From rough to final designs by incremental set–inclusion
5.3 Class partition
The approach of class partition is based on the notion of classification. In the beginning of the design exploration process, we have a concept under which the possible designs as individuals fall5. A collection of individuals is called a class.
A full collection of all possible individuals falling under a concept is called the extensionof the concept. These may, however, not all qualify as proper solutions to the design problem at hand.
The exploration process now divides the class recursively until a satisfactory design concept is reached. In mathematics this is called class partition, or simplypartition.
Definition 5.8 (Class partition) By a class partition, in mathematics, we understand a division of a set of individuals into mutually exclusive and jointly exhaustive subsets. Each such subset is a class of which individuals are intimately associated by means of an equivalence relation [8].
Instead of expressing equivalence relations, it is common to describe individuals of classes by expressions with a predicative nature on individuals. Such
ex-4although not all collections may be proper class extensions.
5Several philosophical problems are here present. Primarily, we have the problem of having collections of artefacts not yet existing. Suggestions for solving this problem include applying a Platonic perspective [84] and introducing possible worlds (see the papers in Chapter 8 and Chapter 9).
5.3 Class partition 117
pressions may refer to properties of the individuals which means that what is expressed is the conditions for individuals to belong to a certain class.
The structure of class partitions spans a tree structure as depicted on Figure 5.2.
Figure 5.2: Classification of individuals.
Design choices here means selecting a certain path among possible alternatives.
Each choice decreases the space of individuals and the process stops when a satisfactory design concept is reached. A partition of the class of bridges is pictorially shown on Figure 5.36.
It is fundamental to the approach, that what is partitioned right from the start is the class of all individuals (perhaps restricting to some basic concept like houseorhospital).
Model–theoretically, the approach is extensional as each stage in the design process associates to it a class of individuals. Going from problem to solution means going from the class of all individuals to more and more narrow classes by means of restriction. Each such restriction corresponds to the incrementally added conditions derived from the design problem.
Although the partition approach complies with the problem–solution under-standing and the mathematical notion of class partition, we claim six objections to it.
6Note, that any pictorial representation of concepts which differ in abstraction is wrong.
Excluding parts of a picture does not correctly express the fact that what is presented is more abstract. However, this is the only graphical means for expression possible.
bridge
free height
suspension bridge with 76m 76m
free height
suspension bridge with 45m 45m
suspension bridge with opening suspension bridge
cable-stayed bridge
Figure 5.3: Partition of the bridge class.
The first objection is that it may not be possible always to state the overall concept under which a final design solution falls. As argued by Schön [149], working with a design may open for other sorts of solutions than first conceived.
Thus, a straight top–down approach is not always possible. As a rule of thumb, we could say that the better we know the domain of discourse, the better can a top–down approach be applied. Solving this problem by starting with a very general concept, may result in loss of the focus instead. One of the characteristic of design is that we do not really know the class of design solutions that well — especially not if we are talking about innovative design; i.e. design of completely new sorts of artefacts.
The second objection is that we are hardly interested in representing the classes of design individuals which have not been selected. Thereby, the whole idea of class partition seems to be lost. In some cases, a design process will then be a series of class name definitions of which the definiendum may be difficult to express simply because we need to express what individuals the classes do not include.
The third objection is that the approach does not give a good account on how to handle removal of design knowledge. We cannot overlook the fact that some design moves are in fact removal of objects, properties of objects or relations between objects, from a design. Basically, there are two problems. The first is that in order to maintain consistency, removal of such design information has
5.3 Class partition 119
to be performed in the opposed order in which it has been added. However, removing one piece of information may also remove other pieces of information, depending on the order in which the classes have been partitioned. In general, we may need to make several such steps upwards the classification structure before the unintended design information has been removed. Along the way we may have removed other design information which we wished to keep. The problem, we call the backtracking problem. The second problem appears if we try to solve the first by allowing removal downwards the tree structure. This may result in obscure situation, in which we start with a general design concept and ends up with this concept again because restrictions have been removed.
The fourth objection is that identical pieces of design information may exists in stages of non–equivalent classes in a partition. This means that we have redundant information. Redundancy, potentially is the cause of inconsistency problems, and should in general be avoided.
The fifth objection is that the partition approach do not offer convenient and easy ways of combining distinct, possibly overlapping design solutions.
The sixth objection is that it may be a problem categorising a design indivi-dual to belong to a certain class. In knowledge engineering, it is common to categorise individuals according to their properties. However, the individuals we are considering may consist of several objects; each having own properties.
Classifying such multi–object constellations is complicated. The reason is that we need to include conditions for whether design concepts have parts of certain kinds. Thereby, the problem of classification becomes a type identity problem based on part–whole relations of objects. This problem is one of the problems in part–whole theory (in general,extensional mereology) for which objections are strong (cf. Chapter 8).
Also, other concept names may better characterise intention and use of the in-dividuals of a class. E.g. several theaters are housed in closed factory buildings.
The first and second objections indicate that the design process should be con-sidered to start with an empty configuration. To this empty configuration, design knowledge is added incrementally, thereby specialising the design and decreasing the solution space. Instead of having concepts for each design stage, we suggest that we focus on properties of objects and relations between objects.
The third, fourth, and fifth objections call for a distinction between two sorts of design moves: (i) those which explore design alternatives, and (ii) those which combine design alternatives. We call these: design moveby aspect and design moveby configuration, respectively. The distinction is convenient in order to represent the many–sorted knowledge of designs.
Together, the five objections suggest that a convenient structure for recording and structuring design processes, is a lattice.
The sixth objection raises more philosophical and epistemological questions of how names can be used to denote classes of artefacts. Among these is the question of whether we can relate concept and use/rôle of objects falling under concepts. However, we shall not treat this issue in this paper (cf. Chapter 8).
5.4 Design lattices
The approach of design lattices does not rely on concepts for denoting individual design solutions. Rather it relies on the properties which aim at characterising the objects in design solutions. The approach is thus intensional, whereas the other is extensional.
Design lattices have the structure of mathematical lattices.
Definition 5.9 (Design lattice) By a design lattice, we understand a collection of design configurations which are partially ordered. Each design configuration corresponds to a specific stage in the design process. The relation stands between such design stages corresponds to design moves which specialise the configurations.
Figure 5.4 shows a design lattice for one design process leading to a bridge design similar to the one depicted on Figure 5.3.
A design lattice is always bounded upwards. The bound is called Top (⊤) and represents the empty design configuration; that which has the empty set of objects. From Top, we can go to stages such that objects, properties of objects, and relations between objects are added — one step at a time, though.
Design configurations (i.e. stages) are brought together by means of the lattice operationmeet. This operation takes two design configurations (partial designs) and gives the configuration with the union set of objects and the union set of properties for equally named objects. A similar account applies to the relations in the two models.
It is not inevitable that design lattices are bound downwards. The reason is that lattice meet can be applied for later being abandoned in favour of the meet of other design stages. However, a design lattice can always be bound downwards by adding to it lattice meet of all lower branches.
5.4 Design lattices 121
84m
12m
84m
12m 340m
bridge slab bridge piers
defining dimensions
combining parts
adding pylons pylons
Figure 5.4: Design lattice leading to a bridge design.
As a general interpretation rule of design stages in design lattices, we can say that objects in a design are required to be existing parts of the artefact in mind.
Similar goes for properties and relations.
Model–theoretically, design lattices have similarities with the approach of class partition. The empty configuration of a design lattice corresponds to a most general design concept; the concept for which we do not require that individuals consist of certain objects nor that certain properties are ascribed the objects. All artefacts satisfy these criteria. Adding objects, properties of objects, and rela-tions between objects introduce restricrela-tions which specialise the design concepts.
Such concepts could be the concept of individuals consisting of two objects of which the former is ascribed the property of being made of steel and the latter the property of being made of wood. Suitable concept names may be difficult to define but with design lattices, we do not have to do so.
Compositionally, the notion of design lattices have similarity with extensional
mereology — i.e. the theory of part–whole relations. The empty configuration corresponds to absurdum — the empty object constellation. Lattice meet of all design configuration corresponds touniverse; although in a very local un-derstanding. In mereology universe is the mereological sum of all objects in the world. In this doctrine, objects are either atomic or sums of other objects, where sum is the inverse operation of performing a partition of the universe or a part of the universe. However, mereology focuses on the opposed direction of going from empty configurations with its emphasis on universe.
In order to overcome the problems stated by the six objections, design lattices make a distinction between two sorts of design moves: design movesby aspect and design movesby configuration.
From lattice theory, we know that redundancy can be eliminated by means of restructuring. This is a possibility, but not a requirement, in design lattices as we aim at recording the design decisions of the practitioner. The partial ordering of lattices ensures that consistency is maintained. Figure 5.5 shows how design lattices handle object removal compared to an approach which does not maintain order of design stages.
remove object
(a)
remove object
(b)
Figure 5.5: Object removal: Design lattice (a), without order (b).
However, it should be stated that design lattices do not give complete freedom in backtracking. Consider the situation where a set of objects have been put together with lattice meet, and then from this stage additional objects are added.
Removing one of the first added objects cannot be done from the resulting stage; only from the stage of the first meet operation. That is, the backtracking problemcan occur with design lattice, but they still add more flexibility (with
5.4 Design lattices 123 respect to order consistency) than the class partition approach.
5.4.1 Design choice by aspect
Design choice by aspect is the class of design moves which add an object, a property of an object, or a relation between two objects, to a design. The choice is made between different incomparable sorts of design knowledge; that is: objects, properties of various domains, and relations of various kinds. The domains of such knowledge are called aspects. That is:
Definition 5.10 (Aspect) By an aspect, we understand the conception of an object or a domain of properties/relations which relate by their kinds.
In a design lattice, design choice by aspect means selecting a certain design path.
That is, exploring possibilities of a certain domain of properties, like colours or material. Figure 5.6 shows the principle of design choice by aspect.
dimensions construction/
material
colour
Figure 5.6: Design choice by aspect.
The notion of aspect is inspired by Gärdenfors’ notion of domains. Besides making partitions of design knowledge into incomparable sorts, design choice by aspect also serves the purpose of making partitions of single property domains.
Not all equivalence classes of a partition need to be represented in the lattice, as indicated on Figure 5.7. Consider a partition of the property of having a length of3mor more into the property of having a length between 3 and 8m, and5and10m, respectively. This partition excludes the possibility of having a length of more than10mand less than3m. It may seem strange mathematically, but there is no rationality in representing design knowledge which will never be considered to be added to the current design. Selecting the colour green for an object does not commit us to also representing all other colours. We simply represent nodes that correspond to actual or previous design stages.
Nat
]10;∞[
[5;8]
[5;10]
[3;8]
[0;3[
Figure 5.7: Partition of length properties.
5.4.2 Design choice by configuration
Design choiceby configurationis the class of design moves which combine exist-ing design stages. That is, they combine different aspects into a configuration.
For properties, this principle is known asmultiple inheritance and — concern-ing representation — it is the point where lattices put distance to trees. For comparable properties like having the colour red and green, the result is the intersection set of the property values. We shall later see how names (called attributes) are used in the identification of property domains.
For incomparable properties like being made of concrete and being shaped as a column, the result is the union set of the property values. An explanation of incomparability is as follows. Consider the property of being transparent and the property of having the colour blue. These properties do not exclude each other; they can co–exist for an object. The reason is that the former is defined as materials capability to let through light, whereas the latter is defined as the property of how light is reflected. The two properties thus instantiate distinct causal relations between the objects in question, and the surrounding.
This understanding of properties is adopted from Shoemaker [153] (see also Chapter 9).
Figure 5.8 shows the principle of design choice by configuration.