Within the last years, the propositional limitations of classical belief revision have been overcome piece by piece. For instance, Boutilier [Boutilier1994] investigated belief revision within a modal framework, and Williams [Williams1994] proposed transmutation schemas for knowledge systems. In general, epistemic states
have moved into the center of interest as representations of belief states of some individual or intelligent agent at a given time. Besides propositional or first-order facts, reflecting certain knowledge, they may contain assumptions, preferences among beliefs, and, as a crucial ingredient, conditional knowledge. They may be represented in different ways, e.g. by gradings of plausibility or disbelief, by making use of epistemic entrenchment, or as a probability distribution.
Epistemic states provide an excellent framework to study iterative revisions which are important to build fully dynamic systems111An interesting approach to iterated revisions of belief sets was proposed quite recently by Lehmann, Magidor and Schlechta in [Lehmann, Magidor, & Schlechta1999].. While propositional AGM theory only observes the results of revisions, considering epistemic states under change allows one to focus on the mechanisms underlying that change, taking conditional beliefs as revision policies explicitly into account (cf. [Boutilier & Goldszmidt1993, Rott1991, Darwiche & Pearl1997]). The connection between epistemic states, , (iterative) revision operators, , and conditionals, , is established by the Ramsey test
Hence revising epistemic states does not only mean to deal with propositional beliefs – it also requires studying how conditional beliefs are changed.
Darwiche and Pearl [Darwiche & Pearl1997] rephrased the AGM postulates for epistemic states. Applying the minimal change paradigm of propositional belief revision in that extended framework, as Goldszmidt and Boutilier did in [Boutilier & Goldszmidt1993], however, may produce unintuitive results [Darwiche & Pearl1997]. So, Darwiche and Pearl [Darwiche & Pearl1997] advanced four postulates as a cautious approach to describe principles of conditional preservation when revising epistemic states by propositional beliefs. In [Kern-Isberner1999c, Kern-Isberner1999b], we extended their approach in considering revisions of epistemic states by conditionals. We proposed a set of axioms outlining conditional revisions which are in accordance with some fundamental postulates of revisions (like, for instance, success) and with propositional AGM theory, and which preserve conditional beliefs in observing conditional interactions. These interactions were specified by two newly introduced relations between conditionals, that of subconditionality and that of perpendicularity (see [Kern-Isberner1999b]).
Earlier, in [Kern-Isberner1998], we defined a principle of conditional preservation in quite a different, namely probabilistic, framework. This principle there was based on the algebraic notion of conditional structures, made use of group theoretical means and postulated the numerical values of the given probability distribution to follow the conditional structures of worlds.
In this paper, we will bring together both approaches to conditional preservation, the qualitative one and the quantitative one, in the semi-quantitative framework of ordinal conditional functions. We will rephrase the probabilistic principle of conditional preservation of [Kern-Isberner1998] for ordinal conditional functions, and we will show, that this quantitative principle of conditional preservation implies the corresponding qualitative postulates of [Kern-Isberner1999b]. Actually, the quantitative version is much stronger than the qualitative one, dealing with sets of conditionals instead of only one revising conditional, and handling interactions of conditionals of arbitrary complexity. Although numerical in nature, the principle is based on a symbolic representation of conditional influences on worlds, called conditional structures. Numbers (rankings or probabilities) are only considered as manifestations of complex conditional interactions. The representation of conditional structures by the aid of group theory provides a rich methodological framework to study conditionals in belief revision and nonmonotonic reasoning.
In the following section, we fix some notations and describe the relationship between conditionals and epistemic states. Then we introduce the crucial notion of conditional structures which the property of conditional indifference is based upon. By applying the concept of indifference to revision functions, we obtain a precise formalization of the principle of conditional preservation, and we characterize ordinal conditional functions observing this principle. Further on, we compare our approach to Goldszmidt, Morris & Pearl’s system-Z and system- in several examples. Finally, we bring together qualitative and quantitative approaches to the principle of conditional preservation, proving the formalization given here to be a most fundamental one. A summary and an outlook conclude this paper.
2 Conditionals and epistemic states
We consider a propositional language over a finite alphabet . Let denote the set of possible worlds for , i.e. is a complete set of interpretations of . Throughout this paper, we will write instead of , and instead of , for formulas .
Conditionals represent statements of the form “If A then B”, expressing a relationship between two (propositional) formulas , the antecedent or premise, and , the consequent. denotes the set of all conditionals with . A conditional with a tautological antecedent is taken to correspond to its (propositional) consequent, . is called a subconditional of , written as
iff and . Typically, subconditionals arise by strengthening the antecedent of a conditional, e.g. is a subconditional of , .
Each possible world either confirms , in case that , or refutes it, if , or does not even satisfy its premise, , and so is of no relevance for it. Although conditionals are evaluated with respect to worlds, they cannot really be accepted (as entities) in single, isolated worlds. To validate conditionals, we need richer epistemic structures than plain propositional interpretations, at least to compare different worlds with regard to their relevance for a conditional (see, for example, [Nute1980, Boutilier1994, Darwiche & Pearl1997]). Epistemic states as representations of cognitive states of intelligent agents provide an adequate framework for conditionals.
An epistemic notion that turned out to be of great importance both for conditionals and epistemic states, in particular in the context of belief revision, is that of plausibility: conditionals are supposed to represent plausible conclusions, and plausibility relations on formulas or worlds, respectively, guide AGM-revisions of belief sets and of epistemic states [Nute1980, Katsuno & Mendelzon1991, Darwiche & Pearl1997].
As Spohn [Spohn1988] emphasized, however, it is not enough to consider the qualitative ordering of propositions according to their plausibility – also relative distances between degrees of plausibility should be taken into account. So he introduced ordinal conditional functions (OCF’s, ranking functions) [Spohn1988] from worlds to ordinals such that some worlds are mapped to the minimal element . Here, we will simply assume that OCF’s are functions from the set of worlds to the natural numbers, extended by and . They specify non-negative integers as degrees of plausibility – or, more precisely, as degrees of disbelief – for worlds. The smaller is, the more plausible the world appears, and what is believed (for certain) in the epistemic state represented by is described precisely by the set . For propositional formulas , we set , so that . In particular, , so that at least one of or is considered mostly plausible. A proposition is believed iff , which is denoted by . A conditional may be assigned a degree of plausibility via . Each OCF induces a (propositional) AGM-revision operator by setting (see [Darwiche & Pearl1997]). The Ramsey test (1) then reads iff . This is in accordance with the plausibility relation imposed by , as the following lemma shows:
Let be a conditional in , let be an ordinal conditional function. Then (by applying the Ramsey test) iff .
So accepts a conditional (via the Ramsey test) iff is more plausible than . The proof of this lemma is immediate.
3 Conditional structures
By observing the behavior of worlds with respect to it, each conditional can be considered as a generalized (namely three-valued) indicator function on worlds:
where stands for undefined [DeFinetti1974, Calabrese1991]). Intuitively, incorporating a conditional as a plausible conclusion in an epistemic state means to make – at least some – worlds confirming the conditional more plausible than the worlds refuting it. In this sense, conditionals to be learned have effects on possible worlds (more exactly, on their degrees of plausibility), shifting them appropriately to establish the intended plausible relationship. (3) then provides a classification of worlds for achieving this: On confirming worlds , i.e. , possibly has a positive effect, while on refuting worlds , possibly has a negative effect; the effects on worlds with is unclear. Which worlds will actually be shifted depends on the chosen revision procedure – for the conditional, all worlds in either of the partitioning sets are indistinguishable.
When we consider (finite) sets of conditionals , we have to modify the representation (3) appropriately to identify the effect of each conditional in on worlds in . This leads to introducing the functions below (see (4)) which generalize (3) by replacing the numbers and by abstract symbols. Moreover, we will make use of a group structure to represent the joint impact of conditionals on worlds.
To each conditional in we associate two symbols . Let
be the free abelian group with generators , i.e. consists of all elements of the form with integers (the ring of integers). Each element of can be identified by its exponents, so that is isomorphic to [Lyndon & Schupp1977]. The commutativity of corresponds to the fact that the conditionals in shall be effective simultaneously, without assuming any order of application. So our way of dealing with conditionals is a symmetric, homogeneous one – we do not need (user-defined) priorities among conditionals. Note that, although we will speak of multiplication and products in , the generators of are merely juxtaposed, like words.
For each , we define a function by setting
represents the manner in which the conditional applies to the possible world . The neutral element of corresponds to the non-applicability of in case that the antecedent is not satisfied. The function ,
describes the all-over effect of on . is called (a representation of) the conditional structure of with respect to . For each world , contains at most one of each or , but never both of them because each conditional applies to in a well-defined way. The next lemma (which is easy to prove) shows that this property characterizes conditional structure functions:
Let be a map from the set of worlds to the free abelian group generated by , such that contains at most one of each or , for each world . Then there is a set of conditionals with such that .
Example 3 Let , where are atoms, and let . We associate with the first conditional, , and with the second one, . The following table shows the values of the function on worlds :
confirms both conditionals, so its conditional structure is represented by . This corresponds to the product (in ) of the conditional structures of the worlds and . Two worlds, namely and , are not affected at all by the conditionals in .
The logical structure of antecedents and consequents of the conditionals in does not really matter, nor do logical relationships between the conditionals. All that we need is a conditional’s partitioning property on the set of worlds (cf. (3) and (4)). labels each world appropriately and allows us to compare different worlds with respect to the impact the conditionals in exert on them. The following example illustrates that also multiple copies of worlds may be necessary to relate conditional structures:
Example 4 Consider the set of conditionals using the atoms . Let be the group generators associated with , , respectively. Then we have
Here two copies of , or of its structure, respectively, are necessary to match the product of the conditional structures of and .
To compare worlds adequately with respect to their conditional structures, we take the worlds as formal generators of the free abelian group
consists of all products , with , and integers. Introducing such a “multiplication between worlds” is nothing but a technical means to comply with the multiplicative structure the effects of conditionals impose on worlds. As in , multiplication in actually means juxtaposition. In [Kern-Isberner1998], where we first developed these ideas, we considered multi-sets of worlds (corresponding to elements in with only positive exponents) and calculated the conditional structure of such a multi-set as the conditional weight it is carrying. Making use of arbitrary elements of as group elements, however, provides a much more convenient and elegant framework to deal with conditional structures. We will usually write instead of .
Now may be extended to in a straightforward manner by setting
yielding a homomorphism of groups . For , we obtain
as a representation of its conditional structure. The exponent of in indicates the number of worlds in which confirm the conditional , each world being counted with its multiplicity, and in the same way, the exponent of indicates the number of worlds that are in conflict with .
By investigating suitable elements of , it is possible to isolate the (positive or negative) net impacts of conditionals in , as the following example illustrates:
Example 5 (continued) In Example 2 above, we have
So reveals the positive net impact of the conditional within , symbolized by .
in Example 2, the element
the negative net impact of the second conditional, :
The following example is taken from [Goldszmidt & Pearl1996, p. 68f]:
Example 6 Consider the set consisting of the following conditionals:
In Table 1, we list the conditional structures of all possible worlds; this table will be helpful in the sequel.
Having the same conditional structure defines an equivalence relation on :
Those elements of that are balanced with respect to the effects of conditionals in are contained in the kernel of , . does not depend on the chosen representation of conditional structures by symbols in and thus, it is an invariant of [Kern-Isberner1999d].
Often, besides the conditionals explicitly given in , implicit normalizing constraints have to be taken into account, like, e.g. for ordinal conditional functions. This can be achieved by focusing on equivalence with repect to . Since simply counts the generators occurring in , two elements are -equivalent, , iff . This means, iff they both are a (cancelled) product of the same number of generators, each generator being counted with its corresponding exponent.
4 Conditional indifference
To study conditional interactions, we now focus on the behavior of OCF’s with respect to the multiplication in . Each such function may be extended to a homomorphism, , by setting
where is the subgroup of generated by the set . This allows us to analyze numerical relationships holding between different . Thereby, it will be possible to elaborate the conditionals whose structures follows, that means, to determine sets of conditionals with respect to which is indifferent:
Definition 7 Suppose is an OCF, and is a set of conditionals such that for all . is indifferent with respect to iff the following two conditions hold:
If then there is such that and for all with .
If is indifferent with respect to , then it does not distinguish between different elements with the same conditional structure with respect to . Normalizing constraints are taken into account by observing -equivalence. Conversely, any deviation can be explained by the conditionals in acting on in a non-balanced way. Condition (i) in Definition 4 is necessary to deal with worlds . Conditional indifference, as defined in Definition 4, captures interactions of conditionals of arbitrary depth by making use of the homomorphism induced by . It also respects, however, indifference on the superficial level of the function itself:
If the ordinal conditional function is indifferent with respect to , then implies for all worlds .
The next theorem gives a simple criteria to check conditional indifference with ordinal conditional functions. Moreover, it provides an intelligible schema to construct conditional indifferent functions.
An OCF is indifferent with respect to a set of conditionals iff for all , and there are rational numbers , , such that for all ,
Sketch of proof. According to Lemma 8, the equivalence relation (5) provides a rough classification of the worlds in with respect to the conditionals in . Obtaining a representation of the form (6) then amounts to checking the solvability of a linear equational system. The proof of this theorem is very similar to the proof of the analogous theorem for probabilistic representation of knowledge given in [Kern-Isberner1998].
5 The principle of conditional preservation
Minimality of change is a crucial paradigm for belief revision, and a “principle of conditional preservation” is to realize this idea of minimality when conditionals are involved in change. Minimizing absolutely the changes in conditional beliefs, as in [Boutilier & Goldszmidt1993], is an important proposal to this aim, but it does not always lead to intuitive results [Darwiche & Pearl1997]. The idea we will develop here rather aims at preserving the conditional structure of knowledge within an epistemic state which we assume to be represented by an OCF .
We just explained what it means for an OCF to follow the structure imposed by on the set of worlds by introducing the notion of conditional indifference (cf. Definition 4). Pursuing this approach further in the framework of belief revision, a revision of by simultaneously incorporating the conditionals in , , can be said to preserve the conditional structure of with respect to if the relative change function is indifferent with respect to 222Why just ? First, it is more accurate than, e.g., , in the sense of taking differences in degrees of plausibility seriously. Second, it makes use of the conditional “”, considering revision as a generalized conditional operation; for more details, see [Kern-Isberner1999d].. Taking into regard the worlds with appropriately, this gives rise to the following definitions:
Definition 10 Let be an OCF, and let be a finite set of conditionals. Let denote the result of revising by . Presuppose further333Note that success is not compulsory for conditional indifference and conditional preservation. We only presuppose for all in order to exclude pathological cases. that for all .
is called -consistent iff implies .
is indifferent with respect to and iff is -consistent and the following two conditions hold:
If then , or there is such that and for all with .
and for , where .
The principle of conditional preservation is now realized as an indifference property:
Definition 11 Let be an OCF, and let be a finite set of conditionals. A revision satisfies the principle of conditional preservation iff is indifferent with respect to and .
So satisfies the principle of conditional preservation if any change in plausibility is clearly and unambigously induced by . The next theorem characterizes revisions of ordinal conditional functions that satisfy the principle of conditional preservation. The theorem is obvious by observing Theorem 9.
Let be OCF’s, and let be a (finite) set of conditionals in . A revision satisfies the principle of conditional preservation iff for all , and there are numbers , such that for all ,
Comparing Theorems 9 and 12 with one another, we see that an OCF is indifferent with respect to a finite set of conditionals iff it can be taken as a revision satisfying the principle of conditional preservation, where for all is the uniform ordinal conditional function.
Up to now, we have not yet taken the success condition into regard, postulating that the revised OCF in fact represents the conditionals in .
Definition 13 Let be OCF’s, and let be a set of conditionals. is called a c-revision iff and satisfies the principle of conditional preservation. is called a c-representation of , iff and is indifferent with respect to .
Theorems 9 and 12 provide simple schemes to construct c-revisions and c-representations. The numbers , then have to be chosen appropriately to ensure that is an ordinal conditional function, and such that for all conditionals (cf. Lemma 1). In the special case that is a representation of , we obtain the following corollary by some easy calculations:
Let be a (finite) set of conditionals in , and let be an OCF.
is a c-representation of iff for all , and there are numbers , such that for all ,
which is equivalent to
This shows (14).
The difference , or the right hand side of (14), respectively, measures the effort needed to establish the i-th conditional. To calculate suitable constants
, we apply the following heuristics: To establish conditional beliefs, one can make confirming worlds more plausible (if required, which amounts to choose), or refuting worlds less plausible (if required, which means ). The normalizing constant then has to be chosen appropriately to ensure that actually an OCF is obtained.
We prefer the second alternative, presupposing
Then (14) reduces to
for . If there are worlds with neutral conditional structure, , we may set . So, we obtain a c-representation of via
where the have to satisfy (11).
Example 15 We will use Corollary 14 and the heuristics (10) to obtain a c-representation (12) of the conditionals of Example 2. To calculate constants according to (11), Table 1 proves to be helpful. We only have to focus on the -labels of worlds, and we obtain
So we set
Since there are also worlds with (cf. Table 1), we set . So we obtain a c-representation of by
6 A comparison with system-Z and system-
A well-known method to represent a (finite) set of conditionals by an OCF is to apply the system-Z of Goldszmidt and Pearl [Goldszmidt & Pearl1992, Goldszmidt & Pearl1996]. The corresponding ranking function is given by
where is an ordering on observing the (logical) interactions of the conditionals (for a detailed description of , see, for instance, [Goldszmidt & Pearl1996]). assigns to each world the lowest possible rank admissible with respect to the constraints in . Comparing (15) with (6), we see that in general, is not a c-representation of , since in its definition (15), maximum is used instead of summation (see Example 6 below). The numbers , however, may well serve to define appropriate constants in (6). Setting , and for , we obtain from a c-representation of via
An even more sophisticated representation is obtained by combining the system-Z approach with the principle of maximum entropy (ME-principle), yielding system- [Goldszmidt, Morris, & Pearl1993]. The corresponding -rankings of the conditionals in have to satisfy the following equation (see equation (16) in [Goldszmidt, Morris, & Pearl1993, p. 225])
and is then calculated by
(see equation (18) in [Goldszmidt, Morris, & Pearl1993, p. 225]). For so-called minimal-core sets – these are sets allowing each conditional to be separable from the other rules by restricting conditional interactions –, a procedure is given to calculate -rankings in [Goldszmidt, Morris, & Pearl1993].
Like our method, system- makes use of summation instead of maximization, as in system-Z. And equations (17) determining the -rankings look similar to our inequality constraints (14). More exactly, if we follow the heuristics (10) and set , then system- turns out to be a special instance of our more general scheme in Corollary 14. In particular, system- yields a c-representation.
This similarity is not accidental – the ME-principle not only provides a powerful base for system-, but also influenced the idea of conditional indifference presented in this paper. In [Kern-Isberner1998], we characterized the ME-principle by four axioms, one of which was the postulate of conditional preservation. Conditional preservation for probability functions there was realized in full analogy to that for OCF’s defined here. So both c-representations and ME-distributions comply with a fundamental principle for representing conditionals, and it is this principle of conditional preservation (or principle of conditional indifference, respectively) that is responsible for a peculiar thoroughness and accuracy when incorporating conditionals. Since we realized this principle completely in a semi-quantitative setting, we did not have to refer to probabilities and to ME-distributions, and we were able to formalize the acceptance conditions, (14), in a purely qualitative manner.
Example 16 Consider once again the conditionals from Example 2. Here we have and (for the details, see [Goldszmidt & Pearl1996, p. 69]). By setting for , we obtain the same constants, (13), as in Example 14. Furthermore, by applying the procedure Z-rank in [Goldszmidt, Morris, & Pearl1993], we calculate