A Practical Comparison of Qualitative Inferences with Preferred Ranking Models
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Abstract
When reasoning qualitatively from a conditional knowledge base, two established approaches are system Z and pentailment. The latter infers skeptically over all ranking models of the knowledge base, while system Z uses the unique paretominimal ranking model for the inference relations. Between these two extremes of using all or just one ranking model, the approach of crepresentations generates a subset of all ranking models with certain constraints. Recent work shows that skeptical inference over all crepresentations of a knowledge base includes and extends pentailment. In this paper, we follow the idea of using preferred models of the knowledge base instead of the set of all models as a base for the inference relation. We employ different minimality constraints for crepresentations and demonstrate inference relations from sets of preferred crepresentations with respect to these constraints. We present a practical tool for automatic cinference that is based on a highlevel, declarative constraintlogic programming approach. Using our implementation, we illustrate that different minimality constraints lead to inference relations that differ mutually as well as from system Z and pentailment.
Keywords
Conditional logic Qualitative conditional Default rule Ranking function Crepresentation Cinference System Z Pentailment1 Introduction
In the area of knowledge representation and reasoning, rules play a prominent role, especially default rules of the form “If A then usually/normally/preferably B”. Sets of such rules, so called knowledge bases, are used to represent the knowledge of a reasoning agent, and the inference relation of the agent depends on this knowledge. A knowledge base usually is incomplete to such an extent that it contains all conditional rules relevant to the agent, but it usually does not contain enough information to represent all preferences, beliefs, and assumptions of the agent, that is, an epistemic state in the sense of [11]. Here, inductive methods come into play that construct a model of the knowledge base. Such models can be representations from various formalisms, encoding, for instance, the probability [20], the (im)possibility [8], or the (im)plausibility [22, 23] of the possible worlds. Based on these models that inductively complete the knowledge given explicitly by the rules of a knowledge base, corresponding inductive inference relations can be constructed.
In this paper, we focus on models based on plausibility as defined by Ordinal Conditional Functions [22, 23] (OCF, also known as ranking functions). Established approaches of inductive inference using OCF include the skeptical inference over all ranking models of a knowledge base, known as pentailment [9], and the inference with the unique, with respect to ranks of worlds, paretominimal OCF with this property, known as System Z [21].
In [12, 13] a criterion when a ranking function respects the conditional structure of a set \({\mathcal{R}}\) of conditionals is defined, leading to the notion of crepresentation for \({\mathcal{R}}\), and it is argued that ranking functions defined by crepresentations are of particular interest for modelbased inference. It has been shown that reasoning inductively with a single crepresentation yields an inference relation of high quality (cf., e.g., [14, 24]). Recent work also shows that the skeptical inference over all crepresentations (called cinference) includes and extends pentailment [2]. To define the inference relation, in this paper, we follow the idea of [18] as well as [21] by using a set of preferred models of the knowledge base \({\mathcal{R}}\) instead of using of the set of all models. We employ different minimality constraints for crepresentations and demonstrate inference relations from sets of preferred crepresentations with respect to these constraints.
The main objective of this paper is to present and illustrate these inference relations and to provide a practical tool for automatic inference and for comparison of the inference results. For this tool, we employ the observation that the definition of crepresentations as solutions of a constraint satisfaction problem \( CR ({\mathcal{R}})\) (see [2, 4]) allows to implement crepresentations in a highlevel, declarative approach using constraint logic programming techniques. In particular, this approach also supports the generation of all minimal solutions, providing a preferred basis for modelbased inference from \({\mathcal{R}}\); previously, no other implementation of minimal cinference has been available.
This article is a revised and largely extended version of [4]. In particular, in this paper we add the comparison to system Z and to pentailment, extend and refine the notions of minimality, introduce corresponding inferences relations, and present a newly developed implementation for computing and comparing different inference relations.
The rest of this paper is organized as follows: After recalling the formal background of conditional logics as far as it is needed here (Sect. 2), we elaborate an illustration for a conditional knowledge base and discuss resulting inference relations based on OCFs in Sect. 3. In Sect. 4, we recall the inductive approaches of System Z and crepresentations and present the constraint satisfaction problem \( CR ({\mathcal{R}})\) whose solutions are computed by the declarative, highlevel CLP program GenOCF (Sect. 5). Section 6 introduces three different notions of minimality for crepresentations, and, in Sect. 7, an implementation of the corresponding inference relations based on GenOCF is presented. Section 8 concludes the paper and points out further work.
2 Background
We start with a propositional language \({\mathcal{L}}\), generated by a finite set \(\Sigma \) of atoms \(a,b,c, \ldots \). The formulas of \({\mathcal{L}}\) will be denoted by uppercase Roman letters \(A,B,C, \ldots \). For conciseness of notation, we will omit the logical andconnective, writing AB instead of \(A \wedge B\), and overlining formulas will indicate negation, i.e. \(\overline{A}\) means \(\lnot A\). Let \(\Omega \) denote the set of possible worlds over \({\mathcal{L}}\); \(\Omega \) 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 \(\Sigma \). For \(\omega \in \Omega \), \(\omega \models A\) means that the propositional formula \(A \in {\mathcal{L}}\) holds in the possible world \(\omega \).
By introducing a new binary operator , we obtain the set \( {({\mathcal{L}}\mid {\mathcal{L}})}= \{ (BA) \mid A,B \in {\mathcal{L}}\} \) of conditionals over \({\mathcal{L}}\). \((B  A)\) formalizes the conditional rule “if A then (normally) B” and establishes a plausible, probable, possible etc. connection between the antecedent A and the consequence B. Here, conditionals are supposed not to be nested, that is, antecedent and consequent of a conditional will be propositional formulas.
Wellknown qualitative, ordinal approaches to represent epistemic states are Spohn’s ordinal conditional functions, OCFs, (also called ranking functions) [22], and possibility distributions [6], assigning degrees of plausibility, or of possibility, respectively, to formulas and possible worlds. In such qualitative frameworks, a conditional (BA) is valid (or accepted), if its confirmation, AB, is more plausible, possible, etc. than its refutation, \({A\overline{B}}\); a suitable degree of acceptance is calculated from the degrees associated with AB and \({A\overline{B}}\).
We call a conditional \((BA)\) with \(A \models B\) selffulfilling since it can not be falsified by any world. Obviously, such conditionals are meaningless from a modeling point of view, and we will not consider them in the following. A set \({\mathcal{R}}\subseteq ({\mathcal{L}}{\mathcal{L}})\) of conditionals is called a knowledge base if it does not contain any selffulfilling conditional. An OCF \(\kappa \) accepts a knowledge base if and only if \(\kappa \) accepts all conditionals in \({\mathcal{R}}\); such an OCF is called a (ranking) model of \({\mathcal{R}}\). A knowledge base \({\mathcal{R}}\) is consistent iff a ranking model of \({\mathcal{R}}\) exists [21].
3 Inference and the Drowning Problem
In order to illustrate the concepts presented in the previous section, we will use a scenario involving a set of some default rules representing commonsense knowledge.
Example 1
Two ranking functions \(\kappa \) and \(\kappa _{{\mathcal{R}}_{ pen }}^Z(\omega )\) accepting the rule set \({\mathcal{R}}_{ pen }\) given in Example 1
\(\omega \)  \(\kappa (\omega )\)  \(\kappa _{{\mathcal{R}}_{ pen }}^Z(\omega )\)  \(\omega \)  \(\kappa (\omega )\)  \(\kappa _{{\mathcal{R}}_{ pen }}^Z(\omega )\) 

pbfwk  2  2  \(\overline{p}bfwk\)  0  0 
\(pbfw\overline{k}\)  2  2  \(\overline{p}bfw\overline{k}\)  0  0 
\(pbf\overline{w}k\)  3  2  \(\overline{p}bf\overline{w}k\)  1  1 
\(pbf\overline{w}\overline{k}\)  3  2  \(\overline{p}bf\overline{w}\overline{k}\)  1  1 
\(pb\overline{f}wk\)  1  1  \(\overline{p}b\overline{f}wk\)  1  1 
\(pb\overline{f}w\overline{k}\)  1  1  \(\overline{p}b\overline{f}w\overline{k}\)  1  1 
\(pb\overline{f}\overline{w}k\)  2  1  \(\overline{p}b\overline{f}\overline{w}k\)  2  1 
\(pb\overline{f}\overline{w}\overline{k}\)  2  1  \(\overline{p}b\overline{f}\overline{w}\overline{k}\)  2  1 
\(p\overline{b}fwk\)  5  2  \(\overline{p}\overline{b}fwk\)  1  1 
\(p\overline{b}fw\overline{k}\)  4  2  \(\overline{p}\overline{b}fw\overline{k}\)  0  0 
\(p\overline{b}f\overline{w}k\)  5  2  \(\overline{p}\overline{b}f\overline{w}k\)  1  1 
\(p\overline{b}f\overline{w}\overline{k}\)  4  2  \(\overline{p}\overline{b}f\overline{w}\overline{k}\)  0  0 
\(p\overline{b}\,\overline{f}wk\)  3  2  \(\overline{p}\overline{b}\,\overline{f}wk\)  1  1 
\(p\overline{b}\,\overline{f}w\overline{k}\)  2  2  \(\overline{p}\overline{b}\,\overline{f}w\overline{k}\)  0  0 
\(p\overline{b}\,\overline{f}\overline{w}k\)  3  2  \(\overline{p}\overline{b}\,\overline{f}\overline{w}k\)  1  1 
\(p\overline{b}\,\overline{f}\overline{w}\overline{k}\)  2  2  \(\overline{p}\overline{b}\,\overline{f}\overline{w}\overline{k}\)  0  0 
4 Inductive Reasoning
In Sect. 2 we recalled that a knowledge base is consistent if and only if there is a ranking model \(\kappa \) for the knowledge base, and Sect. 3 illustrated how to reason with such ranking models. This raises the question how to obtain such a ranking model, if it exists. Also, for any consistent \({\mathcal{R}}\) there may be many different \(\kappa \) accepting \({\mathcal{R}}\), each representing a complete set of beliefs with respect to every possible formula A and every conditional \((BA)\) which we also illustrated in Sect. 3. Thus, every such \(\kappa \) inductively completes the knowledge given by \({\mathcal{R}}\). In this section we recall two established inductive approaches, System Z and crepresentations.
4.1 System Z
The approach of System Z [21] sets up a ranking model of a knowledge base \({\mathcal{R}}\) by inclusionmaximal partitions of \({\mathcal{R}}\) with respect to the notion of tolerance:
Example 2
4.2 CRepresentations

If the world \(\omega \) verifies the conditional \((B_iA_i)\), – i.e., \( \omega \models A_i B_i \) –, then \(\eta _{i}^+\) is used in the summation to obtain the value \(\kappa (\omega )\).

Likewise, if \(\omega \) falsifies the conditional \((B_iA_i)\), – i.e., \( \omega \models A_i \overline{B_i} \) –, then \(\eta _{i}\) is used in the summation instead.

If the conditional \((B_iA_i)\) is not applicable in \(\omega \), – i.e., \( \omega \models \overline{A_i} \) –, then this conditional does not influence the value \(\kappa (\omega )\).
Definition 1
Note that for \(i \in \{1,\ldots ,n\}\), condition (9) expresses that \(\kappa \) accepts the conditional \(R_i = (B_iA_i) \in {\mathcal{R}}\) (cf. the definition of the satisfaction relation in (4)) and that this also implies \(\kappa (A_i) < \infty \).
Thus, finding a crepresentation for \({\mathcal{R}}\) amounts to choosing appropriate values \( \eta _{1}\), ..., \( \eta _{n} \). In [4] this situation is formulated as a constraint satisfaction problem \( CR ({\mathcal{R}})\) whose solutions are vectors of the form \( (\eta _{1}, \ldots , \eta _{n}) \) determining crepresentations of \({\mathcal{R}}\). The development of \( CR ({\mathcal{R}})\) exploits (2) and (8) to reformulate (9) and requires that the \(\eta _{i}\) are natural numbers (and not just rational numbers). In the following, we set \(\min (\emptyset ) = \infty \).
Definition 2
A solution of \( CR ({\mathcal{R}})\) is an ntuple \( (\eta _{1}, \ldots , \eta _{n}) \) of natural numbers, and with \( Sol _ {CR} ({\mathcal{R}})\) we denote the set of all solutions of \( CR ({\mathcal{R}})\).
Proposition 1
Example 3
Verification/falsification behavior of the knowledge base \({\mathcal{R}}'_{ pen }\) and possible worlds used in Example 3
\(\omega \)  verifies  falsifies  \(\omega \)  verifies  falsifies 

\(p\,b\,f\)  \(r_1\), \(r_2\)  \(r_3\)  \(\overline{p}\,b\,f\)  \(r_1\)  – 
\(p\,b\,\overline{f}\)  \(r_2\), \(r_3\)  \(r_1\)  \(\overline{p}\,b\,\overline{f}\)  –  \(r_1\) 
\(p\,\overline{b}\,f\)  –  \(r_2\), \(r_3\)  \(\overline{p}\,\overline{b}\,f\)  –  – 
\(p\,\overline{b}\,\overline{f}\)  \(r_3\)  \(r_2\)  \(\overline{p}\,\overline{b}\,\overline{f}\)  –  – 
Induced ranking function \(\kappa _{(1,2,2)}\) by the solution (1, 2, 2) of the constraints in \( CR ({\mathcal{R}}'_{ pen })\) from Example 3
\(\omega \)  \(\kappa _{(1,2,2)}(\omega )\)  \(\omega \)  \(\kappa _{(1,2,2)}(\omega )\) 

\(p\,b\,f\)  2  \(\overline{p}\,b\,f\)  0 
\(p\,b\,\overline{f}\)  1  \(\overline{p}\,b\,\overline{f}\)  1 
\(p\,\overline{b}\,f\)  4  \(\overline{p}\,\overline{b}\,f\)  0 
\(p\,\overline{b}\,\overline{f}\)  2  \(\overline{p}\,\overline{b}\,\overline{f}\)  0 
 1.
A world \(\omega \) that, ceteris paribus, falsifies less conditionals of a partition than a world \(\omega '\) is ranked to be more plausible, i.e., \(\kappa _{\vec {\eta }}(\omega )<\kappa _{\vec {\eta }}(\omega ')\), and
 2.
A world \(\omega \) that, ceteris paribus, falsifies a conditional of a partition with a lower impact than a world \(\omega '\) is ranked to be more plausible, i.e., \(\kappa _{\vec {\eta }}(\omega )<\kappa _{\vec {\eta }}(\omega ')\).
Example 4
Let \({\mathcal{R}}_{ abcd }=\{(ca),(cb),(\overline{c}d),(ad)\}\). The vector \(\vec {\eta }=(1,1,2,2)\) is a possible solution for the constraint satisfaction problem \( CR ({\mathcal{R}}_{ abcd })\) (cf. Definition 2). Thus, the solution vector \(\vec {\eta }\) partitions the conditionals into the sets \({\mathcal{R}}_0=\{(ca),(cb)\}\), each with an impact of 1, and \({\mathcal{R}}_1=\{(\overline{c}d),(ad)\}\), each with an impact of 2. Now lets consider the possible worlds, \(\omega =ab\overline{c}\overline{d}\), \(\omega '=a\overline{b}cd\) and \(\omega ''=a\overline{b}\overline{c}\overline{d}\).
The world \(\omega \) falsifies both conditionals in \({\mathcal{R}}_0\) and none in \({\mathcal{R}}_1\), and \(\omega '\) falsifies only one conditional in \({\mathcal{R}}_1\) but no other conditionals. The rank of both worlds with respect to \(\kappa _{\vec {\eta }}\) is 2, since \(\kappa _{\vec {\eta }}(\omega )=1+1=2\) and \(\kappa _{\vec {\eta }}(\omega ')=2\), so both worlds are considered equally (im)plausible with respect to this crepresentation. Applying system Z yields the same partitions but ranks of \(\kappa ^Z(\omega )=1\) and \(\kappa _Z(\omega ')=2\), so under system Z, \(\omega \) is considered more plausible than \(\omega '\). This valuation coincides with the lexicographic ordering in the sense of [16], where \(\omega '\) is considered less plausible than \(\omega \) since \(\omega '\) falsifies a conditional in set \({\mathcal{R}}_1\) and \(\omega \) does not.
For the worlds \(\omega \) and \(\omega ''\) we obtain that they are equivalent with respect to their system Z rank, since they both falsify conditionals in \({\mathcal{R}}_0\) and we have \(\kappa (\omega )=1=\kappa (\omega '')\), but for \(\kappa _{\vec {\eta }}\) we have \(\kappa _{\vec {\eta }}(\omega '')=1<2= \kappa _{\vec {\eta }}(\omega )\), so \(\omega ''\) is considered more plausible than \(\omega \). Also, since \(\omega \) falsifies more conditionals in \({\mathcal{R}}_0\) than \(\omega ''\) and no conditionals in a more severe partition, lexicographic ordering in the sense of [16] considers \(\omega ''\) to be more plausible than \(\omega \).
Thus, inference by crepresentations is, in general, different to inference by system Z or lexicographic ordering of the worlds in the sense of [16]. It shares the central ideas of a ceteris paribus ordering with the latter, and shares the property of overcoming the Drowing Problem found for System Z. Since the individual impacts are nonnegative integers, an impact of a conditional can be 0. Since the rank of the worlds is computed by a summation of the impacts of falsified conditionals, falsifying or not falsifying a conditional with a zero impact does not change the rank of a world, which is in accordance with this conditional’s effect already being realised by other conditionals in the knowledge base.
5 A Declarative CLP Program for \( CR ({\mathcal{R}})\)
In this section, we will demonstrate that it is possible to obtain a declarative program, called GenOCF, that solves \( CR ({\mathcal{R}})\) while exploiting the concepts of constraint logic programming in such a way that there is a direct correspondence between the abstract formulation of \( CR ({\mathcal{R}})\) and the executable program code. We will employ finite domain constraints, and from (10) we immediately get 0 as a lower bound for \(\eta _{i}\). Considering that we are interested mainly in minimal solutions, due to (10) we restrict ourselves to n as an upper bound for \(\eta _{i}\), yielding \( 0 \leqslant \eta _{i} \leqslant n \) for all \(i \in \{1,\ldots ,n\}\) with n being the number of conditionals in \({\mathcal{R}}\).
5.1 Input Format and Preliminaries
Example 5
5.2 Generation of Constraints and Solutions
After all constraints have been generated, the final subgoal of kappa/2 (Fig. 1) yields all solutions of \( CR ({\mathcal{R}})\).
Example 6
Using the predicates described in Sect. 5.1, we have presented the complete source code of the constraint logic program GenOCF solving \( CR ({\mathcal{R}})\). In Sect. 7, GenOCF extended to find minimal solutions of \( CR ({\mathcal{R}})\) (cf. [4]) will be used for computing inference relations induced by minimal OCF models of \({\mathcal{R}}\).
6 Minimal CRepresentations
All crepresentations built from (10), (11), and (12) provide an excellent basis for modelbased inference, for instance each crepresentation satisfies System P and none suffers from the drowning problem [12, 13, 14]. However, from the point of view of minimal specificity (see e.g. [6]), those crepresentations with minimal \(\eta _{i}\) yielding minimal degrees of implausibility are most interesting. In [10], an OCF \(\kappa \) accepting \({\mathcal{R}}\) is said to be minimal iff for every other \(\kappa '\) accepting \({\mathcal{R}}\) there exists a world \(\omega \in \Omega \) with \(\kappa (\omega ) < \kappa '(\omega )\). Since in this paper, our focus is on crepresentations, and since for any \({\mathcal{R}}\), the OCFs being crepresentations and accepting \({\mathcal{R}}\) are induced by the solutions of \( CR ({\mathcal{R}})\), we will consider different orderings on \( Sol _ {CR} ({\mathcal{R}})\) proposed in [3, 4], leading to three different minimality notions: The minimal accumulated impact of the conditionals (summinimality), the paretominimal impact of the conditionals (cwminimality), and the paretominimal ranking of the worlds in the induced ranking functions (indminimality).
Definition 3
As we are interested in minimal \(\eta _{i}\)vectors, an important question is whether there is always a unique minimal solution. This is not the case; the following example illustrates that \( Sol _ {CR} ({\mathcal{R}})\) may have more than one summinimal element.
Example 7
Instead of taking the sum of the \(\eta _{i}\), we can also consider the componentwise ordering \(\preccurlyeq _{ cw }\).
Definition 4
Example 8
The two summinimal solution vectors \(\vec {\eta }^{(1)}\) and \(\vec {\eta }^{(2)}\) for \({{\mathcal{R}}_{ birds }}\) from Example 7 are both also cwminimal.
Instead of defining an ordering directly in terms of the solution vectors in \( Sol _ {CR} ({\mathcal{R}})\) as done for \(\preccurlyeq _{+}\) and \(\preccurlyeq _{ cw }\), the following ordering on \( Sol _ {CR} ({\mathcal{R}})\) takes the ordering of the induced ranking functions into account.
Definition 5
Example 9
Consider again the knowledge base \({{\mathcal{R}}_{ birds }}\) from Example 7 and the two solution vectors \(\vec {\eta }^{(1)}\) and \(\vec {\eta }^{(2)}\). Table 4 shows the ranking functions induced by \(\vec {\eta }^{(1)}\) and \(\vec {\eta }^{(2)}\). While both \(\vec {\eta }^{(1)}\) and \(\vec {\eta }^{(2)}\) are summinimal and also cwminimal, only \(\vec {\eta }^{(2)}\) is indminimal because \(\kappa _{\vec {\eta }^{(2)}}(\overline{a}b\overline{f}) = 1 < 2 = \kappa _{\vec {\eta }^{(1)}}(\overline{a}b\overline{f})\) and \(\kappa _{\vec {\eta }^{(2)}}(\omega ) = \kappa _{\vec {\eta }^{(1)}}(\omega )\) for all \(\omega \) with \(\omega \not = \overline{a}b\overline{f}\).
Ranking functions induced by the solution vectors \(\vec {\eta }^{(1)}\) and \(\vec {\eta }^{(2)}\) from Example 7
\(\omega \)  \(\kappa _{\vec {\eta }^{(1)}}(\omega )\)  \(\kappa _{\vec {\eta }^{(2)}}(\omega )\)  \(\omega \)  \(\kappa _{\vec {\eta }^{(1)}}(\omega )\)  \(\kappa _{\vec {\eta }^{(2)}}(\omega )\) 

abf  0  0  \(\overline{a}bf\)  1  1 
\(ab\overline{f}\)  1  1  \(\overline{a}b\overline{f}\)  2  1 
\(a\overline{b}f\)  0  0  \(\overline{a}\overline{b}f\)  0  0 
\(a\overline{b}\overline{f}\)  0  0  \(\overline{a}\overline{b}\overline{f}\)  0  0 
Although for the knowledge base \({{\mathcal{R}}_{ birds }}\) there is a unique indminimal solution of \( CR ({{\mathcal{R}}_{ birds }})\), there are knowledge bases \({\mathcal{R}}\) with multiple indminimal solutions of \( CR ({\mathcal{R}})\) that induce different ranking functions accepting \({\mathcal{R}}\); examples of such knowledge bases are given in [3]. Note that this implies that an indminimal solution of \( CR ({\mathcal{R}})\) does not necessarily induce the unique paretominimal model of \({\mathcal{R}}\) with respect to the ranking of worlds generated with System Z (see Sect. 4.1 and [21]), underpinning the observation that crepresentations and System Z are different in general (see Example 4 and [14, 24]).
Proposition 2
Proof
For proving (20), assume there is a \(\vec {\eta }\in Sol _{\preccurlyeq _{ O }}^{ min }( CR ({\mathcal{R}}))\) with \(\vec {\eta }\not \in Sol _{\preccurlyeq _{ cw }}^{ min }( CR ({\mathcal{R}}))\). Then there is a \(\vec {\eta }'\in Sol _{\preccurlyeq _{ cw }}^{ min }( CR ({\mathcal{R}}))\) with \(\vec {\eta }'\preccurlyeq _{ cw }\vec {\eta }\) and \(\vec {\eta }'\ne \vec {\eta }\). From (14) we get \({\eta '_{i}} \leqslant \eta _{i}\) for all \(i \in \left\{ 1,\ldots ,n\right\} \) and \({\eta '_{s}} < \eta _{s}\) for some \(s \in \left\{ 1,\ldots ,n\right\} \).
7 CInference Based on Preferred Models
Figure 4 shows an example of InfOCF in use. The knowledge base \({\mathcal{R}}_{ pen }\) introduced in Example 1 has been loaded in the top left corner. The computed ranking functions are shown in the top right corner. The lower half of the UI is used for inference where the query is entered in two text fields for checking whether A entails B in the context of the given knowledge base; from these formulas the query conditional \((BA)\) is constructed. In addition to (skeptical) cinference \(\,\!\!{\sim} \,^{\!\!\bullet }\), InfOCF also implements credulous entailment where A credulously entails B (in the context of a knowledge base \({\mathcal{R}}\)) iff there is a \(\bullet \)minimal OCF accepting \({\mathcal{R}}\) that also accepts the conditional \((BA)\).
The results of the last query as well as of the previous queries are listed in the bottom right. The particular queries shown in Fig. 4 are already discussed in Example 1 and demonstrate the drowning problem observable in system Z and the difference between inference over all ranking models (system p) and system Z as well as the different minimal crepresentations.
Since the solutions for \( CR ({\mathcal{R}}_{ pen })\) which are sum, cw or indminimal, respectively, coincide, there is no difference in cinference for the three different notions of minimality. The following example demonstrates that this is not the case in general.
Example 10
For the conditional \((b\overline{p}\overline{f})\) observe that \(\kappa _{\vec {\eta }^{(1)}} \,\,\,{\models }_{\mathcal{O}}\!\!\!\!\!\!\!/ \,\;\; \, (b\overline{p}\overline{f})\) since \(\kappa _{\vec {\eta }^{(1)}}(\overline{p}b\overline{f}) = \kappa _{\vec {\eta }^{(1)}}(\overline{p}\overline{b}\overline{f})\), but \(\kappa _{\vec {\eta }^{(2)}} \, {\models }_{\mathcal{O}} \, (b\overline{p}\overline{f})\) since \(\kappa _{\vec {\eta }^{(2)}}(\overline{p}b\overline{f}) = 1 < 2 = \kappa _{\vec {\eta }^{(2)}}(\overline{p}\overline{b}\overline{f})\). Thus, \(\overline{p}\overline{f} \,\,\,/\!\!\!\!\!\,\!\!{\sim} \,^{\!\! cw }b\) and \(\overline{p}\overline{f} \,\,\,/\!\!\!\!\!\,\!\!{\sim} \,^{\!\! O }b\)ς, but \(\overline{p}\overline{f} \,\!\!{\sim} \,^{+} b\). This shows that skeptical inference over all crepresentations induced by summinimal impact vectors differs both from inference over cwminimal and over indminimal models in general.
Inference with a single crepresentation and inference with system Z are different in general (see [14, 24]), and it also has been shown that skeptical inference over all crepresentations of a given knowledge base (cinference) differs from system Z inference (see [2]). The latter is also the case for skeptical inferences over a set of \(\preccurlyeq _{\bullet }\)preferred crepresentations, as these relations also do not suffer from the Drowning Problem, which does occur for system Z. Compared to skeptical cinference, taking only minimal/preferred models into account relaxes the conditions for inference. Thus, minimal cinference extends skeptical cinference; the exact formal relationships among these different cinference relations have still to be investigated in detail.
8 Conclusions and Further Work
Assigned ranks for worlds in which \(\overline{p}\overline{f}\) holds
\(\omega \)  \(\kappa _{\vec {\eta }^{(1)}}(\omega )\)  \(\kappa _{\vec {\eta }^{(2)}}(\omega )\)  \(\kappa ^{Z}_{{\mathcal{R}}_{ strange }}(\omega )\) 

\(\overline{p}bs\overline{f}\)  1  1  1 
\(\overline{p}b\overline{s}\overline{f}\)  1  1  1 
\(\overline{p}\overline{b}s\overline{f}\)  1  2  1 
\(\overline{p}\overline{b}\overline{s}\overline{f}\)  1  2  1 
Our current work includes the investigation of the formal properties of the inference relations induced by the different notions of minimality, as well as their exact relationship to the inference over all crepresentations. While the focus of our implementation described here was to obtain a highlevel, declarative program close to the abstract problem specification, in future work we will also study the complexity of the inference relations and investigate performance optimizations.
Footnotes
Notes
Acknowledgements
We are grateful to all our students who have been involved in the implementation of the software systems described here, in particular to Karl Södler, Martin Austen, and Fadil Kallat. We also thank the anonymous reviewers of this article for their detailed and helpful comments.
References
 1.Adams E (1965) The logic of conditionals. Inquiry 8(1–4):166–197CrossRefGoogle Scholar
 2.Beierle C, Eichhorn C, KernIsberner G (2016) Skeptical inference based on crepresentations and its characterization as a constraint satisfaction problem. In: Gyssens M, Simari GR (eds) Proceedings of 9th International Symposium on Foundations of Information and Knowledge Systems, FoIKS 2016, Linz, Austria, March 7–11, 2016. LNCS, vol 9616. Springer, New York, pp 65–82Google Scholar
 3.Beierle C, Hermsen R, KernIsberner G (2014) Observations on the minimality of ranking functions for qualitative conditional knowledge bases and their computation. In: Proceedings of the 27th International FLAIRS Conference, FLAIRS2014. AAAI Press, Menlo Park, CA, pp 480–485Google Scholar
 4.Beierle C, KernIsberner G, Södler K (2013) A declarative approach for computing ordinal conditional functions using constraint logic programming. In: 19th International Conference on Applications of Declarative Programming and Knowledge Management, INAP 2011, and 25th Workshop on Logic Programming, WLP 2011, Wien, Austria, revised selected papers. LNAI, vol 7773. Springer, New York, pp 175–192Google Scholar
 5.Benferhat S, Cayrol C, Dubois D, Lang J, Prade H (1993) Inconsistency management and prioritized syntaxbased entailment. In: Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI’93), vol 1. Morgan Kaufmann Publishers, San Francisco, pp 640–647Google Scholar
 6.Benferhat S, Dubois D, Prade H (1992) Representing default rules in possibilistic logic. In: Proceedings 3rd International Conference on Principles of Knowledge Representation and Reasoning KR’92, pp 673–684Google Scholar
 7.DeFinetti B (1974) Theory of probability, vol 1, 2. John Wiley & Sons, New YorkGoogle Scholar
 8.Dubois D, Prade H (2015) Possibility theory and its applications: where do we stand? In: Kacprzyk J, Pedrycz W (eds) Springer handbook of computational intelligence. Springer, Berlin, pp 31–60CrossRefGoogle Scholar
 9.Goldszmidt M, Pearl J (1991) On the consistency of defeasible databases. Artif Intell 52(2):121–149MathSciNetCrossRefzbMATHGoogle Scholar
 10.Goldszmidt M, Pearl J (1996) Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif Intell 84:57–112MathSciNetCrossRefGoogle Scholar
 11.Halpern J (2005) Reasoning about uncertainty. MIT Press, CambridgezbMATHGoogle Scholar
 12.KernIsberner G (2001) Conditionals in nonmonotonic reasoning and belief revision. LNAI, vol 2087. Springer, New YorkGoogle Scholar
 13.KernIsberner G (2002) Handling conditionals adequately in uncertain reasoning and belief revision. J Appl Non Class Log 12(2):215–237MathSciNetCrossRefzbMATHGoogle Scholar
 14.KernIsberner G, Eichhorn C (2014) Structural inference from conditional knowledge bases. In: Unterhuber M, Schurz G (eds) Logic and probability: reasoning in uncertain environments. Studia logica, vol, 102, no 4. Springer Science+Business Media, Dordrecht, pp 751–769Google Scholar
 15.Kraus S, Lehmann DJ, Magidor M (1990) Nonmonotonic reasoning, preferential models and cumulative logics. Artif Intell 44(1–2):167–207MathSciNetCrossRefzbMATHGoogle Scholar
 16.Lehmann D (1995) Another perspective on default reasoning. Ann Math Artif Intell 15(1):61–82MathSciNetCrossRefzbMATHGoogle Scholar
 17.Lehmann DJ, Magidor M (1992) What does a conditional knowledge base entail? Artif Intell 55(1):1–60MathSciNetCrossRefzbMATHGoogle Scholar
 18.Makinson D (1994) General patterns in nonmonotonic reasoning. In: Gabbay DM, Hogger CJ, Robinson JA (eds) Handbook of logic in artificial intelligence and logic programming, vol 3. Oxford University Press, New York, pp 35–110Google Scholar
 19.Carlsson M, Ottosson G, Carlson B (1997) An openended finite domain constraint solver. In: Glaser H, Hartel PH, Kuchen H (eds) Programming languages: implementations, logics, and programs, (PLILP’97). LNCS, vol 1292. Springer, New York, pp 191–206CrossRefGoogle Scholar
 20.Pearl J (1988) Probabilistic reasoning in intelligent systems. Morgan Kaufmann Publishers Inc., San FranciscozbMATHGoogle Scholar
 21.Pearl J (1990) System Z: a natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Parikh R (ed) Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning about Knowledge (TARK1990). Morgan Kaufmann Publishers Inc, San Francisco, pp 121–135Google Scholar
 22.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: Proceedings of the Irvine conference on probability and causation. The Western Ontario Series in Philosophy of Science, vol 42. Springer Science+Business Media, Dordrecht, pp 105–134CrossRefGoogle Scholar
 23.Spohn W (2012) The laws of belief: ranking theory and its philosophical applications. Oxford University Press, OxfordCrossRefGoogle Scholar
 24.Thorn PD, Eichhorn C, KernIsberner G, Schurz G (2015) Qualitative probabilistic inference with default inheritance for exceptional subclasses. In: Beierle C, KernIsberner G, Ragni M, Stolzenburg F (eds) Proceedings of the 5th Workshop on Dynamics of Knowledge and Belief (DKB2015) and the 4th Workshop KI and Kognition (KIK2015). CEUR Workshop Proceedings, vol 1444Google Scholar