ERASURE IN DESIGN SPACE EXPLORATION 1 ...

ERASURE IN DESIGN SPACE EXPLORATION ROBERT WOODBURY, SAMBIT DATTA School of Architecture, Landscape Architecture and Urban Design, The University of Adelaide SA 5005, Australia. AND ANDREW BURROW Department of Computer Science, The University of Adelaide SA 5005, Australia.

Abstract. Design space explorers support designers through the metaphor of exploration, a guided movement through a space of possibilities. As systems, design space explorers present a mixed-initiative environment in which designers engage in their work helped by a computer to an extent both computationally possible and desired by the designer. Design space explorers provide a set of generative operators by which states in a space may be discovered. A subsumption-based design space explorer structures the space in which it navigates by a relation of information specificity. In particular, it conditions its exploration operators so that they move in predictable ways in the underlying space of designs. We have extended the formal system of typed features structures to support subsumption-based design space exploration. This paper discusses the exploration operators available when subsumption-based design space explorers are implemented in typed feature structures. Of particular interest is the handling of erasure, where the constraint on generative operators to be monotonic wrt information specificity leads to a surprisingly clean view of erasure.

1.

Introduction

By definition a design encapsulates information about the designed object. Different designs for the same object encapsulate different amounts and kinds of information about the object. Information specificity is therefore a key ordering amongst designs, as captured in the notion of refinement. However, even the possible space of designs is vast and uncountable in any practical

R. WOODBURY, A. BURROW, AND S. DATTA

sense. Therefore, computer supported design environments must deal foremost with the scale of the design space. Design space explorers are computer environments intended to provide novel and effective means for operating within design spaces. They operate on a subset of the design space implicit in a collection of generative operators, and distinguish between this still vast implicit design space and the fraction that has been visited, the explicit design space. Consequently, the user of a design space explorer interacts with generative operators by following edges in a derivation relation over states, and in doing so constructs an explicit design space recording their actions. Most generative design formalisms and systems use a grammatical mechanism to model the allowable derivations. Such mechanisms are typically cast in terms of operations in an underlying algebra. A typical rule applies by first removing a suitably matched and transformed left hand side (LHS) and then adding a correspondingly transformed right hand side. Since each rule arbitrarily adds and removes design information, the generative system is nonmonotonic wrt design information. Hence, the design space does not organise designs by information specificity, even though there may be such an order implicit in the underlying algebra. A design space explorer which does not organise designs by information specificity can do little to assist the user explore the space of resultant designs, because the notion of refinement is lost to the exploration. Paths in the design space lack a sense of progress that would account for the goal directed aspects of design, and the absence of formal properties relating the design space and the information content of its states precludes algorithms to efficiently search for duplicates or related designs. The result is design spaces that are not easily navigable by the user or by the machine. An alternative is to constrain the generative operators so that they construct states that contain strictly more information than their operands. In such a scheme, generative operators preserve key formal properties in a design state, allowing inferences to be made about the design states that follow. Of course, this comes at a price. It is occasionally convenient to remove information during exploration, hence there is a danger that constrained exploration may be unwieldy or unintuitive. In this paper we describe an approach to information deletion in the context of monotonic generative operators. It is our observation that much of erasure in extant systems is the substitution of a more detailed symbol by a more abstract one. Without generative operators that remove de-

ERASURE IN DESIGN SPACE EXPLORATION

sign information, the implementation of erasure utilises the information ordering. The resulting scheme generalises the simpler notion of replacement by moving erasure from a generative operator acting on design states to an exploration operator acting on the design space. This paper describes how typed feature structure -resolution and operators on the ordered set of partial satisfiers comprise a set of exploration operators within a design space. After a brief review of related work, we introduce the typed feature structure machinery at a level of detail sufficient to frame the exploration operators. We follow this with a description of each of the operators, a sketch of the relevant typed feature structure machinery, and a discussion of the entry points for human-computer interaction.

2.

Related Work

Design space explorers have their genesis in the shape grammar literature (Stiny, 1980; Stiny and March, 1981). Some later researchers focused on the formal systems for states and state spaces as well as the human-computer interfaces required to create effective design space explorers. Notable here are the following projects: Genesis (Heisserman, 1994), Grammatical programming (Carlson, 1993), ABLOOS (Coyne, 1991), discoverForm (Carlson and Woodbury, 1994), Tartan Worlds (Woodbury et al., 1992), discrete and continuous models (Harada et al., 1995; Harada, 1997), SEED-Config (Woodbury and Chang, 1995) and SG-CLIPS (Chien et al., 1998). Others have have restricted their attention to mechanisms for the transformations between design states themselves, and thus give limited insight onto the mechanics of and interaction with exploration. This latter includes the simple interpreter (Gips, 1975), shape grammar interpreter (Krishnamurti, 1980; Krishnamurti, 1981; Krishnamurti and Giraud, 1986), shape grammar system (Chase, 1989), GEdit (Tapia, 1999), shape grammar implementation (Duarte and Simondetti, 1997), shape grammar editor (Piazzalunga and Fitzhorn, 1988), 3D architecture form synthesizer (Wang and Duarte, 1998), coffee-maker grammar (Agarwal and Cagan, 1988), and Shape Schema Grammars (Li et al., 1998). Feature structure theory has its provenance in computational linguistics as described by Carpenter (Carpenter, 1992), but owes much to its application to logic programming by A¨it-Kaci (A¨ıt-Kaci and Podelski, 1993), who was


also the first to realise the potential optimisations for reasoning operations on bounded complete partial orders (A¨ıt-Kaci et al., 1989). The breadth first resolution techniques for recursive type constraints in feature structures also grew out of work on feature structures as a term language for logic programming (A¨ıt-Kaci et al., 1993). The use of order theory for information management and presentation is based on techniques for searching in partial orders (Ellis, 1995) and on the notion of lattices as a media for philosophical discourse as proposed in Formal Concept Analysis (Wille, 1992). There are numerous knowledge level characterisations of design information stressing the role of function. For example, the SEED knowledge level (Flemming and Woodbury, 1995; Woodbury and Chang, 1995) and the SHARED object model (Gorti et al., 1998). Some of these are sufficiently explicit and parsimonious to admit a symbol level interpretation.

3.

Feature Structures

In order to express a design space as an information ordering, the representation of design states must satisfy several formal and operational criteria. Namely, the representation must correspond closely with the knowledge level of the domain — designers must be able to understand what is going on; the representation must be able to express a collection of partial views of a single domain object at varying levels of specificity — each step in a refinement must be expressible; there must exist operators to refine a partial view to a consistent and more specific view — refinement steps themselves must be expressible; there must exist an efficient algorithm to determine whether one view is more specific than another — refinements must be recognisable; and there must exist an efficient algorithm to determine whether two views are consistent — views that possibly represent the same object must be recognisable. Typed feature structures, a generalisation of record-like data structure, are such a representation. They are a form of directed graph, where nodes represent domain objects and edges represent associative connections. Object


identity is modelled by structure sharing and the structure as a whole models partial information. Since feature structures model partial information there must exist an information ordering with an efficient decision procedure, and an operation to combine information where consistent. In addition, descriptions provide a means for calling out collections of feature structures based on satisfaction. Satisfaction has the property that, if a feature structure satisfies a description, then so too does every feature structure which extends the information in it. These properties of feature structures are attractive in design space explorers. As representatives for design states, explicit models for partial information lend legitimacy to intermediate exploration states. In particular, we represent functional decompositions of design states by feature structures, and functional roles in these decompositions by features. In exploration terms, extending a feature structure corresponds to answering “How?” questions which transform the problem to include finer levels of subproblem. This is an attractive process as the surface logic of a design space explorer, because it captures a notion of goal directedness in terms of the problem at hand. -resolution, the fundamental feature structure construction process, is a non-deterministic search for solutions to a query description. A solution is a feature structure, which satisfies a description and the constraint system. It is the result of a sequence of extension steps corresponding to the satisfaction of constraints, which are organised into an inheritance hierarchy of types. Since -resolution proceeds by extension, the resultant search space can be order embedded into the information ordering over feature structures. Since constraints are drawn from an inheritance hierarchy, alternatives may be organised according to notions of abstraction. If feature structures represent functional decompositions, -resolution non-determinism allows exploration in terms of alternative functional decompositions. Given -resolution and a characterisation of the partially resolved feature structures, we have the basis for a design space explorer where the generative operators are monotonic wrt design information taking the form of functional decomposition. The partial satisfiers are the design states, -resolution steps are the derivation steps, and topological search is available to detect duplicates and retrieve the closest consistent designs in the information ordering. To this system we add exploration operators simulating erasure and providing additional novel features.


In this section we define a feature structure language (Carpenter, 1992) in sufficient detail to discuss the exterior mechanisms of -resolution. Since the resolution of type constraints may be described independently of the class of feature structures (Carpenter, 1992, p 228), we do not consider type inference, inequations, or extensional types. Furthermore, we do not consider descriptions in detail except to note their use in representing queries and type constraints. Rather than provide formal definitions for subsumption and unification we consider these mechanisms in terms of functional decomposition. 3.1. FEATURE STRUCTURES DEFINED

Feature structures generalise record-like data structures and represent partial information. They may be depicted as rooted, directed, finite, node and edge labelled graphs, where nodes represent objects, edges represent functional relationships, and structure sharing is taken to model identity. Therefore, cycles represent reflexive relationships, and informational equivalence does not assert identity. Each node in a feature structure may be interpreted as the root of a feature structure, termed a substructure. At nodes, labels are interpreted as types, and on edges, labels name the functional role that the target fills wrt the source. Types are exploited as a means of efficiently encoding information. They are organised into an unambiguous multiple inheritance hierarchy, in which information associated with a type is extended in inheriting types, i.e., an information ordering. It is mandated that this ordering is a bounded complete partial order (BCPO), meaning that for every set of types with a common subtype there is most general common subtype or join. Thus, certain reasoning operations over types are available as the lattice operations t (join, or most general common specialisation) and u (meet, or most specific common generalisation). Rather than formalise this graph definition of feature structures, we provide an informal exposition in terms of their properties. Feature structures may be viewed as recursive structures assigning feature structure values to paths, where paths name the substructure roles. We start by defining an inheritance hierarchy of types hType; Type i, a set of features Feat, and a collection of paths Path Feat , where each path is a sequence of features and the empty path is denoted .

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sfc_house ROOM_ROW

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MASS_EL1 MASS_EL1 SLEEPING

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Figure 1: Feature structures in graph notation.

Definition 1 (Feature Structures) A feature structure is an object F with the following properties F is assigned a type F 2 Type; for every path 2 Path, either F is undefined at , or F defines the substructure F which is itself a feature structure; and F equates a pair of paths ; 0 2 Path, written F 0 , iff F and F 0 are token identical, meaning that they represent the same domain object.

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Figure 1 depicts a pair of simple feature structures in a subsumption relation. The example represents the decomposition of a building design, the nodes denote building entities and the edges denote functional roles in the design. By characterising feature structures as representations of partial information, it is possible to consider both examples as partial representations of some final design. The second feature structure provides additional information about the final design as recognised by the subsumption relation. Central to the representation of partial information in feature structures is the quantification of substructures. Since identity is asserted by structure sharing rather than information equivalence, each substructure is interpreted


as an existential proposition that there exists some object conforming to the substructure’s type. That is, a substructure cannot be interpreted as stating the identity of an object — two distinct substructures can represent the same object, or separate objects. Object identity is specified only wrt the feature paths extending to the representing substructure. Since designs are inherently existential representations that are made more specific by proceeding stages of the design process, this aspect of feature structure partiality is a useful one. In particular, the identification of an object as fulfilling multiple roles is clearly an example of specialisation. 3.2. SUBSUMPTION

The information ordering over types is extended to feature structures by the subsumption relation. A feature structure’s information is represented by the values it assigns to paths. Subsumption is an inclusion ordering over this information. A feature structure F subsumes another F0 , written F v F 0 , if and only if: for every path defined in F , is defined in F0 and the type at in F subsumes the type at in F 0 ; and for every pair of paths and 0 defined in F , if and 0 are equated in F then and 0 are equated in F 0 . If F v F 0 and F 0 v F then F and F 0 are information equivalent alphabetic variants, written F F 0. The example in Fig. 1 demonstrates three sources of ordering in feature structures. The ordering over type labels extends to feature structures — the type of the first feature structure is a supertype of the second feature structure; a feature structure is made more general by the elision of feature paths — the first feature structure does not define the path SKILLION MASS EL2 which is defined in the second; a feature structure is made more specific by equating paths — the second feature structure identifies a single object as fulfilling the roles of both SLEEPING and ROOM ROW MASS EL1, which is information not present in the first. In terms of functional decomposition, the first feature structure in Fig. 1 subsumes the second because every functional role in the first is present in the second, every object fulfilling multiple functional roles in the first fulfils a superset of these roles in the second, and for every functional role identified in the first the object fulfilling this role in the second is the same or a more specific type. Each piece of information represented in a feature structure relates


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Figure 2: Feature structure unification: the feature structure on the right is the unification of the feature structures on the left.

to the existence of a feature path or the type label assigned to a substructure at a feature path. Subsumption can also be related to abstraction. On the structural side abstraction relates to the elision of functional roles and the uncertainty in their identification, while on the type labelling side it relates to the classification of the represented object. The recursive type constraint system discussed in Section 3.5. provides a bridge relating these two sources of information, structural and type labelling. 3.3. UNIFICATION

Given the feature structure subsumption order, we seek a reasoning operation to compute the conjunction of two feature structures. Intuitively, this operation seeks the most general feature structure that is more specific than


either operand. Unification is such a procedure, that either fails if the two feature structures represent inconsistent information, or constructs the most general specialisation otherwise. In doing so unification satisfies two of our criteria. It both tests for consistency and provides a means of refinement. The unification of feature structures F and F0 , written F t F 0 , is a conjunction of the information considered by subsumption. F t F0 must be subsumed by F and F 0 , and must also be the minimal feature structure that does so. This is formalised by a partitioning of the nodes in F and F0 into equivalence classes. F t F 0 exists only if there exists a common specialisation of the types in each equivalence class. The unification algorithm is difficult to intuitively grasp, since it must handle structure sharing and cycles, but its role in -resolution is solely to compute the most general feature structure that is more specific than either operand. Figure 2 depicts three simple feature structures which are the operands and result of a unification operation. The example demonstrates a type join at the root node, the inclusion of new feature path values, and the inclusion of structure sharing. In terms of functional decomposition, the unification of two design representations simultaneously decides whether some object may exist which is consistent with the two functional decompositions and, if so, provides an information minimal representation true only of such an object. Unification is the single constructive mechanism used to reconcile information from a type constraint system with a query description during design space exploration. A critical feature of design space explorers based on typed feature structures is the relationship between operations on design states and the design space. Unification is the procedure that ensures this relationship. While unification is an algorithm acting on feature structure operands, it is described in terms of the information ordering. Generative operators defined in terms of unification preserve this feature. In particular, they are monotonic wrt the information order. However, in a design space explorer the relationship between unification and the design space is complicated by the distinction between the explicit and implicit design space — unification constructs a least upper bound wrt the space of feature structures, but not necessarily wrt the subset that has been visited. This idea is developed in Section 4..


3.4. DESCRIPTIONS

Descriptions are textual objects which call out feature structures through the satisfaction relation. Satisfaction is monotonic: if a feature structure satisfies a description, then so too does every feature structure which it subsumes (Carpenter, 1992, p 55). Therefore, a description may be interpreted as a reference to its most general satisfiers. In fact, every feature structure is the most general satisfier of a disjunction free description (Carpenter, 1992, p 56), thus the describability of feature structures is guaranteed. This relationship between descriptions and feature structures is stated formally in and . Given the set of all feature terms of the functions structures F , for the disjunction free descriptions NonDisjDesc there is a surNonDisjDesc ! F , and for the set of descriptions Desc, jection sets of pairwise incomparable feature structures are identified by the function Desc ! F . In a design space explorer, descriptions act as compact textual specifications of functional decomposition. Since description satisfaction is monotonic wrt feature structure specialisation, descriptions are statements providing a lower bound on information specificity. That is, a description may be regarded as a constraint on minimal information specificity — once a feature structure satisfies a constraint, unification with additional constraint satisfiers only specialises the feature structure and hence preserves the original constraint. The single exception is where the combination of constraints is in fact inconsistent, i.e., there is no meaningful solution to the constraints. With the inclusion of disjunction into descriptions, a conjunction of such constraints is satisfied by an arbitrarily large collection of functional decompositions. In this paper we are proposing that a form of -resolution is a formal framework for a design space explorer. The key feature is the organisation of knowledge about functional decompositions into a system of recursive type constraints. Monotonic satisfaction between descriptions and feature structures allows us to describe a constructive mechanism for functional decomposition. For each description, either an information minimal feature structure exists that satisfies the description, or the description is unsatisfiable. Since every specialisation also satisfies the original description, the unification of all feature structures satisfying a collection of descriptions is a feature structure satisfying every description in the collection and thus satisfying their conjunction. Hence

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we are able to express information about designs by descriptions, and construct feature structures by unification operation as a means of directly processing the descriptions. 3.5. RECURSIVE TYPE CONSTRAINTS

Without additional type information, the type t of a feature structure F asserts only that the represented object is an instance of t or some subtype. By stating restrictions on the feature structures of each type, we assert additional properties. The constraint system expresses constraints on substructures according to their type, and thus places lower bounds on the information associated with a type. This will include structural sources of information recorded in the constraint, so that the constraint system relates the two sources of information ordering: structural and type labelling. Definition 2 (Constraint System (Carpenter, 1992)) A constraint system is a total function Type 7! Desc.

Cons :

A feature structure is resolved if it satisfies the constraint system at every substructure. A substructure satisfies the constraint system if it satisfies the type constraints for its type and all supertypes. Definition 3 (Resolved Feature Structure (Carpenter, 1992)) A feature structure F is resolved if and only if

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8 2 Path; 8t 2 Type t Type F ! F satisfies

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(1)

Satisfaction monotonicity ensures a direct relationship between constraint resolution and unification. A type constraint description is satisfied if a most general satisfier subsumes the constrained substructure. Therefore, constraint resolution is the search for the most general feature structure subsumed by both a constraint satisfier and the current substructure. This is decided by unification. Definition 3 describes the properties of a fully resolved feature structure. However, resolved feature structures are the endpoints in a constraint resolution process. Since the design space is explicitly concerned with partiality, it will include intermediate stages in the resolution process as design states. Resolution constructs and orders these partially resolved feature structures. It is a


complex recursive process — as substructures are resolved new substructures and type labels are introduced that require further resolution steps. 3.6. INCREMENTAL -RESOLUTION

Given a query description D , -resolution is the search across sequences of feature structures F0 v F1 v F2 v : : : v Fk . The initial feature structure in each sequence is a most general satisfier of the query description. The sequence represents the inclusion of type information in the form of constraints — each element extends its predecessor by unification with a type constraint. Since most general satisfiers may occur as collections and unification may fail, the search for resolved feature structures involves a collection of sequences. In a design space explorer based on -resolution, the explicit design space is the union of all such sequences. Definition 4 (Incremental -Resolution) Given a feature structure F , a function recording resolution steps and a type t, we take an incremental -resolution step hF;

, a path ,

i =;t) hF 0; 0 i

if and only if

@ defined (2) t Type (F @ ) (3) 0 0 0 8t 2 Type : t