Upload
ravipande
View
219
Download
0
Embed Size (px)
Citation preview
8/6/2019 Typed Feature Structures and Design Space Exploration
1/16
Typed feature structures and design space exploration
R. WOODBURY,1 A. BURROW,2 S. DATTA,1 and T-W. CHANG1
1School of Architecture, Landscape Architecture and Urban Design, University of Adelaide, SA 5005, Australia2Department of Computer Science, University of Adelaide, SA 5005, Australia
(Received June 23, 1998; Final Revision March 31, 1999; Accepted April 1, 1999!
Abstract
Design space explorers are computer programs that play on an exploration metaphor to support design. They assist
designers in creating alternative designs by structuring the process of design creation in a space of alternatives. Sub-
sidiary metaphors relevant to design space explorers are generation, navigation, and reuse. This paper introduces, in
two sketches, typed feature structures as a formal system in which a design space explorer and its knowledge level
might be implemented. First, informal and abstract properties of typed feature structures suffice to build a sketch of the
behavior of a design space explorer. Second, using an example based on single-fronted cottages ~a common Australian
housing type!, we outline the typed feature structure machinery most relevant to design space exploration.
Keywords: Exploration; Generation; Navigation; Reuse; Design Spaces
1. INTRODUCTION
In studying the phenomenon of design, we use models ~met-
aphors! to envision mechanisms by which computers might
support design. One such mechanism is variously called
search, exploration, discovery ~and other terms!under anyname it is a guided movement through a space of possibil-
ities. Design space explorers are computer media that en-
gage designers in exploration. With them, designers explore
possibilities. This encompasses discovering new designs,
as well as recalling, comparing, and adapting existing de-
signs. In creating design space explorers we represent the
possibilities in the form of symbol structures called states
and seek efficient representations for these states and al-
gorithms over them. We represent spaces of possibilities
as a relation over states and seek definitions of the space-
generating relation, which are relevant to the exploration
metaphor and lend themselves to tractable computations.
A pervasive notion within the exploration metaphor isgoal-directness, which arguably can be traced to the origins
of the exploration metaphor in cognitive accounts of human
rationality. In essence, goals exist independently of the de-
signs that realize them and the process of exploration is
guided by a set of goals that itself may be transformed within
the exploration process. Computing with goals requires their
representation. Because goals may change throughout an ex-
ploration process, design states record the goals toward which
they aim and the artefacts that ~partially! realize those goals.
Another notion is that of nondeterminismthe goal doesnot of itself determine the decisions enacting the explora-
tion. An external agent, often the user, must resolve the in-
determinacy in seeking the goal. In design space explorers,
each decision is expressed in the transformation of one de-
sign state to another. A design space is implicitly defined by
a set of initial design states and a set of state transforming
operators. A design space explorer assists a user to unfold
this compact and implicational description into an explicit
space of recorded possibilities. This exploration history is
necessarily a subset of the complete design space. Design
space explorers provide a mixed initiative environment where
the user and computer play complementary roles.A user ex-
periences the design space explorer inter alia through re-
solving exploration nondeterminism. Support for this role
is therefore critical to the external qualities of a design space
explorer. The careful presentation of alternatives avoids over-
whelming a user, and assists a user to perceive coherent
movements in the design space. Visualization of the design
space and its place in the exploration history provides a con-
text for the presentation of alternatives. A sound organiza-
tion of alternatives manages information and introduces a
perceptible rationale.
Reprint requests to: R. Woodbury, School of Architecture, LandscapeArchitecture and Urban Design, University of Adelaide SA 5005, Austra-lia. Phone: 618-8303-4588; Fax: 618-8303-4377; E-mail:[email protected]
Artificial Intelligence for Engineering Design, Analysis and Manufacturing ~1999!, 13, 287302. Printed in the USA.Copyright 1999 Cambridge University Press 0890-0604099 $12.50
287
8/6/2019 Typed Feature Structures and Design Space Exploration
2/16
The experience from extant spatial grammar based tools
is that a necessary modification to their formalisms and thus
their interpreters is a mechanism to control the availability
of transformations so that construction occurs in coherent
phases. The representation of operators should reflect this
need to structure and organize these alternatives. The set of
transformations available at each design state is the primi-
tive relation describing exploration nondeterminism. How-ever, there are natural structures in this relation that should
be exploited. Alternatives may be mutually exclusive, in-
terdependent, or independent. Without a structuring princi-
ple for operators, these properties will be evident only in
the pattern of discovered alternatives and distributed across
the design space. With a structuring principle for operators
these properties may be expressed in the presentation of al-
ternatives, their grouping, and the ordering of these groups.
In their action of tracing out design spaces, explorers leave
in their trail a record of work, effectively a case-base in the
terms of case-based design. Organizing, accessing, and re-
using this resource of past work is a chief challenge for de-
sign space explorers.A final aspect of the mixed-initiative dialogue between
user and explorer is the genesis and development of opera-
tors. When designer action is interpreted as design space
exploration it is widely recognized that the set of design
moves used are typically developed in the context of de-
sign ~Archea, 1987!. A design space explorer should there-
fore support the discovery and application of new operators
within and between design sessions and design projects.
A design space explorer is then a device that represents
goals, operators, and design possibilities. It organizes these
states into spaces, provides ways of accessing states and ap-
plying operators to extend the explored space, provides ways
of reusing states in the already explored space, and sup-ports the invention of new operators. A designer interacts
with an explorer through states, spaces, operators, and pre-
sentation of their interactions. Systems that can be de-
scribed as design space explorers have at least the
functionality of recognizing how operators apply in a state,
and in most cases can, in some sense, autonomously act upon
such recognition. The design space explorer is an active par-
ticipant in a mixed initiative environment based on the ex-
ploratory unfolding of design spaces.
This paper presents typed feature structures, a formal sys-
tem in which a design space explorer might be realized.
Typed feature structures make the presumption, common but
by no means universal in the design literature, that the en-
tities of interest ~Newsome et al., 1989; March, 1995! can
be structured into a composition of objects and features. Thus
representations such as shapes ~Stiny, 1980! in which a def-
inite representation comprises an indefinite set of parts would
seem, at first glance, to be excluded from the typed feature
structures game.1
A deep understanding of typed feature structures requires
an appreciation of its formal mechanics, which, at least in
the first authors experience, takes fair effort to acquire. In
this paper we approach typed feature structures through two
sketches, one of their behavior as a representation for a de-
sign space explorer and the other of their underlying formal
mechanics and our extensions to them. Carpenter provides
a more complete description of typed feature structures, in-cluding many proofs for those readers wishing to tackle the
formalism in full gory detail.
2. TYPED FEATURE STRUCTURES AS A
REPRESENTATION
As a first approximation, consider a system made out of three
sets of components: types T, structures F, and descriptions
D, put into a representation scheme with a building design
domain as shown in Figure 1. The types comprising Tstand
for expressed knowledge of the domain of interest, in this
case a building design. Structures from Frepresent models
of particular designs, in this case, buildings and0or their com-ponents, either physical or conceptual. Structures from F
are expressed in terms of the information expressed in T.
Descriptions from D are utterances in a formal textual lan-
guage and stand for themselves in the representation scheme,
but correspond symbolically and inter alia to sets of struc-
tures from F. The essential act of generation corresponds to
the discovery of models of buildings that are simulta-
neously compatible with a description and the knowledge
from the domain of interestin typed feature structure ter-
minology generation is the computation of structures com-
patible with a given set of types and a given description. All
of this is ordinary in the realm of current knowledge-based
system theory and can be found, in many variations, in theliterature. In the sequel we describe aspects that typed fea-
ture structures provide to a design space explorer. For now,
as much as possible, we eschew the abstract symbolic terms
of type, structure, and description, relying instead on their
domain counterparts, namely, knowledge, model, and de-
scription.
We use as an example, a house type, common in Austra-
lia, known as a single-fronted cottage, an example of which
is shown in Figure 2. Single-fronted cottages are simple
1See Section 5. Fig. 1. Abstract elements of a typed feature structurerepresentation scheme.
288 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
3/16
houses, comprising a single-loaded hallway, rooms off of
the hallway, and a collection of spaces at the end of the hall-
way. The rooms along the hallway are typically generic in
function and can be used as bedrooms, lounges, dining
spaces, studies, etc. The hallway and its adjacent rooms are
typically housed under a single roof, which is either a gable
or a hip. The collection of rooms at the end of the hallway is
typically configured as a skillion ~space under a shed roof
addition! or historically as separate structures ~typically a
kitchen and toilet!. A porch along the front of the cottage
completes the typical picture. Single-fronted cottages are a
historic housing type, filling a niche of small, mostly inex-
pensive housing. Latterly, many have been upgraded withparticular attention being given to elaborations of the skil-
lion addition, usually into a combination family room, din-
ing room, and kitchen. New houses in the single-fronted
genre are still being constructed in inner suburban areas.
2.1. Structures are both intensional and partial
Intensionality in a representation scheme means that two
symbol structures can be identical in all aspects yet remain
distinct. In an intensional representation, the actual identity
of two structures must be explicitly called out, for example,
by a declaration that two compatible structures are, in fact,
one and the same. From a domain point of view, we wish to
be able to distinguish between models that differ only by
the fact of their separate existences. As shown in Figure 3,
in our example realm of single-fronted cottages, the three
rooms of the room row can be identical in all aspects, yet
remain distinct.
Partialness means that a structure need not contain all in-
formation that might be inferred about it from its associated
types. In effect, a partial representationA stands for all more
complete representations that contain at least the informa-
tion in A. Appealing to a domain perspective, a model may
contain less information than we can infer from our knowl-
edge of it. Figure 4 shows that a model of a single-fronted
cottage with its dining room in the room row can exist with-
out holding any other information on single-fronted cottages.
Partialness and intensionality swim together. We can know
very little of two structures, except that they are differentor
that they are exactly the same structure. In terms of cot-
tages, we may know that the dining room is or is not the
third room in the row. Conversely, within a computation, a
given structure is not necessarily the same as another that
contains consistent information, but might be made so at
some future point in the computation. For example, Fig-
ure 5 shows that two separate cottage models may at some
point become a single composite model, containing all in-
formation from both. In this figure and in all figures dem-
onstrating information inclusion, we depart from the normal
convention in artificial intelligence of drawing specializa-
tion relations with the most general concepts at the top. We
do this for two reasons: ~1! to cohere with the intuition that
a structure containing more information than another is some-
how greater than the other, and ~2! the typed feature struc-
tures literature follows this convention.
2.2. Partialness is in the eye of the beholder
Whether a structure is partial or not depends on the content
of its associated types. In domain terms a model might be
partial against one set of knowledge but complete with re-
spect to a subset of the knowledge. For example, if knowl-
edge of single-fronted cottages is limited to their spatial
Fig. 2. A typical single-fronted cottage.
Fig. 3. Rooms 1, 2, and 3 can be identical in all aspects yet remain dis-
tinct.
Fig. 4. That the dining room is located in Room 3 of the room row is all
that is known in this structure, even though the available knowledge of
single-fronted cottages would allow more to be inferred.
Typed feature structures and design space exploration 289
8/6/2019 Typed Feature Structures and Design Space Exploration
4/16
organization, as might be the case for an urban geographer,
a complete model of a cottage would assign functions to
physical spaces. Such a model would be partial with re-
spect to a larger set of knowledge, containing, for example,
knowledge of how to construct single-fronted cottages.
In terms of design space explorers, partialness in a model
indicates that more exploration is possible from that model.
Conversely, adding new knowledge opens up new explora-
tion possibilities for previously completed models.
In the domain of single-fronted cottages, Figure 6 shows
that a cottage about which we know only the location of the
dining room is consistent with several alternatives for the
placement of other rooms. If our knowledge of the building
type consists of only spatial organization, a model is com-
plete once we have allocated all functions to specific rooms.2
2In this example and those that follow in this section, partialness wouldappear to be based on the subset relation, that is, a partial model is a subsetof a set and the set is the ultimate complete model. With the introductionof the actual typed feature structure mechanism in Section 3 it becomesclear that partialness as a concept has more richness.
Fig. 5. Two cottage models that are consistent yet incomplete can be joined into a single cottage model containing the information
from both models.
Fig. 6. A partial model of a single-fronted cottage that is consistent with multiple complete models.
290 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
5/16
2.3. A generative system without rules
Recall that our formal system comprises three sets: types T,
structures F, and descriptions D, with elements ofD stand-
ing for subsets of F. Further, imagine, first that there is a
relation of satisfaction M linking the structures from Fthat
are consistent with a given description; second, that there is
an efficient procedure P for computing M; and third, that a
type from Titself has a corresponding description in D. Thenthe same procedure that generates structures consistent with
a description can generate structures consistent with the set
of types. In effect, the type hierarchy itself becomes what
Carlson describes as a design space description mechanism~Carlson, 1994!. In domain terms, from knowledge we can
infer designs.
2.4. An exploration system
Much has been made in the literature on the distinction be-
tween search and exploration systems, on the basis of the
ability to consider changing goals within a computationalprocess. The distinction disappears when goals are held in
the same structures as other models in a system, and this is
the case with most extant generative formalisms. The present
mechanism is no different. The set of types need not remain
fixed. Different sets of types define different sets of com-
plete structures. Though we do not discuss how a particular
knowledge level representation that would include, inter alia,
goals and design representations, might be expressed with
the typed feature structures mechanism, like all knowledge,
goals are expressed as types, which like all knowledge, can
be changed.
2.5. Monotonic generation
The fact of partial structures provides, in itself, a notion of
what might comprise generation, namely, the alternative
completions of partial structures into more specific partial
structures and finally to complete structures. This amounts
to a claim that the generating procedure P must act incre-
mentally, that is, it must progressively generate more and
more complete structures as it works through its input de-
scription. P thus stands as the design space generator in this
schemeby its action it traces out a derivation relation from
state to state. Under such a regime, generation is monotonic
with respect to the structures it creates. In other words, if a
fact is known in a model, it remains known in all models
that can be created from that model. It is well-known that
monotonicity is a double-edged sword. While it enables
search strategies based on branch-and-cut, it restricts the set
of allowable operators and possible derivation sequences.
Appealing to our example domain, Figure 7 shows that a
partial model of a single-fronted cottage is consistent with
multiple models, partial and complete, and these models are
related by the action of the procedure that generates one
from the other. In this example, we are clearly recovering
part of the subset relation over the set of rooms in the single-
fronted cottagein the structures we have in mind, mono-
tonic generation is more complex.
2.6. Design spaces are structured by subsumption
Monotonic generation implies that the derivation relation is
a subsumption relation, that is, a model B derived from a
model A contains strictly more information than A. With
respect to single-fronted cottages, each model in the chainin Figure 8 contains strictly more information than its
predecessor.
Derivation defines only a subset of the full subsumption
relation. If we presume an efficient way of comparing struc-
tures from F for their information specificity, that is, for
their relative position in the subsumption relation, then the
full subsumption relation for a set of discovered structures
can be recovered in the process of generation ~Fig. 9!. Fur-
ther, if structures are thought of as themselves comprising
structures this subsumption relation can be recovered for all
the parts of a structure.
Having access to the full subsumption relation in a de-
sign space enables forms of exploration richer than branch-
and-backtrackusual in generative systems. The branch aspect
of generation remains largely unchanged. Backtracking be-
comes more rich, in that the derivation relation along which
backtracking typically takes place is replaced by the full sub-
sumption relationfrom a given state, it is possible to move
to less specific designs that have previously been discov-
ered and to less specific designs for parts, irrespective of
the process by which the designs were generated.
Fig. 7. From a partial model of a single-fronted cottage can be generated
a graph of consistent partial and complete models.
Typed feature structures and design space exploration 291
8/6/2019 Typed Feature Structures and Design Space Exploration
6/16
Last, the ability to find the appropriate place for a givenmodel in a subsumption relation opens the door to the non-
monotonic operator of information removal. Removing in-
formation from a model results in a less-specific model,
which has its own place in the subsumption relation. Thus,
it is possible to move to less specific designs that have not
been previously discovered, though at the cost of losing the
cut part of branch-and-cut search strategies, if the infor-
mation removal operator is included in the generation pro-
cedure.
2.7. Reuse is intrinsic
The aspirations of design space exploration trade heavily
on an ability to reuse information already discovered, for
example, to find and adapt for the present single-fronted de-
sign that clever island-feature kitchen design someone in
your firm did in the Darwin office sometime in the last
2 years.
Together, partialness and the subsumption ordering over
structures provide a seductive view on the reuse of de-
signs, the required actions for which are retrieval and ad-
aptation. Recall that a structure is a partial specification of
an object, that it is consistent with all objects that containat least the information specified in it, and that structures
exist in a subsumption ordering. It follows that any struc-
ture effectively indexes those structures more specific than
it in the subsumption orderingthe same mechanism by
which designs are represented can be used for retrieval. Once
retrieved, the design space explorer operations ~monotonic
generation, enriched backtracking, and informationremoval!
are available to adapt the design to a new context.
In their support of reuse, typed feature structures would
appear to do some of the duties of a case-based design sys-
tem, for which the required mechanisms are representation,
indexing, retrieval, and adaptation. In the typed feature struc-
tures view, both cases and their indexes are indistinguish-able from a structure, case retrieval indistinguishable from
navigation in a subsumption relation of designs, and case
adaptation indistinguishable from design space explora-
Fig. 8. In a derivation chain of partial single-fronted cottages, each suc-
cessive cottage model contains at least the information contained in its
predecessor.
Fig. 9. The derivation relation is a subset of the full subsumption order over models.
292 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
7/16
tion. We believe it worthwhile to draw two distinctions be-
tween cases and the mechanics of typed feature structures
for reuse. First, much of the work in case-based reasoning
is motivated by, or at least plays to, a reproduction of hu-
man reasoning processes. Thus, designs for case-based rea-
soning systems appeal to features such as cognitive closeness
of match in retrieval and a specialized adaptation process.
In contrast, the retrieval and adaptation mechanisms sug-gested by typed feature structures are structured first by ef-
ficient computations in the mechanism and second by an
appeal to elegance in humancomputer interaction marked
by reluctance to introduce new mechanisms. Second, much
of the work in case-based reasoning is situated within weakly
structured symbol systems and therefore requires different
and typically more complex strategies for retrieval and ad-
aptation than does the typed feature structures mechanism,
defined as it is over a strongly structured and ordered uni-
verse. For these two reasons, we believe that casting our
reuse mechanism as case-based reasoning could generate
more heat than light. Nonetheless, below we sketch some
relations between typed feature structures reuse and case-based reasoning.
Structures as cases provide a feature that is commonly
used in case-based systems. By its place in the derivation
relation, a structure provides sets of generation steps by
which a case has been created from scratch, as does, for
example, JULIA ~Hinrichs, 1992!. To this, structures, by their
place in the larger subsumption relation, provide alternative
sets of generation steps by which a case might be created
from scratch.
If cases are structures, then case adaptation is the normal
process of design generation, namely, the combined action
of a monotonic generating procedure, backward movement
in the subsumption relation and removal of content.If we consider a case index as a structure, then indexes,
like structures, have their own place in the subsumption
relation of structures. More specific structures are in-
stances of the case; less specific structures are a form of
partial match of the case. This view on cases corresponds to
the two kinds of matches that Flemming describes for cases
in the SEED system ~Flemming, 1994!. Given the availabil-
ity of the subsumption relation over structures and their parts,
the index vocabulary problem described by Kolodner ~1993,
Chap. 6! would appear to disappearthe index vocabulary
is the full vocabulary from which structures are constructed
and indexes are implicit in the subsumption ordering.
We end this discussion of cases with a disclaimer. Typed
feature structures are seductive as formal devices for case
representation, retrieval, and adaptation. We have yet to dis-
cover if their siren call leads us to founder on some rugged
coast or signals a clear, concise, and new approach.
2.8. Summary
In the light of the above seven features, to the system sketch
in Figure 1 can be added two principal relations correspond-
ing to computational paths among its symbolic components
as shown in Figure 10.
The generating procedure P ~called p-resolution! that cap-
tures a relation from descriptions to structures M:D r F
acts as the main exploration mechanism in the system.
Access to the subsumption relation S : Fr F for struc-
tures and their parts provides a design space structure that
includes the derivation relation as a subpart. Case indexing
and retrieval are essentially search in the subsumption re-
lation. An episode of design space exploration would em-
ploy both P and movement in the subsumption relation.
Several computation paths would appear to be excludedfrom the family of computations available in this scheme.
Types especially seem to be estranged. Later we will show
that a description can call out a type, thus bringing types
back into the fold. Most other paths have corresponding well-
formed relations, but these are subsidiary in design space
exploration and we will only explain the ones on which our
principal paths depend.
3. THE MECHANICS OF TYPED FEATURE
STRUCTURES
The term feature structures denotes a class of record-like
data structures that comprise sets of attribute-value pairs
called features and values, respectively. Feature structures
arose in the areas of computational linguistics and logic
programming and have natural analogues in terminologi-
cal knowledge representationtheir provenance, which is
beyond the scope of this paper, is summarized by Carpen-
ter ~1992!. Here we sketch a particular formalization typed
feature structures as developed in Carpenter ~1992! that
admits a variety of efficient algorithms over these data struc-
tures. Carpenter was motivated by the expression of par-
Fig. 10. Principal computation paths in typed feature structures.
Typed feature structures and design space exploration 293
8/6/2019 Typed Feature Structures and Design Space Exploration
8/16
tial information, so that the absence of a feature in a feature
structure is not taken to imply its absence in an object that
the feature structure might represent.
As a computational system feature structures comprise
four elements: a type hierarchy, a set or sets of feature struc-
tures, a description language, and a set of algorithms over
these. On the first three of these elements are posed condi-
tions that collectively admit the robustness and efficiencyof the algorithms in the fourth component. In the present
context, the type system, feature structures, and utterances
in the description language correspond to the types, struc-
tures, and descriptions of the abstract system described in
the previous section.
We proceed with an informal specification of the princi-
pal algorithms over feature structures as a way of motivat-
ing the necessarily detailed presentation of the type system,
feature structure syntax, and description language. For now,
we take feature structures to be objects comprising parts,
without further specifying what sorts of things these ob-
jects and parts are.
The first algorithm computes if two feature structures Aand B are in a subsumption relation A vB, that is, ifA sub-
sumes B. Informally, subsumption is a relation of informa-
tion specificity: If A subsumes B, then B contains at least
the information contained in A. If two feature structures mu-
tually subsume each other, they are called alphabetic vari-
ants. Subsumption modulo alphabetic variance is a partial
order, under it features structures and thus information may
or may not be ordered, but if ordered display the formal
properties of reflexivity, antisymmetry, and transitivity. In
the design space exploration game, subsumption orders the
design space and the case retrieval process.
The second algorithm, p-resolution, computes the satis-
faction relation from descriptions to feature structures. Fora given description, the algorithm computes a set of consis-
tent feature structures. Satisfaction is monotonic: If a fea-
ture structure satisfies a description, than every feature
structure it subsumes also satisfies the description. Satisfac-
tion is an approximate explanation of p resolution: Later
we explain how and why the p-resolution algorithm com-
putes only the most general satisfiers. In design space ex-
ploration p-resolution is the principal generation process.
A third algorithm, unification, computes a feature struc-
ture satisfying both of two given feature structures. Such a
feature structure is the minimal feature structure subsumed
by both of the given feature structures. Unification is a par-
tial function from F Fr F, where Fis the set of all fea-
ture structures. Unification plays no direct role in design
space exploration; rather, it is an integral component of
p-resolution where it generates consistent structures, and is
used to decide whether two pieces of information are
consistent.
Subsumption and p-resolution are the fundamental fea-
ture structure algorithms for a design space explorer. The
feature structure mechanism, which we now describe, is, in-
ter alia, structured to admit efficient instances of these al-
gorithms. Describing feature structures relies on types and
feature structure descriptions rely on feature structures, so
we specify the parts in the following order: types, feature
structures, and descriptions.
3.1. Types
The finite set of types is organized into a structure accord-
ing to information specificity. We say that type s subsumes
type t ~and write s v t! if type t contains strictly more
information than type s. Type s is said to be a supertype oft; t a subtype ofs. Beyond the partial order properties im-
plied by the type subsumption relation ~which is not the same
relation as the feature structure subsumption relation!, a fea-
ture structure type hierarchy is required to have two addi-
tional properties. First, for any set of types there is at most
one type that is directly subsumed by all types in the set. A
set of types with a common subtype is said to be bounded or
consistent. This condition on the type hierarchy amounts to
saying that there must be a unique most general subtype forany consistent set of types in the hierarchy. Second, there
must be a most general type ~conventionally called Bottom
and written ! at which all types meet. This condition is
implied by the first if the sets of consistent types includes
the empty set. Together with the partial order conditions of
transitivity, antisymmetry, and reflexivity, these conditions
create what is called a bounded complete partial order~BCPO!.
A tree is an example of a bounded complete partial order,
and a tree-structured inheritance relation ~so-called single
inheritance! is therefore admissible as a type hierarchy. More
complex relations are as well, including so-called multiple
inheritance schemes, providing that the inheritance relationremains a BCPO.
With types are associated features, drawn from a finite
set Feat of features. The function Intro : Featr Type de-fines for every feature a unique most general type at which
that feature is introduced into the type hierarchy. All sub-
types of Intro ~ f! contain f and f is said to be appropriate
for Intro ~ f! and its successor subtypes. Subtype feature in-
clusion and being a complete function in Feat implies thatany feature that is multiply inherited from two or more su-
pertypes is, in fact, the same feature; thus some typical am-
biguities of multiple inheritance do not arise. The partial
function Approp : Feat Type r Type gives a type re-striction on the values of a particular feature: Approp ~ f, t!is the most general type that a value of feature f in type t
can have. On Approp are placed the conditions of upward
closure mentioned above and that feature values can only
become more specific in subtypes, that is, if for two types s
and t, Approp ~ f, s! is defined and s v t, then Approp ~ f, t!
is also defined and Approp ~ f, s! v Approp ~ f, t!.
When we discuss descriptions in Section 3.3 below, we
will add a function associating a description with each type
in a type hierarchy. These descriptions serve as type
294 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
9/16
constraintsthey specify further structure in well-formed
instances of a type.
In summary, types are bounded complete partial orders
and impose a strict feature introduction and appropriate-
ness regime on the set of features used. Constraints on types
in the form of descriptions associated with types are dis-
cussed in Section 3.3 below.
3.2. Feature structures
Feature structures are a frame-like representation compris-
ing a set of nodes, each of which is labelled by type infor-
mation, all of which are connected by directed arcs denoting
features and one of which is designated as a root node. Fea-
ture structures have a formal definition ~Carpenter, 1992,
p. 36!, but their essential features can be demonstrated ei-
ther as a graph diagram ~which, despite superficial resem-
blance is neither a Hasse diagram nor a DAG! or in attribute-
value matrix ~AVM! notation, as shown for a fragment of
an abstract boundary solid representation in Figure 11. In
essence, a feature structure comprises:
a set of nodes Q; a root node Sq, a distinguished member of Q;
a total node typing function u : Qr Type that for eachnode assigns a type from Type; a partial feature value function d : FeatQr Q that
assigns a node to features held by nodes in Q.
Feature structures themselves rely on the existence of a
set of types, but these need not be ordered into an inheri-
tance hierarchy until the subsumption and unification con-
ditions over feature structures are considered. In a feature
structure a path, a list of features, identifies a node relative
Fig. 11. Graph diagram and attribute-value matrix
notation for a feature structure.
Typed feature structures and design space exploration 295
8/6/2019 Typed Feature Structures and Design Space Exploration
10/16
to the root node. A node A at the end of a path can itself be
considered as the root node of a feature structure whose nodes
comprise the set of nodes reachable by paths from A. Fea-
ture structures can structure share and be cyclic. The for-
mer, also known as path equation, means that different paths
can pick out the same node; the latter that a path from A can
have A as its value ~thus the transitive closure of the feature
value function is not a partial order!.The relation between feature structures and types does
not follow the common notions of type and instance from
object-oriented programming. Feature structures may exist
which are are consistent with a type but contain additional
information or not well-formed instances of any known type.
3.3. Descriptions
Descriptions are a textual notation for denoting sets of fea-
ture structures. As such they can be specified as a linear
sequence of symbols, which greatly assists their processing
with computers. A given description is satisfied by a set of
feature structures, which is empty if the description is
unsatisfiable.
The description language for feature structures ~Carpen-
ter, 1992, p. 52! is terse and is given over the collection
Type of types and Feat of features as the least set suchthat:
a type is a description; for formal purposes, the absurd type, Top, is a descrip-
tion;
a path comprising features followed by a description isa description;
two paths equated form a description;
two paths disequated form a description; and descriptions may be combined with the logical opera-
tors and and or.
A choice of precedence rules is required over the opera-
tors and these may be notationally circumvented by the in-
troduction of parentheses.
Descriptions serve to call out specific feature structures.
Every feature structure can be picked out by some descrip-
tion that includes no disjunctions. For every description there
is a, possibly empty, set of feature structures that satisfy the
description. Of these there is a subset called out by the func-
tion that comprises pairwise incomparable feature struc-
tures and whose members collectively subsume all feature
structures satisfying the description.
Descriptions also play the role of type constraints. A total
function Cons: Type r Desc maps types to descriptions,
establishing for each type a constraint expressed as a de-
scription. In addition to its own constraint, a type suffers
the constraints inherited from each of its supertypes, namely,
Cons*~t! svt
Cons~s!.
Any feature structure that satisfies a type t must satisfy
the recursive constraint given by Cons*~t!.
With a sketch of the form of feature structures, we now
proceed to the principal relations and algorithms over them.
3.4. Subsumption
Subsumption determines if one feature structure general-izes another and is written A vB if feature structure A sub-
sumes ~or generalizes! feature structure B. Carpenter ~1992,
p. 40! motivates subsumption by the interpretation of fea-
ture structures as representing partial information, that is,
what is known about an object at some stage of a computa-
tion, not as a necessarily complete description of an object.
Thus, determining if a feature structure is more specific than
another can be taken as determining if its represented ob-
ject is more specific than the represented object of the other
feature structure. Such is clearly a sensible motivation in
our context, where we are habitually concerned with incom-
plete designs. We carry this interpretation a step further and
consider that to a feature structure information can be addedto make it an alphabetic variant of any feature structure that
it subsumes. Subsumption is formally described as a mor-
phism, a mapping from the node set of one feature structure
to another in which all of the shared paths and all of the
types at the ends of paths remain consistent. Algorithmi-
cally, subsumption can be decided in time linear to the size
of the subsuming feature structure.
Mutual subsumption A vB and B vA defines the equiv-
alence relation of alphabetic variants written A ; B. Each
element of a set of alphabetic variants carries the same in-
formation as any other in the set. Because feature structures
are intensional, alphabetic variants are distinct objects.
Though it is possible to construct a language free of alpha-betic variance @abstract feature structures ~Carpenter, 1992,
p. 43!#, we take a feature structure to stand for its equiva-
lence class and discipline the construction of feature struc-
tures to ensure disjoint node sets. This is equivalent to
working in the quotient set ~or factor space! of feature struc-
tures modulo alphabetic variance.
3.5. Satisfiability
Descriptions serve to call out feature structures through the
satisfaction relation. Satisfaction is monotonic: If a feature
structure satisfies a description ~Carpenter, 1992, p. 55!, then
so too does every feature structure which it subsumes. There-
fore, a description may be interpreted as a reference to its
most general satisfiers and thus be treated as a name for a
possibly empty set of pairwise incomparable feature struc-
tures. If a description is disjunction free, the named set will
either be empty or contain only mutual alphabetic variants.
Given the set of all disjunction free descriptions NonDisj-
Desc, there is a surjection MGSat : NonDisjDescrF. Forthe set of descriptions Desc, sets of pairwise incomparablefeature structures are identified by the function MGSats :
296 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
11/16
Desc r 2F. Satisfaction : Fr Desc is defined in ref-erence to descriptions ~Carpenter, 1992, p. 53!. Informally,
a feature structure satisfies a type if the type subsumesthe feature structure;
no feature structure satisfies top; a feature structures satisfies a path comprising features
followed by a description if the node at the correspond-
ing path in the feature structure satisfies the description;
a feature structure satisfies two equated paths if bothcorresponding paths in the feature structure pick out
the same node;
a feature structure satisfies two disequated paths if thecorresponding paths in the feature structure do not pick
out the same node and are marked as disequated;
a feature structure satisifies the conjunction of two de-scriptions if it satisfies both of the conjuncts; and
a feature structure satisifies the disjunction of two de-scriptions if it satisfies one of the disjuncts.
3.6. Describability
Satisfiable disjunction free descriptions and feature struc-
tures are interchangeable entities. By the satisfaction re-
lation described above, any satisfiable disjunction free
description corresponds to an equivalence class of mutually
alphabetic variant featurestructures. In addition, any feature
structure corresponds to a disjunction free description.
Formally, corresponding to any feature structure F F
there is a nondisjunctive description Desc ~F! such that F;
MGSat~Desc ~F!!.
The significance of describability is twofold. First, fea-
ture structures can be used directly to specify constraints in
types and queries, suggesting a programming-by-exampleform of interaction with a design space explorer. Second,
feature structures can be converted to a textual form.
3.7. Unification
Given two feature structures, A and B, unification, written
A t B, produces a third feature structure C subsumed by
both A and B if such a feature structure can exist, otherwise
unification fails. Feature structure unification produces the
most general feature structure that contains all of the infor-
mation in its two arguments. Unification is interesting, as it
provides a disciplined way to combine information or to add
information to a structure. In a design context, the result of
a feature structure unification can be taken as a new, more
complete object that is consistent with the objects repre-
sented by the argument feature structures. Unification cre-
ates a set of equivalence classes of nodes by following paths
in the two argument feature structures, A and B. Each node
in A and B begins in a singleton equivalence class, except
for the root nodes whose equivalence classes are merged by
default. Equivalence classes are merged through the feature
value function: Where a feature is shared by nodes in an
equivalence class, the equivalence classes containing the
nodes on the shared features are merged. There exists a lin-
ear time unification algorithm for unification, though in prac-
tice, a near linear algorithm is typically used.
3.8. p-resolution
The p-resolution algorithm maps descriptions into sets offeature structures that satisfy them, taking into account a
type hierarchy, including the recursive t ype constraints de-
fined within it.
Given a query description D, p-resolution is the search
across a sequence of feature structures P0 v P1 v P2 v
. . . v Pk. The initial feature structure in each sequence is a
most general satisfier of the query description. The se-
quence represents the inclusion of type information in the
form of constraintseach element extends its predecessor
by unification with a type constraint. Because most general
satisfiers may occur as collections and unification may fail,
the search for resolved feature structures involves a collec-
tion of sequences.With these relations sketched, Figure 12 adds to the dia-
gram of the principal computation paths in featurestructures
the relations satisfiability, describability, and unification. Be-
cause a type is a primitive description, we also note that sDesc ifsType. Since each type hasa constraint expressed
as a description, we add Cons~u !. Because the root node Sq of
any feature structure has a type, every feature structure calls
out a type u~ Sq! Type. Finally, inheritance orders types.
4. AN EXAMPLESINGLE-FRONTED
COTTAGES
We conclude this paper with an example of types, feature
structures, and descriptions that together denote a corpus of
single-fronted cottages. We include only the physical spaces
identified by name and labels denoting functional use. Both
the expression of a knowledge level representation for a de-
sign space explorer and the consideration of geometric in-
formation are left for a future paper. The language used for
the example is ALE ~Carpenter & Penn, 1997!, which im-
plements typed feature structures in a Prolog environment.
4.1. Type hierarchy
Immediately above the type bottom are three types, corre-
sponding to buildings, briefs, and massings. The models cor-
responding to the type building can be thought of as designs
for buildings, those corresponding to the type brief as ar-
chitectural briefs, and those corresponding to the type mass-
ing as volumes that together compose an overall building
mass. All of these types are abstract, serving to refine the
type bottom and do not introduce any features. We shall
see later how such abstract types can be used in constraint
expressions.
Typed feature structures and design space exploration 297
8/6/2019 Typed Feature Structures and Design Space Exploration
12/16
Massings can recursively contain other massings. The type
massing has a sequence of subtypes massing1, mass-
ing2 . . . in which massingn v massingn1 . Each typemassingn introduces a feature MASS_ELn, which denotes
a submassing ofmassingn. Thus, a feature structure of type
massingn is a representation of a massing of nj submass-
ings, where j 0.
The type brief has a single subtype house at which fea-
tures corresponding to functions are introduced. These fea-
tures carry only functional information, in contrast to the
the features introduced in sfc ~single-fronted cottage!. These
features, corresponding to a porch, hall, row of rooms, and
skillion, denote the particular spatial organization of single-
fronted cottages without making commitment to actual ad-
jacencies or dimensions. Type sfc inherits from both building
and massing4. The type sfc_house inherits features from
both sfc and house. Section 4.2 below describes how the
diverse features relating to function, abstract form, and mass-
ing are related through type constraints.
Figure 13 diagrams these types, showing inheritance, fea-
ture introductionIntro~ f!, and appropriatenessApprop~ f,t!
conditions. Following Carpenter ~1992! and order theorists
such as Davey and Priestley ~1994!, and contrary to stan-
dard practice in artificial intelligence we draw type hierar-
chies with the most general type bottom at the bottom of
the diagram. We do this because type subsumption ex-
presses an information ordering in which s is less than or
equal to t in information content ifs v t.
4.2. Type constraints
Associated with the types above are constraints, Figure 16
shows the types having nonempty constraint sets: house,
sfc, and sfc_house. As is normal in most languages, effi-
ciencies can often be realized through macroexpressions,
and descriptions are no exception. Before explaining the con-
straints, we introduce a bit of notation about constraint mac-
ros. A constraint macro comprises a parameterized head and
a body. The head acts as a name, the body is a description
that substitutes for the name, and the parameters in the head
are substituted into their indicated place in the body. Forexample, the three-argument macro mdq ~mutual dise-
quate! in Figure 14 disequates the node at feature A with
those at features B and C and issues a call @ mdq~B,C! to
the two-argument macro mdq/2.3
Thus, the macro call @ mdq~sleeping, lounge,kitchen! with head mdq, actual parameters sleeping,
3Note the use of the Prolog convention of giving the arity of a macrowhen a parameter list is not given.
Fig. 12. Augmented computation paths in typed feature structures.
298 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
13/16
lounge, and kitchen results in a pairwise inequation
of the functions sleeping, lounge, and kitchen.
The constraints of type massingn simply pairwise dise-
quate the n massing elements inside a massing. This pre-
vents massings from being structure-sharedarchitecturally,
a thoroughly Victorian restriction appropriate with the single-
fronted cottage type.
The constraints of type house pairwise inequate the fea-
tures that compose house. This prevents their ever being
structure-shared in any feature structure consistent with
house.
Type sfc structure-shares the four submassing elements
and the abstract spatial features that compose a single-
fronted cottage. Thus porch, hall, room_row, and
skillion become new names for the elements, mass_el1,
mass_el2, mass_el3, and mass_el4. The reason for this
simple sharing of feature values ~and thus names! is that it
sets the scene for a generic definition of the in/2 macro,
Fig. 13. The type hierarchy for the single-fronted cottage example.
Fig. 14. Definition for mdqmutual disequate.
Fig. 15. Definitions for in1 and in2.
Fig. 16. Type constraints for the single-fronted cottage example. The
share macro shares the nodes at the end of the two given paths. The innmacro constrains the second argument to be structure-shared with one of
the first n massing elements of the first argument.
Typed feature structures and design space exploration 299
8/6/2019 Typed Feature Structures and Design Space Exploration
14/16
which operates over the mass_eln features irrespective of
their other names as architectural elements or functions.
All so far is deterministicthe descriptions in house and
sfc contain no disjunctions, and so have but a single most
general satisfier under p-resolution. The alternatives are all
generated in type sfc_house, in which alternative assign-
ments of functions to spaces are made in four disjuncts as
follows. In all alternatives, the sleeping area is confined tothe first two rooms of the room row.
1. Thedining andthe loungefunctions arebothin theroom
row. The kitchen and bathroom are in the skillion.
2. The dining function is in the room row. The kitchen,
lounge, and bathroom are in the skillion.
3. The lounge function is in the room row. The kitchen,
dining, and bathroom are in the skillion.
4. The kitchen, dining, lounge, and bathroom are in the
skillion.
The macro inn, n 1 makes these assignments of func-
tional spaces into physical spaces by either calling inn1or by structure sharing the functional space with the nth con-
tained massing element of the physical space. Macro in1
simply structure shares the functional space with the first
contained massing element of the physical space. Figure 15
gives the code for in1 and in2the definitions for inn
follow by induction.
Fig. 17. The 32 single-fronted cottage plans produced by the example-type hierarchy. By convention, the layout of the drawings
corresponds to typical abstract layouts for single-fronted cottagesno geometry is included in this example.
300 R. Woodbury et al.
8/6/2019 Typed Feature Structures and Design Space Exploration
15/16
4.3. Feature structures for single-fronted cottages
The type hierarchy and its associated type constraints pro-
vide the entire specification required for generating alter-
native mappings of function into the form of single-fronted
cottages. Given a query comprising only the type sfc_house,
the p-resolution algorithm generates the set of pairwise in-
comparable, most general resolved feature structures con-
sistent with the type hierarchy and its constraints. These areshown in Figure 17.
We return now to the seven behaviors that typed feature
structures lend to a generative design system to interpret
these in the light of the given example.
Intensionality. Each of the feature structures is inten-
sional in standing for a particular representation of a
single-fronted cottage.
Partialness. Any feature structure is partial in that it may
not give all that could be known about the cottage it
represents. Another way of saying this is that there may
be resolved feature structures subsumed by a featurestructure picked out by p-resolution as a most general
resolved feature structure. Figure 19 gives an example
in which feature structure #5 in Figure 17, which has
only two rooms in the room row, in effect stands for
feature structures that have three, four, or more rooms
in the room row.
Adding new knowledge. By extending the type hierar-
chy, say by including massing5, additional resolved fea-
ture structures become possible.
No rules. The type hierarchy itself plays the roles usu-
ally reserved for rules in a generative system. The dis-
juncts in the constraint expressions create choice points
in the derivation graph.
Monotonic generation. We do not discuss the internal
action of the p-resolution algorithm here,4 so we can-
not peek inside its operation to show that each of its
steps is strictly monotonic. However, monotonicity can
be demonstrated indirectly by showing that, for exam-
ple, mgsat~house! v mgsat~sfc-house!.
Subsumption. The 32 most general resolved feature struc-
tures of the query sfc-house form an antechain in
the subsumption relation of feature structures under the
given type hierarchy. Subsuming these feature struc-
tures are the incremental steps of the p-resolution al-
gorithm. Subsumed by these structures are the resolved,
but not most general, feature structures admitted by the
type hierarchy.
Reuse. Consider the first feature structure in Figure 18.
It subsumes only cottages #28 and #31 of Figure 17.
The feature structure for cottage #28 is included in the
figure.
5. SUMMARY
Typed feature structures are a model for design space ex-
ploration in which both the action of exploration and the
structure of a design space are given a sound theoretical
basis. As a representation of designs, typed feature struc-
tures provide well-founded support for an object-and-
relations view of representation and within that view support
intensionality, partialness, structure sharing, and cyclic-
ity. In this paper we have demonstrated that typed feature
structures can tersely express and solve simple, apparently
finite-domain, generation of alternative mappings of func-
tion into form in the building domain. It remains to be
shown how more complex domain structures, such as the4See Burrow and Woodbury ~1998!.
Fig. 18. A feature structure ~a) used as a query to retrieve feature struc-
tures #28 and #31 shown as ~b) from Figure 17.
Typed feature structures and design space exploration 301
8/6/2019 Typed Feature Structures and Design Space Exploration
16/16
SEED5 Knowledge Level can be supported. The problems
posed by geometry also remain. To foreshadow a solution,
consider that well-founded type inference in typed feature
structures requires at least unique joins in a type hierarchy,
and that many operations over point sets, such as union
and intersection, generate structures with stronger formal
properties than this required minimum. Both of these is-
sues, the expression of a knowledge level, and the han-
dling of geometric information, are topics for future papers.
ACKNOWLEDGMENTS
The authors acknowledge the support given by the Australian Re-
search Council Large Grants Scheme; The Department of Indus-
try, Science and Tourism Major Grant Scheme; the Australian
Postgraduate Award Scheme; The Commonwealth of AustraliaOverseas Postgraduate Research Scholarship Scheme; The Key
Centre for the Social Applications of Geographic Information Sys-
tems; andThe University ofAdelaide School of Architecture, Land-
scape Architecture, and Urban Design.
REFERENCES
Akin, O., Aygen, Z., Change, T.-W., Chien, S.-F., Choi, B., Donia, M.,Fenves, S. J., Flemming, U., Garrett, J. H., Gomez, N., Kiliccote, H.,Rivard, H., Sen, R., Snyder, J., Tsai, W.-J., Woodbury, R., & Zhang, Y.~1997!. SEED: A Software Environment to support the Early phases ofbuilding Design. The International Journal of Design Computing. http:00www.arch.usyd.edu.au0kcdc0journal0index.html.
Archea, J. ~1987!. Puzzle-making: What architects do when no one is look-ing. In Computability of Design, Principles of Computer-Aided De-sign, ~Kalay, Y., Ed.!, Chap. 2, pp. 3752. John-Wiley & Sons, NewYork.
Burrow, A., & Woodbury, R. ~1998!. Pi-resolution in design space explo-ration. Technical Report, Department of Architecture, Universityof Adelaide. http:00www.arch.adelaide.edu.au 0;rw0publications 0progress.html.
Carlson, C. ~1994!. Design space description formalisms. Proc. Formal Design Methods for CAD, pp. 121131. North-Holland, Amsterdam.
Carpenter, B. ~1992!. The logic of typed feature structures with applica-tions to unification grammars, logic programs and constraint resolu-
tion. Cambridge University Press, New York.Carpenter, B., & Penn, G. ~1997!. The attribute logic engine users guide.
Computational Linguistics Program, Philosophy Department, Carne-gie Mellon University, Pittsburgh, PA.
Davey, B., & Priestley, H. ~1994!. Introduction to lattices and order. Cam-bridge Mathematical Textbooks. Cambridge University Press, Cam-bridge, UK.
Fenves, S., Rivard, H., Gomez, N., & Chiou, S.-C. ~1995!. Conceptual
structural design in SEED. ASCE Journal of Architectural Engineer-ing 1(4), 179186.
Flemming, U. ~1994!. Case-based design in the SEED system. Automationin Construction 3, 123133.
Flemming, U., & Woodbury, R.F. ~1995!. Software Environment to sup-port Early phases in building Design SEED: Overview. ASCE Journalof Architectural Engineering 1(4), 147152.
Hinrichs, T.R. ~1992!. Problem solving in open worlds: A case study indesign. Erlbaum Associates, Hillsdale, NJ.
Kolodner, J. ~1993!. Case-based reasoning. Morgan Kaufmann Publish-ers, Inc., San Mateo, CA.
March, L. ~1995!. The smallest interesting world? Environment and Plan-ning B: Planning and Design 23 (1), 133142.
Newsome, S., Spillers,W., & Finger, S. ~1989!.Design Theory 88. Springer,New York.
Stiny, G. ~1980!. Introduction to shape and shape grammars. Environmentand Planning B: Planning and Design 7(3), 343352.
Woodbury, R., Fenves, S., Flemming, U., Coyne, R., Chang, T.-W., Chiou,S.-C.,& Gomez,N. ~1994!. SEED-Config requirements analysis. http:00seed.edrc.cmu.edu 0SC0requirements-new0SC-req-new.book.html.
Dr. Robert Woodbury holds a B.Arch from Carleton Uni-
versity and an M.S. and Ph.D. from Carnegie Mellon Uni-
versity. He is presently an Associate Professor ~Reader! at
The University of Adelaide in Australia. His main research
focus is on generative design systems. His work is distinc-
tive in its emphasis on implementation within the research
process, without which results in this area can neither be
falsified nor forcefully demonstrated. He is the author of
over 100 technical papers.
Andrew Burrow is a Ph.D. candidate at the University of
Adelaide in the Department of Computer Science and School
of Architecture, Landscape Architecture and Urban Design.
He holds a Bachelor of Engineering ~Chemical! and a Bach-
elor of Science ~Mathematical and Computer Science! with
Honors from the University of Adelaide. His research inter-
ests include: design applications of typed feature struc-
tures, knowledge visualization, and order theory.
Sambit Datta holds a Dip.Arch from the School of Archi-
tecture, CEPT, India, and an M.Arch from the National Uni-
versity of Singapore. He is presently a doctoral student in
the School of Architecture, Landscape Architecture and Ur-ban Design at the University of Adelaide. His current re-
search is on developing interaction techniques for design
representations.
Mr. Teng-Wen Chang holds a B.Arch. from Tunghai Uni-
versity in Taiwan, an M.Arch. from The University of Penn-
sylvania, and an M.S. from Carnegie Mellon University. He
is presently a Ph.D. student at The University of Adelaide
in Australia. His main research focus is on representing geo-
metric information in generative design systems.
5~Woodbury et al., 1994; Flemming & Woodbury, 1995; Akin et al.,1997!.
Fig. 19. A most general resolved feature structure with respect to a
p-resolution query may subsume other, more specific, resolved feature
structures.
302 R. Woodbury et al.