## 1 Introduction

### 1.1 Overview and related work

The aim of this paper is to investigate semantics and logical properties of theory revisions based on an underlying notion of distance between individual models. In many situations it is indeed reasonable to assume that the agent has some natural way to evaluate the distance between any two models of the logical language of interest. The distance between model and model is a measure of how far appears to be from the point of view of . This distance may measure how different is from under some objective measure: e.g., the number of propositional atoms on which differs from

if the language is propositional and finite, but it may also reflect a subjective assessment of the agent about its own capabilities such as, for example, the probability that, if

is the case, the agent (wrongly) believes that is the case. Any such distance between models may be used to define a procedure for theory revision: both a theory and a formula define a set of models, and , respectively, and the result of revising by , , is the theory of the set of those models of that are closest to .The purpose of this paper is not to suggest specific useful notions of distance. It assumes some abstract notion of a distance is given and studies the properties of the revisions defined by this distance. The distances that will be considered in this paper do not always satisfy the properties generally accepted for distances, they are really pseudo-distances. There is no obvious reason why, in particular, our distance should satisfy the triangular inequality or even be symmetric. It may be the case that, from the perspective of , looks very far away, but, from the perspective of , looks close by. In the terms of one of our examples above: if is the case, our agent may give a very low probability to being the case, but if is the case, our agent may well be hesitant about whether or is the case: assume, for example, that and differ by the value of one atomic proposition that is tested by our agent. The test for may well be very reliable if is the case but quite unreliable if is not the case.

#### 1.1.1 Related work

In [AGM85], Alchourrón, Gärdenfors and Makinson introduced the study of theory revision. Their account of revision is indirect: they describe contractions in terms of maximal non-implying sub-theories and they go on to characterize revisions, reducing revision to contraction via the Levi identity. In [Gro88], Grove gave a direct, semantic, characterization of revision. The result of revising a theory by a proposition is determined by the models of that are, individually, closest to the set, taken collectively, of all models of . It thus uses a relation between individual models and sets of models. It is natural to seek to analyze such closeness in terms of a distance function between models. A first attempt was made by Becher in [Bec95], in view of comparing revision and update in a unified setting. Becher worked with not necessarily symmetric distances and showed that the AGM postulates hold in distance based revision, but gave no representation result. Independently, the authors presented, in [SLM96], a preliminary version of the results given in this paper. There, only the finite case of symmetric distances was treated. We deal here with the infinite case of symmetric distances and with the finite case of non-symmetric distances. We also provide here proofs and counter-examples. We present, first, our results on an abstract level, dealing with abstract sets and, then, specialize our results to the case of sets of models. In recent work (personal communication) Areces and Becher gave a representation result for the arbitrary, i.e. infinite and not necessarily symmetric case. Their conditions are different from ours, based on complete consistent theories, i.e. single models, and partly in an “existential” style, whereas our conditions are “universal” and more in the AGM style. We do not know whether there is an easy direct, i.e. not going via the semantics, proof of the equivalence of the Areces/Becher and our conditions. Thus, their approach is an alternative route, whose relation to our results is a subject of further research. The infinite case for non-symmetric distances with conditions in our style is still open.

#### 1.1.2 Structure of this paper

After a short motivation in Section 1.1, we present and discuss the AGM framework for revision in Section 1.2 and modify it slightly. Section 1.3 introduces pseudo-distances, which are distances weakened to the properties essential in our context. In particular, pseudo-distances are not necessarily symmetric. We formalize revision based on a (pseudo-) distance, and we show that the usual AGM properties hold for such distance-based revisions, at least in the finite case. An additional property (definability preservation) guarantees them to hold in the infinite case, too. Here, we also discuss some properties of distance-based revision going beyond the AGM postulates.

Our main results are algebraic in nature, and work for arbitrary sets, not only for sets of models. The translation to logic is then straightforward.

Section 2 presents the algebraic representation results, which describe the conditions which guarantee that a binary set operator is representable by a pseudo-distance i.e. that is the set of -closest to , formally that

In Section 2.2, we treat the case of symmetric pseudo-distances, the result applies to sets of arbitrary cardinality. Note that this infinite case requires a limit condition: For non-empty sets , there is some with closest distance (among the elements of ) to . In Section 2.3, we treat the not necessarily symmetric case, but our result applies only to the finite situation.

Section 3 finally translates the results of Section 2 to logic. We there describe the conditions which guarantee that a revision operator can be represented by a pseudo-distance between models, i.e. - where is the set of models of the theory and the set of formulas valid in the set of models

Analogously to the algebraic characterization, the logical representation results are for possibly infinite languages in the symmetric case (with some caveat about definability preservation) and for finite ones in the not necessarily symmetric case.

### 1.2 Belief revision

Intelligent agents must gather information about the world, elaborate theories about it and revise those theories in view of new information that, sometimes, contradicts the beliefs previously held. Belief revision is therefore a central topic in Knowledge Representation. It has been studied in different forms: numeric or symbolic, procedural or declarative, logical or probabilistic.

#### 1.2.1 The AGM framework

One of the most successful frameworks in which belief revision has been studied has been proposed by Alchourrón, Gärdenfors and Makinson, and is known as the AGM framework. It deals with operations of revision that revise a theory (the set of previous beliefs) by a formula (the new information). It proposes a set of rationality postulates that any reasonable revision should satisfy. A large number of researchers in AI have been attracted by and have developed this approach further: both in the abstract and by devising revision procedures that satisfy the AGM rationality postulates.

Two remarks should be made immediately. First, the AGM framework presents rationality postulates for revision. It does not choose anyone specific revision among the many possible revisions that satisfy those postulates. Those postulates are justified and defended by the authors, but, recently, some doubts have been expressed as to their desirability, at least for modeling updates, see [KM92] and, more importantly for us, it is not clear that the AGM postulates are all what one would like. A number of authors, in particular [FL94], [DP94], [Leh95], have in fact argued that one would expect some additional postulates to hold. But the consideration of additional postulates has proved slippery and dangerous: the postulates proposed in [DP94] have been shown inconsistent in [Leh95] and have been modified in [DP97]. But this modification forces on us the rejection of one of the basic ontological commitments of the AGM framework, which brings us to a second remark. Secondly, one of the basic ontological commitments of the AGM approach is that what the agent is revising is a belief set. In other terms, epistemic states are belief sets. That this is the AGM position is clear from the formalism chosen: the left hand argument of the star revision operation is a belief set, from the motivation presented, and it is explicitly recognized in [G r88], p. 47.

Over the years, a large number of researchers have moved away from this identification, sometimes without recognizing it [BG93], [Bou93], [DP97], [Wil94], [NFPS95]. Recently, conclusive evidence has been put forward [Leh95], [FH96] to the effect that this identification of epistemic states to belief sets is not welcome in many AI applications. When iterated revisions are considered, it is reasonable to assume that the agent’s epistemic state includes information related to its history of revisions and that all this history, not only the agent’s current belief set, may influence future revisions.

This paper keeps the AGM commitment to identifying epistemic states with belief sets, but proposes additional rationality postulates. Those additional postulates characterize exactly the revisions that are defined by pseudo-distances. They constrain revisions in the way they treat their left argument, the theory to be revised (in this respect the AGM postulates are extremely, probably excessively, liberal) and they imply highly non-trivial properties for iterated revisions. This paper therefore treats iterated revisions within the original AGM commitment to the identification of epistemic states and belief sets. Results related to the ideas of this paper may be found in [BGHPSW97]. Similar ideas in a context in which epistemic states are not belief sets may be found in [BLS99].

This work provides a semantics for theory revision à la AGM, or for a sub-family of such revisions. It is the first such effort to describe semantically the whole revision operation in a unified way. Previous attempts [Gro88], [GM88] describe the revision of each theory by a different structure without any glue relating the different structures: sphere systems or epistemic entrenchment relations, corresponding to different ’s. In this paper, the revisions of the different ’s are obtained from the same pseudo-distance. A tight fit (coherence) between the revisions of different ’s seem crucial for a useful treatment of iterated revisions: it must be the same revision operation that executes the successive revisions for any interesting properties to appear. Our revisions are therefore defined by a polynomial (in the number of models considered) number of pseudo-distances instead of an exponential number of sphere systems or epistemic entrenchment relations (one for each theory ).

Semantics based on a more or less abstract notion of distance is not a new idea in non-classical logics. The best known example is perhaps the Stalnaker/Lewis distance semantics for counterfactual conditionals, see e.g. [Lew73].

The AGM framework, defined in [AGM85], studies revision operations, denoted that operate on two arguments: a set of formulas closed under logical deduction on the left and a formula on the right. Thus is the result of revising theory by formula using revision method

###### Notation 1.1

Our base logic will be classical propositional logic, though our main results are purely algebraic in nature, and therefore carry over to other logics, too.

By abuse of language, we call a language finite whose set of propositional variables is finite.

A theory will be an arbitrary set of formulas, not necessarily deductively closed.

We use the customary notation for the set of all logical consequences of a theory will stand for

will stand for: is (classically) consistent, abbreviates

will be classical validity, and will abbreviate the obvious: for all and for all

Given a propositional language will be the set of its models.

will be the models of a theory (likewise for a formula and the set of formulas valid in a set of models

will be the power set operator.

The logical connectives and and the set connectives and always have precedence over the revision and set operators and

Numbering of conditions: will number conditions common to the symmetric and the not necessarily symmetric set operators and conditions for respectively the symmetric and the not necessarily symmetric set operator will do the same for the theory revision operator

The original AGM rationality postulates are the following, for a deductively closed set of formulas, and formulas.

###### Definition 1.1

is a deductively closed set of formulas.

If then

If is inconsistent then is a logical contradiction.

If , then

If then

#### 1.2.2 Modifications of the AGM framework

We prefer to modify slightly the original AGM formalism on two accounts. First, it seems to us that the difference required in the types of the two arguments of a revision: the left argument being a theory and the right argument being a formula is not founded. The lack of symmetry is twofold: the left-hand argument, being a theory, may be inherently infinite and not representable by a single formula while the right-hand argument is always a single formula, but also the left-hand argument is not an arbitrary set of formulas, but closed under logical implication, whereas the right-hand argument is not deductively closed, thus requiring Postulate to assert invariance under logical equivalence. There is no serious reason for this lack of symmetry. We shall therefore prefer a formalism that is symmetric in both arguments.

Our results for symmetric pseudo-distances are valid for infinite sets, our results for not necessarily symmetric pseudo-distances have been proved only for finite sets - where sets are to be understood as sets of models. The latter are thus proved for languages based on a finite set of propositional variables only. We therefore choose a formalism in which both arguments of the revision operator are theories, which, in the not necessarily symmetric case, will be assumed to be equivalent to single formulas. We thus look at the theory that is the result of revising theory by the new information represented by theory

Secondly, in the AGM formalism, each one of the theory and the formula may be inconsistent. There is no harm in doing so, but the interesting revisions are always revisions of consistent theories by consistent formulas and the consideration of inconsistent arguments makes the treatment unnecessarily clumsy. Therefore, we shall only revise consistent theories by consistent theories, and assume both arguments are consistent.

The AGM postulates may now be rewritten in the following way. We rewrite and in one single postulate, and similarly for and in and are summarized in our general prerequisite and

Remember that are now arbitrary consistent theories.

###### Definition 1.2

If then

is a consistent, deductively closed theory,

If is consistent, then

If is consistent with then

### 1.3 Revision based on pseudo-distances

#### 1.3.1 Pseudo-distances

We will base our semantics for revision on pseudo-distances between models. Pseudo-distances differ from distances in that their values are not necessarily reals, no addition of values has to be defined, and symmetry need not hold. All we need is a totally ordered set of values. If there is a minimal element 0 such that iff we say that respects identity. Pseudo-distances which do not respect identity have their interest in situations where staying the same requires effort.

We first recollect:

###### Definition 1.3

A binary relation on is a preorder, iff is reflexive and transitive. If is in addition total, i.e. iff or then is a total preorder.

A binary relation on is a total order, iff is transitive, irreflexive, i.e. for all and for all or or

##### Note 1.1:

If is a total preorder on the corresponding equivalence relation defined by iff and [x] the -equivalence class of and we define iff but not then is a total order on

###### Definition 1.4

is called a pseudo-distance on iff (d1) holds:

(d1) is totally ordered by a relation

If, in addition, has a -smallest element 0, and (d2) holds, we say that respects identity:

(d2) iff

If, in addition, (d3) holds, then is called symmetric:

(d3)

(For any

Let stand for .

Note that we can force the triangle inequality to hold trivially (if we can choose the values in the real numbers): It suffices to choose the values in the set i.e. in the interval from 0.5 to 1, or as 0.

Recall that our main representation results are purely algebraic, and apply to arbitrary sets which need not necessarily be sets of models. Intuitively however, is to be understood as the set of models for some language and the distance from to represents the “cost” or the “difficulty” of a change from the situation represented by to the situation represented by

M. Dalal [Dal88] has considered one such distance: the distance between two propositional worlds is the number of atomic propositions on which they differ, i.e., the Hamming distance between worlds considered as binary k-dimensional vectors, where

is the number of atomic propositional variables. A. Borgida [Bor85] considered a similar but different distance, based on set inclusion. His distances are not totally ordered and therefore the framework presented here does not fit his work.Another example of such a distance is the trivial distance: is 0 if and 1 otherwise.

Both those distances satisfy the triangular inequality. In applications dealing with reasoning about actions and change, one may want to consider the distance between two models to represent how difficult, or unexpected, the transition is. In such a case, a natural pseudo-distance may well not be symmetric.

We give the formal definition of the elements of d-closest to :

###### Definition 1.5

Given a pseudo-distance , let for .

Definition 1.5 may be presented in a slightly different light. Put iff . Let be the set of all minimal elements (under ) of the set . Then, is nothing else than the right projection (on ) of .

Thus, is the subset of consisting of all that are closest to . Note that, if or is infinite, may be empty, even if and are not empty. A condition assuring non-emptiness will be imposed when necessary.

The aim of Section 2 of this article is to characterize those operators for which there is a pseudo-distance such that We call such representable:

###### Definition 1.6

An operation is representable iff there is a pseudo-distance such that

(1)

#### 1.3.2 Revision based on pseudo-distances

The representation results of [AGM95], the semantics of Grove [Gro88] and the very close connection with the rational relations of [LM92], showed in [GM94], all leave essentially unanswered the question of the nature of the dependence of the revision on its left argument, . Since we, like most researchers in Artificial Intelligence, are mostly interested in iterated revisions, proper understanding, and semantics, for this dependence is crucial. The purpose of this paper is to answer the question by proposing a suitable semantics. We completely characterize the semantics by a set of postulates. We do not claim that the semantics proposed are the most general ones, we present one family of reasonable semantics, based on pseudo-distances between models.

The following is the central definition, it describes the way a revision is attached to a pseudo-distance on the set of models.

###### Definition 1.7

is called representable iff there is a pseudo-distance on the set of models s.t.

The main goal of this work is to characterize the properties, i.e., rationality postulates satisfied by revisions representable by pseudo-distances.

#### 1.3.3 Revision based on pseudo-distances and the AGM postulates

##### The AGM postulates hold for revision based on pseudo-distances in the finite case

###### Definition 1.8

An operation on the sets of models of some logic is called definability preserving iff is again the set of models of some theory for all theories

Abstractly, definability preservation strongly couples proof theory and semantics. To obtain the same kind of results without definability preservation, we would have to allow a “decoupling” on a “small” set of exceptions. This is illustrated e.g. by the results in [Sch97] for the definability preservation case, and in [Sch98] for the unrestricted case of representation results for preferential structures. A similar problem arose already in a finite situation in [ALS98] in the context of partial and total orders, and is treated there by an inductive process.

A first easy result is: any such revision defined for a language satisfies the AGM postulates if respects identity. (We use to abbreviate

The same proof shows that the AGM postulates also hold in the infinite case, if the operation is definability preserving, and if we impose a limit condition for postulate .

is evident, as we work with models.

holds in the finite case, we will impose it, i.e. a limit condition, in the infinite case.

trivial by definition.

this holds, as is minimal for all , by respect of identity.

Note that and that By prerequisite, so Let “ ”: Let By prerequisite, there is Thus As but too. “ ” : Let Thus so by We conclude thus that

##### The AGM postulate may fail in the infinite not definability preserving case

The importance of definability preservation is illustrated by the following example, which shows that already the AGM properties may fail when the distance between models does not preserve definability. Essentially the same example will show in Section 3 (Example 3.1 there) that our Loop Condition may fail when the distance is not definability preserving. We see here that this is not related to our stronger conditions, but happens already in the general AGM framework.

###### Example 1.1

Consider an infinite propositional language

Let be complete (consistent) theories, a theory with infinitely many models, Assume further so is not definable by a theory.

Arrange the models of in the real plane s.t. all have the same distance (in the real plane) from has distance 2 from and has distance 3 from

(See Figure 1.1.)

Then but so is
consistent with and
But
and

##### AGM revisions are not all definable by pseudo-distances

But any revision defined by a pseudo-distance also satisfies some properties that do not follow from the AGM postulates. We note again for

Consider, for example, the set where and are finite sets. will be

If then If we have If then we have It follows that any revision defined by a pseudo-distance satisfies (for a finite language): is equal to to or to

This property does not follow from the AGM postulates, as will be shown below, but seems a very natural property. Indeed, when revising a disjunction by a formula there are two possibilities. First, it may be the case that our indecision concerning or persists after the revision, and, in this case, the revised theory is naturally the disjunction of the revisions. But it may also be the case that the new information makes us revise backwards and conclude that it must be the case that or, respectively, was (before the new information) the better theory and, in this case, the revised theory should be or Notice that this last property of revisions generated by pseudo-distances is the left argument analogue of AGM’s Ventilation Principle which concerns the argument on the right. The Ventilation Principle follows from the AGM postulates and states that: is equal to , to or to .

One can conclude that any revision defined by a pseudo-distance satisfies the following properties, that deal with iterated revisions:

if and then

and

if then, either or

Those properties seem intuitively right. If after any one of two sequences of revisions that differ only at step (step being in one case and in the other), one would conclude that holds, then one should conclude after the sequence of revisions that differ from the two revisions only in that step is a revision by the disjunction since knowing which of or is true cannot be crucial. This property is an analogue for the left argument of the Or property of [KLM90]. Similarly, if one concludes from a revision by a disjunction, one should conclude it from at least one of the disjuncts. This property is an analogue for the left argument of the Disjunctive Rationality property of [KLM90], studied in [Fre93]. It is easy to see that the property (C1) of Darwiche and Pearl [DP94], i.e., is not satisfied by all revisions defined by pseudo-distances. Section 2 will precisely characterize those revisions that are defined by pseudo-distances.

Notice that in each of the AGM postulates, the left-hand side argument of the revision operation is the same all along: all revisions have the form Since, as has been shown above, every revision defined by a pseudo-distance satisfies the AGM postulates, if, for each theory we define by some pseudo-distance, then the revision defined will satisfy the AGM postulates, even if we use different pseudo-distances for different ’s.

Consider, for a simple example, 4 points in the real plane, , to be interpreted as the models of a propositional language of two variables. Let have the coordinates (0,1), (0,-1), (1,0), (2,0), and define by the natural distance the revisions with any except for on the left hand side. As seen above, they will satisfy the AGM postulates. To define the revisions with on the left hand side, interchange the positions of and This, too, satisfies the AGM postulates. As the AGM postulates say nothing about coherence between different ’s, all these revisions together satisfy the AGM postulates.

But we will then have but , so such a system of revisions cannot be defined by a pseudo-distance.

## 2 The algebraic representation results

### 2.1 Introduction

First, a generalized abstract nonsense result. This result is certainly well-known and we claim no priority. It will be used repeatedly below to extend a relation to a relation . The equivalence classes under will be used to define the abstract distances.

###### Lemma 2.1

Given a set and a binary relation on there exists a total preorder on that extends such that

(2)

where is the reflexive and transitive closure of

##### Proof:

Define iff and .
The relation is an equivalence relation.
Let be the equivalence class of under .
Define iff .
The definition of does not depend on the representatives
and chosen.
The relation on equivalence classes is a partial order:
reflexive, antisymmetric and transitive.
A partial order may always be extended to a total order.
Let be any total order on these equivalence classes
that extends .
Define iff .
The relation is total (since is total) and transitive
(since is transitive): it is a total preorder.
It extends by the definition of
and the fact that extends .
Let us show that it satisfies Equation (2) of Lemma 2.1.
Suppose and .
We have and
and therefore by antisymmetry
( is an order relation).
Therefore and .

The algebraic representation results we are going to demonstrate in this Section 2 are independent of logic, and work for arbitrary sets not only for sets of models. On the other hand, if the (propositional) language is defined from infinitely many propositional variables, not all sets of models are definable by a theory: There are s.t. there is no with Moreover, we will consider only consistent theories. This motivates the following:

Let and let contain all singletons, be closed under finite non-empty and finite and consider an operation (For our representation results, finite suffices.)

We are looking for a characterization of representable operators. We first characterize those operations which can be represented by a symmetric pseudo-distance in Section 2.2, and then those representable by a not necessarily symmetric pseudo-distance in Section 2.3.

###### Notation 2.1

For will stand for etc.

### 2.2 The result for symmetric pseudo-distances

We work here with possibly infinite, but nonempty

Both Example 2.1 and Example 2.2 show that revision operators are relatively coarse instruments to investigate distances. The same revision operation can be based on many different distances. Consequently, in the construction of the distance from the revision operation, one still has a lot of freedom left. Example 2.2 will show that, in the case one does not require symmetric distances, the freedom is even greater. The reader should note that the situation described in Example 2.2 corresponds to the remark in the proof of Proposition 2.5, that the constructed distance does not necessarily satisfy i.e., may behave strangely on the left hand side. But even when the pseudo-distance is a real distance, the resulting revision operator does not always permit reconstructing the relations of the distances.

Distances with common start (or end, by symmetry) can always be compared by looking at the result of revision:

iff

iff

iff

This is not the case with arbitrary distances and as the following example will show.

###### Example 2.1

We work in the real plane, with the standard distance, the angles have 120 degrees. is closer to than is to a is closer to than is to but is farther away from than is from Similarly for b,b’. But we cannot distinguish the situation and the situation through (See Figure 2.1.)

##### Proof:

Seen from , the distances are in that order: .

Seen from the distances are in that order: .

Seen from the distances are in that order: .

Seen from the distances are in that order: .

Seen from the distances are in that order: .

Seen from the distances are in that order: .

Seen from the distances are in that order: .

Seen from the distances are in that order: .

Thus, any will be the same in both situations (with interchanged with with ). The same holds for any where has two elements.

Thus, any will be the same in both situations, when we
interchange with
and with So we cannot determine by
whether or not.

###### Proposition 2.2

Let be closed under finite non-empty and finite

Let

Let and consider the conditions

(Loop): …. imply

(a) is representable by a symmetric pseudo-distance iff satisfies and

(b) is representable by an identity-respecting symmetric pseudo-distance iff satisfies and

Note that corresponds to to will hold trivially, holds by definition of and will be a consequence of representation. corresponds to: … and, by symmetry, … i.e. transitivity of , or to absence of loops involving .

We first show the hard direction via a number of auxiliary definitions and lemmas (up to Fact 2.4). We assume all A,B etc. to be in and to hold from now on.

We first define a precursor to the pseudo-distance between A and and a relation on these We then prove some elementary facts about and in Fact 2.3. We extend to a total preorder using Lemma 2.1, the pseudo-distances will be S-equivalence classes of the It remains to show that the revision operation defined by this pseudo-distance is the same as the operation we started with, this is shown in Fact 2.4.

###### Definition 2.1

Set iff

set iff but not

is to be read as the pseudo-distance between and or between and . Recall that the pseudo-distance will be symmetric, so operates on the unordered pair Note that by definition of the function

Let be the transitive closure of we write also if it involves Write for etc.

The loop condition reads in the -notation as follows:

###### Fact 2.3

(1) iff

(2)

(3) There are no cycles of the forms or involving (The difference between the two cycles is that the first contains possibly only variations on one side, of the form the second one possibly only alternating variations, of the form

(4)

(5)

(6)

(7) implies (a) (b)

(8)

(9) Then

If holds, then

(10)

(11)

##### Proof:

(1) and (2) are trivial.

(3) We prove both variants simultaneously. Case 1, length of so contradiction. Case 2: length Let e.g. be such a cycle. If the cycle is not yet in the form of the loop condition, we can build a loop as in the loop condition by repeating elements, if necessary. E.g.: can be transformed to By Loop, we conclude contradicting (1).

(4) and (5) are trivial.

(6) contradicting by (3).

(7) (a) By (6), it suffices to show that But (b) Let we show By (6), it suffices to show

(8)

(9)

(10) as by Likewise Moreover, by (2).

(11) We show first that
implies
so
Thus,

We define:

###### Definition 2.2

Let by Lemma 2.1, be a total preorder on extending s.t. and imply

Let iff and and be the set of -equivalence classes and define iff but not This is a total order on Define for

If holds, let for any A. This is then well-defined by Fact 2.3, (10).

Note that by abuse of notation, we use also between equivalence classes.

###### Fact 2.4

(1) The restriction to singletons of as just defined is a symmetric pseudo-distance; if holds, then respects identity.

(2)

##### Proof:

(1)

(d1) Trivial. If then but not contradicting Fact 2.3, (10).

(d2) iff iff by Fact 2.3, (10) and (11).

(d3) is trivial.

(2)

Let Then there is By Fact 2.3, (8), for all So for all and

Let Take Then by Fact 2.3, (9) so

It remains to show the easy direction of Proposition 2.2.

All conditions but are trivial.
Define for two sets where and
Then by for all . Loop amounts thus to
which is now obvious.
(Proposition 2.2)

### 2.3 The result for not necessarily symmetric pseudo-distances

Note that we work here with finite only, will be