Operational Rationality through Compilation of Anytime Algorithms

description of the domain In a hierarchical (quad-trees) representation, the level of abstraction corresponds to a certain coarse grid in which every position, , is an abstraction of a 2 !#" 2 ! matrix of base-level positions. Each high level position has a certain degree of “obstacleness” associated with it which is simply the proportion of the matrix that is covered by obstacles. A general position in this two dimensional domain has therefore three components: $&%(' : where $ and % are the coordinates and ' is the level of abstraction. The position 3 3 1 , for example, corresponds to the following set of base level positions: 6 6 0 6 7 0 7 6 0 7 7 0 . If one of these positions is blocked by an obstacle and the rest are free, then the “obstacleness” of 3 3 1 is 0 ) 25. The anytime planning algorithm The interruptible anytime planner (ATP) constructs a series of plans whose quality improves over time. It starts with a plan generated by performing best-first search at the highest level of abstraction. Then, it repeatedly refines the plan created so far by selecting the worst segment of the plan, dividing it into two segments (of identical length), and replacing each one of those segments by more detailed plans at a lower abstraction level. The worst segment of the plan is selected according to the degree to which the segment is blocked by obstacles and according to its abstraction level. A special data structure, called a multi-path, is used in order to keep intermediate results. It is a list of successive path segments of arbitrary abstraction level. The algorithm is shown in Figure 8.3. CHAPTER 8. APPLICATION AND EVALUATION 124 ATP(start, goal, domain-description) 1 multi-path [SEGMENTIZE(start), PATH-FINDER(PROJECT(start, ), PROJECT(goal, ), domain-description), SEGMENTIZE(goal)] 2 REGISTER-RESULT(multi-path) 3 while REFINABLE(multi-path) do 4 REFINE(WORST-SEGMENT(multi-path), domain-description) 5 REGISTER-RESULT(multi-path) 6 SIGNAL(TERMINATION) 7 HALT Figure 8.3: The anytime planning algorithm PATH-FINDER(start, goal, domain-description) 1 level LEVEL-OF(start) 2 size ACTUAL-DOMAIN-SIZE 2 3 domain NTH-ABSTRACTION-LEVEL(level, domain-description) 4 close UNVISITED(domain) 5 open [MAKE-STATE(start)] 6 best-path best-first-search(domain, open, close, goal) 7 return [start best-path] Figure 8.4: The path finder Note that start and goal are the start and goal positions, and is the maximal abstraction level. The length of each segment of an intermediate plan is invariant. It depends only on the length of the initial path at the highest level of abstraction. As a result, the run-time of the refinement step is approximately the same for any segment of the plan regardless of its level of abstraction. The PATH-FINDER is a search procedure that returns the best path between any two positions in the same abstraction level. The path is represented as a list of positions at the same abstraction level as the start and goal positions. A base-level path must be obstacle-free and hence, it is a route that the robot can follow. A path at a higher level of abstraction, on the other hand, is the result of a best-first search that minimizes the length as well as the obstacleness of the result. It does not correspond to a particular list of base-level positions that the robot can follow. The particular positions are determined at execution time. The algorithm is shown in Figure 8.4. Plan execution In order to follow an abstract path, the robot must use an obstacle avoidance procedure that controls its movement whenever the planned route is blocked. The robot can sense that the planned route is blocked CHAPTER 8. APPLICATION AND EVALUATION 125 Figure 8.5: Optimal path as it reaches an obstacle (using a different kind of sensing method). Navigation using obstacle avoidance alone is not efficient and may lengthen the route. In this implementation, as long as there exists a path that connects the start and goal positions, the obstacle avoidance procedure alone can bring the robot to its destination. Therefore, any abstract plan is executable. Obstacle avoidance is clearly not a smart navigation method, but it can always substitute for missing details in an abstract plan. The quality of an abstract plan is defined as follows: route-length route-length where route-length is the length of the route generated when the robot is guided by the plan , and is the optimal plan. Note that the higher the level of abstraction, the lower the quality of the plan. At the same time, high-level abstract planning reduces the search space and hence it is performed much faster. The notion of executable abstract plans – regardless of their level of detail – is made possible by using plans as suggestions that direct the base level execution mechanism but do not impel a particular behavior. This idea was promoted by Agre and Chapman [1990] and was experimentally supported by Gat [1992]. Uncertainty alone makes it impossible to use plans except as a guidance mechanism. Performance with perfect vision What is the performance of the abstract planner? First, I will examine the performance under the assumption of perfect domain description. Then, I will examine the effect of degradation in vision on the quality of planning. Figure 8.5 shows the path found by the path finder when activated with start and goal positions being the lower left and upper right positions respectively. The search level is zero (base level) hence the path shown is optimal (i.e. it is a shortest path). CHAPTER 8. APPLICATION AND EVALUATION 126 (a) Level 3 plan (b) Level 2 plan (c) Level 2/1 plan (d) Level 1/0 plan Figure 8.6: Abstract plans with perfect vision CHAPTER 8. APPLICATION AND EVALUATION 127 0.40 0.50 0.60 0.70 0.80 0.90