[Leishman, 1989a]. Deborah Leishman. A Principled Analogical Tool. Masters thesis. University of Calgary
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1 [Coyne et al., 1990]. R.D. Coyne, M.A. Rosenman, A.D. Radford, M. Balachandran and J.S. Gero. Knowledge-Based Design Systems. Reading, Massachusetts, Addison-Wesley [Garey and Johnson, 1979]. Michael Garey and David Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness. New York, W.H. Freeman and Company [Gentner, 1987]. Dedre Gentner. Mechanisms of Analogical Learning. University of Illinois. Computer Science Department. Number UIUCDCS-R [Kambhampati, 1989]. Subbarao Kambhampati. Flexible Reuse and Modification in Hierarchical Planning: A Validation Structure-Based Approach. Ph.D. thesis. The University of Maryland [Leishman, 1989a]. Deborah Leishman. A Principled Analogical Tool. Masters thesis. University of Calgary [Leishman, 1989b]. Deborah Leishman. Analogy as a Constrained Partial Correspondence Over Conceptual Graphs. Proceedings of the First International Conference on Principles of Knowledge REpresentation and Reasoning, Toronto, Canada. pp , [Leishman, 1994]. Deborah Leishman. Design Reuse by Relational Analogy. PhD thesis. Unviersity of Calgary [Veloso, 1992]. Manuela Veloso. Learning by Analogical Reasoning in General Problem Solving. PhD thesis. Carnegie Mellon University
2 Matching Heuristics The front end matching performed in RELAY is guided by three basic heuristics. The first is to try to obtain the maximal match between parts of the graphs that are the most similar. At any level of the matching process, the maximum number of nodes are matched together and matches with the highest scores are then used to constrain the next lower levels of matching if these levels exist. The evaluation of the matches is based on attributes of the nodes and edges being matched. These include node valences, the size of the structure represented by a node at the next lower level, the type of structure represented (linear or non-linear) and whether an edge is incident or not. The second heuristic is to maintain connectivity in the graphs being matched. This means that one common subgraph is formed as opposed to several disconnected subgraphs. The connectivity is maintained not only within a single level of the source and target graphs but is also maintained across the various graph levels represented by the dual graphs. Thus, the connections formed at one level of the graph abstractions are maintained at the next lower level by matching incident edges together. The third heuristic used in the matching algorithms is to constrain the matches formed as quickly as possible. In the ordering of the matches prior to switching graph levels, the more internal, more constraining nodes/structures are matched first. These are then used to constrain the matches on nodes and edges of neighboring structures. As well, potential correspondences are removed as early a possible by a limitconstraints function. This eagerness helps ensure efficient matching algorithms. 3.2 The Back End Plan Formation The above algorithms find the structural similarity between the two problems. This structural similarity is then translated into problem solving process information through formation of a fully instantiated partial plan and subsequent plan completion for those goals that must be solved from scratch. Four heuristics are used to decide which of the plan steps from the source plan are applicable for solution of the new target plan. These are used to form a fully instantiated partial plan that consists of completed plan steps and goal ordering. These plan steps as shown in Figure 1 specify placement of rooms relative to other rooms, and the ordering of that placement. The reuse of entire previous plan steps is possible because the front end matcher is able to identify independent or nearly-independent sub-problems at the structural level which then map down to the process information in a particular plan. Once the fully instantiated partial plan is completed, an extension of the base level planner is used to complete the layout plans in all possible ways. Several plans may result because each goal (room to be laid out) may be achieved in more than one way. Each of the layouts produced as a result of the plan completion will be similar in terms of the independent or nearly independent sub-problem found, but will differ in how the extra rooms are placed. 4.0 Results of Using RELAY In several tests with RELAY, it produces consistent sets of room correspondences and produces fewer plans in less time than planning from scratch, while also maintaining as much of a previous plan as possible. In all tests run, the maximal common subgraph between the source and target problems was found, resulting in reusability of from 47%-100%. Where all of the previous plan steps were reusable, only one layout was produced. These results show that RELAY has achieved its goals of finding as maximal as possible common subgraphs of the source and target problems that are independent or nearly independent and allow for reuse of a common sub-plan. The small number of layouts produced through reuse will allow a designer to fully evaluate the layouts using several objectives and provide support for the iterative, exploratory nature of design. 5.0 Summary This paper has presented a description of a reuse based problem solving system that is based on global assessment of similarity between problem structures. A system called RELAY was described that uses analogies between problem structures to find independent or nearly independent reusable sub-plans which are completed by a base level planner. The front end matcher that performs the global similarity assessment was described in detail. Computational efficiency of the matcher was shown to be derived by utilizing abstractions of the problem structure. An important consideration in design of the RELAY system was the need to deal with complex multi-goal interactions common in layout and design problems. The type of reuse described in this paper is applicable to many areas of spatial layout design and also for reuse in other domains such as object oriented software systems consisting of groups of collaborating objects. The requirement for other systems to be able to perform reuse based problem solving through global assessment in the manner presented here, is the need for problem abstractions to exist. These abstractions may be structural in nature as for spatial layout, or there may be other types of abstractions. As well as a requirement for problem abstractions, there must also be traceability between these abstractions. This is necessary in order for correspondences made between parts of higher level abstraction to be used to constrain lower level correspondences. This is necessary in order to deal with matching complexity. Within software, the recent advancements in object oriented systems, architecture, and reusable frameworks is beginning to provide the necessary requirements for implementation of a reuse system similar to RELAY. Current research is concentrating in this area. References [Bhansali, 1991]. Sanjay Bhansali. Domain-Based Program Synthesis Using Planning and Derivational Analogy. Ph.D. thesis. University of Illinois at Urbana-Champaign [Coyne, 1988]. Richard Coyne. Logic Models of Design. Long Acre, London, Pitman Publishing
3 reuse that is appropriate to the spatial layout domain is the need to support iterative or incremental design. The design reuse system described here supports this by maintaining as much as possible of a previous design iteration. 3.0 RELAY: REusing LAYouts The RELAY system for design reuse of spatial layouts addresses the problems mentioned above by reducing the number of layouts produced and the time taken to produce them, and by retaining as much as possible of a previous design. RELAY accomplishes this through a reuse process that is driven by the formation of global relational analogies between high level problem specifications. The correspondences formed between the high level goals in source and target problems are used to constrain the portions of a previous layout plan that are reused, and to identify what portions of a problem must be solved from scratch. There are two parts to the RELAY program; a front and back end. The front end of RELAY takes as input high level specification of source and target problems and produces the input to the RELAY back end in the form of consistent correspondences between goals, and a particular layout plan that met the requirements of the source problem specification. This part of RELAY then produces a fully instantiated partial plan that reuses as much as possible of the previous source plan. From there, final plans are produced which maintain those completed parts of the fully instantiated partial plan, and solve any new goals. 3.1 Front End Matcher The front end of the RELAY program forms correspondences between rooms in source and target adjacency graphs that represent connection constraints between rooms. An important requirement on the RELAY front end is that it be efficient enough so as not to overcome any gains made by the RELAY reuse back end. Given two graphs representing source and target analogues, what is desired, is to find correspondences that provide the largest common subgraph of the two given some correspondence relation. A related matching problem of subgraph isomorphism is know to be NP complete [Garey and Johnson, 1979]. The matching algorithms in RELAY use heuristics to provide efficient methods for formation of relevant common subgraphs. RELAY utilizes an efficient method of forming consistent relational analogies as common subgraphs and relies on the mathematical properties of graphical structures. RELAY uses dual graphs (Figure 1) as the method of problem abstraction that leads to efficient matching algorithms and a method of forming consistent correspondences for goal conflict avoidance. A dual graph is formed by placing a node within each enclosed structure of the adjacency graph and joining the nodes across each edge. A node is also placed to represent any outlier structures such as room f in Figure 1. Dual graphs can be formed at higher levels of abstraction as well, finishing with a linear graph that contains no cycles. A dual then, is a graph that specifies connection information at increasing levels of abstraction, where the graph at each level is directly related to a lower level graph. This dual graph representation is based on mathematical dual graphs with the exception that the outer face on the planar surface is not fully represented. Given the dual graph representation, correspondence between rooms in the source and target examples is derived by first forming correspondences between the highest common level of dual graphs. This set of correspondences between nodes in the dual graphs then serves to constrain matches between nodes at the next lower level of abstraction. This continues down to the adjacency graph, where final correspondences between rooms is made. Thus, a node at one level of generalization becomes a structure at the next lower level and the nodes within that structure are matched with nodes in the corresponding structure. The complexity of the matching algorithms is now based in the worst case on the number of nodes in the largest structure within a graph rather than on the number of nodes in the entire adjacency graph as is the case in other analogical reasoning systems [Gentner, 1987]. Levels of Interaction Another important reason for using dual graphs as part of an example s representation is to provide a method for dealing with the multiple goal interactions that take place in complex problem domains such as layout. Dual graphs allow for representation of the three levels of constraints that exist in these types of problems. Each of the constraint levels deal with adjacency constraints but between different levels of structure in a graph. The three types of constraints are local, global, and meta. Local constraints are constraint between nodes within a structure. These rooms must be laid out such that there is a path from one connecting node to the next. Local constrains are represented by an adjacency graph. Global constraints are constraints between structures in a graph. Structures must be laid out such that for each structure there is a path from one structure to the next across incident edges or singularly connecting nodes in the case of linear structures. This means that groups of rooms must be laid next-to adjacent groups of rooms. A first dual graph represents global constraints. Meta constraints are constraints between groups of structures in a graph, such that there is a path from one group of structures to another across incident edges or singularly connecting nodes for linear structures. Higher level dual graphs represent meta constraints. Solutions to layout problems deal with all three of these constraint types. A reuse system must reconcile all three levels in order to find a set of correspondences that is consistent across the levels. This is necessary so that portions of a plan that are reused will fit together properly and allow for maximal reuse. By matching the highest levels of dual graphs first and allowing them to constrain the lower level matches, RELAY is actually satisfying meta constraints followed by satisfying meta and global constraints, followed by satisfying meta, global and local constraints at the lowest level matches. Thus, higher level constraints are maintained while new constraints are added and satisfied at each level.
4 f A d e B a C D g f h b c g a c b h d e High Level Layout Specification Physical Layout Plan: anchor (b) put (a) east (b) put (g) west (a) north (b) put (c) south (b) west (a) put (e) west (a) south (c) put (h) north (b) w put (d) south (c) west (e) put (f) north (d) west (c) Dual Graph A B C D Figure 1 The spatial layout system described by Coyne [1988] has been re-implemented to serve as the base level planner for a design reuse system. To deal with the potentially infinite number of ways of locating, orienting and dimensioning objects in space, a grammar for producing layouts is used. The grammar is called rectangular dissection and only allows for layout of rectangular shaped objects, but is sufficient for many spatial layout tasks. This system uses a rule-based hierarchical planning approach, where layout transformation rules are divided into three independent hierarchical sub-tasks. The first four rules are ordering rules that take an input specification of goals such as put room a, put room b, etc. and transforms the unordered goals into an ordered, linear sequence of goals to be achieved. The valency of rooms is used for this ordering. The second set of three rules establish adjacencies between the rooms and thus turn a high level plan into a more specific detailed plan. These rules also add new goals to a plan by specifying that a room is not only to be placed, but that it be placed adjacent to other rooms. For example, put room a is transformed into put room a next-to room b and next-to room c. The third set of rules establish very specific orientations between the rooms in the existing plan and contains thirty-two rules. All rules contain pre-conditions which when satisfied allow for a rule to be triggered and later fired. Most rules contain conflict avoidance knowledge to reduce interaction problems, and the last set of rules reflect the rectangular dissection rules. The transformation from the input graph into final layouts is not a trivial task as many conflicts can occur when searching for how to lay the rooms out, and for large numbers of rooms many layouts will conform to a single specification. A more detailed explanation of the system can be found in Leishman [1994]. The complexity of performing room layout is potentially infinite, but with the use of the rectangular dissection grammar discussed above becomes in the order of 8 n where eight is the branching factor of the search tree and corresponds to the number of possible rectangular placements and n is the number of objects to be placed in the layout. Empirical studies of the complexity of the implemented system, show that there are on average two to four rules that apply for placement of each object, resulting in a search complexity on average of 3 n. This search size is greatly reduced from 8 n but is still exponential in the number of objects to be placed and thus layouts for large numbers of objects are not feasible with the system as implemented. For example, placement of twenty objects may produce over one million alternative layouts, far too many for a human to evaluate. One way to provide a layout system that can scale up to larger numbers of objects and can reduce the search time for problems, is to reduce or eliminate the exponent corresponding to the depth of the search. The next section describes how reuse through relational analogy achieves such a reduction. As well, reuse of previous designs provides a means to build on previous successful designs and to eliminate errors or bad designs. Another aspect of design
5 Relational Analogy and Abstraction for Design Reuse Deborah Leishman 750 Fielding Drive Ottawa, Ontario K1V 7G4 Canada (613) Abstract Systems that focus on reuse of previous problem solving efforts either utilize independent sub-problems in a domain [Veloso, 1992; Bhansali, 1991] or determine what parts of an entire similar problem-solution pair is reusable [Leishman, 1994]. This distinction is true in many domains including planning, design and software and can be described in terms of systems that perform global versus local similarity assessment for reuse. This paper discusses the importance and applicability of global similarity assessment for problem solving by describing RELAY, a system that reuses spatial layout designs and focussing on its global matching portion. The mechanisms used in RELAY show that similarities in problem structure can result in full reuse of related process information. The use of abstractions of the problem structure also provides the computational efficiency to make the mechanism viable. 1.0 Local and Global Assessment for Reuse Reuse based problem solving systems in the domains of planning and design can be characterized as to whether they emphasize local or global assessment of problems and their solutions. For example, Veloso [1992] uses derivational analogy techniques which reuse a problem solving process in a justified play as you go method. Here, local justifications are used to decide what to reuse. To make this system computationally feasible, the system also learns reusable subproblems. The local assessment of what is reusable is then performed on these similar sub-problems during problem solving. Other systems such as Kambhampati [1989] may globally assess a new problem to retrieve a similar case, but then perform local assessment of what to reuse for problem solving. What is lost in the process is the global view of why the retrieved and new cases were similar. This information is thus rediscovered during problem solving. Few if any systems in these domains perform retrieval or matching based on a global assessment of problem similarity as a controlling mechanism to guide subsequent problem solving. A reasoning mechanism that supports a global assessment of problem similarity is that of relational analogy. This refers to analogical reasoning systems that utilize a similarity based syntactic approach. Examples of such systems include the structure matching engine of Gentner [1987] and analogy as a minimal common generalization [Leishman, 1989]. These systems form correspondences between analogues by concentrating on forming systems of relations. These systems give no added importance for one type of relation over another, but stress the importance of connected systems of relations as the force behind correspondence formation and reasoning. 2.0 Design Reuse for Spatial Layout The problem within design considered in this paper is that of spatial layout, particularly, architectural room layout. As in many design problems, preliminary work in the problem domain is performed to produce a high level specification in terms of both goals to be satisfied and constraints between the goals. In room layout as exemplified by Coyne [1988], this corresponds to specifying the rooms to be placed and the connection constraints between the rooms. The requirement then is to formulate a plan, that when executed, will result in a layout that satisfies the problem specification. A typical action in such a plan is: place room a north of room b and east of room d. A series of such actions will produce an entire layout as shown in Figure 1. The specification can be seen as a goal state that specifies a problem in terms of goals to be achieved and interactions between the goals. Complexity in this domain is dominated by these multi-goal interactions that are visible through the problem specification. These interactions are complex in nature because for example, the placement of two objects may prohibit placement of several other objects. In addition, it is not feasible in this domain to learn sets of goal interactions that form independent sub-problems as chunks to be stored in memory and reused in subsequent problems as is done in Veloso s derivational analogy system [Veloso, 1992]. Instead, an entire past case is reused as a focussing mechanism to find problem specific independent or nearly independent sub-problems. These sub-problems can then be totally reused, producing a fully instantiated partial plan that a base level planner can complete. The method of reuse that is appropriate for the complexities of this domain is that of relational analogy.
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