Generating a Compact Representation of an Exponentially Large Solution Space

exit states Abstract State Both techniques eliminate distinctions within unconstrained regions of the solution space by abstracting these regions into a single state within the search tree. (The gure above is a generalized representation of the search space explored during simulation.) By identifying the boundaries of the unconstrained region, both algorithms are able to generate an abstract state describing the region (a) in addition to identifying the states that exit the region. The states that exit the region must be combined to eliminate the distinctions that are made within the abstracted region. Figure 5: Abstracting unconstrained regions of the state space and 4). Thus, it leverages the inference capabilities already provided by the simulation algorithm and can be shown to be both sound and complete with respect to the inferences made by the unmodi ed simulation algorithm. The problem with this technique is that the complexity of the recursive call to the simulation algorithm is exponential in the number of unconstrained variables within the region. Thus, as the size of a model grows, the technique becomes prohibitively expensive. Dynamic chatter abstraction identi es the unconstrained region by a dynamic analysis of the constraints within the model and the current state. Thus, as opposed to using brute force search (as chatter box abstraction does), dynamic chatter abstraction reformulates the problem into a CSP that simply attempts to answer the question: \Is variable x unconstrained within the current region of the state space." Thus, dynamic chatter abstraction provides a more focused search of the state space as opposed to enumerating all possible paths through the unconstrained region as chatter box abstraction does. The chatter box abstraction algorithm is fairly straight{forward. Dynamic chatter abstraction algorithm, on the other hand, must address a number of detailed issues that makes it much more complicated. In particular, it must reason not only about the values within the current state, but it also must reason about how these values might change over time as variables begin to chatter. For example, suppose that the derivative of a variable x is unconstrained within the current qualitative state. Thus, its direction of change is free to chatter. Once the variable begins to change in a new direction, however, a variable y that was previously constrained may now become unconstrained and thus begin to chatter. Both variables must be identi ed as chattering. Understanding the nuances of the dynamic chatter algorithm requires a more detailed understanding of qualitative simulation and the manner in which chatter is manifested within a simulation. In general, the dynamic chatter abstraction algorithm can be viewed as a special purpose inference engine that has been designed to answer the focused question: \Does a given variable chatter following the current qualitative state before a non{chattering variable changes?" The algorithm encodes knowledge not only about how the constraints that restrict a given variable at a single point in time, but it also encodes information about how these values can change across time without explicitly computing all of the behaviors. Comparison of these techniques Both chatter box abstraction and dynamic chatter abstraction provide unique bene ts. Chatter box abstraction provides a simple algorithm that leverages the inference capabilities of the simulation algorithm thus enabling the seamless integration of extensions to the simulation algorithm with the abstraction technique. Dynamic chatter abstraction, on the other hand, provides a much more e cient algorithm that scales to larger problems. As the simulation algorithm is extended, however, addi#of Cascade Chained Loop Tanks Qsim DecS Qsim DecS Qsim DecS 2 0.20 0.815 3.07 6.79 0.757 5.58 3 0.62 1.6 10.9 19.90 16.14 8.14 4 2.2 3.12 37.5 25.98 89.41 12.6 5 7.09 5.49 139 36.71 493.8 23.2 6 21.9 6.32 676 62.4