In the area of knowledge representation and reasoning, conditionals play a prominent role. Nonmonotonic reasoning investigates qualitative conditionals of the form “If A then usually B”. Various semantical approaches for inferences based on sets of such conditionals as well as criteria and postulates for evaluating the obtained inference relations have been proposed (cf. [1, 20, 24, 18, 22, 12, 19, 6, 21, 9, 13, 7, 14, 10]). Among the different semantical models of conditional knowledge bases are Spohn’s ordinal conditional functions (OCFs) [24, 23], also called ranking functions. An OCF assigns a degree of surprise to each world , the higher the value assigned to , the more surprising . Each that accepts a set of conditionals, called a knowledge base, induces a nonmonotonic inference relation that inductively completes the explicit knowledge given in .
Two inference relations which are defined based on specific OCFs obtained from a knowledge base have received some attention: system Z [22, 13] and c-representations [14, 15], or the induced inference relations, respectively, both show excellent inference properties. System Z is based upon the ranking function , which is the unique Pareto-minimal OCF that accepts . The definition of crucially relies on the notions of tolerance and of inclusion-maximal ordered partition of obtained via the tolerance relation [22, 13]. Among the OCF models of , c-representations are special models obtained by assigning an individual impact to each conditional and generating the world ranks as the sum of impacts of falsified conditionals [14, 15]. While for each consistent , the system Z ranking function is uniquely determined, there may be many different c-representations of . Skeptical c-inference [2, 4] is the inference relation obtained by taking all c-representations of into account.
It is known that system Z and skeptical c-inference both satisfy system P [18, 13, 4] and other desirable properties. Furthermore, there are system Z inferences that are not obtained by skeptical c-inference, and on the other hand, there are skeptical c-inferences that are not system Z inferences . Another notable difference between system Z and skeptical c-inference is that the single unique system Z model  can be computed much easier than skeptical c-inference which involves many models obtained from the solutions of a complex constraint satisfaction problem . In recently published work , we showed that the exponential lower bound is needed as possible impact factor for c-representations to fully realize skeptical c-inference, supporting the observation that skeptical c-inference is less tractable than system Z inference (cf. [13, 4]).
Inspired by our findings in , here we develop the preferred structure relation on worlds and propose the new nonomonotonic system W inference based on it. The main contributions of this paper are:
We introduce the preferred structure relation on worlds based on the notions of tolerance and verification/falsification behavior of a knowledge base .
By employing , we develop a new inference relation, called system W inference, which is as tractable as system Z.
We prove that system W inference captures and strictly extends both system Z inference and skeptical c-inference.
We show that system W inference exhibits desirable inference properties like satisfying the axioms of system P and avoiding the drowning problem.
The rest of the paper is organized as follows. After briefly recalling the required in Section 2, we introduce the preferred structure on worlds and prove several of its properties in Section 3. In Section 4, we give the formal definition of system W, illustrate it with various examples and show its main properties. In Section 5, we conclude and point out future work.
2 Conditional logic, system Z, and c-Representations
Let be a propositional alphabet. A literal is the positive () or negated () form of a propositional variable, stands for either or . From these we obtain the propositional language as the set of formulas of closed under negation , conjunction , and disjunction . For shorter formulas, we abbreviate conjunction by juxtaposition (i.e., stands for ), and negation by overlining (i.e., is equivalent to ). Let denote the set of possible worlds over ; will be taken here simply as the set of all propositional interpretations over and can be identified with the set of all complete conjunctions over ; we will often just write instead of . For , means that the propositional formula holds in the possible world . With , we denote the set of all worlds in which holds.
An ordinal conditional function (OCF, ranking function) [24, 25] is a function that assigns to each world an implausibility rank : the higher , the more surprising is. OCFs have to satisfy the normalization condition that there has to be a world that is maximally plausible, i.e., . The rank of a formula is defined by . An OCF accepts a conditional , denoted by , if the verification of the conditional is less surprising than its falsification, i.e., iff . This can also be understood as a nonmonotonic inference relation between the premise and the conclusion : Basically, we say that -entails , written , if accepts ; formally this if given by
Note that the reason for including the disjunctive condition in (2) is to ensure that satisfies supraclassicality, i.e., implies , also for the case as it is required, for instance, by the reflexivity axiom of system P [1, 18]. Let us remark that -entailment is based on the total preorder on possible worlds induced by a ranking function and can be expressed equivalently by:
The acceptance relation is extended as usual to a set of conditionals, called a knowledge base, by defining iff for all . This is synonymous to saying that is admissible with respect to , or that is a ranking model of . is consistent iff it has a ranking model.
Two inference relations which are defined by specific OCFs obtained from a knowledge base have received some attention: system Z  and c-representations [14, 15], or the induced inference relations, respectively, both show excellent inference properties. We recall both approaches briefly.
System Z  is based upon the ranking function , which is the unique Pareto-minimal OCF that accepts . The definition of crucially relies on the notion of tolerance. A conditional is tolerated by a set of conditionals if there is a world such that and , i.e., iff verifies and does not falsify any conditional in . For every consistent knowledge base, the notion of tolerance yields an ordered partition of , where each is tolerated by . The inclusion-maximal partition of , in the following denoted by , is the ordered partition of where each is the (with respect to set inclusion) maximal subset of that is tolerated by . This partitioning is unique due to the maximality and can be computed using the consistency test algorithm given in ; for an inconsistent knowledge base , does not exist. Using , the system Z ranking function is defined by
where the function is given by if .
Let be a knowledge base and let , be formulas. We say that is a system Z inference of in the context of , denoted by , iff holds.
Among the OCF models of , c-representations are special models obtained by assigning an individual impact to each conditional and generating the world ranks as the sum of impacts of falsified conditionals. For an in-depth introduction to c-representations and their use of the principle of conditional preservation ensured by respecting conditional structures, we refer to [14, 15]. The central definition is the following:
A c-representation of a knowledge base is a ranking function constructed from with integer impacts assigned to each conditional such that accepts and is given by:
We will denote the set of all c-representations of by .
As every ranking model of , each c-representation gives rise to an inference relation according to (2). While for each consistent , the system Z ranking function is uniquely determined, there may be many different c-representations of . C-inference [2, 4] is an inference relation taking all c-representations of into account.
Let be a knowledge base and let , be formulas. is a (skeptical) c-inference from in the context of , denoted by , iff holds for all c-representations for .
In  a modeling of c-representations as solutions of a constraint satisfaction problem is given and shown to be sound and complete with respect to the set of all c-representations of .
Let . The constraint satisfaction problem for c-representations of , denoted by , on the constraint variables ranging over is given by the conjunction of the constraints, for all :
A solution of is an -tuple For a constraint satisfaction problem , the set of solutions is denoted by . Thus, with we denote the set of all solutions of .
Let be a knowledge base. With as in (5), we then have:
Example 1 ()
To illustrate the definitions and concepts presented in this paper let us consider an instance of the well known penguin bird example. This example will be a running example and will be continued and extended throughout the paper. Consider the propositional alphabet representing whether something is a penguin , whether it is a bird , or whether it can fly . Thus, the set of worlds is . The knowledge base contains the conditionals
For and we have the ordered partitioning such that every conditional in is tolerated by and every conditional in is tolerated by . For instance, is tolerated by since there is, for example, the world with as well as . Furthermore is indeed the inclusion-maximal partition of . Therefore, is consistent. An OCF that accepts is:
For instance, we have since and and therefore .
3 Preferred Structure on Worlds
Aiming at developing a nonmonotonic inference relation combining the advantages of system Z and skeptical c-inference, we first introduce the new notion of preferred structure on worlds with respect to a knowledge base . The idea is to take into account both the tolerance information expressed by the ordered partition of and the structural information which conditionals are falsified.
Definition 5 (, , preferred structure on worlds)
Consider a consistent knowledge base with . For , and are the functions mapping worlds to the set of falsified conditionals from the tolerance partition and from , respectively, given by
The preferred structure on worlds is given by the binary relation defined by, for any ,
|there exists such that|
Thus, if and only if falsifies strictly less conditionals than in the partition with the biggest index where the conditionals falsified by and differ. The preferred structure on worlds will be the basis for defining a new inference relation induced by . Before formally defining this new inference relation and elaborating its properties, we proceed by illustrating the preferred structure on worlds for a knowledge base , relating it to c-representations of , and proving a set of its properties that will be useful for investigating the characteristics and properties of the resulting inference relation.
Example 2 ()
Let us determine the preferred structure on worlds for the knowledge base from Example 1 whose verification/falsification behavior is shown in Table 1. The inclusion-maximal partition is given by and . Figure 1 shows the preferred structure on worlds for the knowledge base . An edge between two worlds indicates that . The full relation is obtained from the transitive closure of in Figure 1.
The following proposition can be seen as a generalization of a result from . It extends [5, Proposition 15] to the relation and to arbitrary knowledge bases, not just knowledge bases only consisting of conditional facts as in [5, Proposition 15]. It tells us that the set of c-representations is rich enough to guarantee the existence of a particular c-representation fulfilling the ordering constraints given in the proposition.
Let be a consistent knowledge base, let and let . Assume that for all . Then there exists a solution and thus a c-representation, such that, for all , we have:
(Sketch) Due to lack of space, we give a sketch of the proof. The claim follows by combining the following two statements:
If , satisfy
for all where with then is a solution of and so defined as in (5) is a c-representation of .
A complete proof that (i), (ii) hold is given in the full version of this paper. ∎
The rest of this section is dedicated to the investigation of further properties of the relation . Let us start with a lemma that tells us that worlds falsifying the same sets of conditionals are equivalent with respect to .
Let be a knowledge base, and let falsify the same sets of conditionals, i.e., for all , we have iff . Then behave exactly the same way with respect to , i.e., for all , the following equivalences hold:
The claim follows from for all . ∎
In general, the relation cannot be obtained from a ranking function.
There exists a knowledge base such that there is no ranking function with iff .
The proof is by contradiction. Assume there is a ranking function with iff for . For (cf. Figure 1) we have and and furthermore and . Therefore, we obtain and . Thus, which is a contradiction to . ∎
Let us end this subsection by proving that defines a strict partial order.
The relation is irreflexive, antisymmetric and transitive, meaning that is a strict partial order.
Condition (11) immediately yields that is irreflexive and antisymmetric. It remains to show that is transitive. Define and . Then and is equivalent to and .
If then and and so . If then and and so . If then for all and ; therefore and and so . ∎
4 System W
The preferred structure on worlds for a knowledge base is defined using both the tolerance information provided by the inclusion-maximal ordered partition and information on the set of falsified conditionals. Inference based on is called system W inference and is defined as follows.
Definition 6 (system W, )
Let be a knowledge base and be formulas. Then is a system W inference from (in the context of ), denoted
A consequence of this definition is that system W inference is as tractable as system Z because the preferred structure on worlds is obtained directly from the ordered partition of and the verification/falsification behavior of . We apply the definition of system W to our running example.
Example 3 (, cont.)
Note that , i.e., this inference is also a skeptical c-inference (cf. [4, Example 5]). Therefore, Example 3 presents a c-inference that is also a system W inference. The following proposition tells us that always implies .
Proposition 3 (system W captures c-inference)
Let be a consistent knowledge base. Then we have for all formulas
Furthermore, every system Z inference is also a system W inference.
Proposition 4 (system W captures system Z)
Let be a consistent knowledge base. Then we have for all formulas
In , a preference relation on worlds is defined that is based on structural information by preferring a world to a world if falsifies fewer conditionals than and falsifies at least all conditionals falsified by . Using this preference relation, the following entailment relation along the scheme as given by (3) is obtained; we present the definition from  in a slightly modified form adapted to our notion for the set of conditionals from falsified by .
Definition 7 (-structural inference )
Let with for be a knowledge base, formulas, and let be the relation on worlds given by iff . Then can be structurally inferred, or -inferred, from , written as
We can show that every -structural inference is also a system W inference.
Proposition 5 (system W captures -structural inference)
Let be a consistent knowledge base. Then we have for all formulas
The following proposition summarizes Propositions 3, 4, and 5 and shows aditionally that system W strictly extends skeptical c-inference, system Z, and structural inference by licensing more entailments than each of these three inference modes.
Proposition 6 (system W)
For every consistent knowledge base
Furthermore, all inclusions in (20) are strict, i.e., there are with:
For proving the strictness part of (21), consider the knowledge base whose verification/falsification behavior is given by Table 2. First, due to , we obtain . Making use of the verification/falsification behavior stated in Table 2, for we obtain and
. Now consider the solution vector. For the associated c-representation (see Table 2) we then obtain and thus .
For proving the strictness part of (22), consider the knowledge base from Example 1. Let us show that for and we have , i.e., that penguins which are no bird usually do not fly. According to Example 2, we have . Therefore, since and it follows from (14) that indeed . Looking at Table 1, we observe and thus .