Comparison of Inference Relations Defined over Different Sets of Ranking Functions

Abstract

Skeptical inference in the context of a conditional knowledge base $\mathcal R$ can be defined with respect to a set of models of $\mathcal R$. For the semantics of ranking functions that assign a degree of surprise to each possible world, we develop a method for comparing the inference relations induced by different sets of ranking functions. Using this method, we address the problem of ensuring the correctness of approximating c-inference for $\mathcal R$ by constraint satisfaction problems (CSPs) over finite domains. While in general, determining a sufficient upper bound for these CSPs is an open problem, for a sequence of simple knowledge bases investigated only experimentally before, we prove that using the number of conditionals in $\mathcal R$ as an upper bound correctly captures skeptical c-inference.

1 Introduction

For a knowledge base $\mathcal R$ containing conditionals of the form If A then usually B, various semantics have been proposed, e.g. [4, 9]. Here, we will consider the approach of ranking functions (or Ordinal Conditional Functions (OCF) [10]), assigning a degree of surprise to each possible world. The models of $\mathcal R$ are then OCFs accepting all conditionals in $\mathcal R$, and every OCF model of $\mathcal R$ induces a nonmonotonic inference relation (e.g. [4, 9, 10]). For any set $O$ of models of $\mathcal R$, skeptical inference with respect to $O$ takes all elements of $O$ into account. C-representations are particular ranking functions exibiting desirable inference properties [7], and c-inference is skeptical inference with respect to all c-representations of $\mathcal R$ [1].

The two main objectives of this paper are (1) to develop an approach for comparing the inference relations with respect to two different sets of OCFs $O$ and $O'$, and (2) to illustrate how this approach can be used for proving that in the context of c-representations [7], particular upper bounds in a finite domain constraint system are sufficient for correctly modeling skeptical c-inference [1] so that only a subset of all c-representations have to be taken into account.

For checking that the inference relations with respect to $O$ and $O'$ are identical, we introduce the notion of merged order compatibility and show that it suffices to check that their inference cores coincide if $O$ and $O'$ are merged order compatible. We demonstrate that there are knowledge bases $\mathcal R$ such that the set of all ranking modes of $\mathcal R$ is not merged order compatible, while at the same time the set of all c-representations of $\mathcal R$ is merged order compatible. We then investigate how this approach can be employed for c-representations [7] and skeptical c-inference [1]. For the sequence of knowledge bases $\mathcal R_n$ considered in [2] that contain only conditional facts of the form $(a|\top )$ we formally prove upper bounds that are sufficient for skeptical c-inference. This indicates that the concepts developed here may be helpful for addressing the open problem of determining upper bounds for general knowledge bases $\mathcal R$ that are sufficient for modelling skeptical c-inference for $\mathcal R$.

2 Background: Conditional Logic and OCFs

Let $\varSigma =\{v_1,...,v_m\}$ be a propositional alphabet. A literal is the positive ($v_i$) or negated ($\overline{v_i}$) form of a propositional variable, $\dot{v}_i$ stands for either $v_i$ or $\overline{v_i}$. From these we obtain the propositional language $\mathcal {L}$ as the set of formulas of $\varSigma$ closed under negation $\lnot$, conjunction $\wedge$, and disjunction $\vee$. For shorter formulas, we abbreviate conjunction by juxtaposition (i.e., AB stands for $A\wedge B$), and negation by overlining (i.e., $\overline{A}$ is equivalent to $\lnot A$). Let $\varOmega _{\varSigma }$ denote the set of possible worlds over $\mathcal {L}$; $\varOmega _{\varSigma }$ will be taken here simply as the set of all propositional interpretations over $\mathcal {L}$ and can be identified with the set of all complete conjunctions over $\varSigma$; we will often just write $\varOmega$ instead of $\varOmega _{\varSigma }$. For $\omega \in \varOmega$, $\omega \,\models \, A$ means that the propositional formula $A \in \mathcal {L}$ holds in the possible world $\omega$. For any propositional formula A let $\varOmega _A = \{\omega \in \varOmega \mid \omega \,\models \, A\}$ be the set of all possible worlds satisfying A.

A conditional (B|A) with $A,B\in \mathcal {L}$ encodes the defeasible rule “if A then usually B” and is a trivalent logical entity with the evaluation [5, 7]

An Ordinal Conditional Function (OCF, ranking function) [10] is a function $\kappa :\varOmega \rightarrow \mathbb {N}_0\cup \{\infty \}$ that assigns to each world $\omega \in \varOmega$ an implausibility rank $\kappa (\omega )$: the higher $\kappa (\omega )$, the more surprising $\omega$ is. OCFs have to satisfy the normalization condition that there has to be a world that is maximally plausible, i.e., $\kappa ^{-1}(0)\ne \emptyset$. The rank of a formula A is defined by $\kappa (A)=\min \{\kappa (\omega ) \mid \omega \,\models \, A\}$. An OCF $\kappa$ accepts a conditional $(B|A)$, denoted by $\kappa \,\models \,(B|A)$, iff the verification of the conditional is less surprising than its falsification, i.e., iff $\kappa (AB) < \kappa (A\overline{B})$. This can also be understood as a nonmonotonic inference relation between the premise A and the conclusion B: We say that A $\kappa$ -entails B, written , iff $\kappa$ accepts the conditional $(B|A)$: $\kappa \,\models \, (B|A)$ iff $\kappa (AB) \ < \ \kappa (A\overline{B})$ iff

Note that $\kappa$-entailment is based on the total preorder on possible worlds induced by a ranking function $\kappa$ as iff for all $\omega ' \in \varOmega _{A\overline{B}}$, there is a $\omega \in \varOmega _{AB}$ such that $\kappa (\omega ) < \kappa (\omega ')$.

The acceptance relation is extended as usual to a set $\mathcal R$ of conditionals, called a knowledge base, by defining $\kappa \,\models \, \mathcal R$ iff $\kappa \,\models \, (B|A)$ for all $(B|A) \in \mathcal R$. This is synonymous to saying that $\kappa$ is admissible with respect to $\mathcal R$ [6], or that $\kappa$ is a ranking model of $\mathcal R$; the set of all ranking models of $\mathcal R$ is denoted by $Mod(\mathcal R)$. $\mathcal R$ is consistent iff it has a ranking model.

3 Skeptical Inference and Merged Order Inference

While each OCF $\kappa$ accepting $\mathcal R$ induces a nonmonotonic inference relation, also each set O of such ranking functions induces an inference relation determined by taking all elements of O into account.

Definition 1

(skeptical inference). Let $\mathcal R$ be a knowledge base, $O \subseteq Mod(\mathcal R)$, and $A,B \in \mathcal L$. Skeptical Inference over O in the context of $\mathcal R$, denoted by , is defined by

Thus, holds if every $\kappa \in O$ accepts $(B|A)$. The skeptical inference relations defined over two different sets of OCFs may be identical. Instead of having to check the acceptance of all possible conditionals $(B|A)$ with respect to both sets of OCFs, we will investigate conditions under which it suffices to check only so-called base conditionals.

Definition 2

(base conditional). A base conditional over the signature $\varSigma$ is a conditional of the form $(\omega _1|\omega _1 \vee \omega _2)$ with $\omega _1$, $\omega _2 \in \varOmega _{\varSigma }$ and $\omega _1 \ne \omega _2$.

Note that a base conditional $(\omega _1|\omega _1 \vee \omega _2)$ is accepted by a ranking model $\kappa$, iff $\kappa (\omega _1) < \kappa (\omega _2)$. To characterize the behavior of an inference relation for these base conditionals, we define the inference core of an inference relation as the reduction of from pairs of formulas to pairs of possible worlds.

Definition 3

(inference core, ). Let be an inference relation. The inference core of , denoted by , is the set of all pairs $(\omega _1,\omega _2) \in \varOmega \times \varOmega$ with $\omega _1 \ne \omega _2$, such that , i.e., .

The notion of the inference core is based on an inference relation. The corresponding concept of a merged order is based solely on a set of ranking models:

Definition 4

(merged order, $<_{O}$ ). Let O be a set of OCFs. The merged order $<_{O}$ is given by $<_{O}\;= \{ (\omega _1,\omega _2) \mid \omega _1 \ne \omega _2, \kappa (\omega _1) < \kappa (\omega _2) \text{ for } \text{ all } \kappa \in O\}$.

Note that in general $<_{O}$ is a strict weak ordering, i.e. it is irreflexive, asymmetric, and transitive. The inference core of skeptical inference over a set of OCFs O coincides with the merged order induced by O:

Proposition 1

(inference core and merged order). For any knowledge base $\mathcal R$ and any set $O \subseteq Mod(\mathcal R)$ it holds that .

Proof

We now define an inference relation with respect to $<_{O}$ in a similar way to inference with respect to the total pre-order on worlds induced by an OCF.

Definition 5

(inference relation induced by merged order, ). Let $\mathcal R$ be a knowledge base, $O \subseteq Mod(\mathcal R)$, and $A,B \in \mathcal L$. Then

Proposition 2

For any two sets of ranking models O and $O'$ of $\mathcal R$ it holds that if $<_{O} \; = \; <_{O'}$ then .

The inference relation induced by the merged order of a set of OCFs O approximates skeptical inference over O.

Proposition 3

For any knowledge base $\mathcal R$ and $O \subseteq Mod(\mathcal R)$ it holds that

(1)

Proof

While it is always the case that an inference over the merged order of a set O is also a skeptical inference over that set, the other direction of (1) does not hold in general.

Proposition 4

There is a knowledge base $\mathcal R$ and a set $O \subseteq Mod(\mathcal R)$ with

(2)

Proof

Consider $\mathcal R= \{(a|\top )\}$ over $\varSigma = \{a,b\}$. Let $\kappa _1$ and $\kappa _2$ be defined as:

$$\begin{aligned} \kappa _1(\omega ) = {\left\{ \begin{array}{ll} 0 &{} \text{ if } \omega = ab \\ 1 &{} \text{ otherwise } \end{array}\right. } \;\;\;\;\;\;\; \kappa _2(\omega ) = {\left\{ \begin{array}{ll} 0 &{} \text{ if } \omega = a\overline{b} \\ 1 &{} \text{ otherwise } \end{array}\right. } \end{aligned}$$

Both $\kappa _1$ and $\kappa _2$ accept $\mathcal R$, but for $O = \{\kappa _1,\kappa _2\}$ it holds that $<_{O}= \emptyset .$ Thus, since both OCFs accept $\mathcal R$ it holds that , but since $<_{O}$ is empty,    $\square$

Since (1) holds for all sets of OCF models, but the reverse direction does not hold in general, we introduce the notion of merged order compatibility, classifying the sets of OCFs for which the other direction of (1) holds.

Definition 6

(merged order compatible). Let $\mathcal R$ be a knowledge base and $O \subseteq Mod(\mathcal R)$. O is called merged order compatible iff .

Thus, for merged order comaptible $O$ we immediately get:

Proposition 5

If $O \subseteq Mod(\mathcal R)$ is merged order compatible, then .

Since the merged order of a set of ranking models is equal to the inference core of the skeptical inference over that set of models, merged order compatibility ensures that equivalence of skeptical inference relations coincides with equivalence of inference cores.

Proposition 6

For any two merged order compatible sets of ranking models O and $O'$ of a knowledge base $\mathcal R$ it holds that:

(3)

Proof

The direction from right to left trivially holds since base conditionals are a subset of all conditionals. For the other direction we have:

$\square$

Note that according to Proposition 6, merged order compatibility provides a sufficient condition for reducing the question of skeptical inference equivalence to the equality of the inference cores.

4 C-Inference and Merged Order Compatibility

We will now illustrate merged order compatibility for a special kind of ranking models. C-Representations are special ranking models of a knowledge base $\mathcal R$, obtained by assigning individual impacts to the conditionals in $\mathcal R$. The rank of a possible world is then defined as the sum of impacts of falsified conditionals.

Definition 7

(c-representation [7, 8]). A c-representation of a knowledge base $\mathcal R$ is a ranking function $\kappa$ constructed from integer impacts $\eta _{i}\in \mathbb {N}_0$ assigned to each conditional $(B_i|A_i)$ such that $\kappa$ accepts $\mathcal R$ and is given by:

$$\begin{aligned} \kappa (\omega ) =\sum \limits _{\begin{array}{c} 1 \leqslant i \leqslant n\\ \omega \models A_i\overline{B}_i \end{array}}\eta _{i} \end{aligned}$$

(4)

Every c-representation exibits desirable inference properties, and two c-representations induce the same inference relation if they induce the same total pre-order on worlds. In [1], a modeling of c-representations as solutions of a constraint satisfaction problem $CR (\mathcal R)$ is given and shown to be correct and complete with respect to the set of all c-representations of $\mathcal R$. Recently, it has been suggested to take inferential equivalence of c-representations into account and to sharpen $CR (\mathcal R)$ by introducing an upper bound for the impact values $\eta _{i}$.

Definition 8

( $\varvec{ CR ^{u}(\mathcal R)}$ [3]). Let $\mathcal R= \{(B_1|A_1),\ldots ,(B_n|A_n)\}$ and $u\in \mathbb {N}$. The finite domain constraint satisfaction problem $CR ^{u}(\mathcal R)$ on the constraint variables $\{\eta _{1}, \ldots , \eta _{n}\}$ ranging over $\mathbb {N}$ is given by the conjunction of the constraints, for all $i \in \{1,\ldots ,n\}$:

$$\begin{aligned}&\eta _{i} \geqslant 0 \end{aligned}$$

(5)

$$\begin{aligned}&\eta _{i} > \min _{\omega \models A_i B_i} \sum _{\begin{array}{c} j \ne i \\ \omega \models A_j \overline{B_j} \end{array}} \eta _{j} - \min _{\omega \models A_i \overline{B}_i} \sum _{\begin{array}{c} j \ne i \\ \omega \models A_j \overline{B_j} \end{array}} \eta _{j} \end{aligned}$$

(6)

$$\begin{aligned}&\eta _{i} \leqslant u\end{aligned}$$

(7)

A solution of $CR ^{u}(\mathcal R)$ is an $n$-tuple $(\eta _{1}, \ldots , \eta _{n})$ of natural numbers, its set of solutions is denoted by $Sol ( CR ^{u}(\mathcal R))$. For and $\kappa$ as in Eq. (4), $\kappa$ is the OCF induced by , denoted by , and the set of all induced OCFs is denoted by . The constraint satisfaction problem $CR (\mathcal R)$, given in [1], is obtained by removing the constraints (7) from $CR ^{u}(\mathcal R)$.

C-inference is skeptical inference over the set of all c-representations.

Definition 9

(c-inference, [1]). Let $\mathcal R$ be a knowledge base and let A, B be formulas. B is a (skeptical) c-inference from A in the context of $\mathcal R$, denoted by , iff holds for all c-representations $\kappa$ for $\mathcal R$.

We will now illustrate c-representations, c-inference, and how the inference over the merged order of the set of all c-representations accepting a knowledge base can coincide with c-inference in the context of that knowledge base.

Example 1

( $\mathcal R_{lw}$ ). Consider $\varSigma _{lw}= \left\{ l,w\right\}$ and $\mathcal R_{lw}= \left\{ r_1,r_2,r_3\right\}$ with

representing some default knowledge about vehicles in a country like Germany.

Table 1. Verification ($v$), falsification ($f$), impacts ($\eta _{i}$), solution vectors , induced OCFs , and merged order of for $CR (\mathcal R_{lw})$ in Example 1.

Using the verification and falsification behavior of the four possible worlds reveals that as given in Table 1 are solutions to $CR (\mathcal R_{lw})$. Furthermore, there are no other solutions of $CR (\mathcal R_{lw})$ inducing an ordering on worlds that is different from every of the orderings induced by ; for example, the solution induces the same ordering on worlds as and thus allows for exactly the same inferences.

Therefore, the merged order for , given in the lower right corner of Table 1, coincides with the merged order over all c-representations of $\mathcal R_{lw}$. Checking all pairs of formulas over $\varSigma _{lw}$ shows that for $\mathcal R_{lw}$ there is no difference between merged order inference over O and skeptical c-inference.

The following example illustrates an interesting difference between the set of all ranking models of a knowledge base and the set of its c-representations and shows that there are knowledge bases $\mathcal R$ such that the former set is not merged order compatible while the latter set is merged order compatible.

Example 2

Consider the knowledge base $\mathcal R$ and $\varSigma$ from the proof of Proposition 4, and let $P = Mod(\mathcal R)$ and let O be the set of all c-representations accepting $\mathcal R$. For both P and O, a can be inferred skeptically from $\top$ in the context of $\mathcal R$, i.e. and . The two ranking functions $\kappa _1$ and $\kappa _2$ used in the proof of Proposition 4 both accept $\mathcal R$ and are thus elements of P. Since there are no two distinct worlds $\omega$ and $\omega '$ with $\kappa _1(\omega ) < \kappa _1(\omega ')$ and $\kappa _2(\omega ) < \kappa _2(\omega ')$, the merged order $<_{P}$ is empty, and therefore On the other hand, for every c-representation $\kappa$ accepting $\mathcal R$ it holds that $\kappa (ab) = \kappa (a\overline{b})$ and $\kappa (\overline{a}b) = \kappa (\overline{a}\overline{b})$ and $\kappa (a\dot{b}) < \kappa (\overline{a}\dot{b})$. Thus, $<_{O} = \left\{ (ab,\overline{a}b),(ab,\overline{a}\overline{b}),(a\overline{b},\overline{a}b),(a\overline{b},\overline{a}\overline{b})\right\}$ and hence . In fact, the set O of all c-representations accepting $\mathcal R$ is merged order compatible, while and shows that the set P of all ranking models of $\mathcal R$ is not merged order compatible.

For studying the exact relationship between $CR (\mathcal R)$ and $CR ^{u}(\mathcal R)$, the concept of a sufficient $CR ^{u}(\mathcal R)$ was introduced in [3] to capture the idea that only a finite number of c-representations is needed for modeling c-inference.

Definition 10

(sufficient). Let $\mathcal R$ be a knowledge base and let $u\in \mathbb {N}$. Then $CR ^{u}(\mathcal R)$ is called sufficient (for skeptical inference) iff for all formulas $A,B$ we have

If $CR ^{u}(\mathcal R)$ is sufficient, we will also call $u$ sufficient for $\mathcal R$.

In terms of the classical skeptical inference relation over a set of ranking models given in Definition 1 this means that $CR ^{u}(\mathcal R)$ is sufficient iff

(8)

For various $\mathcal R$ and $u$, we will now use merged order compatibility for proving (8).

5 Proving Sufficient Upper Bounds

In this section, we continue the investigation from [2] and use the concepts from the previous section to formally prove an experimental result from [2].

Definition 11

( $\varSigma _{n}, \mathcal R_{n}$ ). For $n \geqslant 1$ and $\varSigma _{n} = \{a_1,\ldots ,a_n\}$, $\mathcal R_{n} = \{(a_1|\top ),\ldots ,(a_n|\top )\}$ is called the knowledge base of n conditional facts.

Note that from the constraints in $CR (\mathcal R_{n})$ and $CR ^{u}(\mathcal R_{n})$ it follows that for all impacts in c-representations accepting $\mathcal R_{n}$ it holds that $\eta _i \geqslant 1$. In the rest of this section, we investigate how the concepts of merged order compatibility and inference cores can be used to prove that for $\mathcal R_{n}$ the CSP $\mathcal {O}( CR ^{n-1}(\mathcal R_{n}))$ is indeed sufficient. In [2] this was solely illustrated by means of some examples.

Because the structure of knowledge bases $\mathcal R_{n}$ is very simple, the rank of a world $\omega$ over $\varSigma _{n}$ assigned by a c-representation depends on the set of falsified atoms in $\omega$ in a very predictable way.

Definition 12

( $f(\omega )$, $<_f$ ). For , is the set of indices of the negated literals in $\omega$. The ordering $<_f$ on $\varOmega _{\varSigma _{n}}$ is defined such that for two worlds $\omega , \omega ' \in \varOmega _{\varSigma _{n}}$, $\omega <_f\omega ' \text{ iff } f (\omega ) \subsetneqq f (\omega ')$.

As the ordering $<_{O}$ on worlds, also $<_f$ induces an inference relation.

Definition 13

( ). For $n > 1$ and formulas $A,B \in \mathcal L_{\varSigma _{n}}$

The following proposition generalizes a proposition from [2] regarding the ranking of worlds $\omega$ and $\omega '$ incomparable in $<_f$.

Proposition 7

Let $n > 1$, $\omega ' \in \varOmega _{\varSigma _{n}}$ and $\varOmega _V= \left\{ \omega _1,\ldots ,\omega _m \right\} \subseteq \varOmega _{\varSigma _{n}}$. If for all $i \in \left\{ 1,\ldots ,m\right\}$, $f (\omega ') \not \subseteq f (\omega _i)$ and $f (\omega _i) \not \subseteq f (\omega '),$ then there exists a c-representation $\kappa$ accepting $\mathcal R_{n}$ such that for all $i \in \left\{ 1,\ldots ,m\right\}$, $\kappa (\omega ') \leqslant \kappa (\omega _i).$

Proof

Let $I$ be $I = (\bigcup _{i = 1}^{m} f (\omega _i)) \setminus f (\omega ')$. Note that because of the precondition $f (\omega ') \not \subseteq f (\omega _i)$ and $f (\omega _i) \not \subseteq f (\omega '),$ it holds that $I \ne \emptyset$. Let with

$$ \eta _i = {\left\{ \begin{array}{ll} 1 &{} i \not \in I\\ n-1 &{} i \in I \end{array}\right. } $$

Since for every $i \in f (\omega ')$ the impact vector assigns 1 to the corresponding conditional $(a_i|\top ) \in \mathcal R_{n}$ and because we know that $\omega ' \ne \overline{a_1}\ldots \overline{a_n}$, we get Because I is not empty, for every $i \in \left\{ 1,\ldots ,m\right\}$, there is some $k \in f (\omega _i)$ such that $\eta _k = n-1$. Thus, we get Therefore, it holds that for every $i \in \left\{ 1,\ldots ,m\right\}$.    $\square$

We now use Proposition 7 to show that the inference relation defined over the ordering on worlds $<_f$ is equal to the skeptical inference over all c-representations accepting $\mathcal R_{n}$.

Proposition 8

For $n > 1$ and $O = \mathcal {O}( CR (\mathcal R_n))$,

Proof

Let A and B be arbitrary formulas from $\mathcal L_{\varSigma _{n}}$. If then for all $\omega ' \in \varOmega _{A\overline{B}}$ there is a $\omega \in \varOmega _{AB}$ such that $\omega <_f\omega '$. Thus $f (\omega ) \subsetneqq f (\omega ')$, and because $\kappa (\omega )$ for a c-representation $\kappa$ is defined by the sum of all impacts of negative literals in $\omega$, it also holds that $\kappa (\omega ) < \kappa (\omega ')$ for $\kappa \in O$. Thus holds for all $\kappa \in O$, implying that

To show the other direction, we assume that and show that If , then there is a world $\omega ' \in \varOmega _{A\overline{B}}$, such that for all worlds $\omega \in \varOmega _{AB}$ $\omega \not <_f\omega '$ holds. If $\omega ' <_f\omega ,$ then $\kappa (\omega ') \leqslant \kappa (\omega )$ for every c-representation $\kappa$ with $\kappa \,\models \, \mathcal R_n$ and therefore If $\omega ' \not <_f\omega$ then for all worlds $\omega \in \varOmega _{AB}$ $f (\omega ') \not \subseteq f (\omega )$ and $f (\omega ) \not \subseteq f (\omega '),$ and we use Proposition 7 by setting $\varOmega _V= \varOmega _{AB}$ and construct a c-representation $\kappa$ such that $\kappa (\omega ') \leqslant \kappa (\omega )$ for all $\omega \in \varOmega _{AB}$. Thus, $min\left\{ \kappa (\omega ) \mid \omega \,\models \, A\overline{B}\right\} \leqslant \min \left\{ \kappa (\omega ) \mid \omega \,\models \, AB\right\}$, implying and therefore    $\square$

Since both $<_f$ and $<_{O}$ are orderings of worlds and and are defined in the same way, it is now straightforward to show that $\mathcal {O}( CR (\mathcal R_n))$ is merged order compatible for any $\mathcal R_{n}$.

Proposition 9

For $n > 1$, $\mathcal {O}( CR (\mathcal R_n))$ is merged order compatible for $\mathcal R_{n}$.

Proof

To show that $O = \mathcal {O}( CR (\mathcal R_n))$ is merged order compatible for $\mathcal R_{n}$, we need to show that Since we already know (Proposition 8), due to Proposition 2 it suffices to show that $<_f= \;<_{O}.$ If $\omega <_f\omega '$, then $f (\omega ) \subsetneqq f (\omega ')$. As was already pointed out in the proof of Proposition 8, this means that for all c-representations $\kappa$ we have $\kappa (\omega ) < \kappa (\omega ')$ and thus $\omega <_{O}\omega '.$ We now have and $\mathcal {O}( CR (\mathcal R_n))$ is merged order compatible.    $\square$

Since we do not make use of impacts $\eta _i > n-1$, the proofs of Propositions 8 and 9 also work for $O = \mathcal {O}( CR ^{n-1}(\mathcal R_{n}))$, implying:

Proposition 10

For $n\! > \!1$, $\mathcal {O}( CR ^{n-1}(\mathcal R_{n}))$ is merged order compatible for $\mathcal R_{n}$.

These results now enable us to prove that $n-1$ is sufficient for $\mathcal R_{n}$, implying that the inference relation induced by the solutions of $CR ^{n-1}(\mathcal R_{n})$ is equal to the skeptical inference over all c-representations for $\mathcal R_{n}$.

Proposition 11

For $n\!> \!1$, $CR ^{n-1}(\mathcal R_{n})$ is sufficient for $\mathcal R_{n}$.

Proof

We need to show that . Since both $\mathcal {O}( CR ^{n-1}(\mathcal R_n))$ and $\mathcal {O}( CR (\mathcal R_n))$ are merged order compatible, it suffices to show that the inference cores are equal, i.e. .

It is easy to see that if a pair of possible worlds $(\omega , \omega ')$ is in , then it is also in since $\,|\sim \,^{\mathcal {O}( CR ^{n-1}(\mathcal R_n))}_{\mathcal R_{n}}$ allows for possibly more inferences. To show the other direction, we assume that and show that .

If $(\omega , \omega ')$ is not in the inference core of the unbounded skeptical c-inference, it means that there is a c-representation $\kappa$ in which $\kappa (\omega ) \geqslant \kappa (\omega ').$ If $f (\omega ') \subseteq f (\omega )$, then for it holds that If $f (\omega ) \subseteq f (\omega ')$, then there is no c-representation $\kappa$ such that $\kappa (\omega ) \geqslant \kappa (\omega ')$, contradicting the assumption. If neither $f (\omega ') \subseteq f (\omega )$ nor $f (\omega ) \subseteq f (\omega ')$ holds, the precondition of Proposition 7 is met for $\omega '$ and $\varOmega _V= \left\{ \omega \right\}$, and we can construct a c-representation $\kappa$ in $\mathcal {O}( CR ^{n-1}(\mathcal R_n))$ such that $\kappa (\omega ) \geqslant \kappa (\omega ')$; hence .    $\square$

6 Conclusions and Further Work

We introduced the notion of inference core of a nonmonotonic inference relation taking only so called base conditionals into account. By showing that a set of ranking models is merged order compatible, we can reduce the question of equality of inference relations to equivalence of inference cores. We illustrated arising differences between the set of all ranking models of a knowledge base $\mathcal R$ and the set of all c-representations of $\mathcal R$, and we applied our approach to skeptical c-inference for proving that for certain knowledge bases a maximal impact of $|\mathcal R|-1$ is sufficient to fully capture the behavior of skeptical c-inference.

In our current work, we employ the concepts of inference cores and merged order compatibility for extending our investigations on sufficient upper bounds for $CR (\mathcal R)$ to more general kinds of knowledge bases, and for addressing the open problems of characterizing knowledge bases whose set of c-representations is merged order compatible or whether this property holds for all knowledge bases. This goes along with finding a suitable characterization of merged order compatible sets of ranking models, and exploring relationships to approaches employing e.g. possibilistic or probabilistic semantics.