Syntax Splitting for Iterated Contractions, Ignorations, and Revisions on Ranking Functions Using Selection Strategies

Abstract

For characterizing belief sets consisting of independent parts, Parikh introduced the notion of syntax splitting. Corresponding postulates have been developed for the reasoning from and for the revision of belief bases with respect to syntax splitting. Kern-Isberner and Brewka introduced syntax splitting for epistemic states and iterated belief revision. Only recently, syntax splitting has also been studied for contractions and iterated contractions of epistemic states; however, all of the evaluated contractions proposed in the literature failed to fulfil the full syntax splitting postulates. In this paper, we study syntax splitting for iteratively contracting and revising epistemic states, represented by ranking functions, not only with respect to a set of formulas, but with respect to a set of conditionals. Using a framework of belief change governed by the principle of conditional preservation, we employ the concept of selection strategies. We develop axioms for selection strategies ensuring that the induced contractions and revisions fully obey the desired syntax splitting properties. Furthermore, we transfer our approach to ignorations and prove a theorem showing how selection strategies satisfying the axioms can effectively be constructed.

1 Introduction

As intelligent agents live in a dynamic environment they must be able to adapt their state of mind if they receive new information. This process is called belief change and has been investigated intensively in the literature, researchers studied e.g. the revision of belief sets (i.e. sets of formulas, e.g. [1]), preorders (e.g. [8]), and ranking functions (e.g. [14]).

In 1999, Parikh introduced the notion of syntax splitting for belief sets [20]. In the same paper, he developed a postulate ($\text {P}$) for belief revision, the basic idea being that if a belief base splits into separate sub-bases over disjoint sub-signatures, revisions of one of the sub-bases should be independent from the other sub-bases. The concept of syntax splitting has been investigated further, e.g., by Peppas et al. [22]. More recently, the notion of syntax splitting was extended to other representations of epistemic states, total preorders and ranking functions, by Kern-Isberner and Brewka [17]. Again, postulates for belief revision of total preorders and ranking functions in the presence of a syntax splitting were introduced in the same paper [17]. Another major belief change operation besides revision is belief contraction. While already introduced by AGM, contraction of beliefs gained more interest only recently (e.g. [7, 18, 19, 23, 25]). In [11], we developed syntax splitting postulates for belief contraction on belief sets, epistemic states with total preorders, and ranking functions, and evaluated different contraction operations with respect to syntax splitting, namely moderate contraction [23], natural contraction [23], lexicographic contraction [23], and c-contractions which are special kinds of operations on ranking functions [6, 16] that are based on the principle of conditional preservation [13, 14]. It was shown that none of the evaluated contraction operators is fully compatible with syntax splitting; even the (with respect to syntax splitting) well-behaved c-contractions do not fulfil all the syntax splitting postulates in general [11].

In this paper, we refine the notion of c-contraction as used in [11] in such a way that all required syntax splitting properties are ensured. For this, we employ the concept of selection strategies that has been proposed for reasoning with c-representations and c-revisions recently [5, 15]. We extend selection strategies to general belief change operations in the c-change framework based on the principle of conditional preservation [6, 16]. In this way, our approach covers not only iterated contractions but also iterated revisions both of which are fully compatible with syntax splitting. Furthermore, we will show that this transfers also to iterated ignorations where an ignoration is a specific contraction where the agent gives up the judgement on a belief, see e.g. [6]. Thus, here we will focus on the syntax splitting for the revision, contraction, and ignoration of ranking functions. Since the corresponding postulates with respect to syntax splitting considered here are structurally very similar, we generalize them to postulates for belief changes. We also extend the syntax splitting postulates to cover belief change with sets of conditionals instead of sets of formulas. Note that this extension goes far beyond the classic AGM framework [1]. The most general postulate ($\text {P}^{\textit{ocf}}_\circ$) developed in this paper entails all syntax splitting postulates for revision and contraction in [11, 17]. In our general framework, each selection strategy for belief change induces a belief change operator. We develop a very natural postulate $( IP ^{ cc })$ for selection strategies for c-changes and show that each change operator fulfils ($\text {P}^{\textit{ocf}}_\circ$) and therefore all considered syntax splitting postulates if it is induced by a selection strategy that fulfils $( IP ^{ cc })$. We also prove a theorem yielding an effective method for constructing selection strategies satisfying $( IP ^{ cc })$.

In summary, the main contributions of this paper are:

• Introduction of syntax splitting postulates ($\text {P}^{\textit{ocf}}_\circ$), ($\text {MR}^{ocf}_\circ$), and ($\text {P}^{it-ocf}_\circ$) for belief change on ranking functions that generalize syntax splitting postulates introduced in [17] and [11] and cover changes with sets of conditionals

• Introduction of selection strategies for c-contractions and c-ignorations

• Introduction of a new postulate $( IP ^{ cc })$ for selection strategies for contractions, revisions, and ignorations

• Proof that if a selection strategy fulfils $( IP ^{ cc })$ then the induced c-change fulfils ($\text {P}^{\textit{ocf}}_\circ$) and therefore ($\text {MR}^{ocf}_\circ$) and ($\text {P}^{it-ocf}_\circ$)

• Effective construction of selection strategies satisfying $( IP ^{ cc })$

• Iterative contraction, revision, and ignoration of ranking functions by sets of conditionals fully compatible with syntax splitting.

2 Background

Let $\varSigma$ be a (propositional) signature. The set of all propositional formulae over $\varSigma$ is denoted by $\mathrm {Form}(\varSigma )$. We will use $\bar{A}$ as shorthand for $\lnot A$ and AB as shorthand for $A \wedge B$ with $A, B \in \mathrm {Form}(\varSigma )$. The set of all interpretations, also called worlds, of $\varSigma$ will be denoted as $\mathrm {Int}(\varSigma )$ or $\varOmega$. An interpretation $\omega \in \mathrm {Int}(\varSigma )$ is a model for $A \in \mathrm {Form}(\varSigma )$, denoted as $\omega \models A$, if A holds in $\omega$. The set of models for a formula is ${ Mod}\,_\varSigma (A) = \{ \omega \in \mathrm {Int}(\varSigma ) \mid \omega \models A \}$. A formula with at least one model is called consistent, otherwise inconsistent. For $A, B \in \mathrm {Form}(\varSigma )$ we say A entails B if ${ Mod}\,_\varSigma (A) \subseteq { Mod}\,_\varSigma (B)$. The concepts of models, consistency and entailment are analogously used for sets of formulae. For $M \subseteq \mathrm {Form}(\varSigma )$, the deductive closure of M is $\mathrm {Cn}_\varSigma (M) = \{ A \in \mathrm {Form}(\varSigma ) \mid M \models A \}$. If $M = \mathrm {Cn}_\varSigma (M)$ then M is called deductively closed. Based on propositional logic, we define the set of conditionals $\mathrm {Cond}(\varSigma ) = \{ (B|A) \mid A,B \in \mathrm {Form}({\varSigma })\}$. A conditional $(B|A)$ formalizes that the antecedent $A$ plausibly entails the consequent $B$. Propositional logic can be embedded in conditional logic by using conditionals $(A|\top )$ with tautological antecedents. A conditional $(B|A)$ is verified by a world $\omega \in \mathrm {Int}(\varSigma )$ if $\omega \models AB$ and is falsified by $\omega$ if $\omega \models A\overline{B}$. If $\omega \not \models A$, then the conditional is not applicable to $\omega$ (see [9]). A conditional $(B|A)$ is called self-fulfilling if ${ Mod}\,(A) \subseteq { Mod}\,(B)$ and contradictory if ${ Mod}\,(A) \cap { Mod}\,(B) = \emptyset$. The counter-conditional of a conditional $(B|A)$ is the conditional $(\overline{B}|A)$. The counter-conditional of a self-fulfilling conditional is contradictory and vice versa.

There are many approaches to model the epistemic state of a reasoning agent. In this paper, we consider agents whose epistemic state is completely represented by a ranking function over a fixed signature. A ranking function or ordinal conditional function (OCF), introduced (in a more general form) in [26], is a function $\kappa : \varOmega \rightarrow \mathbb {N}_0$ with $\kappa ^{-1}(0) \ne \emptyset$. The rank of $\omega \in \varOmega$ is $\kappa (\omega )$. The lower the rank of a world, the more plausible it is. The most plausible worlds are those with rank 0. The rank of a formula A is $\kappa (A) = \min _{\omega \in { Mod}\,(A)} \kappa (\omega )$ and $A$ is accepted by $\kappa$, denoted as $\kappa \models A$, if $\kappa (A) = 0$. An OCF $\kappa$ accepts a conditional $(B|A)$, denoted as $\kappa \models (B|A)$, if $\kappa (AB) < \kappa (A \overline{B})$, i.e., if the verification of the conditional is more plausible than its falsification. If $\kappa$ models the epistemic state of an agent, she considers the formulas and conditionals accepted by $\kappa$ to be (plausibly) true. A set $\mathcal R$ of conditionals is called consistent if there is at least one OCF that accepts every conditional in $\mathcal R$. Otherwise, $\mathcal R$ is called inconsistent.

A reasoning agent is usually not in a static environment and needs to adapt her beliefs in order to account for incoming information. An operation that maps an epistemic state and some given input to an epistemic state is called a belief change. While in the AGM framework, the new input is only a formula, we will use c-changes that are a special kind of change operations on ranking functions taking a set of conditionals as new input.

Definition 1

(c-change, $\kappa _{\vec {\gamma }}$ [16]). Let $\kappa$ be a ranking function and $\varDelta = \{(B_1|A_1), \dots , (B_n|A_n)\}$ be a set of conditionals. For $\vec {\gamma } = (\gamma _1^+, \gamma _1^-, \dots , \gamma _n^+, \gamma _n^-) \in \mathbb {Q}^{2n}$ we define $\kappa _{\vec {\gamma }}$ by $\kappa _{\vec {\gamma }}(\omega ) =\kappa _0 + \kappa (\omega ) + \sum _{\begin{array}{c} i=1\\ \omega \models A_i B_i \end{array}}^n \gamma _i^+ + \sum _{\begin{array}{c} i=1\\ \omega \models A \overline{B_i} \end{array}}^n \gamma _i^-$ where $\kappa _0 \in \mathbb {Q}$ is a normalization factor ensuring that the minimal worlds have rank 0. A change from $\kappa$ with $\varDelta$ to $\kappa ^\circ$ is a c-change if there is $\vec {\gamma } \in \mathbb {Q}^{2n}$, called impact vector, such that $\kappa ^\circ = \kappa _{\vec {\gamma }}$.

The impacts $\gamma _i^+, \gamma _i^-$ are values that are added to the rank of a world $\omega$ if $\omega$ verifies or falsifies $(B_i|A_i)$, respectively. The idea of c-changes is based on the principle of conditional preservation; a detailed motivation and explanation is given in [13, 14].

3 Contractions, Revisions, and Ignorations

In this section, we will define the belief change operations contraction, revision, and ignoration on ranking functions for sets of conditionals. Furthermore, we will discuss the realization of these belief change operations with c-changes.

Definition 2

(Revision $\kappa *\varDelta$). A belief change operator $*$ is a revision operator if for any ranking function $\kappa$ and consistent set of conditionals $\varDelta$ we have $\kappa *\varDelta \models (B|A)$ for all $(B|A) \in \varDelta$.

A belief revision introduces new information to the epistemic state and changes the existing knowledge to resolve conflicts. A contraction operator on the other hand removes beliefs from the epistemic state.

Definition 3

(Contraction $\kappa -\varDelta$). A belief change operator $-$ is a contraction operator if for any ranking function $\kappa$ and set of conditionals $\varDelta$ that does not contain self-fulfilling conditionals we have $\kappa -\varDelta \not \models (B|A)$ for all $(B|A) \in \varDelta$.

Note that there are several approaches to contraction of multiple statements. Definition 3 represents a “package contraction” approach. While c-contractions for single conditionals and c-change for sets of conditionals in general have been investigated before [16], c-contractions for sets of conditionals have not been considered so far. A third kind of belief change is ignoration. While ignoration was introduced for single conditionals in [6, 16], it can be defined in a more general way for sets of conditionals.

Definition 4

(Ignoration $\kappa \div \varDelta$). A belief change operator $\div$ is an ignoration operator if for any ranking function $\kappa$ and set of conditionals $\varDelta$ that does not contain self-fulfilling or contradictory conditionals, we have $(\kappa \div \varDelta ) \not \models (B|A)$ and $(\kappa \div \varDelta ) \not \models (\overline{B}|A)$ for all $(B|A) \in \varDelta$.

Thus, an ignoration “forgets” both the conditionals and their counter-conditionals. The three operations defined above are called c-revision, c-contraction, and c-ignoration, respectively, if they are c-changes (Definition 1). Similar to c-representations and c-inference they can each be characterized by a constraint satisfaction problem (CSP), cf. [4].

Definition 5

($CR ^{*}(\kappa , \varDelta ), CR ^{-}(\kappa , \varDelta ), CR ^{\div }(\kappa , \varDelta )$). Let $\kappa$ be a ranking function and $\varDelta = \{(B_1|A_1), \dots , (B_n|A_n)\}$ be a set of conditionals. The constraint satisfaction problem $CR ^{\circ }(\kappa , \varDelta )$ with $\circ \in \{*, -, \div \}$ for constraint variables $\gamma _1^+, \gamma _1^-, \dots , \gamma _n^+, \gamma _n^-$ taking values in $\mathbb {Q}$ is given by the set of constraints

for $i = 1, \dots , n$ where $\sim _\circ$ is > for $\circ = *$, or $\leqslant$ for $\circ = -$, or $=$ for $\circ = \div$.

The CSP for c-revisions is given by $CR ^{*}(\kappa , \varDelta )$, the CSP for c-contractions is given by $CR ^{-}(\kappa , \varDelta )$, and the CSP for c-ignorations is given by $CR ^{\div }(\kappa , \varDelta )$.

The set of solutions of a CSP $CR$ is denoted by $Sol ( CR )$.

Proposition 1

(Soundness and completeness of $CR ^{*}(\kappa , \varDelta )$, $CR ^{-}(\kappa , \varDelta )$ and $CR ^{\div }(\kappa , \varDelta )$). Let $\kappa$ be a ranking function and $\varDelta = \{(B_1|A_1), \dots , (B_n|A_n)\}$ be a set of conditionals.

• 1. If $\vec {\gamma } \in Sol ( CR ^{*}(\kappa , \varDelta )$, then the change from $\kappa$ with $\varDelta$ to $\kappa _{\vec {\gamma }}$ is a c-revision. Conversely, if $\varDelta$ is consistent and the change from $\kappa$ with $\varDelta$ to $\kappa ^*$ is a c-revision, then there is a $\vec {\gamma } \in Sol ( CR ^{*}(\kappa , \varDelta ))$ such that $\kappa ^*= \kappa _{\vec {\gamma }}$.

• 2. If $\vec {\gamma } \in Sol ( CR ^{-}(\kappa , \varDelta ))$, then the change from $\kappa$ with $\varDelta$ to $\kappa _{\vec {\gamma }}$ is a c-contraction. If $\varDelta$ does not contain self-fulfilling conditionals and the change from $\kappa$ with $\varDelta$ to $\kappa ^-$ is a c-contraction, there is a $\vec {\gamma } \in Sol ( CR ^{-}(\kappa , \varDelta ))$ such that $\kappa ^-= \kappa _{\vec {\gamma }}$.

• 3. If $\vec {\gamma } \in Sol ( CR ^{\div }(\kappa , \varDelta ))$, the change from $\kappa$ with $\varDelta$ to $\kappa _{\vec {\gamma }}$ is a c-ignoration. If $\varDelta$ does not contain self-fulfilling or contradictory conditionals and the change from $\kappa$ with $\varDelta$ to $\kappa ^\div$ is a c-ignoration, there is a $\vec {\gamma } \in Sol ( CR ^{\div }(\kappa , \varDelta ))$ such that $\kappa ^\div = \kappa _{\vec {\gamma }}$.

Proof

A proof for (1.) is given in [4, 14], and (2) and (3.) can be shown by analogous derivations.    $\square$

All three mentioned types of c-change exist if not prohibited by $\varDelta$.

Proposition 2

Let $\kappa$ be a finite ranking function and $\varDelta$ be a set of conditionals.

• 1. $Sol ( CR ^{*}(\kappa , \varDelta )) \ne \emptyset$ iff $\varDelta$ is consistent.

• 2. $Sol ( CR ^{-}(\kappa , \varDelta )) \ne \emptyset$ iff $\varDelta$ does not contain self-fulfilling conditionals.

• 3. $Sol ( CR ^{\div }(\kappa , \varDelta )) \ne \emptyset$ iff $\varDelta$ does not contain self-fulfilling or contradictory conditionals.

Proof

We first consider the $\Leftarrow$-direction in the three statements. The theorems given in [14] imply (1.). For proving (2.) and (3.) let $\kappa$ be a ranking function and $\varDelta = \{(B_1|A_1), \dots , (B_n|A_n)\}$ a set of non-self-fulfilling conditionals. The impacts of a c-contraction can be constructed by the following algorithm.

If all conditionals in $\varDelta$ are neither self-fulfilling nor contradictory, then the algorithm yields a c-ignoration. This approach shows that c-contraction and c-ignoration operators exist.

For the $\Rightarrow$-direction, if $\varDelta$ is not consistent, there is no ranking function accepting all conditionals in $\varDelta$. Therefore, (1.) holds. If $\varDelta$ contains self-fulfilling conditionals, there is no ranking function that contracts all conditionals in $\varDelta$. Therefore, (2.) holds. Analogously, (3.) holds because there is no ranking function that can ignore a self-fulfilling or a contradictory conditional.    $\square$

All change operations in this section can be applied to a set of formulas $\{A_1, \dots , A_n\}$ by representing the formulas with conditionals $(A_1|\top ), \dots , (A_n|\top )$.

4 Syntax Splitting on Ranking Functions

The concept of syntax splitting and corresponding postulates for belief change were originally developed by Parikh [20] for belief revision on belief sets. The basic idea is that for a belief set that contains independent information over different parts of the signature, the revision with a formula that contains only variables from one of such parts should only affect the information about this part of the signature. The notion of syntax splitting was later extended to other representations of epistemic states such as ranking functions [17]. Considering that Parikh’s (P) is incompatible with the Darwiche-Pearl-Postulates [8] as stated in [2, 21], it might seem problematic to investigate the combination of syntax splitting and frameworks for iterated belief revision. But while (P) only focusses on the belief set, the syntax splitting postulates for ranking functions considered here require syntax splittings on the whole ranking function. Therefore, the mentioned incompatibility results do not apply here.

Definition 6

(syntax splitting for ranking functions [17]). Let $\varSigma$ be a signature and $\kappa$ a ranking function over $\varOmega = \mathrm {Int}(\varSigma )$. Let $\omega ^j$ be the variable assignment of the variables in $\varSigma _j \subseteq \varSigma$ as in $\omega$. A partitioning $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ is a syntax splitting for $\kappa$ if there are ranking functions $\kappa _i: \varSigma _i \mapsto \mathbb {N}_0$ for $i = 1, \dots , n$ such that $\kappa (\omega ) = \kappa _1(\omega ^1) + \dots + \kappa _n(\omega ^n)$, denoted as $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$.

The following proposition shows that syntax splitting for ranking functions respects conditional knowledge.

Proposition 3

Let $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ be a ranking function with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $r = (B|A)$ a conditional with $A, B \in \varSigma _j$ for any $j \in \{1, \dots , n\}$. Then $\kappa \models (B|A)$ iff $\kappa _j \models (B|A)$.

Proof

Let $\kappa$ and $(B|A)$ be as in the proposition. Because of $\kappa (\omega ) = \sum _{1 \leqslant i \leqslant n} \kappa _i(\omega _i)$ we have that $\kappa (C) = \kappa _i(C)$ if $C \in \varSigma _i$ for $i = 1, \dots , n$. Therefore, it holds that .    $\square$

For the definition of some syntax splitting postulates for ranking functions, the concept of the marginalisation of a ranking function is important. Marginalisation formalizes the restriction of a ranking function to a sub-signature.

Definition 7

(marginalisation on ranking functions [3, 17]). Let $\varSigma$ be a signature and $\kappa$ be an OCF over $\varOmega = \mathrm {Int}(\varSigma )$. Let $\varTheta \subseteq \varSigma$. The marginalisation of $\kappa$ to $\varTheta$ is the function $\kappa {}_{|\varTheta }: \varTheta \mapsto \mathbb {N}_0$ with $\kappa {}_{|\varTheta }(\omega ) = \kappa (\omega )$ for $\omega \in \varOmega _\varTheta$.

For an OCF $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ it holds that $\kappa {}_{|\varSigma _i} =\kappa _i$ for $i = 1, \dots , n$. Note that the marginalization of OCFs presented above is a special case of a general forgetful functor $Mod (\varrho )$ from $\varSigma$-models to $\varSigma '$-models given in [3] where $\varSigma ' \subseteq \varSigma$ and $\varrho$ is the inclusion from $\varSigma '$ to $\varSigma$. Informally, a forgetful functor forgets everything about the interpretation of the symbols in $\varSigma {\setminus }\varSigma '$ when mapping a $\varSigma$-model to a $\varSigma '$-model.

All syntax splitting postulates for ranking functions proposed so far can only deal with revision or contraction of a ranking function with a set of formulas. The postulate (MR$^{ocf}$) describes that a revision of an OCF with syntax splitting should only depend on the relevant part of the OCF and the relevant formula.

Postulate

($\mathbf {MR}^{ocf}$) ([17]). Let $*$ be a revision operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $C =\{C_1, \dots , C_n\}$ such that $C_i \in \mathrm {Form}(\varSigma _i)$ for $i = 1, \dots , n$ it holds that $(\kappa *C){}_{|\varSigma _i} = \kappa {}_{|\varSigma _i} *C_i = \kappa _i *C_i \quad {\text {for}}\,\,$i = 1, ..., n$.$

Another postulate ($\text {P}^{it-ocf}$) states that a syntax splitting of a ranking function should survive a revision under certain circumstances.

Postulate

(${\mathbf {P}^{it-ocf}}$) ([11]). Let $*$ be a revision operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $C =\{C_1, \dots , C_n\}$ such that $C_i \in \mathrm {Form}(\varSigma _i)$ for $i = 1, \dots , n$ the partitioning $\varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ is a syntax splitting for $\kappa *C$.

Both postulates can be combined into one postulate. It can be shown that (MR$^{ocf}$) and ($\text {P}^{it-ocf}$) together are equivalent to (P$^{ocf}$).

Postulate

($\mathbf {P}^{ocf}$) ([17]). Let $*$ be a revision operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $C =\{C_1, \dots , C_n\}$ such that $C_i \in \mathrm {Form}(\varSigma _i)$ for $i = 1, \dots , n$ it holds that $\kappa *C = (\kappa _1 *C_1) \oplus \dots \oplus (\kappa _n *C_n)$.

These postulates have been transferred to contractions of OCFs [11].

Postulate

(${\mathbf {P}^{ocf}_-}$) ([11]). Let $-$ be a contraction operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $C =\{C_1, \dots , C_n\}$ such that $C_i \in \mathrm {Form}(\varSigma _i)$ for $i = 1, \dots , n$ it holds that $\kappa -C = (\kappa _1 -C_1) \oplus \dots \oplus (\kappa _n -C_n)$.

Postulate

(${\mathbf {P}^{it-ocf}_-}$) ([11]). Let $-$ be a contraction operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $C =\{C_1, \dots , C_n\}$ such that $C_i \in \mathrm {Form}(\varSigma _i)$ for $i = 1, \dots , n$ the partition $\varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ is a syntax splitting for $\kappa -C$.

Postulate

(${\mathbf {MK}^{ocf}}$) ([11]). Let $-$ be a contraction operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $C =\{C_1, \dots , C_n\}$ such that $C_i \in \mathrm {Form}(\varSigma _i)$ for $i = 1, \dots , n$ it holds that $(\kappa -C){}_{|\varSigma _i} =\kappa _i -C_i = \kappa {}_{|\varSigma _i} -C_i \,\, \text {for }i = 1, \dots , n$.

A contraction operator $-$ fulfils ($\text {P}^{ocf}_-$) iff it fulfils ($\text {MK}^{ocf}$) and ($\text {P}^{it-ocf}_-$). As the postulates for revision and contraction are structurally similar, we can generalize these postulates to cover both revisions and contractions with sets of conditionals. Furthermore, the following generalized postulate ($\text {P}^{\textit{ocf}}_\circ$) also fully covers ignorations by set of conditionals.

Postulate

(${\mathbf {P}^{{\varvec{ocf}}}_\circ }$). Let $\circ$ be a revision, contraction, or ignoration operator on ranking functions with sets of conditionals. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma _i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$ it holds that $\kappa \circ \varDelta = (\kappa _1 \circ \varDelta _1) \oplus \dots \oplus (\kappa _n \circ \varDelta _n)$.

The postulate ($\text {P}^{\textit{ocf}}_\circ$) is a generalisation of (P$^{ocf}$) and ($\text {P}^{ocf}_-$) in several ways: First, as mentioned, ($\text {P}^{\textit{ocf}}_\circ$) covers both revision and contraction, and furthermore, also ignorations. Second, ($\text {P}^{\textit{ocf}}_\circ$) allows for revision, contraction, and ignorations with respect to conditionals instead of formulas. This is a generalization, as a change with a formula $A$ can be realized by a change with the conditional $(A|\top )$. Third, ($\text {P}^{\textit{ocf}}_\circ$) covers changes where the number of partitions in the knowledge base does not equal the number of conditionals in the set that the knowledge base is changed with. Similarly, we can generalize (MR$^{ocf}$) and ($\text {MK}^{ocf}$) to ($\text {MR}^{ocf}_\circ$) as well as ($\text {P}^{it-ocf}$) and ($\text {P}^{it-ocf}_-$) to ($\text {P}^{it-ocf}_\circ$).

Postulate

(${\mathbf {MR}^{ocf}_\circ }$). Let $\circ$ be a revision, contraction, or ignoration operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma _i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$ it holds that $(\kappa \circ \varDelta ){}_{|\varSigma _i} = \kappa {}_{|\varSigma _i} \circ \varDelta _i = \kappa _i \circ \varDelta _i\,\, {\text {for}}\$i = 1, ..., n$.$

Postulate

(${\mathbf {P}^{it-ocf}_\circ }$). Let $\circ$ be a revision, contraction, or ignoration operator on ranking functions. For every ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma _i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$ the partitioning $\varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ is a syntax splitting for $\kappa \circ C$.

Proposition 4

A revision, contraction, or ignoration operator fulfils ($\text {P}^{\textit{ocf}}_\circ$) iff it fulfils both ($\text {MR}^{ocf}_\circ$) and ($\text {P}^{it-ocf}_\circ$).

Proof

The direction $\varvec{\Rightarrow }$ is clear.

Direction $\varvec{\Leftarrow }$: Let $\circ$ be a change operator that fulfils ($\text {MR}^{ocf}_\circ$) and ($\text {P}^{it-ocf}_\circ$). Let $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ be a ranking function with syntax splitting $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma _i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$. ($\text {P}^{it-ocf}_\circ$) implies that $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ is a syntax splitting for $\kappa ^\circ = \kappa \circ \varDelta$, i.e., there are ranking functions $\kappa _1^\circ , \dots , \kappa _n^\circ$ such that $\kappa ^\circ = \kappa _1^\circ \oplus \dots \oplus \kappa _n^\circ$. ($\text {MR}^{ocf}_\circ$) implies that $\kappa _i^\circ = \mathrel {{\kappa ^\circ }{}_{|\varSigma _i}} = \kappa _i \circ \varDelta _i$ for $\i = 1, \dots , n$. Therefore, $\circ$ fulfils ($\text {P}^{\textit{ocf}}_\circ$).    $\square$

Fig. 1.

Overview of the syntax splitting postulates for ranking functions mentioned in this paper. The arrows indicate that a postulate or a combination of postulates implies another postulate.

An overview of the different conditionals is given in Fig. 1.

5 Selection Strategies for c-Changes

For a given ranking function $\kappa$ and a set of conditionals $\varDelta$, the definition of c-changes does not determine the output of the change. In fact, the constraint systems $CR ^{*}(\kappa , \varDelta )$, $CR ^{-}(\kappa , \varDelta )$, and $CR ^{\div }(\kappa , \varDelta )$ may have multiple solutions that lead to different outcomes of the belief change. A c-revision, c-contraction, or c-ignoration operator has to select one of the possible solutions. A similar situation occurs when a c-representation is determined [4]. In [15], the selection of an impact vector for c-representations is formalized by introducing selection strategies that select one of the possible solutions of a constraint system as an impact vector. Selection strategies for c-revisions are introduced in [5].

We adapt this idea to the case of c-changes here.

Definition 8

(selection strategy). A selection strategy for c-revisions (c-contractions, or c-ignorations, respectively) is a function $\sigma : (\kappa , \varDelta ) \mapsto \vec {\gamma }$ mapping an OCF $\kappa$ and a set of conditionals $\varDelta$ to an impact vector $\vec {\gamma }$ such that $\vec {\gamma } \in Sol ( CR ^{*}(\kappa , \varDelta ))$ ($\vec {\gamma } \in Sol ( CR ^{-}(\kappa , \varDelta ))$, or $\vec {\gamma } \in Sol ( CR ^{\div }(\kappa , \varDelta ))$, resp.).

The selection of a solution takes both $\varDelta$ and $\kappa$ into account, as the sets $Sol ( CR ^{*}(\kappa , \varDelta ))$, $Sol ( CR ^{-}(\kappa , \varDelta ))$, and $Sol ( CR ^{\div }(\kappa , \varDelta ))$ of possible solutions depend on $\varDelta$ and $\kappa$. Each selection strategy induces a change operator.

Definition 9

($*_{\sigma }$, $-_{\sigma }$, $\div _{\sigma }$). The c-revision (c-contraction, or c-ignoration, respectively) of a ranking function $\kappa$ with a set of conditionals $\varDelta$ induced by a selection strategy $\sigma$, denoted by $\kappa *_{\sigma } \varDelta$ (or $\kappa -_{\sigma } \varDelta$ or $\kappa \div _{\sigma } \varDelta$, respectively) is given by $\kappa _{\vec {\gamma }}$ with $\vec {\gamma } = \sigma (\kappa , \varDelta )$.

Now we can formalize desirable properties of selection strategies. A natural property is that the impacts chosen for two independent subproblems should be preserved when choosing impacts for the combination of the two subproblems.

Definition 10

(impact preserving, $( IP ^{ cc })$). A selection strategy $\sigma$ for c-revisions, c-contractions, or c-ignorations fulfils $( IP ^{ cc })$ and is called impact preserving if for any ranking function $\kappa = \kappa _1 \oplus \kappa _2$ with syntax splitting $\varSigma _1 \dot{\cup } \varSigma _2$ and set of conditionals $\varDelta = \varDelta _1 \cup \varDelta _2$ such that $A, B \in \mathrm {Form}(\varSigma _i)$ for every $(B|A) \in \varDelta _i$, $i = 1,2$ it holds that $\sigma (\kappa , \varDelta ) = (\sigma (\kappa _1, \varDelta _1), \sigma (\kappa _2, \varDelta _2)).$

While $( IP ^{ cc })$ is defined only for syntax splittings with two partitions, it implies the described property also for syntax splittings with more partitions.

Proposition 5

If a selection strategy $\sigma$ for c-revisions, c-contractions, or c-ignorations fulfils $( IP ^{ cc })$, then for any ranking function $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ with syntax splitting $\varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ and set of conditionals $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ such that $A, B \in \mathrm {Form}(\varSigma _i)$ for every $(B|A) \in \varDelta _i$, $i = 1, \dots , n$ it holds that:

$$\begin{aligned} \sigma (\kappa , \varDelta ) = (\sigma (\kappa _1, \varDelta _1), \dots , \sigma (\kappa _n, \varDelta _n)). \end{aligned}$$

(1)

Proof

Let $\sigma$ be a selection strategy for c-revisions, c-contractions, or c-ignorations that fulfils $( IP ^{ cc })$. Let $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ be a ranking function with a syntax splitting $\varSigma = \varSigma _1 \dot{\cup } \dots \dot{\cup } \varSigma _n$. Let $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma _i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$. The proof is by induction over $n \geqslant 2$.

Base case:   For $n=2$ this is $( IP ^{ cc })$.

Induction step:   For $n > 2$ we have that $\varSigma = \varSigma _1 \dot{\cup } (\bigcup _{2 \leqslant l \leqslant n}\varSigma _l)$ is a syntax splitting for $\kappa = \kappa _1 \oplus (\bigoplus _{2 \leqslant l \leqslant n} \kappa _l)$. The induction hypothesis implies that $\sigma (\bigoplus _{2 \leqslant l \leqslant n} \kappa _l, \bigcup _{2 \leqslant l \leqslant n}\varDelta _l) = (\sigma (\kappa _2, \varDelta _2), \dots , \sigma (\kappa _n, \varDelta _n))$ because $\bigcup _{2 \leqslant l \leqslant n}\varSigma _l$ is a syntax splitting for $\bigoplus _{2 \leqslant l \leqslant n} \kappa _l$. Because of $( IP ^{ cc })$, we know that

$$\begin{aligned} \sigma (\kappa , \varDelta ) =&\big (\sigma (\kappa _1, \varDelta _1), \sigma (\bigoplus _{1 \leqslant l \leqslant n}\kappa _l, \bigcup _{1 \leqslant l \leqslant n}\varSigma _l) \big )\\ \overset{\begin{array}{c} \text {induction}\\ \text {hypothesis} \end{array}}{=}&(\sigma (\kappa _1, \varDelta _1), \sigma (\kappa _2, \varDelta _2), \dots , \sigma (\kappa _n, \varDelta _n)). \end{aligned}$$

   $\square$

In the next section, we show how $( IP ^{ cc })$ relates to syntax splitting.

6 Selection Strategies and Syntax Splitting

In this section, we will connect property $( IP ^{ cc })$ on selection strategies with ($\text {P}^{\textit{ocf}}_\circ$) for belief change.

Proposition 6

(${( IP ^{ cc })}$ ensures (${\mathbf {P}^{{\varvec{ocf}}}_\circ }$)). If a selection strategy $\sigma$ for revision, contractions, or ignorations, respectively, fulfils $( IP ^{ cc })$, then the induced c-revision $*_{\sigma }$, c-contraction $*_{\sigma }$, or c-ignoration $*_{\sigma }$, respectively, fulfils ($\text {P}^{\textit{ocf}}_\circ$).

Proof

Let $\sigma$ be a selection strategy for c-revisions that fulfils $( IP ^{ cc })$. Let $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ be a ranking function with a syntax splitting $\varSigma = \varSigma _1 \dot{\cup } \dots \dot{\cup } \varSigma _n$. Let $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma _i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$. Let $\gamma = (\gamma _{1,1}^+, \gamma _{1,1}^-, \dots , \gamma _{n, k_n}^+, \gamma _{n, k_n}^-) = \sigma (\kappa , \varDelta )$. Let $\kappa ^* = \kappa *_{\sigma } \varDelta$. We have:

$$\begin{aligned} \kappa ^*(\omega )&= \kappa _0 + \kappa (\omega ) + \sum _{\begin{array}{c} 1 \leqslant i \leqslant n \\ 1\leqslant j \leqslant k_i \\ \omega \models A_{i,j} B_{i,j} \end{array}} \gamma _{i,j}^+ + \sum _{\begin{array}{c} 1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} \overline{B_{i,j}} \end{array}} \gamma _{i,j}^- \nonumber \\&= \kappa _0 + \sum _{1 \leqslant i \leqslant n} \kappa _i(\omega ) + \sum _{\begin{array}{c} 1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} B_{i,j} \end{array}} \gamma _{i,j}^+ + \sum _{\begin{array}{c} 1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} \overline{B_{i,j}} \end{array}} \gamma _{i,j}^- \nonumber \\&= \kappa _0 + \sum _{1 \leqslant i \leqslant n} \Big (\kappa _i(\omega ) + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} B_{i,j} \end{array}} \gamma _{i,j}^+ + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} \overline{B_{i,j}} \end{array}} \gamma _{i,j}^- \Big ) \end{aligned}$$

(2)

Now, consider a revision of $\kappa _i$ with $\varDelta _i$. This is revision is properly defined, as $\kappa _i$ is defined for the signature $\varSigma _i$ and $\varDelta _i$ only contains variables from $\varSigma _i$. Because of Proposition 5, we know that $\sigma (\kappa _i, \varDelta _i) = (\gamma _{i,1}^+, \gamma _{i,1}^-, \dots , \gamma _{i,k_i}^+, \gamma _{i,k_i}^-)$. Let $\kappa _i^* = \kappa _i *_{\sigma } \varDelta _i$. We have $\kappa _i^*(\omega ) = \kappa _{i,0} + \kappa _i(\omega ) + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} B_{i,j} \end{array}} \gamma _{i,j}^+ + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} \overline{B_{i,j}} \end{array}} \gamma _{i,j}^-$. The combination of all $\kappa _i^*$ is

$$\begin{aligned}&(\bigoplus _{1 \leqslant i \leqslant n} \kappa _i^*) (\omega ) = \sum _{1 \leqslant i \leqslant n} \kappa _i^*(\omega _i) \nonumber \\ =&\sum _{1 \leqslant i \leqslant n} \Big (\kappa _{i,0} + \kappa _i(\omega _i) + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega _i \models A_{i,j} B_{i,j} \end{array}} \gamma _{i,j}^+ + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega _i \models A_{i,j} \overline{B_{i,j}} \end{array}} \gamma _{i,j}^-\Big ) \nonumber \\ \overset{*}{=}&\underbrace{\sum _{1 \leqslant i \leqslant n} \kappa _{i,0}}_{\kappa _0} + \sum _{1 \leqslant i \leqslant n} \Big (\kappa _i(\omega _i) + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} B_{i,j} \end{array}} \gamma _{i,j}^+ + \sum _{\begin{array}{c} 1 \leqslant j \leqslant k_i \\ \omega \models A_{i,j} \overline{B_{i,j}} \end{array}} \gamma _{i,j}^-\Big ) \end{aligned}$$

(3)

with $\omega = \omega _1 \dots \omega _n$. Equation $*$ holds because $A_{i,j}, B_{i,j} \in \mathrm {Form}(\varSigma _i)$ and therefore $\omega _i \models A_{i,j} B_{i,j}$ iff $\omega \models A_{i,j} B_{i,j}$ for any $i = 1, \dots , n, j = 1, \dots , k_n$.

Now, let us compare term (3) with (2). They are identical except for the term $\kappa _0$ or $\sum _{1 \leqslant i \leqslant n} \kappa _{i,0}$. We know that (3) is a ranking function, because it is the combination of the ranking functions $\kappa _1^*, \dots , \kappa _n^*$. As $\kappa _0$ is chosen such that (2) is a ranking function, we know that $\kappa _0 = \sum _{1 \leqslant i \leqslant n} \kappa _{i,0}$ and that (2) and (3) are equal. This implies that $*_{\sigma }$ fulfils ($\text {P}^{\textit{ocf}}_\circ$). Taking the variations in $CR ^{-}(\kappa , \varDelta )$ and $CR ^{\div }(\kappa , \varDelta )$ into account, the proofs for $-_{\sigma }$ and $\div _{\sigma }$ are similar.    $\square$

We illustrate the connection between $( IP ^{ cc })$ and ($\text {P}^{\textit{ocf}}_\circ$).

Example 1

Let $\kappa$ be the ranking function over $\varSigma =\{a, b\}$ illustrated in Fig. 2. $\kappa = \kappa _1 \oplus \kappa _2$ has a syntax splitting $\{a\} \mathbin {\dot{\cup }}\{b\}$. Let $r_1 = (a|\top ), r_2 = (b|\top )$ and $\varDelta = \{r_1, r_2\}$. Let $\sigma _1, \sigma _2$ be two selection strategies for contraction with

$$\begin{aligned} \sigma _1(\kappa , \varDelta )&= (0, -2, 0, -2)&\sigma _1(\kappa _1, \{r_1\})&= (0, -2)&\sigma _1(\kappa _2, \{r_2\})&= (0, -1)\\ \sigma _2(\kappa , \varDelta )&= (0, -2, 0, -1)&\sigma _2(\kappa _1, \{r_1\})&= (0, -2)&\sigma _2(\kappa _2, \{r_2\})&= (0, -1). \end{aligned}$$

Thus, $\gamma _1^+ = \gamma _2^+ = 0$ in these six impact vectors. The selection strategy $\sigma _2$ fulfils $( IP ^{ cc })$, the selection strategy $\sigma _1$ does not fulfil $( IP ^{ cc })$. The contractions induced by $\sigma _1$ and $\sigma _2$ are displayed in Fig. 2. We can see that $-_{\sigma _1}$ does not fulfil ($\text {MR}^{ocf}_\circ$) while $-_{\sigma _2}$ fulfils ($\text {P}^{\textit{ocf}}_\circ$).

Fig. 2.

Comparison of the selection strategies $\sigma _1$ and $\sigma _2$ from Example 1. The OCFs $\kappa$, $\kappa _1$, and $\kappa _2$ from Example 1 are displayed in the first row (Figs. 2a to 2c). The boxes represent worlds; their vertical alignment corresponds to their rank. The further up a box is placed, the higher its rank is. The second row (Figs. 2d to 2f) illustrates the results of contractions of these OCFs using selection strategy $\sigma _1$. All impacts $\vec {\gamma }$ are given in the caption of the Figs. 2d to 2i. Note that for the normalization constant $\kappa _0$, in Fig. 2d we have $\kappa _0 = 1$, and $\kappa _0 = 0$ in all other cases. The third row (Figs. 2g to 2i) illustrates contractions of these OCFs using selection strategy $\sigma _2$.

Finally, it is left to show that selection strategies exists that fulfil $( IP ^{ cc })$.

Proposition 7

There are selection strategies for contraction, ignoration, and revision that fulfil $( IP ^{ cc })$.

Proof

We first prove the statement about revisions by constructing a selection strategy for belief revision. Let $\kappa$ be a ranking function over $\varSigma$ and $\varDelta$ be a set of conditionals in $\mathrm {Cond}(\varSigma )$. We can distinguish two cases.

Case 1: $\kappa$ and $\varDelta$ have no common syntax splitting.    In this case we choose $\sigma (\kappa , \varDelta )$ arbitrarily from $Sol ( CR ^{*}(\kappa , \varDelta ))$. This is allowed as $( IP ^{ cc })$ does not state anything about such situations. Proposition 2 ensures that there is at least one impact vector to choose.

Case 2: $\kappa$ and $\varDelta$ have a common syntax splitting.    Let $\varSigma = \varSigma _1 \mathbin {\dot{\cup }}\dots \mathbin {\dot{\cup }}\varSigma _n$ be the finest common splitting of $\kappa$ and $\varDelta$, i.e., $\kappa = \kappa _1 \oplus \dots \oplus \kappa _n$ and $\varDelta = \varDelta _1 \cup \dots \cup \varDelta _n$ with $\varDelta _i = \{(B_{i,1}|A_{i,1}), \dots , (B_{i,k_i}|A_{i,k_i})\}$ with $A_{i, j}, B_{i, j} \in \mathrm {Form}(\varSigma '_i)$ for $j = 1, \dots , k_i$ for every $i = 1, \dots , n$. Choose impacts $\sigma (\kappa _i, \varDelta _i)$ for each of the revisions $\kappa _i *\varDelta _i$ as in Case 1. Let $\sigma (\kappa , \varDelta ) = (\sigma (\kappa _1, \varDelta _1), \dots , \sigma (\kappa _n, \varDelta _n))$ according to $( IP ^{ cc })$ and Eq. (1). We have $\kappa -_{\sigma } \varDelta = (\kappa _1 -_{\sigma } \varDelta _1) \oplus \dots \oplus (\kappa _n -_{\sigma } \varDelta _n)$. The revised ranking function $\kappa -_{\sigma } \varDelta$ accepts $\varDelta$ because of Proposition 3.

The proofs for contractions and ignorations can be done analogously.    $\square$

An implication of Proposition 7 is that there are revisions, contractions, and ignorations that fulfil ($\text {P}^{\textit{ocf}}_\circ$), and therefore also the specific postulates (P$^{ocf}$) and ($\text {P}^{ocf}_-$) in the case of revisions and contractions, respectively. Thus, all three belief change operations are fully compatible with syntax splitting, while no such operators having this property have been known before.

7 Conclusion

We generalized syntax splitting postulates from [17] and [11]. Our new generalized postulates cover not only the iterated contraction, ignoration, and revision of ranking functions, representing the epistemic state of an agent, with sets of formulas, but also with sets of conditionals. Using selection strategies for c-changes, we showed that all contractions, ignorations, and revisions fulfil the generalized postulates (and therefore the original postulates) if they are induced by a selection strategy that fulfils the newly developed property $( IP ^{ cc })$.

Our current work includes generalizing the concept of selection strategies to further belief changes like, for instance, complex belief change operations based on descriptor revision [12] over a conditional logic [10, 24].