σ𝑖 ∪ Σ3 , we have, for any minimal world 𝜔 satisfying any of these formulas, that 𝜔 falsifies a conditional in Δ𝑘 , iff 𝜔 falsifies a conditional in Δ𝑘𝑖 . Thus System Z satisfies (CRelg ). Because System Z suffers from the drowning problem it does not satisfy (CInd) (Heyninck et al., 2023) and thus does not satisfy (CIndg ) and therefore not (CSynSplitg ). Thus System Z does not comply with (CSynSplitg ), but does satisfy (CRelg ). In the following, we will look at two inference operators extending System Z.

5.2. Lexicographic Inference

Lexicographic inference (Lehmann, 1995) is another inductive inference operator based on the tolerance partition 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ). It extends System Z by taking into account also the number of falsified conditionals per partition. For the definition of lexicographic inference, we use the following functions 𝜉 𝑙 and 𝜉 which map worlds to the set of falsified conditionals from the set Δ𝑙 in the tolerance partition and from Δ, respectively, given by 𝜉Δ𝑙 (𝜔) ∶= {(𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙 ∣ 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 },

(17)

𝜉Δ (𝜔) ∶= {(𝐵𝑗 |𝐴𝑗 ) ∈ Δ ∣ 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 }.

(18)

Additionally we will use the lexicographic ordering on two vectors in ℕ𝑛0 defined by (𝑣1 , … , 𝑣𝑛 ) <𝑙𝑒𝑥 (𝑤1 , … , 𝑤𝑛 ) iff there is a 𝑘 ∈ {1, … , 𝑛} such that 𝑣𝑘 < 𝑤𝑘 and 𝑣𝑗 = 𝑤𝑗 for 𝑗 = 𝑘 + 1, … , 𝑛. Utilizing these notions, we obtain the following definition of lexicographic inference.

L.Spiegel et al.: Preprint submitted to Elsevier

Page 11 of 34

Broadening the Applicability of Conditional Syntax Splitting

Definition 19 (<𝑙𝑒𝑥 , lexicographic inference (Lehmann, 1995)). The binary relation <𝑙𝑒𝑥 ⊆ Ω × Ω on worlds induced Δ Δ 0 𝑘 ′ by a belief base Δ with 𝑂𝑃 (Δ) = (Δ , … , Δ ) is defined by, for any 𝜔, 𝜔 ∈ Ω, ′ 𝜔 <𝑙𝑒𝑥 Δ 𝜔

(|𝜉Δ0 (𝜔)|, … , |𝜉Δ𝑘 (𝜔)|) <𝑙𝑒𝑥 (|𝜉Δ0 (𝜔′ )|, … , |𝜉Δ𝑘 (𝜔′ )|).

The order <𝑙𝑒𝑥 is lifted to consistent formulas by letting, for 𝐹 , 𝐺 ∈ , Δ 𝐹 <𝑙𝑒𝑥 Δ 𝐺

min <𝑙𝑒𝑥 Mod (𝐹 ) <𝑙𝑒𝑥 Δ min <𝑙𝑒𝑥 Mod (𝐺) Δ

Then, for formulas 𝐴, 𝐵, 𝐴 lexicographically entails 𝐵 given Δ, denoted as 𝐴 |∼ 𝑙𝑒𝑥 Δ 𝐵

𝐴𝐵 ≡ ⊥

𝐴𝐵, 𝐴𝐵 ≢ ⊥ and 𝐴𝐵 <𝑙𝑒𝑥 Δ 𝐴𝐵.

We illustrate lexicographic inference with an example. Example 20 (Δ𝑏 ). Let Σ = {𝑏, 𝑝, 𝑓 , 𝑤} represent birds, penguins, flying entities and winged entities, and let Δ𝑏 = {(𝑓 |𝑏), (𝑓 |𝑝), (𝑏|𝑝), (𝑤|𝑏)}. Then 𝑂𝑃 (Δ𝑏 ) = (Δ0 , Δ1 ) with Δ0 = {(𝑓 |𝑏), (𝑤|𝑏)} and Δ1 = {(𝑓 |𝑝), (𝑏|𝑝)}. From the ordering <𝑙𝑒𝑥𝑏 shown in Figure 1, we can see that min <𝑙𝑒𝑥 Mod (𝑝𝑏𝑤) = 𝑏𝑝𝑓 𝑤 <𝑙𝑒𝑥𝑏 𝑏𝑝𝑓 𝑤 = min <𝑙𝑒𝑥 Mod (𝑝𝑏𝑤) Δ

𝑤. and thus 𝑝𝑏 |∼ 𝑙𝑒𝑥 Δ𝑏

Before showing that lexicographic inference fully complies with (CSynSplitg ) we state some useful lemmata relating generalized safe conditional syntax splitting to the lexicographic ordering. The first lemma states that the set of falsified conditionals in some partition Δ𝑙 is equal to the unification of the falsified conditionals in the partitions Δ𝑙1 , Δ𝑙2 of the subbases. ⋃𝗀𝗌 Lemma 21. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 with 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ), 𝑂𝑃 (Δ1 ) = (Δ01 , … , Δ𝑛1 ) and 𝑂𝑃 (Δ2 ) = 1

(Δ02 , … , Δ𝑚 ). Then, for 𝑙 ∈ {0, … , 𝑘} and all 𝜔 ∈ Ω, we have 2 𝜉Δ𝑙 (𝜔) = 𝜉Δ𝑙 (𝜔) ∪ 𝜉Δ𝑙 (𝜔). 1

Proof. We show both set inclusions separately. “⊆”: Let (𝐵𝑗 |𝐴𝑗 ) ∈ 𝜉Δ𝑙 (𝜔). Then (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙 and 𝜔 ⊧ 𝐴𝐵. Due to Proposition 17 we then have (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙1 or

Proposition 17 we then have (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙 . Due to 𝜔 ⊧ 𝐴𝐵 this also means (𝐵𝑗 |𝐴𝑗 ) ∈ 𝜉Δ𝑙 (𝜔) and we are done. The next lemma states that the number of falsified conditionals in a partition Δ𝑙 can be calculated by summing up the number of falsified conditionals in the partitions of the subbases Δ𝑙1 and Δ𝑙2 , taking double counting for Δ3 into account. ⋃𝗀𝗌 Lemma 22. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 with 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ). Then, for 𝑙 ∈ {0, … , 𝑘} and all 𝜔 ∈ Ω, 1

(19)

|𝜉Δ𝑙 (𝜔)| = |𝜉Δ𝑙 (𝜔)| + |𝜉Δ𝑙 (𝜔)| − |𝜉Δ𝑙 (𝜔)| 1

(20)

= |𝜉Δ𝑙 (𝜔1 𝜔3 )| + |𝜉Δ𝑙 (𝜔2 𝜔3 )| − |𝜉Δ𝑙 (𝜔3 )| 1

(𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙1 ∩ Δ𝑙2 and 𝜔 ⊧ 𝐴𝐵. With Proposition 17 we have (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑙3 and thus with 𝜔 ⊧ 𝐴𝐵 we have (𝐵𝑗 |𝐴𝑗 ) ∈ 𝜉Δ𝑙 (𝜔). The other direction is analogous. 3

Page 12 of 34

Broadening the Applicability of Conditional Syntax Splitting

Finally, the next lemma states that the lexicographic ordering of worlds that coincide on the signature of one subbase is preserved in the lexicographic ordering of the entire belief base and vice versa. ⋃𝗀𝗌 ′ ′ Lemma 23. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . Let 𝜔1 , 𝜔2 ∈ Ω and let 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ . If 𝜔𝑖1 𝜔31 = 𝜔𝑖2 𝜔32 then 𝜔1 <𝑙𝑒𝑥 𝜔2 Δ 1

iff 𝜔𝑖1 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔32 . Δ

𝑖

Proof. Let Δ = Δ1

⋃𝗀𝗌

Δ ∣ Σ3 . Let 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ), 𝑂𝑃 (Δ1 ) = (Δ01 , … , Δ𝑛1 ), 𝑂𝑃 (Δ2 ) = (Δ02 , … , Δ𝑚 ) and Σ1 ,Σ2 2 2 ′ 3 ′ 3 𝑝 𝑖 𝑖 0 𝑂𝑃 (Δ3 ) = (Δ3 , … , Δ3 ). Let 𝜔1 , 𝜔2 ∈ Ω such that 𝜔1 𝜔1 = 𝜔2 𝜔2 . With Lemma 22 we have for any 𝑙 ∈ {0, … , 𝑘}: |𝜉Δ𝑙 (𝜔1 )| < |𝜉Δ𝑙 (𝜔2 )| ′

′

iff |𝜉Δ𝑙 (𝜔𝑖1 𝜔31 )| + |𝜉Δ𝑙 ′ (𝜔𝑖1 𝜔31 )| − |𝜉Δ𝑙 (𝜔31 )| < |𝜉Δ𝑙 (𝜔𝑖2 𝜔32 )| + |𝜉Δ𝑙 ′ (𝜔𝑖2 𝜔32 )| − |𝜉Δ𝑙 (𝜔32 )| 𝑖

𝑖

′

iff |𝜉Δ𝑙 (𝜔𝑖1 𝜔31 )| + |𝜉Δ𝑙 ′ (𝜔𝑖1 𝜔31 )| − |𝜉Δ𝑙 (𝜔31 )| < |𝜉Δ𝑙 (𝜔𝑖2 𝜔32 )| + |𝜉Δ𝑙 ′ (𝜔𝑖1 𝜔31 )| − |𝜉Δ𝑙 (𝜔31 )| 𝑖

𝑖

iff |𝜉Δ𝑙 (𝜔𝑖1 𝜔31 )| < |𝜉Δ𝑙 (𝜔𝑖2 𝜔32 )| 𝑖

𝑖

the definition of lexicographic inference, we obtain 𝜔1 <𝑙𝑒𝑥 𝜔2 iff 𝜔𝑖1 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔32 . Δ Δ

𝑖

Now we are ready to prove that lexicographic inference satisfies generalized conditional syntax splitting. Proposition 24. Lexicographic inference satisfies (CRelg ) and (CIndg ) and thus (CSynSplitg ). ⋃𝗀𝗌 Proof. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . Let 𝑖 ∈ {1, 2} and let 𝐴, 𝐵 ∈ (Σ𝑖 ), 𝐷 ∈ (Σ𝑖′ ), and let 𝐸 be a complete 1 2 conjunction over Σ3 . We show (CRelg ) (I) and (CIndg ) (II) separately. 𝑙𝑒𝑥 𝑙𝑒𝑥 𝑙𝑒𝑥 (I) We show (CRelg ) first. We have to show 𝐴𝐸 |∼ 𝑙𝑒𝑥 Δ 𝐵 iff 𝐴𝐸 |∼ Δ𝑖 𝐵, i.e. 𝐴𝐸𝐵 <Δ 𝐴𝐸𝐵 iff 𝐴𝐸𝐵 <Δ𝑖 𝐴𝐸𝐵. We show both directions of the iff separately. “⇒”: Let 𝐴𝐸𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐵. Let 𝜔1 be a minimal model of 𝐴𝐸𝐵, i.e. 𝜔1 ⊧ 𝐴𝐸𝐵 and there is no 𝜔′1 with 𝜔 ⊧ 𝐴𝐸𝐵 Δ

𝜔2 . Now choose and 𝜔′1 <𝑙𝑒𝑥 𝜔1 . Similarly, let 𝜔2 be a minimal model with 𝜔2 ⊧ 𝐴𝐸𝐵. Then we have 𝜔1 <𝑙𝑒𝑥 Δ Δ ′ 𝑖 𝑖 3 𝜔3 = 𝜔1 𝜔1 𝜔2 . Because 𝐸 is a full conjunction, notice that 𝜔31 = 𝜔32 . With Lemma 23 we then have 𝜔𝑖1 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔32 . Δ 𝑖

with this property because Due to (8) we have 𝜔𝑖1 𝜔31 ⊧ 𝐴𝐸𝐵 and 𝜔𝑖2 𝜔32 ⊧ 𝐴𝐸𝐵 and both worlds are minimal in <𝑙𝑒𝑥 Δ 𝑖

𝐴𝐸𝐵. 𝜔1 and 𝜔2 are minimal in <𝑙𝑒𝑥 with this property. Thus 𝐴𝐸𝐵 <𝑙𝑒𝑥 Δ Δ 𝑖

𝐴𝐸𝐵. Let 𝜔𝑖1 𝜔31 , 𝜔𝑖2 𝜔32 ∈ Ω(Σ𝑖 ∪Σ3 ) with 𝜔𝑖1 𝜔31 ⊧ 𝐴𝐸𝐵 and 𝜔𝑖2 𝜔32 ⊧ 𝐴𝐸𝐵 be minimal in <𝑙𝑒𝑥 “⇐”: Let 𝐴𝐸𝐵 <𝑙𝑒𝑥 Δ Δ 𝑖

𝑖

′

with this property. Now choose some 𝜔∗ such that 𝜔3∗ 𝜔𝑖∗ falsifies no conditional outside Δ𝑖 and such that 𝜔3∗ = 𝜔31 = 𝜔32 . ′ ′ Such an 𝜔∗ must exist due to the generalized safety property. Now set 𝜔1 = 𝜔𝑖1 𝜔𝑖∗ 𝜔3∗ and 𝜔2 = 𝜔𝑖2 𝜔𝑖∗ 𝜔3∗ . Note that ′ the falsification of conditionals outside Δ𝑖 is only dependent on the 𝜔𝑖 𝜔3 part of any world 𝜔 and thus neither 𝜔1 nor 𝜔2 falsify any conditionals outside Δ𝑖 . With Lemma 23 we have 𝜔1 <𝑙𝑒𝑥 𝜔2 and because neither world falsifies a Δ 𝑙𝑒𝑥 conditional outside of Δ𝑖 their minimality in <Δ follows from the minimality of 𝜔𝑖1 𝜔31 and 𝜔𝑖2 𝜔32 in <𝑙𝑒𝑥 . Δ 𝑖

(II) We show (CIndg ) next. We have to show 𝐴𝐸 |∼ 𝑙𝑒𝑥 𝐵 iff 𝐴𝐸𝐷 |∼ 𝑙𝑒𝑥 𝐵, i.e., 𝐴𝐸𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐵 iff Δ Δ Δ 𝑙𝑒𝑥 𝐴𝐸𝐷𝐵 <Δ 𝐴𝐸𝐷𝐵. “⇒”: Assume 𝐴𝐸𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐵. Note that 𝐴𝐸𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐷𝐵. Now choose some 𝜔1 , 𝜔2 with 𝜔1 ⊧ 𝐴𝐸𝐵 and Δ Δ 𝑙𝑒𝑥 𝜔2 ⊧ 𝐴𝐸𝐷𝐵 and such that 𝜔1 and 𝜔2 are minimal in <Δ with this property. ′ Note that 𝜔31 = 𝜔32 because 𝐸 is a complete conjunction. Now choose 𝜔∗ = 𝜔𝑖2 𝜔𝑖1 𝜔31 . Then 𝜔∗ ⊧ 𝐴𝐸𝐵 and 𝜔1 <𝑙𝑒𝑥 𝜔∗ per assumption. With Lemma 23 we have 𝜔𝑖1 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔31 . Again with Lemma 23 we have Δ Δ ′

′

𝑖

′

𝜔𝑖1 𝜔𝑖2 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔𝑖2 𝜔31 . Observe that 𝜔𝑖1 𝜔𝑖2 𝜔31 ⊧ 𝐴𝐸𝐷𝐵. Because 𝜔1 = 𝜔𝑖2 𝜔𝑖2 𝜔31 is a minimal model of 𝐴𝐸𝐷𝐵 Δ

we have shown 𝐴𝐸𝐷𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐷𝐵. Δ “⇐”: Assume 𝐴𝐸𝐷𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐷𝐵. Choose 𝜔1 , 𝜔2 such that 𝜔1 ⊧ 𝐴𝐸𝐷𝐵 and 𝜔2 ⊧ 𝐴𝐸𝐵 and that both are Δ ′ 𝑙𝑒𝑥 minimal in <Δ with this property. Observe again 𝜔31 = 𝜔32 . Now choose 𝜔∗ = 𝜔𝑖2 𝜔𝑖1 𝜔31 . Then 𝜔∗ ⊧ 𝐴𝐸𝐷𝐵 L.Spiegel et al.: Preprint submitted to Elsevier

Page 13 of 34

Broadening the Applicability of Conditional Syntax Splitting

and 𝜔1 <𝑙𝑒𝑥 𝜔∗ per assumption. With Lemma 23 we have 𝜔𝑖1 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔31 . Again with Lemma 23 we have Δ Δ ′

′

𝑖

′

𝜔𝑖1 𝜔𝑖2 𝜔31 <𝑙𝑒𝑥 𝜔𝑖2 𝜔𝑖2 𝜔31 . Observe that 𝜔𝑖1 𝜔𝑖2 𝜔31 ⊧ 𝐴𝐸𝐵 and that 𝜔2 = 𝜔𝑖2 𝜔𝑖2 𝜔31 is a minimal model of 𝐴𝐸𝐵 per Δ assumption. Thus we have shown 𝐴𝐸𝐵 <𝑙𝑒𝑥 𝐴𝐸𝐵. Δ

Hence, lexicographic inference fully complies with (CSynSplitg ). In the next section, we evaluate another inference operator employing the tolerance partition with respect to generalized conditional syntax splitting.

5.3. System W

System W (Komo and Beierle, 2020, 2022) is an inference operator also using the tolerance partition 𝑂𝑃 (Δ). While System Z considers only which parts of 𝑂𝑃 (Δ) contain falsified conditionals, and lexicographic inference only considers the number of conditionals falsified, System W also takes into account the structural information about which conditionals are falsified. The results and proofs we give in this subsection are based on results and proofs published in the PhD Dissertation (Haldimann, 2024) for safe conditional syntax splittings. Here, we adapt these results and proofs to generalized safe conditional syntax splittings. System W utilizes the preferred structure on worlds <𝗐 which compares worlds according to the set of conditionals Δ in Δ they falsify, giving preference to the more specific conditionals according to 𝑂𝑃 (Δ). Definition 25 (preferred structure <𝗐 on worlds (Komo and Beierle, 2022)). Let Δ be a belief base with 𝑂𝑃 (Δ) = Δ (Δ0 , … , Δ𝑘 ). The preferred structure on worlds is given by the binary relation <𝗐 ⊆ Ω × Ω defined by, for any Δ 𝜔, 𝜔′ ∈ Ω, ′ 𝜔 <𝗐 Δ 𝜔

iff

there exists 𝑙 ∈ {0 , … , 𝑘} such that 𝜉Δ𝑚 (𝜔) = 𝜉Δ𝑚 (𝜔′ )

∀𝑚 ∈ {𝑙 + 1 , … , 𝑘}, and (21)

𝜉Δ𝑙 (𝜔) ⊊ 𝜉Δ𝑙 (𝜔′ ) .

Thus, 𝜔 <𝗐 𝜔′ if and only if 𝜔 falsifies a strict subset of the conditionals that 𝜔′ falsifies in the partition with the Δ largest index 𝑙 where the conditionals falsified by 𝜔 and 𝜔′ differ. Definition 26 (System W, |∼𝗐 Δ (Komo and Beierle, 2022)). Let Δ be a belief base and 𝐴, 𝐵 be formulas. Then 𝐵 is a System W inference from 𝐴 (in the context of Δ), denoted 𝐴 |∼𝗐 Δ 𝐵, if we have: 𝐴 |∼𝗐 Δ𝐵

iff

𝜔′ for every 𝜔′ ∈ Ω with 𝜔′ ⊧ 𝐴𝐵 there is an 𝜔 ∈ Ω with 𝜔 ⊧ 𝐴𝐵 s.t. 𝜔 <𝗐 Δ

(22)

We illustrate the above definitions with an example. Example 27 (Δ𝑏 cont.). Consider again 𝑂𝑃 (Δ𝑏 ) = (Δ0 , Δ1 ) with Δ0 = {(𝑓 |𝑏), (𝑤|𝑏)} and Δ1 = {(𝑓 |𝑝), (𝑏|𝑝)}. Utilizing <w 𝑏 , shown in Figure 2, we can see that 𝑝𝑏 |∼𝗐 𝑤 holds, because for every 𝜔′ ∈ Ω with 𝜔 ⊧ 𝑝𝑏𝑤 there is the Δ𝑏 Δ

world 𝜔 = 𝑏𝑝𝑓 𝑤 such that 𝜔 <𝗐𝑏 𝜔′ . Δ

Before showing that System W fully complies with (CSynSplitg ), we show some lemmata regarding generalized safe conditional syntax splitting and the preferred structure on worlds first. The first lemma states that the order between two worlds must be reflected by the order of one of the subbases. ⋃𝗀𝗌 Lemma 28. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 and let 𝜔, 𝜔′ ∈ Ω. If 𝜔 <𝗐 𝜔′ then 𝜔 <𝗐 𝜔′ or 𝜔 <𝗐 𝜔′ . Δ Δ Δ 1

Proof. Let 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ). Let 𝜔, 𝜔′ ∈ Ω with 𝜔 <𝗐 𝜔′ . By definition there is 𝑛 ∈ {0, … , 𝑘} such that Δ 𝜉Δ𝑗 (𝜔) = 𝜉Δ𝑗 (𝜔′ ) for all 𝑗 ∈ {𝑛 + 1, … , 𝑘} and 𝜉Δ𝑛 (𝜔) ⊊ 𝜉Δ𝑛 (𝜔′ ). Then there is 𝛿 ∈ 𝜉Δ𝑛 (𝜔′ ) with 𝛿 ∉ 𝜉Δ𝑛 (𝜔). We have 𝛿 ∈ Δ, implying 𝛿 ∈ Δ1 or 𝛿 ∈ Δ2 . Let 𝑖 ∈ {1, 2} such that 𝛿 ∈ Δ𝑖 and let 𝑂𝑃 (Δ𝑖 ) = (Δ1𝑖 , … , Δ𝑚 𝑖 ). With 𝑗 𝑗 𝑗 𝑗 𝑗 𝑖 ′ ′ Proposition 17 we have that Δ𝑖 ⊆ Δ . Thus due to 𝜉Δ (𝜔) = 𝜉Δ (𝜔 ) we also have 𝜉Δ (𝜔) = 𝜉Δ (𝜔 ) and because 𝑖 𝑖 𝜉Δ𝑛 (𝜔) ⊊ 𝜉Δ𝑛 (𝜔′ ) and 𝛿 ∈ Δ𝑖 we have 𝜉Δ𝑛 (𝜔) ⊊ 𝜉Δ𝑛 (𝜔′ ). Thus 𝜔 <𝗐 𝜔′ . Δ 𝑖 𝑖 𝑖 The next lemma states that the order between two worlds in the preferred structure of Δ𝑖 is preserved in the preferred structure of Δ if the two worlds coincide on the signature of Δ𝑖′ . L.Spiegel et al.: Preprint submitted to Elsevier

Page 14 of 34

Broadening the Applicability of Conditional Syntax Splitting

⋃𝗀𝗌 Lemma 29. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 , let 𝜔, 𝜔′ ∈ Ω and let 𝑖, 𝑖′ ∈ {1, 2} with 𝑖 ≠ 𝑖′ . If 𝜔 <𝗐 𝜔′ and Δ𝑖 1 2 𝗐 ′ ′ 𝜔∣Σ𝑖′ ∪Σ3 = 𝜔∣Σ ′ ∪Σ then 𝜔 <Δ 𝜔 . 3

𝑖

Proof. Let 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ , and let 𝑂𝑃 (Δ) = (Δ0 , … , Δ𝑘 ). Let 𝜔, 𝜔′ ∈ Ω with 𝜔 <𝗐 𝜔′ and 𝜔∣Σ𝑖′ ∪Σ3 = 𝜔′∣Σ ′ ∪Σ . Δ𝑖 3 𝑖 𝑗 𝑗 ′ Let 𝑂𝑃 (Δ𝑖 ) = (Δ0𝑖 , … , Δ𝑚 𝑖 ). Per definition there is 𝑛 ∈ {0, … , 𝑚} such that 𝜉Δ𝑖 (𝜔) = 𝜉Δ𝑖 (𝜔 ) for 𝑗 ∈ {𝑛+1, … , 𝑚} and 𝜉Δ𝑛 (𝜔) ⊊ 𝜉Δ𝑛 (𝜔′ ). With Lemma 21 we have 𝜉Δ𝑙 (𝜔∗ ) = 𝜉Δ𝑙 (𝜔∗ ) ∪ 𝜉Δ𝑘 ′ (𝜔∗ ) for all worlds 𝜔∗ and 𝑙 ∈ {0, … , 𝑚}. Because 𝑖 𝑖 𝑖 𝑖 𝜔∣Σ𝑖′ ∪Σ3 = 𝜔′∣Σ ′ ∪Σ we have 𝜉Δ𝑖′ (𝜔) = 𝜉Δ𝑖′ (𝜔′ ). Then 𝜉Δ𝑗 (𝜔) = 𝜉Δ𝑗 (𝜔′ ) implies 𝜉Δ𝑗 (𝜔) = 𝜉Δ𝑗 (𝜔′ ) and 𝜉Δ𝑗 (𝜔) ⊊ 𝜉Δ𝑗 (𝜔′ ) 𝑖 𝑖 𝑖 𝑖 3 𝑖 ′ implies 𝜔 <𝗐 𝜔′ . implies 𝜉Δ𝑗 (𝜔) ⊊ 𝜉Δ𝑗 (𝜔′ ), and thus 𝜔 <𝗐 𝜔 Δ𝑖 Δ Finally, the next lemma shows that the order of worlds with respect to a subbase is only dependent on the signature relevant to that subbase. ⋃𝗀𝗌 Lemma 30. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 , let 𝜔, 𝜔′ , 𝜔∗ ∈ Ω and let 𝑖 ∈ {1, 2} with 𝜔∣Σ𝑖 ∪Σ3 = 𝜔′∣Σ ∪Σ . Then we have 1 2 𝑖 3 𝜔 <𝗐 𝜔∗ iff 𝜔′ <𝗐 𝜔∗ and 𝜔∗ <𝗐 𝜔 iff 𝜔∗ <𝗐 𝜔′ . Δ Δ Δ Δ 𝑖

𝑖

is defined solely Proof. Let 𝜔, 𝜔′ , 𝜔∗ ∈ Ω as above. Then 𝜔 and 𝜔′ falsify the same conditionals in Δ𝑖 . Because <𝗐 Δ𝑖 based on the falsification of conditionals in Δ𝑖 the lemma follows.

Now we are ready to show that system W fully complies with generalized conditional syntax splitting. Proposition 31. System W satisfies (CRelg ) and (CIndg ) and thus (CSynSplitg ). Proof. Let 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ , and let 𝐴, 𝐵 ∈ (Σ𝑖 ), 𝐷 ∈ (Σ𝑖′ ), and let 𝐸 be a complete conjunction over Σ3 . We show (CRelg ) (I) and (CIndg ) (II) separately. (I) We show (CRelg ) first. We need to show 𝐴𝐸 |∼𝗐 Δ𝐵

iff 𝐴𝐸 |∼𝗐 Δ 𝐵. 𝑖

Thus, we need to show that for every 𝜔′ ∈ Mod Σ (𝐴𝐸𝐵) there is an 𝜔 ∈ Mod Σ (𝐴𝐸𝐵) with 𝜔 <𝗐 𝜔′ iff for every Δ 𝗐 ′ ′ 𝜔 ∈ Mod Σ (𝐴𝐸𝐵) there is an 𝜔 ∈ Mod Σ (𝐴𝐸𝐵) with 𝜔 <Δ 𝜔 . 𝑖

𝗐 𝗐 ′ "⇒": Assume that 𝐴𝐸 |∼𝗐 Δ 𝐵, i.e., 𝐴𝐸𝐵 <Δ 𝐴𝐸𝐵. We need to show 𝐴𝐸𝐵 <Δ 𝐴𝐸𝐵. Let 𝜔 be any world with 𝑖

𝜔′ ⊧ 𝐴𝐸𝐵. Choose 𝜔′𝑚𝑖𝑛 such that ′ 𝜔′𝑚𝑖𝑛 ⩽𝗐 Δ 𝜔 ,

(23)

𝜔 | Σ ∪Σ = 𝜔′𝑚𝑖𝑛 | Σ ∪Σ , and 𝑖 3 𝑖 3 ′ there is no 𝜔′𝑚𝑖𝑛2 with 𝜔′𝑚𝑖𝑛2 <𝗐 Δ 𝜔𝑚𝑖𝑛 fulfilling (23) and (24).

(24)

′

(25)

Such an 𝜔′𝑚𝑖𝑛 exists because 𝜔′ exists. With (24) and 𝜔′ ⊧ 𝐴𝐸𝐵 we have 𝜔′𝑚𝑖𝑛 ⊧ 𝐴𝐸𝐵. Then there must be 𝜔 with 𝜔′𝑚𝑖𝑛 or 𝜔 <𝗐 𝜔′𝑚𝑖𝑛 . The second case is not possible, if it 𝜔 ⊧ 𝐴𝐸𝐵 and 𝜔 <𝗐 𝜔′ . With Lemma 28 either 𝜔 <𝗐 Δ′ Δ 𝑚𝑖𝑛 Δ 𝑖

𝑖

𝜔′ . But with Lemma 29 and were the case that 𝜔 <𝗐 𝜔′ then Lemma 30 implies 𝜔′𝑚𝑖𝑛2 = (𝜔′𝑚𝑖𝑛 | Σ 𝜔| Σ ′ ∪Σ ) <𝗐 Δ ′ 𝑚𝑖𝑛 Δ ′ 𝑚𝑖𝑛 𝑖

𝑖

𝜔′ which contradicts (25). Therefore 𝜔 <𝗐 𝜔′𝑚𝑖𝑛 . the fact that 𝜔′𝑚𝑖𝑛2 | Σ ∪Σ = 𝜔′𝑚𝑖𝑛 | Σ ∪Σ we would have 𝜔′𝑚𝑖𝑛2 <𝗐 Δ 𝑚𝑖𝑛 Δ 𝑖

𝑖

Then with (24) and Lemma 30 we get 𝜔 <𝗐 𝜔′ and thus we have 𝐴𝐸𝐵 <𝗐 𝐴𝐸𝐵. Δ Δ 𝑖

𝑖

𝗐 𝗐 ′ "⇐": Assume that 𝐴𝐸 |∼𝗐 Δ 𝐵, i.e., 𝐴𝐸𝐵 <Δ 𝐴𝐸𝐵. We need to show 𝐴𝐸𝐵 <Δ 𝐴𝐸𝐵. Let 𝜔 be any world with 𝑖

𝑖

𝜔′ ⊧ 𝐴𝐸𝐵. 𝜔∗ with 𝜔∗ ⊧ 𝐴𝐸𝐵 and 𝜔∗ <𝗐 𝜔′ . Let 𝜔 = (𝜔∗ | Σ ∪Σ 𝜔′ | Σ ′ ). Then 𝜔 ⊧ 𝐴𝐸𝐵. With Lemma 30 and the Δ 𝑖

𝑖

fact that 𝜔| Σ ∪Σ = 𝜔∗ | Σ ∪Σ we have 𝜔 <𝗐 𝜔′ . With 𝜔| Σ ′ ∪Σ = 𝜔′ | Σ ′ ∪Σ and with Lemma 29 we get 𝜔 <𝗐 𝜔′ and Δ Δ 𝑖

𝑖

therefore 𝐴𝐸𝐵 <𝗐 𝐴𝐸𝐵. Δ (II) Next we show (CIndg ). We need to show 𝐴𝐸 |∼𝗐 Δ𝐵

iff 𝐴𝐸𝐷 |∼𝗐 Δ 𝐵.

L.Spiegel et al.: Preprint submitted to Elsevier

Page 15 of 34

Broadening the Applicability of Conditional Syntax Splitting

Thus, we need to show that for every 𝜔′ ∈ Mod Σ (𝐴𝐸𝐵) there is an 𝜔 ∈ Mod Σ (𝐴𝐸𝐵) with 𝜔 <𝗐 𝜔′ iff for every Δ 𝗐 ′ ′ 𝜔 ∈ Mod Σ (𝐴𝐸𝐷𝐵) there is an 𝜔 ∈ Mod Σ (𝐴𝐸𝐷𝐵) with 𝜔 <Δ 𝜔 . 𝗐 𝗐 ′ "⇒: Assume that 𝐴𝐸 |∼𝗐 Δ 𝐵, i.e., 𝐴𝐸𝐵 <Δ 𝐴𝐸𝐵. We need to show 𝐴𝐸𝐷𝐵 <Δ 𝐴𝐸𝐷𝐵. Let 𝜔 be any world with 𝜔′ ⊧ 𝐴𝐸𝐷𝐵. Define 𝜔′𝑚𝑖𝑛 again, such that equations (23), (24) and (25) are satisfied. Because 𝜔′ ⊧ 𝐴𝐸𝐷𝐵 and (24) we have that 𝜔′𝑚𝑖𝑛 ⊧ 𝐴𝐸𝐵. Due to 𝐴𝐸𝐵 <𝗐 𝐴𝐸𝐵 there is 𝜔∗ with 𝜔∗ ⊧ 𝐴𝐸𝐵 and 𝜔∗ <𝗐 𝜔′𝑚𝑖𝑛 . Then, Δ Δ 𝗐 𝗐 ′ ′ ∗ ∗ with Lemma 28, either 𝜔 <Δ 𝜔𝑚𝑖𝑛 or 𝜔 <Δ ′ 𝜔𝑚𝑖𝑛 . Utilizing the same arguments as the ⇒ direction of the proof 𝑖

𝑖

for (CRelg ), 𝜔∗ <𝗐 𝜔′𝑚𝑖𝑛 is not possible due to (25). Now let 𝜔 = (𝜔∗ | Σ ∪Σ 𝜔′ | Σ ′ ). Then also 𝜔| Σ ′ ∪Σ = 𝜔′ | Σ ′ ∪Σ Δ′ 𝑖

𝑖

because 𝜔 ⊧ 𝐸 and 𝜔′ ⊧ 𝐸. With (24) and Lemmata 29 and 30 we get 𝜔 <𝗐 𝜔′ . Because 𝜔 ⊧ 𝐴𝐸𝐷𝐵 and 𝜔′ ⊧ 𝐴𝐸𝐷𝐵 Δ we are done. 𝐴𝐸𝐵. Let 𝜔′ be any world "⇐": Assume that 𝐴𝐸𝐷 <𝗐 𝐵, i.e., 𝐴𝐸𝐷𝐵 <𝗐 𝐴𝐸𝐷𝐵. We need to show 𝐴𝐸𝐵 <𝗐 Δ Δ Δ ′ ′ with 𝜔 ⊧ 𝐴𝐸𝐵. Choose 𝜔𝑚𝑖𝑛 such that 𝜔′𝑚𝑖𝑛 ⊧ 𝐷,

(26)

𝜔 | Σ ∪Σ = 𝜔′𝑚𝑖𝑛 | Σ ∪Σ , and 𝑖 3 𝑖 3 ′ there is no 𝜔′𝑚𝑖𝑛2 with 𝜔′𝑚𝑖𝑛2 <𝗐 Δ 𝜔𝑚𝑖𝑛 fulfilling (26) and (27).

(27)

′

(28)

Again such a 𝜔′𝑚𝑖𝑛 exists because 𝐶 ≢ ⊥, <𝗐 is irreflexive and transitive and Σ is finite. With (26), (27) and 𝜔′ ⊧ 𝐴𝐸𝐵 Δ we have 𝜔′𝑚𝑖𝑛 ⊧ 𝐴𝐸𝐷𝐵. Due to 𝐴𝐸𝐷𝐵 <𝗐 𝐴𝐸𝐷𝐵 there is then 𝜔∗ with 𝜔∗ ⊧ 𝐴𝐸𝐷𝐵 and 𝜔∗ <𝗐 𝜔′𝑚𝑖𝑛 . With Δ Δ 𝗐 𝗐 ′ ′ ∗ ∗ Lemma 28 again either 𝜔 <Δ 𝜔𝑚𝑖𝑛 or 𝜔 <Δ ′ 𝜔𝑚𝑖𝑛 . Utilizing the same arguments as the ⇒ direction of the proof 𝑖

𝑖

for (CRelg ), 𝜔∗ <𝗐 𝜔′𝑚𝑖𝑛 is not possible due to (28). Now let 𝜔 = (𝜔∗ | Σ ∪Σ 𝜔′ | Σ ′ ). Then also 𝜔| Σ ′ ∪Σ = 𝜔′ | Σ ′ ∪Σ Δ′ 𝑖

𝑖

because 𝜔 ⊧ 𝐸 and 𝜔′ ⊧ 𝐸. With (28) and Lemmata 29 and 30 we get 𝜔 <𝗐 𝜔′ . Because 𝜔 ⊧ 𝐴𝐸𝐵 and 𝜔′ ⊧ 𝐴𝐸𝐵 Δ we are done. Thus, while System Z satisfies (CRelg ) but not (CIndg ), both lexicographic inference and also System W fully comply with (CSynSplitg ). In the next section we will evaluate several inference operators based on a special subclass of ranking functions.

6. Evaluating Inductive Inference Operators based on c-Representations This section addresses several inductive inference operators employing c-representations (Kern-Isberner, 2001, 2004). In Sect. 6.1, we present propositions and lemmata regarding c-representations and present the notion of conditional 𝜅-independence that will be helpful for the following sections. In Sect. 6.2, we show that inference with respect to single c-representations selected by an appropriate strategy (Beierle and Kern-Isberner, 2021) satisfies generalized conditional syntax splitting. In Sect. 6.3, we show that c-core closure inference (Wilhelm et al., 2024) fully complies with generalized conditional syntax splitting. In Sect. 6.4, we show that also inference with respect to all c-representations (Beierle et al., 2018) satisfies generalized conditional syntax splitting.

6.1. c-Representations and 𝜅-Independence

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 (KernIsberner, 2001, 2004). Definition 32 (c-representation (Kern-Isberner, 2001, 2004)). A c-representation of Δ = {(𝐵1 |𝐴1 ), … , (𝐵𝑛 |𝐴𝑛 )} is an OCF 𝜅 constructed from non-negative impacts 𝜂𝑗 ∈ ℕ0 assigned to each (𝐵𝑗 |𝐴𝑗 ) such that 𝜅 accepts Δ and is given by: ∑ 𝜅(𝜔) = 𝜂𝑗 (29) 1⩽𝑗⩽𝑛 𝜔⊧𝐴𝑗 𝐵 𝑗

c-Representations can conveniently be specified using a constraint satisfaction problem (for detailed explanations, see (Kern-Isberner, 2001, 2004)): L.Spiegel et al.: Preprint submitted to Elsevier

Page 16 of 34

Broadening the Applicability of Conditional Syntax Splitting

Definition 33 (𝐶𝑅(Δ), (Kern-Isberner, 2001; Beierle et al., 2018)). The constraint satisfaction problem 𝐶𝑅(Δ) for c-representations of Δ = {(𝐵1 |𝐴1 ), … , (𝐵𝑛 |𝐴𝑛 )} is given by the conjunction of the constraints, for all 𝑗 ∈ {1, … , 𝑛}: 𝜂𝑗 ⩾ 0 𝜂𝑗 > min

∑

𝜔⊧𝐴𝑗 𝐵𝑗

𝜂𝑘 − min 𝜔⊧𝐴𝑗 𝐵 𝑗

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘

∑

(30) (31)

𝜂𝑘

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘

Note that (30) expresses that falsification of conditionals should make worlds not more plausible, and (31) ensures that 𝜅 as specified by (29) accepts Δ. A solution of 𝐶𝑅(Δ) is a vector #» 𝜂 = (𝜂1 , … , 𝜂𝑛 ) of natural numbers. 𝑆𝑜𝑙(𝐶𝑅(Δ)) denotes the set of all solutions of 𝐶𝑅(Δ). For #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) and 𝜅 as in Equation (29), 𝜅 is the OCF induced by #» 𝜂 and is denoted by 𝜅#» 𝜂 . 𝐶𝑅(Δ) is sound and complete (Kern-Isberner, 2001; Beierle et al., 2018): For every #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)), 𝜅#» is a c-representation with 𝜅#» 𝜂 𝜂 ⊧ Δ, and for every c-representation 𝜅 with 𝜅 ⊧ Δ, there is #» #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) such that 𝜅 = 𝜅#» . For an impact vector 𝜂 , we will simply write #» 𝜂 1 and #» 𝜂 2 for the corresponding 𝜂 #» #» #» #» 1 2 projections 𝜂 |Δ1 and 𝜂 |Δ2 , and ( 𝜂 , 𝜂 ) for their composition. We illustrate these notions with an example. Example 34 (Δ𝑏 cont.). For the belief base Δ𝑏 from Example 20, 𝐶𝑅(Δ𝑏 ) contains 𝜂𝑖 ⩾ 0 for 𝑖 ∈ {1, 2, 3, 4} as well as the following constraints: ∑ ∑ 𝜂𝑗 − min 𝜂𝑗 𝜂1 > min 𝜔∈ΩΣ 𝜔⊧𝑏𝑓

𝜂2 >

𝜂3 >

𝜂4 >

min

𝜔∈ΩΣ 𝜔⊧𝑝𝑓

min

𝜔∈ΩΣ 𝜔⊧𝑝𝑏

min

𝜔∈ΩΣ 𝜔⊧𝑏𝑤

𝜔∈ΩΣ 𝜔⊧𝑏𝑓

𝑗≠1 𝜔⊧𝐴𝑗 𝐵𝑗

∑

−

𝜂𝑗

𝑗≠2 𝜔⊧𝐴𝑗 𝐵𝑗

∑

−

𝜂𝑗

𝑗≠3 𝜔⊧𝐴𝑗 𝐵𝑗

∑

−

𝜂𝑗

𝑗≠4 𝜔⊧𝐴𝑗 𝐵𝑗

min

𝜔∈ΩΣ 𝜔⊧𝑝𝑓

min

𝜔∈ΩΣ 𝜔⊧𝑝𝑏

min

𝜔∈ΩΣ 𝜔⊧𝑏𝑤

𝑗≠1 𝜔⊧𝐴𝑗 𝐵𝑗

∑

𝜂𝑗

𝑗≠2 𝜔⊧𝐴𝑗 𝐵𝑗

∑

𝜂𝑗

𝑗≠3 𝜔⊧𝐴𝑗 𝐵𝑗

∑

𝜂𝑗

𝑗≠4 𝜔⊧𝐴𝑗 𝐵𝑗

Table 1 shows some solutions for Δ𝑏 as well as their corresponding induced c-representations. For example #» 𝜂1 = 𝜂 21 = (1) ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑏2⧵3 )). (1, 2, 2, 1) ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑏 )), #» 𝜂 11 = (1, 2, 2) ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑏1⧵3 )) and #» ⋃ A fundamental property of c-representations is that for any syntax splitting Δ = Δ1 Δ2 the composition of any Σ1 ,Σ2

impact vectors for the subbases yields an impact vector for Δ, and vice versa (Kern-Isberner et al., 2020). This property was also shown to extend to safe conditional syntax splittings (Beierle et al., 2024b). However, a key part in showing this was Lemma 5 which no longer holds for generalized safe conditional syntax splittings. Indeed the composition property no longer holds, as the impacts assigned to the conditionals in Δ3 can be vastly different between the two subbases. Thus, we show a slightly weaker property here. While it still states that any impact vector for Δ can be split into impact vectors for the subbases, impact vectors for the subbases may only yield an impact vector for Δ if they match on the impacts assigned to the conditionals in Δ3 . ⋃𝗀𝗌 Proposition 35. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . The following two properties hold for 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ : 1

• For every #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) there are #» 𝜇 𝑖 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖 )), #» 𝜇 𝑖′ ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖′ )) and #» 𝜇 3 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ3 )) with #» #» #» #» #» #» #» , 𝜇 ), i.e., #» 𝜂 | = #» 𝜇 and #» 𝜂 | = #» 𝜇 . 𝜇 | = 𝜇 ′ | = 𝜇 , such that 𝜂 = ( 𝜇 | , 𝜇 ′ | 𝑖 Δ3

𝑖 Δ3

𝑖 Δ𝑖⧵3

𝑖 Δ𝑖′ ⧵3

Δ𝑖

𝑖

Δ𝑖′

𝑖

Page 17 of 34

Broadening the Applicability of Conditional Syntax Splitting 𝜔

𝛿1∶ (𝑓 |𝑏)

𝛿2∶ (𝑓 |𝑝)

𝛿 3∶ (𝑏|𝑝)

𝛿4∶ (𝑤|𝑏)

impact on 𝜔

𝜅#» 𝜂1 (𝜔)

𝜅#» 𝜂2 (𝜔)

𝜅#» 𝜂3 (𝜔)

𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤

v v f f v v f f − − − − − − − −

f f v v − − − − f f v v − − − −

v v v v − − − − f f f f − − − −

v f v f v f v f − − − − − − − −

𝜂2 𝜂2 + 𝜂 4 𝜂1 𝜂1 + 𝜂 4 0 𝜂4 𝜂1 𝜂1 + 𝜂4 𝜂 2 + 𝜂3 𝜂2 + 𝜂3 𝜂3 𝜂3 0 0 0 0

2 3 1 2 0 1 1 2 4 4 2 2 0 0 0 0

4 7 3 6 0 3 3 6 8 8 4 4 0 0 0 0

5 12 4 11 0 7 4 11 11 11 6 6 0 0 0 0

impacts: #» 𝜂1 #» 𝜂2 #» 𝜂3

𝜂1 1 3 4

𝜂2 2 4 5

𝜂3 2 4 6

𝜂4 1 3 7

Table 1 Verification and falsification with induced impacts for Δ𝑏 in Example 34. The impact vectors #» 𝜂 1 , #» 𝜂 2 , and #» 𝜂 3 are solutions #» #» of 𝐶𝑅(Δ𝑏 ) and 𝜅#» , 𝜅 , 𝜅 are their induced ranking functions according to Definition 32. 𝜂1 𝜂2 𝜂3

#» 𝜇 3 as #» 𝜂 |Δ3 . Then clearly #» 𝜇 𝑖 |Δ3 = #» 𝜇 𝑖′ |Δ3 = #» 𝜇 3 . For the second property we define #» 𝜂 = ( #» 𝜇 𝑖 |Δ𝑖⧵3 , #» 𝜇 𝑖′ |Δ𝑖′ ⧵3 , #» 𝜇 3 ). Then #» #» it is sufficient to show that 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) iff 𝜇 𝑙 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑙 )), for 𝑙 ∈ {1, 2, 3} for both points of Proposition 35. We show the hard direction ⇐ first: Assume #» 𝜇 𝑙 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑙 )), for 𝑙 ∈ {1, 2, 3}, towards a contradiction, now assume #» 𝜂 ∉ 𝑆𝑜𝑙(𝐶𝑅(Δ)). Then there is some conditional (𝐵𝑗 |𝐴𝑗 ) ∈ Δ whose constraints in 𝐶𝑅(Δ) are not satisfied by #» 𝜂: 𝜂𝑗 ⩾ 0 𝜂𝑗 > min

𝜔⊧𝐴𝑗 𝐵𝑗

∑

𝜂𝑘 − min

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ

𝜔⊧𝐴𝑗 𝐵 𝑗

(32) 𝜂𝑘

(33)

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝑉𝑚𝑖𝑛

𝐹𝑚𝑖𝑛

The unsatisfied constraint must either be of form (32) or (33). If it is of form (32), then the solution vector for the subbase containing (𝐵𝑗 |𝐴𝑗 ) also falsifies this constraint, leading to a contradiction. So the violated constraint is of form (33). Assume that (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑖 , the case where (𝐵𝑗 |𝐴𝑗 ) ∉ Δ𝑖 implies (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑖′ and is then analogous. There must be a corresponding constraint ∑ ∑ 𝜇𝑘𝑖 − min 𝜇𝑘𝑖 (34) 𝜇𝑗𝑖 > min 𝜔⊧𝐴𝑗 𝐵𝑗

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

𝜔⊧𝐴𝑗 𝐵 𝑗

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝑖 𝑉𝑚𝑖𝑛

𝑖 𝐹𝑚𝑖𝑛

L.Spiegel et al.: Preprint submitted to Elsevier

Page 18 of 34

Broadening the Applicability of Conditional Syntax Splitting

in 𝐶𝑅(Δ𝑖 ). Since the impact vectors are constructed such that #» 𝜇 𝑖 |Δ3 = #» 𝜇 𝑖′ |Δ3 = #» 𝜇 3 , we have that #» 𝜂 |Δ𝑖 = #» 𝜇𝑖 = #» #» #» #» #» #» #» 𝑖 𝑖 ( 𝜇 𝑖 |Δ𝑖⧵3 , 𝜇 3 ) and 𝜂 |Δ𝑖′ = 𝜇 𝑖′ = ( 𝜇 𝑖′ |Δ𝑖′ ⧵3 , 𝜇 3 ). This way, by construction of 𝜂 , we have that 𝜂𝑗 = 𝜂𝑗 = 𝜇𝑗 for all (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑖 . Now we show that, if (33) is not satisfied, then (34) is not satisfied either. Because (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑖 , we have that (𝐵𝑗 |𝐴𝑗 ) ∈ ((Σ𝑖 ∪ Σ3 )|(Σ𝑖 ∪ Σ3 )). Thus there is some 𝜔𝑖 𝜔3 with 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 iff 𝜔𝑖 𝜔3 ⊧ 𝐴𝑗 𝐵𝑗 . Due to the generalized ′ safety property we have that any 𝜔𝑖 𝜔3 can be extended by an 𝜔𝑖 such that no conditional in Δ𝑖′ ⧵3 is falsified. This means that the worlds minimizing 𝑉𝑚𝑖𝑛 and 𝐹𝑚𝑖𝑛 do not falsify any conditional in Δ𝑖′ ⧵3 and that they falsify as few 𝑖 and 𝐹 𝑖 . Therefore, the restrictions imposed on conditionals in Δ𝑖 as possible. Thus these worlds also minimize 𝑉𝑚𝑖𝑛 𝑚𝑖𝑛 𝑖 𝑖 𝜂𝑗 and 𝜇𝑗 are the same. Since 𝜂𝑗 = 𝜇𝑗 for all (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑖 this means that if (33) is not satisfied, then (34) is also not satisfied. Thus, #» 𝜇 𝑖 ∉ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖 )), contrary to our assumption. The other direction of the proof uses similar arguments. We give an example illustrating Proposition 35. Example 36 (Δ𝑏 cont.). Consider #» 𝜂 1 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑏 )) with #» 𝜂 1 = (1, 2, 2, 1) from Example 34. Because Δ = ⋃𝗀𝗌 𝜂 1 for Δ𝑏 by combining the {(𝑓 |𝑏), (𝑓 |𝑝), (𝑏|𝑝)} {𝑝,𝑓 },{𝑤} {(𝑤|𝑏)} ∣ {𝑏} and Δ3 = ∅ we can obtain the solution #» #» #» 𝑏 𝑏 1 2 solutions 𝜂 1 = (1, 2, 2) and 𝜂 1 = (1) for Δ1 and Δ2 utilizing Proposition 35. Vice versa, we can also utilize Proposition 35 to split #» 𝜂 = (4, 5, 6, 7) into #» 𝜂 1 = (4, 5, 6) and #» 𝜂 2 = (7), obtaining solutions for Δ𝑏 and Δ𝑏 from a 3

solution for Δ𝑏 .

To show that nonmonotonic reasoning with c-representations satisfies (CSynSplitg ) we employ the concept of conditional 𝜅-independence. Definition 37 (conditional 𝜅-independence (Heyninck et al., 2023),(Spohn, 2012)). Let Σ1 , Σ2 , Σ3 ⊆ Σ where Σ1 , Σ2 and Σ3 are pairwise disjoint and let 𝜅 be an OCF. Σ1 , Σ2 are conditionally 𝜅-independent given Σ3 , in symbols Σ1 ⟂ ⟂𝜅 Σ2 |Σ3 , if for all 𝜔1 ∈ Ω(Σ1 ), 𝜔2 ∈ Ω(Σ2 ), and 𝜔3 ∈ Ω(Σ3 ), it holds that 𝜅(𝜔1 |𝜔2 𝜔3 ) = 𝜅(𝜔1 |𝜔3 ). Given a c-representation and a safe conditional syntax splitting, the subsignatures defined by this splitting are 𝜅independent (Beierle et al., 2024b). This result is extended to the case of generalized safe conditional syntax splitting in the following proposition; its proof largely follows the proof of (Beierle et al., 2024b, Proposition 26), but has been adapted in the last few steps to hold also for generalized safe splittings. ⋃𝗀𝗌 Proposition 38. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 , and 𝜅 a c-representation with 𝜅 ⊧ Δ. Then Σ1 ⟂⟂𝜅 Σ2 |Σ3 . 1

⋃𝗀𝗌

Proof. Let Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . Let 𝜔 = 𝜔1 𝜔2 𝜔3 and let #» 𝜂 ∈ 𝑆𝑜𝑙(Δ) such that 𝜅 = 𝜅#» 𝜂 . Recall the definition of 1 2 c-representations (29). We can rewrite (29) to ∑ ∑ ∑ 𝜂𝑗 + 𝜂𝑗 + 𝜂𝑗 (35) 𝜅#» 𝜂 (𝜔) = 𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ1⧵3

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ2⧵3

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ3

By simply adding and subtracting the last sum of (35) we obtain the following equation. ∑ ∑ ∑ ∑ ∑ 𝜅#» 𝜂𝑗 + 𝜂𝑗 + 𝜂𝑗 + 𝜂𝑗 − 𝜂𝑗 𝜂 (𝜔) = 𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ1⧵3

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ2⧵3

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ3

(36)

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ3

Then we can combine the sums for Δ1⧵3 and Δ2⧵3 with the sum for Δ3 to obtain sums for Δ1 and Δ2 respectively. 𝜅#» 𝜂 (𝜔) =

∑

𝜂𝑗 +

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ1

∑

𝜂𝑗 −

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ2

L.Spiegel et al.: Preprint submitted to Elsevier

∑

𝜂𝑗

(37)

𝜔⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ3

Page 19 of 34

Broadening the Applicability of Conditional Syntax Splitting

Since Δ1 is in (Σ1 ∪ Σ3 ), Δ2 is in (Σ2 ∪ Σ3 ) and Δ3 is in (Σ3 ) we can use (8) to simplify (37). ∑ ∑ ∑ 𝜅#» 𝜂𝑗 + 𝜂𝑗 − 𝜂𝑗 𝜂 (𝜔) = 𝜔1 𝜔3 ⊧𝐴𝑗 𝐵 𝑗

(𝐵𝑗 |𝐴𝑗 )∈Δ1

𝜔2 𝜔3 ⊧𝐴𝑗 𝐵 𝑗

(𝐵𝑗 |𝐴𝑗 )∈Δ2

(38)

𝜔3 ⊧𝐴𝑗 𝐵 𝑗

(𝐵𝑗 |𝐴𝑗 )∈Δ3

Due to the generalized safety property 𝜔𝑖 𝜔3 can not falsify any conditional in Δ ⧵ Δ𝑖 for 𝑖, ∈ 1, 2. Thus (38) can be rewritten to ∑ ∑ ∑ 𝜂𝑗 + 𝜂𝑗 − 𝜂𝑗 (39) 𝜅#» 𝜂 (𝜔) = 𝜔1 𝜔3 ⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ

𝜔2 𝜔3 ⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ

𝜔3 ⊧𝐴𝑗 𝐵 𝑗 (𝐵𝑗 |𝐴𝑗 )∈Δ

By applying the definition of c-representations again, we obtain (40)

1 2 3 1 3 2 3 3 𝜅#» 𝜂 (𝜔 𝜔 𝜔 ) = 𝜅#» 𝜂 (𝜔 𝜔 ) + 𝜅#» 𝜂 (𝜔 𝜔 ) − 𝜅#» 𝜂 (𝜔 ) 1 2 3 1 3 which is equivalent to 𝜅#» 𝜂 (𝜔 |𝜔 𝜔 ) = 𝜅#» 𝜂 (𝜔 |𝜔 ), completing the proof.

The following useful lemma will aid us in proofs. Lemma 39 ((Beierle et al., 2024b)). Let Σ1 , Σ2 , Σ3 ⊆ Σ where Σ1 , Σ2 and Σ3 are pairwise disjoint and let 𝜅 be an OCF. Σ1 , Σ2 are conditionally 𝜅-independent given Σ3 iff for all 𝐴 ∈ (Σ1 ), 𝐵 ∈ (Σ2 ) and complete conjunctions 𝐸 ∈ (Σ3 ) it holds that 𝜅(𝐴𝐵𝐸) = 𝜅(𝐴𝐸) + 𝜅(𝐵𝐸) − 𝜅(𝐸). Lemma 39 allows us to use the arithmetics provided by OCFs to calculate the ranks of formulas over disjoint and 𝜅-independent subsignatures which we will exploit in the following subsections.

6.2. Inference with Single c-Representations

In this section we look at inference with respect to a single c-representation, obtained by assigning one crepresentation to each belief base, yielding an OCF-based inductive inference operator. For this, it will be useful to introduce an alternative characterization of (CIndg ) and (CRelg ) for OCF-based inductive inference operators. Corresponding propositions for safe conditional syntax splittings have been introduced by Heyninck et al. (Heyninck et al., 2023); here we extend them to generalized safe conditional syntax splittings. Proposition 40. An inductive inference operator for OCFs 𝐂𝑜𝑐𝑓 ∶ Δ ↦ 𝜅Δ satisfies (CIndg ) if for any Δ = ⋃𝗀𝗌 Δ1 Σ ,Σ Δ2 ∣ Σ3 we have Σ1 ⟂ ⟂𝜅Δ Σ2 |Σ3 . 1

Proof. Let Δ be a belief base, 𝐶 𝑜𝑐𝑓 ∶ Δ ↦ 𝜅Δ be an inductive inference operator for OCFs, and let Δ = Δ1

⋃𝗀𝗌

Δ ∣

Σ1 ,Σ2 2 Σ3 . Assume Σ1 ⟂ ⟂𝜅Δ Σ2 |Σ3 . W.l.o.g. we assume 𝑖 = 1, 𝑖′ = 2, the other case is analogous. We need to show that 𝐶 𝑜𝑐𝑓 satisfies (CIndg ), i.e., for all 𝐴, 𝐵 ∈ 1 , 𝐷 ∈ 2 and every complete conjunction 𝐸 ∈ 3 we have 𝜅Δ (𝐴𝐵𝐸) <

𝜅Δ (𝐴𝐵𝐸) iff 𝜅Δ (𝐴𝐵𝐷𝐸) < 𝜅Δ (𝐴𝐵𝐷𝐸). With Lemma 39 we have 𝜅Δ (𝐴𝐵𝐷𝐸) = 𝜅Δ (𝐴𝐵𝐸)+𝜅Δ (𝐷𝐸)−𝜅Δ (𝐸). Then clearly 𝜅Δ (𝐴𝐵𝐸) < 𝜅Δ (𝐴𝐵𝐸) implies 𝜅Δ (𝐴𝐵𝐷𝐸) < 𝜅Δ (𝐴𝐵𝐷𝐸). On the other hand we can rearrange 𝜅Δ (𝐴𝐵𝐷𝐸) = 𝜅Δ (𝐴𝐵𝐸)+𝜅Δ (𝐷𝐸)−𝜅Δ (𝐸) to 𝜅Δ (𝐴𝐵𝐸) = 𝜅Δ (𝐴𝐵𝐷𝐸)−𝜅Δ (𝐷𝐸)+𝜅Δ (𝐸). Thus 𝜅Δ (𝐴𝐵𝐷𝐸) < 𝜅Δ (𝐴𝐵𝐷𝐸) implies 𝜅Δ (𝐴𝐵𝐸) < 𝜅Δ (𝐴𝐵𝐸) and we are done. Thus, an inductive inference operator for OCFs satisfies generalized conditional independence if the subsignatures of any generalized safe conditional syntax splitting are 𝜅-independent with respect to the conditional pivot. Proposition 41. An inductive inference operator for OCFs 𝐂𝑜𝑐𝑓 ∶ Δ ↦ 𝜅Δ satisfies (CRelg ) if for any Δ = ⋃𝗀𝗌 Δ1 Σ ,Σ Δ2 ∣ Σ3 and 𝑖 ∈ {1, 2} we have 𝜅Δ𝑖 = 𝜅Δ| Σ ∪Σ . 1

L.Spiegel et al.: Preprint submitted to Elsevier

𝑖

Page 20 of 34

Broadening the Applicability of Conditional Syntax Splitting 𝑜𝑐𝑓 ∶ Δ ↦ 𝜅 be an inductive inference operator for OCFs, and let Δ = Proof. Δ ⋃𝗀𝗌 Let Δ be a belief base, 𝐶 Δ1 Σ ,Σ Δ2 ∣ Σ3 . Assume 𝜅Δ𝑖 = 𝜅Δ| Σ ∪Σ . We need to show that 𝐶 𝑜𝑐𝑓 satisfies (𝐶𝑅𝑒𝑙g ), i.e., for 𝑖 ∈ {1, 2}, for 1

𝑖

all 𝐴, 𝐵 ∈ 𝑖 and every complete conjunction 𝐸 ∈ 3 we have 𝜅Δ (𝐴𝐵𝐸) < 𝜅Δ (𝐴𝐵𝐸) iff 𝜅Δ𝑖 (𝐴𝐵𝐸) < 𝜅Δ𝑖 (𝐴𝐵𝐸). Since 𝐴𝐵𝐸 ∈ Σ𝑖 ∪Σ3 we have that 𝜅Δ (𝐴𝐵𝐸) = 𝜅Δ| Σ ∪Σ (𝐴𝐵𝐸) which means 𝜅Δ (𝐴𝐵𝐸) = 𝜅Δ𝑖 (𝐴𝐵𝐸) because 𝑖 3 𝜅Δ𝑖 = 𝜅Δ| Σ ∪Σ per our assumption. 𝑖

Thus, an inductive inference operator for OCFs satisfies generalized conditional relevance if, for every generalized safe conditional syntax splitting, the marginalization of the operator’s image of a conditional belief base to the language of one subbase coincides with applying the operator to that subbase directly. We will now define model-based inductive inference operators assigning a c-representation 𝜅 to each Δ, by employing the concept of selection strategies. Definition 42 (selection strategy 𝜎, (Beierle and Kern-Isberner, 2021)). A selection strategy (for c-representations) is a function 𝜎 ∶ Δ ↦ #» 𝜂 assigning to each conditional belief base Δ an impact vector #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)). c-rep

Each selection strategy yields an inductive inference operator 𝐂𝜎 ∶ Δ ↦ 𝜅𝜎(Δ) where |∼𝜅𝜎(Δ) is obtained via c-rep Equation (1) from 𝜅𝜎(Δ) . Note that 𝐂𝜎 is an inductive inference operator because each |∼𝜅𝜎(Δ) satisfies both (Direct Inference) and (Trivial Vacuity). A recent example for a specific selection strategy are minimal core c-representations (Wilhelm et al., 2024) which we will investigate in Section 6.3. In principle, for every Δ, a selection strategy may choose some impact vector independently from the choices for all other belief bases. The following property generalizes a corresponding postulate (IP-CSP) for safe conditional splittings (Beierle et al., 2024b) and characterizes selection strategies that preserve the impacts chosen for subbases of a generalized safe conditional syntax splitting. (IP-CSPg ) A selection ⋃ strategy 𝜎 is impact preserving with respect to generalized safe conditional syntax splitting if, 𝗀𝗌 for every Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 , for 𝑖 ∈ {1, 2}, we have 𝜎(Δ𝑖 ) = 𝜎(Δ)|Δ𝑖 . 1

It has been shown that any inductive inference operator based on a selection strategy that is impact reserving according to (IP-CSP) satisfies (CSynSplit) (Beierle et al., 2024b); we extend this result to (IP-CSPg ) and (CSynSplitg ). c-rep

Proposition 43. Let 𝜎 be a selection strategy satisfying (IP-CSPg ). Then 𝐂𝜎 (CSynSplitg ).

satisfies (CRelg ) and (CIndg ) and thus

Proof. The proof is obtained by adapting the proof of the proposition for (IP-CSP) and (CSynSplit) (Beierle et al., 2024b, Proposition 27) by observing that the prerequisites of the steps in the proof are also satisfied by generalized safe conditional syntax splittings. Thus, inference based on a single c-representation satisfies (CSynSplitg ) if the underlying selection strategy satisfies (IP-CSPg ). In the next section we give an example of an inference operator based on a specific selection strategy.

6.3. c-Core closure Inference

A special subclass of c-representations are core c-representations (Wilhelm et al., 2024). Core c-representations stand out from the class of all c-representations in the fact that each strongly consistent belief base always has a uniquely determined minimal core c-representation. Choosing this minimal core c-representation yields an OCF-based inductive inference operator via Equation (1). The definition of core c-representations makes use of a constraint reduction system in the form of transformation rules, simplifying the set of constraints 𝐶𝑅(Δ) without altering it’s solutions. To express these rules compactly, an alternative notation of the constraint system 𝐶𝑅(Δ) is used by employing, for each conditional (𝐵𝑖 |𝐴𝑖 ) ∈ Δ, the following sets of sets of verified and falsified conditionals (Beierle, Kutsch and Sauerwald, 2019): 𝑉𝑖 = {{(𝐵𝑗 |𝐴𝑗 ) ∈ Δ ⧵ {𝛿𝑖 } ∣ 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 } ∣ 𝜔 ∈ 𝑣𝑒𝑟(𝐵𝑖 |𝐴𝑖 )}

(41)

𝐹𝑖 = {{(𝐵𝑗 |𝐴𝑗 ) ∈ Δ ⧵ {𝛿𝑖 } ∣ 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 } ∣ 𝜔 ∈ 𝑓 𝑎𝑙(𝐵𝑖 |𝐴𝑖 )}

(42)

Employing the constraint-inducing sets (41) and (42), the constraint satisfaction problem 𝐶𝑅(Δ) = {𝐶1 , … , 𝐶𝑛 } can then be specified as follows for all (𝐵𝑖 |𝐴𝑖 ) ∈ Δ: ∑ ∑ 𝐶𝑖 ∶ 𝜂𝑖 > min{ 𝜂𝑗 ∣ 𝑆 ∈ 𝑉𝑖 } − min{ 𝜂𝑗 ∣ 𝑆 ∈ 𝐹 𝑖 } (43) 𝛿𝑗 ∈𝑆

L.Spiegel et al.: Preprint submitted to Elsevier

𝛿𝑗 ∈𝑆

Page 21 of 34

Broadening the Applicability of Conditional Syntax Splitting

The positive and negative parts of (43) correspond to the positive and negative parts of (29). Figure 3 shows the set {𝑅1, … , 𝑅6} of transformation rules for simplifying 𝐶𝑅(Δ) employed for the definition of core c-representations (Wilhelm et al., 2024). These rules were first given in (Beierle et al., 2019) for speeding up the computation of c-representations and of c-inference (Beierle et al., 2018) and then extended in (Beierle, Haldimann and Kern-Isberner, 2021b; Wilhelm, Sezgin, Kern-Isberner, Haldimann, Beierle and Heyninck, 2023). Since they only make use of general arithmetic properties of the minimum, they do not change the set of solutions 𝑆𝑜𝑙(𝐶𝑅(Δ)). Furthermore, {𝑅1, … , 𝑅6} is terminating and confluent; thus, exhaustive application of {𝑅1, … , 𝑅6} yields a uniquely determined constraint system. For a constraint system 𝐶𝑅, a constraint 𝐶, and constraint inducing sets 𝑉 and 𝐹 , we ̂ 𝐶, ̂ 𝑉̂ , and 𝐹̂ the constraint system, the constraint, and the constraint inducing sets, respectively, after denote with 𝐶𝑅, applying {𝑅1, … , 𝑅6} exhaustively. Definition 44 (Core c-Representation (Wilhelm et al., 2024)). Let Δ be a belief base, let 𝑖 ∈ {1, … , 𝑛} and let 𝐶𝑅+ (Δ) = {𝐶̂1+ , … , 𝐶̂𝑛+ } where 𝐶̂𝑖+ ∶

𝜂𝑖 > min{

∑

(44)

𝜂𝑗 ∣ 𝑆 ∈ 𝑉̂𝑖 }

𝛿𝑗 ∈𝑆

#» If #» 𝜂 ∈ ℕ𝑛0 is a solution of 𝐶𝑅+ (Δ), then the c-representation 𝜅#» 𝜂 determined from 𝜂 via equation (29) is called a core c-representation of Δ. Thus core c-representations are defined by first applying the transformation rules {𝑅1 , … , 𝑅6 } exhaustively and then focusing only on the positive part of the reduced constraint system. While in general 𝐶𝑅(Δ) can have different pareto-minimal solutions, 𝐶𝑅+ (Δ) always has a unique pareto-minimal solution #» 𝜂 𝑚𝑐 (Wilhelm et al., 2024). The cΔ #» 𝑚𝑐 𝑚𝑐 𝑚𝑐 representation 𝜅Δ = 𝜅#» induced by 𝜂 is called the minimal core c-representation of Δ (Wilhelm et al., 2024). 𝜂Δ Δ A method for constructing the minimal core c-representation involving multiple other concepts, such as generalized tolerance partitions and the notion of base functions is given in (Wilhelm et al., 2024). According to (Wilhelm, KernIsberner and Beierle, 2026), 𝜅Δ𝑚𝑐 can be characterized as in the following proposition. Proposition 45 ((Wilhelm et al., 2026), adapted). Let Δ be a belief base and let #» 𝜂 𝑚𝑐 = (𝜂1𝑚𝑐 , … , 𝜂𝑛𝑚𝑐 ) be the impact Δ vector inducing the minimal core c-representation of Δ. Then ∑ (45) 𝜂𝑗𝑚𝑐 ∣ 𝑆 ∈ 𝑉̂𝑖 } + 1. 𝜂𝑖𝑚𝑐 = min{ 𝛿𝑗 ∈𝑆

Proof. The proof is obtained by applying a result from Wilhelm et. al. (Wilhelm et al., 2026, Proposition 9) to the original definition of core c-representations (Wilhelm et al., 2026, Definition 11). can be computed in a stratified manner such that the right-hand-side Crucial to Proposition 45 is the fact that #» 𝜂 𝑚𝑐 Δ of (45) only mentions those impacts that have already been determined in a previous stratum; for details we refer to (Wilhelm et al., 2026). Because the minimal core c-representation is uniquely determined (Wilhelm et al., 2024), it yields the OCF-based inductive inference operator c-core closure. Definition 46 (c-Core closure (Wilhelm et al., 2024)). Let Δ be a belief base, and let 𝐴, 𝐵 ∈ (Σ). The c-core closure inference operator 𝐶 𝑚𝑐 ∶ Δ ↦ |∼ Δ𝑚𝑐 is defined by 𝐴 |∼ Δ𝑚𝑐 𝐵 iff 𝐴 |∼ 𝜅 𝑚𝑐 𝐵 where 𝜅Δ𝑚𝑐 is the minimal core c-representation Δ of Δ. We illustrate these notions with an example. Example 47 (Δ𝑏 cont.). Table 2 shows the sets 𝑉𝑖 and 𝐹𝑖 and their reductions 𝑉̂𝑖 and 𝐹̂𝑖 of Δ𝑏 . Thus 𝐶𝑅+ (Δ𝑏 ) consists of the following constraints: 𝐶̂1+ ∶ 𝜂1 > 0

𝐶̂2+ ∶ 𝜂2 > 𝜂1

𝐶̂3+ ∶ 𝜂3 > 𝜂1

𝐶̂4+ ∶ 𝜂4 > 0

L.Spiegel et al.: Preprint submitted to Elsevier

Page 22 of 34

Broadening the Applicability of Conditional Syntax Splitting 𝑉𝑖 {∅, {𝛿2 }, {𝛿4 }, {𝛿2 , 𝛿4 }} {{𝛿1 }, {𝛿3 }, {𝛿1 , 𝛿4 }} {{𝛿1 }, {𝛿2 }, {𝛿1 , 𝛿4 }, {𝛿2 , 𝛿4 }} {∅, {𝛿1 }, {𝛿2 }}

𝛿1∶ (𝑓 |𝑏) 𝛿2∶ (𝑓 |𝑝) 𝛿3∶ (𝑏|𝑝) 𝛿4∶ (𝑤|𝑏)

𝑉̂𝑖 {∅} {{𝛿1 }} {{𝛿1 }} {∅}

𝐹𝑖 {∅, {𝛿4 }} {∅, {𝛿3 }, {𝛿4 }} {∅, {𝛿2 }} {∅, {𝛿1 }, {𝛿2 }}

𝐹̂𝑖 {∅} {∅} {∅} {∅}

Table 2 Sets 𝑉𝑖 and 𝐹𝑖 and their reductions 𝑉̂𝑖 and 𝐹̂𝑖 for Δ𝑏 in Example 47.

To compute the minimal core c-representation 𝜅 𝑚𝑐𝑏 of Δ𝑏 , we need to find the pareto-minimal solution #» 𝜂 𝑚𝑐𝑏 = Δ Δ (𝜂1𝑚𝑐 , 𝜂2𝑚𝑐 , 𝜂3𝑚𝑐 , 𝜂4𝑚𝑐 ) of 𝐶𝑅+ (Δ𝑏 ). By utilizing Proposition 45, we obtain 𝜂1𝑚𝑐 = 1

𝜂2𝑚𝑐 = 𝜂1𝑚𝑐 + 1

𝜂3𝑚𝑐 = 𝜂1𝑚𝑐 + 1

𝜂4𝑚𝑐 = 1.

Now #» 𝜂 𝑚𝑐𝑏 can be computed in a stratified manner: First, we obtain 𝜂1𝑚𝑐 = 1 and 𝜂4𝑚𝑐 = 1 immediately. Then we can Δ determine 𝜂2𝑚𝑐 = 2 and 𝜂2𝑚𝑐 = 2, yielding #» 𝜂 𝑚𝑐𝑏 = (1, 2, 2, 1). Note that 𝜂2𝑚𝑐 and 𝜂3𝑚𝑐 can be determined by taking only Δ 𝑚𝑐 the previously computed impacts 𝜂1 and 𝜂4𝑚𝑐 into account. The minimal core c-representation 𝜅 𝑚𝑐𝑏 of Δ𝑏 is then given Δ #»𝑚𝑐 #» 𝑚𝑐 𝑚𝑐 by 𝜅 𝑚𝑐𝑏 = 𝜅#» 𝜂 𝑚𝑐 and can be seen in Table 1, where 𝜂 𝑏 = 𝜂 1 . We have 𝜅 𝑏 (𝑝𝑏𝑤) = 1 and 𝜅 𝑏 (𝑝𝑏𝑤) = 2 and thus Δ

𝑝𝑏 |∼ Δ𝑚𝑐𝑏 𝑤.

Δ𝑏

In order to show that c-core closure fully complies with generalized conditional syntax splitting, we first first show that the selection strategy assigning to each belief base Δ the impact vector #» 𝜂 𝑚𝑐 satisfies (IP-CSPg ). Δ Proposition 48. The selection strategy 𝜎 𝑚𝑐 ∶ Δ → #» 𝜂 𝑚𝑐 satisfies (IP-CSPg ). Δ Proof. Let 𝜎 𝑚𝑐 be the selection strategy assigning to each belief base Δ the impact vector #» 𝜂 𝑚𝑐 , yielding the minimal Δ ⋃𝗀𝗌 𝑚𝑐 𝑚𝑐 core c-representation 𝜅Δ = 𝜅#» Δ ∣ Σ3 and let of Δ. Let Δ = {(𝐵 |𝐴 ), … , (𝐵 |𝐴 )} with Δ = Δ 𝜂Δ 1 1 𝑛 𝑛 1 Σ1 ,Σ2 2 𝑚𝑐 ′ ′ 𝑖, 𝑖 ∈ {1, 2}, 𝑖 ≠ 𝑖 . Let 𝜅Δ be the minimal core c-representation of Δ based on the impact vector #» 𝜂 𝑚𝑐 , i.e., Δ #» #» #» 𝑚𝑐 𝑚𝑐 𝑚𝑐 𝑚𝑐 𝑚𝑐 𝑚𝑐 𝑚𝑐 𝜎 (Δ) = 𝜂 Δ and 𝜅#» = 𝜅Δ . We need to show 𝜎 (Δ)| Δ = 𝜎 (Δ𝑖 ), i.e., 𝜂 Δ | Δ = 𝜂 Δ . 𝜂 𝑚𝑐 Δ 𝑖 𝑖 𝑖 ̂ ̂ First we show 𝐶𝑅(Δ) = 𝐶𝑅(Δ | 𝑖 ). Consider the constraint 𝐶𝑗 for the conditional (𝐵𝑗 |𝐴𝑗 ) ∈ Δ: Δ𝑖

𝐶𝑗 ∶ 𝜂𝑗 > min

𝜔⊧𝐴𝑗 𝐵𝑗

∑

𝜂𝑘 − min

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ

𝜔⊧𝐴𝑗 𝐵 𝑗

∑

𝜂𝑘

(46)

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ

Assume now (𝐵𝑗 |𝐴𝑗 ) ∈ Δ𝑖 and consider the constraint 𝐶𝑗𝑖 corresponding to 𝐶𝑗 : 𝐶𝑗𝑖 ∶ 𝜂𝑗 > min

𝜔⊧𝐴𝑗 𝐵𝑗

∑ 𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

𝜂𝑘 − min 𝜔⊧𝐴𝑗 𝐵 𝑗

∑

𝜂𝑘

(47)

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

We show that 𝐶̂𝑗 and 𝐶̂𝑗𝑖 are equivalent. We have 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 iff 𝜔𝑖 𝜔3 ⊧ 𝐴𝑗 𝐵𝑗 , and 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 iff 𝜔𝑖 𝜔3 ⊧ 𝐴𝑗 𝐵𝑗 . Then due to the generalized safety property, for each 𝜔 with 𝜔 ⊧ 𝐴𝑗 𝐵𝑗 there is 𝜔2 with 𝜔𝑖 𝜔3 = 𝜔𝑖2 𝜔32 such that 𝜔2 falsifies no conditional outside of Δ𝑖 . Because 𝜔𝑖 𝜔3 = 𝜔𝑖2 𝜔32 , we have that 𝜔 and 𝜔2 falsify exactly the same conditionals in Δ𝑖 . Thus 𝜔2 falsifies only a subset of conditionals that 𝜔 falsifies and therefore ∑ ∑ 𝜂𝑘 ⩽ 𝜂𝑘 . 𝑘≠𝑗 𝜔2 ⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ

L.Spiegel et al.: Preprint submitted to Elsevier

Page 23 of 34

Broadening the Applicability of Conditional Syntax Splitting

Because (46) utilizes only the minimal worlds and 𝜔2 only falsifies conditionals in Δ𝑖 , the transformation rules R1 and R2 can be used to transform (46) into ∑ ∑ 𝜂𝑗𝑖 > min 𝜂𝑘 − min 𝜂𝑘 (48) 𝜔⊧𝐴𝑗 𝐵𝑗

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

𝜔⊧𝐴𝑗 𝐵 𝑗

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

which is the definition of 𝐶𝑗𝑖 . Therefore 𝐶𝑗 can be transformed into 𝐶𝑗𝑖 by applying transformation rules R1 and R2 to each constraint 𝐶𝑗 . Thus, 𝐶𝑅(Δ)| Δ can be transformed into 𝐶𝑅(Δ𝑖 ) by applying R1 and R2. Hence, because the set 𝑖 ̂ ̂ of transformation rules {𝑅1 , … 𝑅6 } is confluent and terminating, this means that 𝐶𝑅(Δ) and 𝐶𝑅(Δ | 𝑖 ) coincide and Δ𝑖

also that 𝐶𝑅+ (Δ)| Δ and 𝐶𝑅+ (Δ𝑖 ) coincide. 𝑖 Because the positive parts of the relevant constraint systems after applying transformation rules {𝑅1 , … 𝑅6 } are the same, 𝜎 𝑚𝑐 assigns the same value to 𝜂𝑗 and to 𝜂𝑗𝑖 (cf. Definition 45) and thus 𝜎 𝑚𝑐 (Δ)| Δ = 𝜎 𝑚𝑐 (Δ𝑖 ). Thus, 𝜎 𝑚𝑐 𝑖 satisfies (IP-CSPg ).

Utilizing this selection strategy we can now show that c-core closure fully complies with generalized conditional syntax splitting. Proposition 49. c-Core closure satisfies (CRelg ) and (CIndg ) and thus (CSynSplitg ). Proof. The proposition follows immediately from Propositions 43 and 48. Thus we have shown that c-core closure is an example of an inference operator based on a single c-representation that satisfies (IP-CSPg ) and thus fully complies with our generalized version of conditional syntax splitting. Next, we will look at an inference operator taking not a single, but all c-representations of a belief base into account.

6.4. c-Inference

c-Inference was introduced in (Beierle, Eichhorn and Kern-Isberner, 2016; Beierle et al., 2018) as the skeptical inference relation obtained by taking all c-representations of a belief base Δ into account. Definition 50 (c-inference, |∼c-sk , (Beierle et al., 2016)). Let Δ be a belief base and let 𝐴, 𝐵 be formulas. 𝐵 is a 𝚫 (skeptical) c-inference from 𝐴 in the context of Δ, denoted by 𝐴 |∼c-sk Δ 𝐵, iff 𝐴 |∼𝜅 𝐵 holds for all c-representations 𝜅 of Δ, yielding the inductive inference operator 𝐂c-sk ∶ Δ ↦ |∼c-sk Δ Before proving that c-inference satisfies conditional syntax splitting, we recall the following: Consider a safe #» conditional syntax splitting of Δ into Δ1 and Δ2 , and a c-representation 𝜅#» 𝜂 determined by a solution vector 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) together with its projections 𝜅#» 𝜂 1 and 𝜅#» 𝜂 2 to Δ1 and Δ2 , respectively. Then the rank of any formula 𝐹𝑖 over the language (Σ𝑖 ∪ Σ3 ) of Δ𝑖 under the projection 𝜅#» 𝜂 𝑖 coincides with the rank of the formula rank determined by 𝜅#» 𝜂 (Beierle et al., 2024b). We extend this result to generalized safe conditional syntax splittings in the next proposition. ⋃𝗀𝗌 Proposition 51. For any Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 , for all #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)), and for 𝑖 ∈ {1, 2}, 𝐹𝑖 ∈ (Σ𝑖 ∪ Σ3 ), we 1 2 #» have 𝜅#» (𝐹 ) = 𝜅 (𝐹 ). 𝑖 𝜂 𝑖 𝜂 𝑖 Proof. The proof is obtained by adapting the corresponding proof for safe splittings (Beierle et al., 2024b, Proposition 29) by observing that the prerequisites of the steps in the proof are also satisfied by generalized safe conditional syntax splittings. A related proposition for safe splittings additionally states that, for 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ , it holds that 𝜅#» 𝜂 𝑖′ (𝐹𝑖 ) = 0, i.e., formulas defined over the language of one subbase get assigned the rank 0 in models of the other subbase (Beierle et al., 2024b). However, this result cannot be extended to generalized safe splittings because conditionals in Δ3 can be falsified by 𝐹𝑖 and thus it is possible that 𝜅#» 𝜂 𝑖′ (𝐹𝑖 ) > 0. Next we can show that for every generalized safe conditional syntax splitting and every solution vector for Δ𝑖 , we can actually find matching solution vectors for Δ𝑖′ and Δ3 . L.Spiegel et al.: Preprint submitted to Elsevier

Page 24 of 34

Broadening the Applicability of Conditional Syntax Splitting

⋃𝗀𝗌 Proposition 52. Let Δ be a belief base with Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 . Then for 𝑖 ∈ {1, 2}, and for every 1 2 ′ ′ #» 𝜂 𝑖 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖 )) there are #» 𝜂 𝑖 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖′ )) and #» 𝜂 3 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ3 )), such that #» 𝜂 𝑖 |Δ3 = #» 𝜂 𝑖 |Δ3 = #» 𝜂 3. Proof. Consider some constraint 𝜂𝑗𝑖 ∈ 𝐶𝑅(Δ3 ): ∑

𝜂𝑗𝑖 > min

𝜔⊧𝐴𝑗 𝐵𝑗

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

∑

𝜂𝑘 − min 𝜔⊧𝐴𝑗 𝐵 𝑗

(49)

𝜂𝑘

𝑘≠𝑗 𝜔⊧𝐴𝑘 𝐵𝑘 (𝐵𝑘 |𝐴𝑘 )∈Δ𝑖

because Δ3 ∈ ((Σ3 )|(Σ3 )), the only relevant part for the verification of (𝐵𝑗 |𝐴𝑗 ) is 𝜔3 for any 𝜔. With the general ′ ′ safety property this means that there is an extension 𝜔3 𝜔𝑖 𝜔𝑖 such that 𝜔3 𝜔𝑖 𝜔𝑖 falsifies no conditional in Δ ⧵ Δ3 . Clearly this must then also hold for the world minimizing this expression. This means, that the solution of 𝐶𝑅(Δ3 ) is independent of the conditionals in Δ𝑖⧵3 or Δ𝑖′ ⧵3 . Therefore we have 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖 ))|Δ3 = 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖′ ))|Δ3 = ′ 𝜂 3 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ3 )), 𝜂 𝑖 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖′ )) and #» 𝑆𝑜𝑙(𝐶𝑅(Δ3 )). Since Δ is (strongly) consistent we can then always find #» ′ ′ ′ #» #» #» #» #» #» #» #» 𝑖 𝑖 3 3 𝑖 𝑖 𝑖 𝑖 such that 𝜂 |Δ3 = 𝜂 |Δ3 = 𝜂 by choosing 𝜂 = 𝜂 |Δ3 and 𝜂 such that 𝜂 |Δ3 = 𝜂 |Δ3 . With Propositions 35, 51, and 52 we can show: Proposition 53. c-Inference satisfies (CRelg ) and (CIndg ) and thus (CSynSplitg ). ⋃ Proof. Let Δ = Δ1 𝗌Σ ,Σ Δ2 ∣ Σ3 . W.l.o.g. assume 𝐴, 𝐵 ∈ (Σ1 ), 𝐷 ∈ (Σ2 ), 𝐵̇ ∈ {𝐵, 𝐵} and assume 𝐸 ∈ (Σ3 ) 1 2 is a complete conjunction with 𝐷𝐸 ≢ ⊥. We show that c-inference satisfies both (CRelg ) (I) and (CIndg ) (II). c-sk (I) To prove that 𝐂c-sk satisfies (CRelg ) we need to show that 𝐴𝐸 |∼c-sk Δ 𝐵 iff 𝐴𝐸 |∼Δ 𝐵. By applying the definition 1

of |∼c-sk Δ we obtain: ∀ #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) ∶ 𝜅#» 𝜂 (𝐴𝐵𝐸) < 𝜅#» 𝜂 (𝐴𝐵𝐸)

(50)

iff ∀ #» 𝜂 1 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ1 )) ∶ 𝜅#» 𝜂 1 (𝐴𝐵𝐸) < 𝜅#» 𝜂 1 (𝐴𝐵𝐸)

(51)

Direction ⇒: Due to Proposition 35 and Lemma 52 we have that every #» 𝜂 1 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ𝑖 )) has an extension #» #» #» #» 1 1 2 𝜂 such that ( 𝜂 , 𝜂 |Δ𝑖′ ⧵3 ) ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)). Due to (50) we know that 𝜅#» 𝜂 (𝐴𝐵𝐸) < 𝜅#» 𝜂 (𝐴𝐵𝐸) holds for 𝜂 = ( #» 𝜂 1 , #» 𝜂 2| ). We need to show that 𝜅#»1 (𝐴𝐵𝐸) < 𝜅#»1 (𝐴𝐵𝐸). With Lemma 51 this follows directly because Δ𝑖′ ⧵3

𝜂

̇ ̇ 𝜅#» 𝜂 (𝐴𝐵𝐸) = 𝜅#» 𝜂 1 (𝐴𝐵𝐸) since 𝐴, 𝐵 ∈ (Σ1 ), 𝐸 ∈ (Σ3 ) and Δ𝑖 ⊆ ((Σ1 ∪ Σ3 )|(Σ1 ∪ Σ3 )). Direction ⇐: With Proposition 35 we have that every #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) can be split into ( #» 𝜂 1 , #» 𝜂 2 |Δ𝑖′ ⧵3 ) ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)). Due to (51) we know that 𝜅#» 𝜂 1 (𝐴𝐵𝐸) holds. Then with Lemma 51 it follows directly 𝜂 1 (𝐴𝐵𝐸) < 𝜅#» that 𝜅#» 𝜂 1 (𝐴𝐵𝐸) as above. 𝜂 1 (𝐴𝐵𝐸) < 𝜅#» c-sk (II) Next we prove that 𝐂c-sk satisfies (CInd). We need to show 𝐴𝐸 |∼c-sk Δ 𝐵 iff 𝐴𝐷𝐸 |∼Δ 𝐵. By applying the definition of |∼c-sk Δ we obtain: ∀ #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) ∶ 𝜅#» 𝜂 (𝐴𝐵𝐸) < 𝜅#» 𝜂 (𝐴𝐵𝐸)

(52)

iff ∀ #» 𝜂 ∈ 𝑆𝑜𝑙(𝐶𝑅(Δ)) ∶ 𝜅#» 𝜂 (𝐴𝐵𝐷𝐸) < 𝜅#» 𝜂 (𝐴𝐵𝐷𝐸)

(53)

̇ Direction ⇒: Due to Proposition 38 we know that Σ1 ⟂⟂𝜅#»𝜂 Σ2 |Σ3 . With Lemma 39 we then have 𝜅#» 𝜂 (𝐴𝐵𝐷𝐸) = ̇ 𝜅#» 𝜂 (𝐷𝐸) − 𝜅#» 𝜂 (𝐸). With (52) it is clear that (53) must also hold. 𝜂 (𝐴𝐵𝐸) + 𝜅#» ̇ Direction ⇐: Due to Proposition 38 we know that Σ1 ⟂⟂𝜅#»𝜂 Σ2 |Σ3 . With Lemma 39 we then have 𝜅#» 𝜂 (𝐴𝐵𝐷𝐸) = ̇ ̇ ̇ 𝜅#» 𝜂 (𝐴𝐵𝐸) + 𝜅#» 𝜂 (𝐷𝐸) − 𝜅#» 𝜂 (𝐸). This is equivalent to 𝜅#» 𝜂 (𝐴𝐵𝐸) = 𝜅#» 𝜂 (𝐴𝐵𝐷𝐸) − 𝜅#» 𝜂 (𝐷𝐸) + 𝜅#» 𝜂 (𝐸). With (53) it is clear that (52) must also hold. Thus also the inference taking all c-representations into account fully complies with (CSynSplitg ).

L.Spiegel et al.: Preprint submitted to Elsevier

Page 25 of 34

Broadening the Applicability of Conditional Syntax Splitting

7. (CSynSplitg ) properly strengthens (CSynSplit) While (CSynSplit) takes into account all safe conditional syntax splittings of a belief base, (CSynSplitg ) takes into account all generalized safe splittings. Because every safe splitting is generalized safe, but not vice versa (cf. Proposition 9), this means that (CSynSplitg ) is harder to satisfy than (CSynSplit). In this section we formalize this observation by providing a proof showing that (CSynSplitg ) implies (CSynSplit) but not the other way around. First, we first introduce the following lemma, stating that System Z complies with conditional independence when restricted to simple (non-genuine) and safe conditional syntax splittings. We will afterwards exploit this lemma to show that there are inductive inference operators satisfying (CSynSplit) but not (CSynSplitg ). Lemma 54. Let 𝐂𝐳 be the System Z induced⋃inductive inference operator and Δ a belief base. Then, for every simple, 𝗀𝗌 safe conditional syntax splitting Δ = Δ1 Σ ,Σ Δ2 ∣ Σ3 , 𝐂𝐳 satisfies, for all 𝐴, 𝐵 ∈ (Σ𝑖 ), 𝐷 ∈ (Σ𝑖′ ), with 1 2 𝑖, 𝑖′ ∈ {1, 2}, 𝑖 ≠ 𝑖′ , and a full conjunction 𝐸 ∈ (Σ3 ) with 𝐷𝐸 ≢ ⊥, that 𝐴𝐸 |∼𝑧Δ 𝐵

iff

𝐴𝐷𝐸 |∼𝑧Δ 𝐵.

Proof. W.l.o.g. assume that 𝑖 = 1, 𝑖′ = 2, the other case is analogous. Because the splitting is not genuine, we have that either Δ1 ⊆ Δ2 or Δ2 ⊆ Δ1 . Because the splitting is also safe, we have that Δ1 ∩ Δ2 = ∅. Thus, either Δ1 = ∅ or Δ2 = ∅. First assume Δ1 = ∅. Then Δ2 = Δ. We deal with the border cases first. Assume either 𝐴 ≡ ⊥, or 𝐵 ≡ ⊥. If 𝐴 ≡ ⊥, then 𝐴𝐸 ≡ 𝐴𝐷𝐸 ≡ ⊥ and the equation holds. If 𝐵 ≡ ⊥, then we need to show, that 𝐴𝐸 |∼𝑧Δ ⊥ iff 𝐴𝐸𝐷 |∼𝑧Δ ⊥.. Both sides of the iff are false, unless 𝐴𝐸 ≡ ⊥ or 𝐴𝐸𝐷 ≡ ⊥. Neither 𝐸 ≡ ⊥ nor 𝐷 ≡ ⊥ are allowed as per our assumption and if 𝐴 ≡ ⊥ we obtain the first case. So assume 𝐴 ≢ ⊥ and 𝐵 ≢ ⊥. Because Δ1 ⊆ Δ2 , there exists no signature element in Σ1 that appears in any conditional in Δ, because the existence of such an element would mean, that there is some conditional (𝐵|𝐴) ∈ Δ with (𝐵|𝐴) ∈ Δ1 but (𝐵|𝐴) ∉ Δ2 , contrary to our assumption. This means that no formula 𝐹1 ∈ (Σ1 ), 𝐹1 ≢ ⊥ can cause the falsification of any conditional in Δ, i.e., 𝜅Δ𝑧 (𝐺) = 𝜅Δ𝑧 (𝐺𝐹1 ) for any 𝐺 ∈ (Σ). With this we have, for 𝐴, 𝐵 ≢ ⊥, 𝜅Δ𝑧 (𝐴𝐸𝐵) = 𝜅Δ𝑧 (𝐴𝐸𝐵) = 𝜅Δ𝑧 (𝐸) and 𝜅Δ𝑧 (𝐴𝐸𝐷𝐵) = 𝜅Δ𝑧 (𝐴𝐸𝐷𝐵) = 𝜅Δ𝑧 (𝐸𝐷) and thus 𝐴𝐸 |∼𝑧Δ 𝐵 iff 𝐴𝐷𝐸 |∼𝑧Δ 𝐵. Next assume Δ2 = ∅. Then Δ1 = Δ. The case 𝐷 ≡ ⊥ is not possible as per our assumption. Using the same arguments as above, we obtain for any 𝐹2 ∈ (Σ2 ) and any 𝐺 ∈ (Σ), that 𝜅Δ𝑧 (𝐺) = 𝜅Δ𝑧 (𝐺𝐹2 ). Thus we have 𝜅Δ𝑧 (𝐴𝐸𝐵) = 𝜅Δ𝑧 (𝐴𝐷𝐸𝐵) and 𝜅Δ𝑧 (𝐴𝐸𝐵) = 𝜅Δ𝑧 (𝐴𝐷𝐸𝐵) and therefore 𝐴𝐸 |∼𝑧Δ 𝐵 iff 𝐴𝐷𝐸 |∼𝑧Δ 𝐵. Now we show that the postulate (CSynSplitg ) is indeed harder to satisfy than (CSynSplit). Proposition 55. The following relationships hold: 1. (CSynSplitg ) implies (CSynSplit). 2. (CSynSplit) does not imply (CSynSplitg ). Proof. 1.: This follows immediately from the fact that every safe conditional syntax splitting is also generalized safe (cf. Lemma 9). 2.:We construct an inductive inference Operator 𝐂𝐳𝐰 in the following manner: { |∼𝗐 Δ , if Δ has a genuine safe conditional syntax splitting 𝐂𝐳𝐰 (Δ) = 𝑧 |∼Δ , otherwise 𝐂𝐳𝐰 satisfies (CRel), because both 𝐂𝐰 and 𝐂𝐳 satisfy (CRel). Furthermore, whenever 𝐂𝐳𝐰 (Δ) = 𝐂𝐰 (Δ), then

𝐂𝐳𝐰 satisfies (CInd) because 𝐂𝐰 satisfies (CInd). Whenever 𝐂𝐳𝐰 (Δ) = 𝐂𝐳 (Δ), then all safe splittings of Δ are simple

splittings. Therefore, with Lemma 54, (CInd) is also satisfied whenever 𝐂𝐳𝐰 (Δ) = 𝐂𝐳 (Δ), meaning 𝐂𝐳𝐰 satisfies both (CRel) and (CInd). We now show that 𝐂𝐳𝐰 violates (CIndg ) by constructing a belief base with no genuine safe conditional syntax splitting, but a generalized safe conditional syntax splitting. Consider the signature Σ = {𝑏, 𝑝, 𝑓 , 𝑤, 𝑘} meaning that an entity is a bird, is a penguin, flies, has wings, and is a kiwi. The belief base Δ𝑘 = {(𝑓 |𝑏), (𝑓 |𝑝), (𝑏|𝑝), (𝑓 |𝑘), (𝑏|𝑘), (𝑤|𝑏), (𝑤|𝑘)} L.Spiegel et al.: Preprint submitted to Elsevier

Page 26 of 34

Broadening the Applicability of Conditional Syntax Splitting

makes use of this signature. The belief base Δ𝑘 does not have a genuine safe conditional syntax splitting; the list of all conditional syntax splitting of Δ𝑘 can be found in the appendix. Thus, 𝐂𝐳𝐰 (Δ𝑘 ) coincides with 𝐂𝐳 (Δ𝑘 ). However, Δ𝑘 has a generalized safe conditional syntax splitting, given by Δ𝑘 = {(𝑓 |𝑏), (𝑓 |𝑝), (𝑏|𝑝)}

𝗀𝗌 ⋃

{(𝑓 |𝑏), (𝑓 |𝑘), (𝑏|𝑘), (𝑤|𝑘), (𝑤|𝑏)} ∣ {𝑓 , 𝑏}.

{𝑝},{𝑘,𝑤}

Regarding this splitting however, we have that 𝑏𝑓 |∼𝑧Δ𝑘 𝑤 while 𝑝𝑏𝑓 |̸ ∼𝑧Δ𝑘 𝑤. Since 𝑏, 𝑓 ∈ Σ3 , 𝑝 ∈ Σ1 and 𝑤 ∈ Σ2 this is a violation of (CIndg ). Thus 𝐂𝐳𝐰 does not satisfy (CSynSplitg ). Thus, Proposition 55 shows that, because (CSynSplitg ) covers a broader notion of safety and thus more splittings, (CSynSplitg ) implies (CSynSplit) but not the other way around.

8. Conclusions and Future Work In this article we generalized the notion of safety for conditional syntax splittings, allowing the subbases to share non-trivial conditionals. This is achieved by introducing a more relaxed notion of safety, significantly broadening the application scope of the beneficial splitting techniques. Moreover, we identified genuine splittings as the subclass of meaningful conditional syntax splittings, separating them from the class of simple splittings which provide no advantage for inductive inference. Thus, we have made two major steps towards utilizing conditional syntax splitting postulates for inductive inference applications. First, we have significantly broadened the applicability of conditional syntax splitting postulates by adapting them to a more relaxed notion of safety, allowing them to be applied to significantly more splittings. This includes making them applicable to belief bases where previous postulates were not able to be meaningfully applied at all (cf. Example 7). Second, we have classified splittings beneficial for inductive inference as genuine splittings, filtering out the large class of simple splittings in the process (cf. Example 10). Thus we were able to identify those splittings where conditional syntax splitting postulates can be meaningfully applied as exactly the genuine, generalized safe splittings. We introduced adapted conditional syntax splitting postulates to fit with our generalized notion of conditional syntax splitting. The new postulate (CSynSplitg ) covers more splittings than (CSynSplit) and is thus more relevant, but harder to satisfy, i.e., there exist inductive inference operators complying with conditional syntax splitting but not with generalized conditional syntax splitting. We showed that lexicographic inference, System W, inductive inference with a single c-representation based on an adequate selection strategy, c-core closure, and c-inference all fully comply with generalized conditional syntax splitting. While System Z fails to satisfy generalized conditional syntax splitting, we showed that it complies with generalized conditional relevance. In future work we will study the exact relationship of our approach to syntactic contextual filtering (Dupin de Saint-Cyr and Bisquert, 2024) and to propositional forgetting (Lang, Liberatore and Marquis, 2003; Sauerwald, KernIsberner, Becker and Beierle, 2022; Sauerwald, Beierle and Kern-Isberner, 2024). We will exploit the beneficial properties of splitting techniques for implementations of inductive inference (Beierle, Haldimann, Sanin, Schwarzer, Spang, Spiegel and von Berg, 2024a; Beierle, Haldimann, Sanin, Spang, Spiegel and von Berg, 2025), and we will adapt the concepts shown here to include also belief bases that satisfy a weaker notion of consistency (cf. (Haldimann, Beierle, Kern-Isberner and Meyer, 2023; Haldimann, Beierle and Kern-Isberner, 2024)).

L.Spiegel et al.: Preprint submitted to Elsevier

Page 27 of 34

Broadening the Applicability of Conditional Syntax Splitting |𝜉Δ1 𝑏 (𝜔)| 2 1 1 0 0 0

<𝑙𝑒𝑥 Δ𝑏

|𝜉Δ0 𝑏 (𝜔)| 0 1 0 2 1 0

𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤, 𝑏𝑝𝑓 𝑤

over worlds from Example 20. The corresponding values of |𝜉Δ0 𝑏 (𝜔)|, |𝜉Δ1 𝑏 (𝜔)| are indicated on the Figure 1: The order <𝑙𝑒𝑥 Δ𝑏 left.

𝑏𝑝𝑓 𝑤

𝑏𝑝𝑓 𝑤 𝑏𝑝𝑓 𝑤

𝑏𝑝𝑓 𝑤

Figure 2: The preferred structure on worlds <wΔ𝑏 in Example 27. An edge 𝜔 → 𝜔′ indicates that 𝜔 <wΔ𝑏 𝜔′ ; edges that can

be obtained from transitivity are omitted.

R1 subset-V ∶

⟨𝑉 ∪ {𝑆, 𝑆 ′ }, 𝐹 ⟩𝑖 ⟨𝑉 ∪ {𝑆}, 𝐹 ⟩𝑖

𝑆 ⊊ 𝑆′

R2 subset-F ∶

⟨𝑉 , 𝐹 ∪ {𝑆, 𝑆 ′ }⟩𝑖 ⟨𝑉 , 𝐹 ∪ {𝑆}⟩𝑖

𝑆 ⊊ 𝑆′

{ } { } ⟨ 𝑉1 ∪ {𝛿}, … , 𝑉𝑝 ∪ {𝛿} , 𝐹1 ∪ {𝛿}, … , 𝐹𝑞 ∪ {𝛿} ⟩𝑖 R3 element ∶ { } { } ⟨ 𝑉1 , … , 𝑉𝑝 , 𝐹1 , … , 𝐹𝑞 ⟩𝑖 R4 trivial ∶

R5 subsets ∶

R6 circle ∶

⟨𝑉 , 𝐹 ⟩𝑖 ⟨{∅}, {∅}⟩𝑖

𝑉 =𝐹

{ } { } ⟨ 𝑆1 ∪̇ 𝑇 , … , 𝑆𝑝 ∪̇ 𝑇 , 𝑆1 ∪̇ 𝑇 ′ , … , 𝑆𝑝 ∪̇ 𝑇 ′ ⟩𝑖 ⟨{𝑇 } , {𝑇 ′ }⟩𝑖 ⟨ ∪̇ {{𝛿𝑗 }}, {∅}⟩𝑖 ⟨, {∅}⟩𝑖

⟨ ∪̇ {{𝛿𝑖 }}, {∅}⟩𝑗 ⟨, {∅}⟩𝑗

𝑖≠𝑗

Figure 3: Transformation rules {𝑅1, … , 𝑅6} for simplifying (the constraint-inducing sets of) 𝐶𝑅(Δ). A pair ⟨𝑉 , 𝐹 ⟩𝑖 represents the sets of constraint variables in the minimum expressions associated to the verification and the falsification, respectively, of the 𝑖-th conditional 𝛿𝑖 ∈ Δ in the constraint 𝐶𝑖 ∈ 𝐶𝑅(Δ) modeling the acceptance condition of 𝛿𝑖 .

L.Spiegel et al.: Preprint submitted to Elsevier

Page 28 of 34

Broadening the Applicability of Conditional Syntax Splitting

Acknowledgments 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. Jonas Haldimann’s work was supported in part by the National Research Foundation of South Africa (REFERENCE NO: SAI240823262612).

References Adams, E.W., 1975. The Logic of Conditionals: An Application of Probability to Deductive Logic. Synthese Library, Springer Science+Business Media, Dordrecht, NL. Beierle, C., Eichhorn, C., Kern-Isberner, G., 2016. Skeptical inference based on c-representations and its characterization as a constraint satisfaction problem, in: Gyssens, M., Simari, G. (Eds.), Foundations of Information and Knowledge Systems - 9th International Symposium, FoIKS 2016, Linz, Austria, March 7–11, 2016. Proceedings, Springer. pp. 65–82. doi:10.1007/978-3-319-30024-5_4. Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S., 2018. Properties of skeptical c-inference for conditional knowledge bases and its realization as a constraint satisfaction problem. Ann. Math. Artif. Intell. 83, 247–275. doi:10.1007/s10472-017-9571-9. Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S., 2021a. Properties and interrelationships of skeptical, weakly skeptical, and credulous inference induced by classes of minimal models. Artificial Intelligence 297, 103489. doi:10.1016/j.artint.2021.103489. Beierle, C., Haldimann, J., Kern-Isberner, G., 2021b. Semantic splitting of conditional belief bases, in: Raschke, A., Riccobene, E., Schewe, K. (Eds.), Logic, Computation and Rigorous Methods - Essays Dedicated to Egon Börger on the Occasion of His 75th Birthday, Springer. pp. 82–95. doi:10.1007/978-3-030-76020-5_5. Beierle, C., Haldimann, J., Sanin, A., Schwarzer, L., Spang, A., Spiegel, L., von Berg, M., 2024a. Scaling up reasoning from conditional belief bases, in: Destercke, S., Martinez, M.V., Sanfilippo, G. (Eds.), Scalable Uncertainty Management - 16th International Conference, SUM 2024, Proceedings, Springer. pp. 29–44. doi:10.1007/978-3-031-76235-2_3. Beierle, C., Haldimann, J., Sanin, A., Spang, A., Spiegel, L., von Berg, M., 2025. The InfOCF library for reasoning with conditional belief bases, in: Casini, G., Dundua, B., Kutsia, T. (Eds.), Logics in Artificial Intelligence, 19th European Conference (JELIA 2025), Proceedings, Part II, Springer. pp. 19–27. doi:10.1007/978-3-032-04590-4_2. Beierle, C., Kern-Isberner, G., 2012. Semantical investigations into nonmonotonic and probabilistic logics. Annals of Mathematics and Artificial Intelligence 65, 123–158. doi:10.1007/S10472-012-9310-1. Beierle, C., Kern-Isberner, G., 2021. Selection strategies for inductive reasoning from conditional belief bases and for belief change respecting the principle of conditional preservation, in: Bell, E., Keshtkar, F. (Eds.), Proceedings of the 34th International Florida Artificial Intelligence Research Society Conference (FLAIRS-34). doi:10.32473/flairs.v34i1.128459. Beierle, C., Kutsch, S., Sauerwald, K., 2019. Compilation of static and evolving conditional knowledge bases for computing induced nonmonotonic inference relations. Ann. Math. Artif. Intell. 87, 5–41. doi:10.1007/s10472-019-09653-7. Beierle, C., Spiegel, L.P., Haldimann, J., Wilhelm, M., Heyninck, J., Kern-Isberner, G., 2024b. Conditional splittings of belief bases and nonmonotonic inference with c-representations, in: Marquis, P., Ortiz, M., Pagnucco, M. (Eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the 21st International Conference, KR 2024, pp. 106–116. doi:10.24963/kr.2024/10. Benferhat, S., Dubois, D., Prade, H., 1993. Argumentative inference in uncertain and inconsistent knowledge bases, in: Heckerman, D., Mamdani, E.H. (Eds.), Proceedings Ninth Annual Conference on Uncertainty in Artificial Intelligence, UAI-93, Morgan Kaufmann. pp. 411–419. Delgrande, J.P., 2017. A knowledge level account of forgetting. J. Artif. Intell. Res. 60, 1165–1213. doi:10.1613/jair.5530. Dubois, D., Prade, H., 1994. Conditional objects as nonmonotonic consequence relationships. Special Issue on Conditional Event Algebra, IEEE Transactions on Systems, Man and Cybernetics 24, 1724–1740. Goldszmidt, M., Pearl, J., 1996. Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artificial Intelligence 84, 57–112. Haldimann, J., Beierle, C., 2024. Approximations of system W for inference from strongly and weakly consistent belief bases. Int. J. Approx. Reason. 175, 109295. doi:10.1016/j.ijar.2024.109295. Haldimann, J., Beierle, C., Kern-Isberner, G., 2024. Syntax splitting and reasoning from weakly consistent conditional belief bases with c-inference, in: Meier, A., Ortiz, M. (Eds.), Foundations of Information and Knowledge Systems - 13th International Symposium, FoIKS 2024, Springer. pp. 85–103. doi:10.1007/978-3-031-56940-1_5. Haldimann, J., Beierle, C., Kern-Isberner, G., Meyer, T., 2023. Conditionals, infeasible worlds, and reasoning with system W, in: Chun, S.A., Franklin, M. (Eds.), Proceedings of the Thirty-Sixth International Florida Artificial Intelligence Research Society Conference. doi:10.32473/ flairs.36.133268. Haldimann, J.P., 2024. Nonmonotonic Reasoning with Defeasible Rules on Feasible and Infeasible Worlds - Exploring a Landscape of Inductive Inference Operators. volume 355 of Diss. Artif. Intell. IOS Press. doi:10.3233/DAI355. Heyninck, J., Kern-Isberner, G., Meyer, T., Haldimann, J.P., Beierle, C., 2023. Conditional syntax splitting for non-monotonic inference operators, in: Williams, B., Chen, Y., Neville, J. (Eds.), Proceedings of the 37th AAAI Conference on Artificial Intelligence, pp. 6416–6424. doi:10.1609/aaai.v37i5.25789. Kern-Isberner, G., 2001. Conditionals in Nonmonotonic Reasoning and Belief Revision – Considering Conditionals as Agents. Number 2087 in Lecture Notes in Computer Science, Springer Science+Business Media, Berlin, DE. Kern-Isberner, G., 2004. A thorough axiomatization of a principle of conditional preservation in belief revision. Ann. Math. Artif. Intell. 40(1-2), 127–164.

L.Spiegel et al.: Preprint submitted to Elsevier

Page 29 of 34

Broadening the Applicability of Conditional Syntax Splitting Kern-Isberner, G., Beierle, C., Brewka, G., 2020. Syntax splitting = relevance + independence: New postulates for nonmonotonic reasoning from conditional belief bases, in: Calvanese, D., Erdem, E., Thielscher, M. (Eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the 17th International Conference, KR 2020, IJCAI Organization. pp. 560–571. doi:10.24963/kr.2020/56. Kern-Isberner, G., Brewka, G., 2017. Strong syntax splitting for iterated belief revision, in: Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19-25, 2017, pp. 1131–1137. Komo, C., Beierle, C., 2020. Nonmonotonic inferences with qualitative conditionals based on preferred structures on worlds, in: Schmid, U., Klügl, F., Wolter, D. (Eds.), KI 2020: Advances in Artificial Intelligence - 43rd German Conference on AI, Springer. pp. 102–115. doi:10.1007/978-3-030-58285-2_8. Komo, C., Beierle, C., 2022. Nonmonotonic reasoning from conditional knowledge bases with system W. Ann. Math. Artif. Intell. 90, 107–144. doi:10.1007/s10472-021-09777-9. Lang, J., Liberatore, P., Marquis, P., 2003. Propositional independence: Formula-variable independence and forgetting. J. Artif. Intell. Res. 18, 391–443. doi:10.1613/jair.1113. Lehmann, D., 1995. Another perspective on default reasoning. Ann. Math. Artif. Intell. 15, 61–82. Lehmann, D., Magidor, M., 1992. What does a conditional knowledge base entail? Artificial Intelligence 55, 1–60. Parikh, R., 1999. Beliefs, belief revision, and splitting languages. Logic, Language, and Computation 2, 266–278. Pearl, J., 1990. System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning, in: Proc. of the 3rd Conf. on Theoretical Aspects of Reasoning About Knowledge (TARK’1990), Morgan Kaufmann Publ. Inc., San Francisco, CA, USA. pp. 121–135. Peppas, P., Williams, M.A., Chopra, S., Foo, N.Y., 2015. Relevance in belief revision. Artificial Intelligence 229, 126–138. Dupin de Saint-Cyr, F., Bisquert, P., 2024. The form and the content: Non-monotonic reasoning with syntactic contextual filtering, in: ECAI 2024. IOS Press, pp. 1309–1316. Sauerwald, K., Beierle, C., Kern-Isberner, G., 2024. Propositional variable forgetting and marginalization: Semantically, two sides of the same coin, in: FoIKS 2024, Proceedings, Springer. pp. 144–162. doi:10.1007/978-3-031-56940-1_8. Sauerwald, K., Kern-Isberner, G., Becker, A., Beierle, C., 2022. From forgetting signature elements to forgetting formulas in epistemic states, in: de Saint-Cyr, F.D., Öztürk-Escoffier, M., Potyka, N. (Eds.), Scalable Uncertainty Management - 15th International Conference, SUM 2022, Springer. pp. 92–106. doi:10.1007/978-3-031-18843-5_7. Spiegel, L.P., Haldimann, J., Heyninck, J., Kern-Isberner, G., Beierle, C., 2025. Generalized safe conditional syntax splitting of belief bases, in: Kwok, J. (Ed.), Proceedings of the 34th International Joint Conference on Artificial Intelligence, IJCAI-25, IJCAI Organization. pp. 4678–4686. doi:10.24963/ijcai.2025/521. Spohn, W., 1988. Ordinal conditional functions: a dynamic theory of epistemic states, in: Harper, W., Skyrms, B. (Eds.), Causation in Decision, Belief Change, and Statistics, II. Kluwer Academic Publishers, pp. 105–134. Spohn, W., 2012. The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford University Press. Weydert, E., 1998. System JZ - How to build a canonical ranking model of a default knowledge base, in: Cohn, A., Schubert, L., Shapiro, S. (Eds.), Proc. of the Sixth International Conference on Principles of Knowledge Representation and Reasoning (KR’98), Morgan Kaufmann. pp. 190–201. Wilhelm, M., Kern-Isberner, G., Beierle, C., 2024. Core c-representations and c-core closure for conditional belief bases, in: Meier, A., Ortiz, M. (Eds.), Foundations of Information and Knowledge Systems - 13th International Symposium, FoIKS, Springer. pp. 104–122. doi:10.1007/ 978-3-031-56940-1_6. Wilhelm, M., Kern-Isberner, G., Beierle, C., 2026. c-core closure and syntax splitting for conditional belief bases. The Knowledge Engineering Review , to appear. Wilhelm, M., Sezgin, M., Kern-Isberner, G., Haldimann, J., Beierle, C., Heyninck, J., 2023. Splitting techniques for conditional belief bases in the context of c-representations, in: Gaggl, S.A., Martinez, M.V., Ortiz, M. (Eds.), Logics in Artificial Intelligence - 18th European Conference, JELIA 2023, Dresden, Germany, September 20-22, 2023, Proceedings, Springer. pp. 462–477. doi:10.1007/978-3-031-43619-2_32.

L.Spiegel et al.: Preprint submitted to Elsevier

Page 30 of 34

Broadening the Applicability of Conditional Syntax Splitting

A. List of all conditional syntax splitting of belief base Δ𝑟𝑎𝑖𝑛 from Example 14. All genuine, generalized safe splittings are marked with boxes. Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗌 ⋃

∅∣∅

Σ,∅

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗌 ⋃

∅ ∣ {𝑏}

{𝑠,𝑟,𝑜,𝑢},∅ 𝗌 ⋃ {𝑏,𝑟,𝑜,𝑢},∅ 𝗌 ⋃ {𝑏,𝑠,𝑜,𝑢},∅ 𝗌 ⋃ {𝑏,𝑠,𝑟,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑠}

∅ ∣ {𝑟}

∅ ∣ {𝑜}

∅ ∣ {𝑢}

{𝑏,𝑠,𝑟,𝑜},∅ 𝗌 ⋃

∅ ∣ {𝑏, 𝑠}

{𝑟,𝑜,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑏, 𝑟}

{𝑠,𝑜,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑏, 𝑜}

{𝑠,𝑟,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑏, 𝑢}

{𝑠,𝑟,𝑜},∅ 𝗀𝗌 ⋃

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)}

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟}

{𝑏},{𝑜,𝑢}

𝑟𝑎𝑖𝑛

=Δ

𝗀𝗌 ⋃

{(𝑠|𝑟), (𝑟|𝑠)} ∣ {𝑠, 𝑟}

{𝑏,𝑜,𝑢},∅

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

∅ ∣ {𝑠, 𝑜}

{𝑏,𝑟,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑠, 𝑢}

{𝑏,𝑟,𝑜},∅

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)}

𝗀𝗌 ⋃

{(𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑟, 𝑜}

{𝑏,𝑠},{𝑢}

𝑟𝑎𝑖𝑛

=Δ

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗀𝗌 ⋃

{(𝑜|𝑟)} ∣ {𝑟, 𝑜}

{𝑏,𝑠,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑟, 𝑢}

{𝑏,𝑠,𝑜},∅

L.Spiegel et al.: Preprint submitted to Elsevier

Page 31 of 34

Broadening the Applicability of Conditional Syntax Splitting

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗌 ⋃

{(𝑏|𝑘)} ∣ {𝑜, 𝑢}

{𝑏,𝑠,𝑟},∅ 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)} ∣ {𝑏, 𝑠, 𝑟}

{𝑜,𝑢},∅

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

∅ ∣ {𝑏, 𝑠, 𝑜}

{𝑟,𝑢},∅ 𝗌 ⋃

∅ ∣ {𝑏, 𝑠, 𝑢}

{𝑟,𝑜},∅

𝑟𝑎𝑖𝑛

𝗀𝗌 ⋃

= {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)}

{(𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑏, 𝑟, 𝑜}

{𝑠},{𝑢}

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛 Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗀𝗌 ⋃

{(𝑜|𝑟)} ∣ {𝑏, 𝑟, 𝑜}

{𝑠,𝑢},∅ 𝗌 ⋃

∅} ∣ {𝑏, 𝑟, 𝑢}

{𝑠,𝑜},∅ 𝗌 ⋃

∅ ∣ {𝑏, 𝑜, 𝑢}

{𝑠,𝑟},∅ 𝗀𝗌 ⋃

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)}

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟, 𝑜}

{𝑏},{𝑢}

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗀𝗌 ⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟)} ∣ {𝑠, 𝑟, 𝑜}

{𝑏,𝑢},∅

Δ𝑟𝑎𝑖𝑛 = {(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)}

𝗀𝗌 ⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟, 𝑢}

{𝑏},{𝑜}

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗀𝗌 ⋃

{(𝑠|𝑟), (𝑟|𝑠)} ∣ {𝑠, 𝑟, 𝑢}

{𝑏,𝑜},∅

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

∅ ∣ {𝑠, 𝑜, 𝑢}

{𝑏,𝑟},∅ 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

{(𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑟, 𝑜, 𝑢}

{𝑏,𝑠},∅ 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟), (𝑜|𝑠𝑟), (𝑜|𝑟)} ∣ {𝑏, 𝑠, 𝑟, 𝑜}

{𝑢},∅ 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑏|𝑠𝑟)} ∣ {𝑏, 𝑠, 𝑟, 𝑢}

{𝑜},∅

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

∅ ∣ {𝑏, 𝑠, 𝑜, 𝑢}

{𝑟},∅ 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

{(𝑜|𝑟), (𝑢|𝑜𝑟)} ∣ {𝑏, 𝑟, 𝑜, 𝑢}

{𝑠},∅

L.Spiegel et al.: Preprint submitted to Elsevier

Page 32 of 34

Broadening the Applicability of Conditional Syntax Splitting

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

𝗀𝗌 ⋃

{(𝑠|𝑟), (𝑟|𝑠), (𝑜|𝑠𝑟), (𝑜|𝑟), (𝑢|𝑟𝑜)} ∣ {𝑠, 𝑟, 𝑜, 𝑢}

{𝑏},∅ 𝗀𝗌

Δ𝑟𝑎𝑖𝑛 = Δ𝑟𝑎𝑖𝑛

⋃

Δ𝑟𝑎𝑖𝑛 ∣ Σ

∅,∅

B. List of all conditional syntax splittings of belief base Δ𝑘 from the proof of Proposition 55. All genuine, generalized safe splittings are marked with boxes. Δ𝑘 = Δ𝑘

𝗌 ⋃

∅∣∅

Σ,∅ 𝗀𝗌

Δ𝑘 = Δ𝑘

⋃ ∅,∅

𝑘

Δ =Δ

Δ𝑘 = Δ𝑘 Δ𝑘 = Δ𝑘

Δ𝑘 ∣ Σ

⋃

∅ ∣ {𝑝}

{𝑓 ,𝑘,𝑏,𝑤},∅ 𝗌 ⋃ {𝑝,𝑓 ,𝑘,𝑤},∅ 𝗌 ⋃

∅ ∣ {𝑏}

∅ ∣ {𝑓 }

{𝑝,𝑘,𝑏,𝑤},∅

Δ𝑘 = Δ𝑘 Δ𝑘 = Δ𝑘

⋃

∅ ∣ {𝑘}

{𝑝,𝑓 ,𝑏,𝑤},∅ 𝗌 ⋃

∅ ∣ {𝑤}

{𝑝,𝑓 ,𝑘,𝑏},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑝)} ∣ {𝑝, 𝑓 }

{𝑘,𝑏,𝑤},∅

Δ𝑘 = Δ𝑘

⋃

∅ ∣ {𝑝, 𝑘}

{𝑓 ,𝑏,𝑤},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑏|𝑝)} ∣ {𝑝, 𝑏}

{𝑓 ,𝑘,𝑤},∅

Δ𝑘 = Δ𝑘

⋃

∅ ∣ {𝑝, 𝑤}

{𝑓 ,𝑘,𝑏},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑘)} ∣ {𝑓 , 𝑘}

{𝑝,𝑏,𝑤},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑤|𝑘)} ∣ {𝑘, 𝑤}

{𝑝,𝑓 ,𝑏},∅ 𝗀𝗌

Δ𝑘 = Δ𝑘 Δ𝑘 = Δ𝑘

⋃

{(𝑤|𝑏)} ∣ {𝑏, 𝑤}

{𝑝,𝑓 ,𝑘},∅ 𝗌 ⋃

∅ ∣ {𝑓 , 𝑤}

{𝑝,𝑘,𝑏},∅ 𝗀𝗌

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑏)} ∣ {𝑓 , 𝑏}

{𝑝,𝑘,𝑤},∅

L.Spiegel et al.: Preprint submitted to Elsevier

Page 33 of 34

Broadening the Applicability of Conditional Syntax Splitting

Δ𝑘 = Δ𝑘

⋃

{(𝑏|𝑘)} ∣ {𝑘, 𝑏}

{𝑝,𝑓 ,𝑤},∅

Δ𝑘 = {(𝑓 |𝑏), (𝑓 |𝑝), (𝑏|𝑝)}

𝗀𝗌 ⋃

{(𝑓 |𝑏), (𝑤|𝑏), (𝑓 |𝑘), (𝑏|𝑘), (𝑤|𝑘)} ∣ {𝑓 , 𝑏}

{𝑝},{𝑘,𝑤}

⋃

Δ𝑘 = {(𝑏|𝑘), (𝑓 |𝑏), (𝑏|𝑝), (𝑓 |𝑝), (𝑓 |𝑘)} Δ𝑘 = Δ𝑘

{(𝑏|𝑘), (𝑤|𝑏), (𝑤|𝑘)} ∣ {𝑘, 𝑏}

{𝑝,𝑓 },{𝑤}

⋃

{(𝑓 |𝑘), (𝑓 |𝑝)} ∣ {𝑝, 𝑓 , 𝑘}

{𝑏,𝑤},∅ 𝗀𝗌

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑏), (𝑏|𝑝), (𝑓 |𝑝)} ∣ {𝑝, 𝑓 , 𝑏}

{𝑘,𝑤},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑝)} ∣ {𝑝, 𝑓 , 𝑤}

{𝑘,𝑏},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑏|𝑝), (𝑏|𝑘)} ∣ {𝑝, 𝑘, 𝑏}

{𝑓 ,𝑤},∅

⋃

Δ𝑘 = {(𝑏|𝑝), (𝑏|𝑘), (𝑓 |𝑏), (𝑓 |𝑝), (𝑓 |𝑘)} Δ𝑘 = Δ𝑘

{(𝑏|𝑝), (𝑏|𝑘), (𝑤|𝑏), (𝑤|𝑘)} ∣ {𝑝, 𝑘, 𝑏}

{𝑓 },{𝑤}

⋃

{(𝑤|𝑘)} ∣ {𝑝, 𝑘, 𝑤}

{𝑓 ,𝑏},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑏|𝑝), (𝑤|𝑏)} ∣ {𝑝, 𝑏, 𝑤}

{𝑓 ,𝑘},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑘), (𝑏|𝑘), (𝑓 |𝑏)} ∣ {𝑓 , 𝑘, 𝑏}

{𝑝,𝑤},∅

⋃

Δ𝑘 = {(𝑓 |𝑘), (𝑏|𝑘), (𝑓 |𝑏), (𝑏|𝑝), (𝑓 |𝑝)} Δ𝑘 = Δ𝑘

{(𝑓 |𝑘), (𝑏|𝑘), (𝑓 |𝑏), (𝑤|𝑏), (𝑤|𝑘)} ∣ {𝑓 , 𝑘, 𝑏}

{𝑝},{𝑤}

⋃

{(𝑓 |𝑘), (𝑤|𝑘)} ∣ {𝑓 , 𝑘, 𝑤}

{𝑝,𝑏},∅

Δ𝑘 = {(𝑓 |𝑏), (𝑤|𝑏), (𝑓 |𝑝), (𝑏|𝑝)} Δ𝑘 = Δ𝑘

𝗀𝗌 ⋃

{(𝑓 |𝑏), (𝑤|𝑏), (𝑓 |𝑘), (𝑏|𝑘), (𝑤|𝑘)} ∣ {𝑓 , 𝑏, 𝑤}

{𝑝},{𝑘}

⋃

{(𝑤|𝑘), (𝑏|𝑘), (𝑤|𝑏)} ∣ {𝑘, 𝑏, 𝑤}

{𝑝,𝑓 },∅

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑘), (𝑏|𝑘), (𝑓 |𝑏), (𝑏|𝑝), (𝑓 |𝑝)} ∣ {𝑝, 𝑓 , 𝑘, 𝑏}

{𝑤},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑘), (𝑤|𝑘), (𝑓 |𝑝)} ∣ {𝑝, 𝑓 , 𝑘, 𝑤}

{𝑏},∅ 𝗀𝗌

Δ𝑘 = Δ𝑘

⋃

{(𝑤|𝑏), (𝑓 |𝑏), (𝑏|𝑝), (𝑓 |𝑝)} ∣ {𝑝, 𝑓 , 𝑏, 𝑤}

{𝑘},∅

Δ𝑘 = Δ𝑘

⋃

{(𝑤|𝑘), (𝑏|𝑘), (𝑤|𝑏), (𝑏|𝑝)} ∣ {𝑝, 𝑘, 𝑏, 𝑤}

{𝑓 },∅ 𝗀𝗌

Δ𝑘 = Δ𝑘

⋃

{(𝑓 |𝑘), (𝑏|𝑘), (𝑓 |𝑏), (𝑤|𝑏), (𝑤|𝑘)} ∣ {𝑓 , 𝑘, 𝑏, 𝑤}

{𝑝},∅

L.Spiegel et al.: Preprint submitted to Elsevier

Page 34 of 34

Record · ID 13155 · SHA-256 a64b9da2d8c79b5b
Conceptio Open Knowledge Archive — every document is proof-bundled with source, license, and retrieval metadata.