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Operations on Partial Orders

09/27/2018
by   Karl Schlechta, et al.
0

We define analogues of Boolean operations on not necessarily complete partial orders, the height of an element as the maximal length of chains going from BOTTOM to that element, and use height to define probability measures.

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1 Introduction

1.1 Overview

We define here Boolean operations on not necessarily complete partial orders, and then probability measures on such orders.

In a way, this is a continuation of work in [Leh96] and [DR15].

  1. Boolean operators

    In the rest of Section 1 (page 1), we discuss in general terms how to find and judge a generalizing definition.

    In Section 2 (page 2), we first give the basic definitions, see Definition 2.3 (page 2.3), they are quite standard, but due to incompleteness, the results may be sets of several elements, and not single elements (or singletons). This forces us to consider operators on sets of elements, which sometimes complicates the picture, see Definition 2.4 (page 2.4). An alternative definition for sets is given in Definition 2.5 (page 2.5), but Fact 2.4 (page 2.4) shows why we will not use this definition.

    We then discuss basic properties of our definitions in Fact 2.5 (page 2.5).

    We conclude in Section 2.3 (page 2.3) by defining a - to our knowledge - novel way to treat absent elements, by giving an indication of what is really meant.

  2. Height, size, and probability

    In Section 3 (page 3), we introduce the “height” of an element, as the maximal length of a chain from to that element, see Definition 3.1 (page 3.1).

    In Section 3.2 (page 3.2), we argue that the situation for sequences of partial orders may be more complicated than their product - basically as a warning about perhaps unexpected problems.

    Section 3.3 (page 3.3) introduces two notions of size for sets of elements, one, Definition 3.3 (page 3.3), by the maximal height of its elements, the other, Definition 3.6 (page 3.6), as the sum of their heights. We also discuss some basic properties of these definitions.

This is a rough draft, and mainly intended to present ideas.

1.2 Adequacy of a definition

We are not perfectly happy with our generalizations of the usual operations of and to not necessarily complete partial orders. We looked at a few alternative definitions, but none is fully satisfactory.

There are a number of possible considerations when working on a new definition, here a generalization of a standard definition:

  • Do we have a clear intuition?

  • Is there a desired behaviour?

  • Are there undesirable properties, like trivialisation in certain cases?

  • Can we describe it as an approximation to some ideal? Perhaps with some natural distance?

  • How does the new definition behave for the original situation, here complete partial orders, etc.?

In Section 2.3 (page 2.3), we discuss in preliminary outline a (new) approach, by adding supplementary information to the results of the operations, which may help further processing. Thus, it may help to improve definitions of the operators. The operators now do not only work on elements or sets of elements, but also the additional information, e.g., instead of considering we consider etc., where “inf” and “sup” is the supplementary information.

1.3 Motivation

In reasoning about complicated situations, e.g. in legal reasoning, see for instance [Haa14]

, the chapter on legal probabilism, classical probability theory is often criticised for imposing comparisons which seem arbitrary. Our approach tries to counter such criticism by a more flexible approach.

2 Boolean operations in partial orders

Assume a finite partial order with TOP, and BOTTOM, and i.e. has at least two elements. is assumed transitive. We do not assume that the order is complete.

We will not always detail the order, so if we do not explicitly say that or we will assume that they are incomparable - with the exception for any and transitivity is always assumed to hold.

2.1 Definitions

Definition 2.1

  1. For set iff and implies

  2. For define

Fact 2.1

(Trivial by transitivity.)

Definition 2.2

We define

  1. iff

  2. iff

Remark 2.2

  1. (trivial).

  2. (trivial).

  3. The alternative definition:

    iff

    does not seem right, as the example and shows, as then

We want to define analogues of the usual boolean operators, written here

We will see below that the result of a simple operation will not always give a simple result, i.e. an element or a singleton, but a set with several elements as result. Consequently, we will, in the general case, have to define operations on sets of elements, not only on single elements. Note that we will often not distinguish between singletons and their element, what is meant will be clear from the context.

Definition 2.3

  1. Let The ususal might not exist, as the order is not necessarily complete. So, instead of a single “best” element, we might have only a set of “good” elements.

    Define

    1. and

      This is not empty, as

      If is a set, we define

      for all

      In particular,

    2. We may refine, and consider

      Usually, also will contain more than one element.

      We will consider in the next section a subset of but may still contain several elements.

  2. Consider now The same remark as for applies here, too.

    Define

    1. and Note that

      If is a set, we define

      for all

      In particular,

    2. Next, we define

      Again, we will also define some later.

  3. Consider now

    Define

    1. Unary

      1. note that

        If is a set, we define

        for all

      2. Define

        Again, we will also define some later.

      It is not really surprising that the seemingly intuitively correct definition for the set variant of behaves differently from that for and negation often does this. We will, however, discuss an alternative definition in Definition 2.5 (page 2.5), (3), and show that it seems inadequate in Fact 2.4 (page 2.4), (3).

    2. Binary We may define either by or directly:

      1. and note again that

        and

      For a comparison between direct and indirect definition, see Fact 2.5 (page 2.5), (4.4).

We turn to the set operations, so assume are sets of elements, and we define

The natural idea seems to consider all pairs

Definition 2.4

We define the set operators:

  1. and were already defined. We do not define but see it as an abbreviation for

See Definition 2.5 (page 2.5) and Fact 2.4 (page 2.4) for an alternative definition for sets, and its discussion.

2.2 Properties

We now look at a list of properties, for the element and the set versions.

Fact 2.3

Consider with We compare with with and with

  1. so but “almost”, and

  2. so but “almost”, and

  3. and by Definition 2.3 (page 2.3), (3.1.1), and

  4. Consider

    Then by Definition 2.4 (page 2.4), (1), and

  5. Consider again

    Then by Definition 2.4 (page 2.4), (2), and

Thus, seem the better variants.

Definition 2.5

We define alternative set operators, and argue in Fact 2.4 (page 2.4) that they do not seem the right definitions.

  1. : for some

Fact 2.4

Consider with and

    1. Then by Definition 2.5 (page 2.5), (1.1),

    2. Then by Definition 2.5 (page 2.5), (1.2),

    1. Then by Definition 2.5 (page 2.5), (2.1),

    2. Then by Definition 2.5 (page 2.5), (2.2),

    1. (Trivial)

    2. In particular, which seems doubtful.

    3. By and the above,

    4. fails in general.

      Consider a, with

      Then so

    5. so defined is not antitone

      Consider then

Thus, the variants in Definition 2.5 (page 2.5) do not seem adequate.

Fact 2.5

Commutativity of and is trivial. We check simple cases like show that associativity holds, but distributivity fails. Concerning we see that is not well-behaved, and neither is the combination of with

  1. and

    1. ?

    2. ?

      for some

      - which is not necessarily (if there are with

    3. ?

    4. ?

      Note that for all thus for some

      - which is not necessarily

    5. ?

      Let

      Set

      1. We have to show

        by Definition 2.4 (page 2.4), (1).

        If then moreover so

        Let then there is As so so by transitivity.

      2. Set

        Let so there is and by so

        Suppose there is Then by (1.5.1) contradicting maximality of a.

        Conversely, let then by (1.5.1). Suppose there is so we may assume then as we just saw, contradiction.

        Thus, it works for too.

  2. and

    1. ?

    2. ?

      for some

      - which is not necessarily

    3. ?

    4. ?

      Note that for all thus for some

      - which is not necessarily

    5. ?

      Let

      Set

      1. We have to show

        by Definition 2.4 (page 2.4), (2).

        If then moreover so

        Let then there is As so so by transitivity.

      2. Set

        Let so there is and by so

        Suppose there is Then by (2.5.1) contradicting minimality of a.

        Conversely, let then by (2.5.1). Suppose there is so we may assume then as we just saw, contradiction.

        Thus, it works for too.

  3. Distributivity for

    1. ?

      Let

      Then so

      so

      so

      so

      So distributivity fails for both versions.

    2. ?

      Let

      So it fails again for both versions.

  4. and

    1. ?

    2. ?

    3. ?

      Consider with

      Then so it fails for both versions.

    4. ?

      and

      and for some and by Fact 2.1 (page 2.1).

    5. ?

      by Let for then and so

    6. ?

      for all by for all Conclude as for (4.5).

    7. ?

      and

    8. ?

      Consider with

      Then and

      so it fails for both versions.

    9. is antitone:

      for all so for all

    10. for all

      Let By so

    11. fails in general.

      Consider a, with

      Then so

2.3 Sets with a sign

We will outline here a - to our knowledge, new - approach, and code the last operation into the result, so the “same” result of two different operations will look differently, and the difference will be felt in further processing the result, see the following Example 2.1 (page 2.1). See also Definition 3.3 (page 3.3).

Basically, we give not only the result, as well as we can, but also an indication, what the intended result is, “what is really meant”, the ideal - even if we are unable to formulate it, for lack of an suitable element.

More precisely, if the result is a set but what we really want is which does not exist in the structure we will have the result with the sign sup, i.e., likewise inf and and further processing may take this into consideration.

In a way, it is a compromise. The full information gives all arguments and operators, the basic information gives just the result, we give the result with an indication how to read it.

Example 2.1

Let with

see Diagram 2.1 (page 2.1). (The relations involving and are not shown in the diagram,

Then:

  1. more precisely - which does not exist, but we do as if, i.e., we give a “label” to

    Reason:

    We have is the smallest such that thus but this does not exist.

  2. more precisely - which does not exist, but we do as if, i.e., we give a “label” to

    Reason:

    We have is the biggest such that thus but this does not exist.

To summarize, we have and - but and need not exist.

Consider now and to see the difference:

Basically, we remember the last operation resulting in an intermediate result, but even this is not always sufficient as the example in Fact 2.5 (page 2.5), (3.1), failure of distributivity, shows: The intermediate results are singletons, so our idea has no influence.

One could try to write everything down without intermediate results, but one has to find a compromise between correctness and simplicity.

Diagram 2.1
Diagram Inf/Sup Recall that and do not existecdxx’e’abyf’f

3 Height

3.1 Basic definitions

Definition 3.1

Let

Set the length of the longest chain from to - where we count the number of in the chain.

Let

Define

and

In Section 3.3.2 (page 3.3.2), we give an alternative definition of a probability using height.

Fact 3.1

  1. for all

  2. We have for all

  3. If and are -incomparable, it does not necessarily follow that

    (This is trivial, as seen e.g. in the example with so and are incomparable.)

Definition 3.2

We might also have chosen instead of etc., by Fact 3.1 (page 3.1), (5).

Example 3.1

Consider with

Then and so we lose important information, in particular, if we want to continue with Boolean operations.

For this reason, the versions should be used with caution.

3.2 Sequences

Example 3.2

In the second example, we compensate a loss in the second coordinate by a bigger gain in the first. Thus, the situation in the product might be more complex that the combined situations of the elements of the product.

  1. Consider and with the natural orders. In in