Abstracting Irrelevant Distinctions in Qualitative Simulationt

ing Irrelevant Distinctions in Qualitative Simulationt Pierre Fouche( 1 ) & Benjamin Kuipers(2 ) (1 )Universitd de Technologie de Compibgne C.N .R.S . U.R.A. 817 B.P . 649 60206 Compiegne, France Email : fouche@ a frutc5l .bitnet (2)Department of Computer Sciences The University of Texas at Austin Austin TX 78712, USA Email : kuipers@ a cs .utexas.edu Abstract : One main problem in qualitative simulation is that it often produces too detailed qualitative descriptions of a system's possible behaviors, or produce a lot of behaviors that differ very slightly . This paper addresses the problem of summarizing the result of a qualitative simulation to make it more perspicuous. Two causes of behavior proliferation have been identified and two algorithms to aggregate behaviors that do not differ significantly are presented. Simple -xamples are used to illustrate the functionning of each algorithm . One main problem in qualitative simulation is that it often produces too detailed qualitative descriptions of a system's possible behaviors, or produce a lot of behaviors that differ very slightly . This paper addresses the problem of summarizing the result of a qualitative simulation to make it more perspicuous. Two causes of behavior proliferation have been identified and two algorithms to aggregate behaviors that do not differ significantly are presented. Simple -xamples are used to illustrate the functionning of each algorithm . t This work has taken place in the Qualitative Reasoning Group at the Artificial Intelligence Laboratory, The University of Texas at Austin, and in the Al Group in CNRS-URA 817, The University of Compibgne . Research of the Qualitative Reasoning Group is supported in part by NSF grants IRI-8602665, IRI-8905494, and IRI8904454, by NASA grants NAG 2-507, and by the Texas Advanced Research Program under grant no . 003658175 . Pierre Fouchd holds a grant from Rh6ne-Poulenc . 1 . Introduction The goal of qualitative simulation is to produce qualitative descriptions of a system's possible behaviors from a qualitative model. Two main problems in qualitative simulation can be distinguished : " Due to the interaction of qualitative representations and local reasoning techniques, qualitative simulation sometimes produces behaviors that are not actually possible . Such behaviors are called spurious behaviors . " Qualitative simulation may give too many details compared to what is expected, or produce a lot of behaviors that differ very slightly . It is then very difficult to extract the behavioral features of a system . This papers deals with the second problem. In practice, analyzing a large behavior tree or a large envisionment graph is a difficult task and tools to summarize the result of a qualitative simulation to make it more perspicuous are necessary to make qualitative simulation useable . At first sight, there are many ways to aggregate similar behaviors, and finding general similarity criteria that could be successfully applied on a large variety of simulation results is critical . After having analyzed many envisionment graphs, it turned out that behavior proliferation could generally be attributed to two main causes : chatter and occurrence branching . Chatter happens when the derivative of a variable is unconstrained ; occurrence branching happens when the temporal ordering of two or more events cannot be determined. For each of these causes, we will present an algorithm to aggregate behaviors that do not differ significantly and apply it to a simple example to illustrate its functionning . The algorithms presented here can be used or adapted to any system that produce an envisionment graph . They have been implemented in Common Lisp as extensions of Kuipers's QS IM [11],[12], and tested on TI Explorers, MicroExplorers and on VAX Workstations running Xwindows. We will first recall some basics definitions necessary to qualitative simulation . 1 .1 . Basic Concepts and Definitions A physical system (p is modelled by a set of state variables and a set of constraints that relate them . Variables are real-valued, continuously differentiable functions of time. The qualitative magnitude or qmag of a variable is defined with respect to a set of values, called landmarks, that represent real qualitative distinctions for the behavior of the variable . A qualitative magnitude is either a landmark value or an open interval between two adjacent landmarks . The direction of change or gdir of a variable is the sign of its derivative with respect to time, and can be either dec, std, or inc (for decreasing, steady, or increasing). The qualitative value or qval of a variable is the pair (gmag,qdir) A qualitative state of a system is a set of qualitative values of all the system's variables, which are consistent with the set of constraints . We distinguish states that are valid only for an instant, called P-states, and states that are valid over an interval of time, called 1-states. Qualitative simulators that use states to describe a system's behavior are called state-based simulators . Simulators that describe a system's behavior in terms of behaviors of its variables, considered individually, are called history-based simulators (see paragraph 3.2) . 1 .2 . Behavior Generation VS Envisionment A qualitative behavior (in a state-based representation) is a sequence of states which alternates P-states and I-states . Qualitative behaviors can be produced in two ways: " directly from an initial state ; the result of a simulation is a tree ofpossible behaviors. " from a graph, called envisionment, whose vertices are possible states and edges are valid transitions between them . A behavior is then a path in the graph . An envisionment is said to be total if it contains all the possible states of the system, or attainable if it only contains states that can be reached from an initial state. Kuipers' QSIM [11],[12] adopts the first strategy, while Forbus' QPE [4] and de Kleer & Brown's ENVISION [10] produce envisionments . On one hand, an envisionment provide a compact, finite description of a set of possible behaviors. On the other hand, reasoning over behaviors allows one to use stronger reasoning techniques, such as reasoning in a phase space representation, or reasoning about energy (see [6]) . An attractive feature of behavior generation is that it allows one to introduce new distinctions during simulation, by dynamically creating new landmarks, for instance, when a variable reaches a critical value. The nature of oscillations of oscillatory systems can be determined using this mechanism, but this cannot be done using an envisionment graph. However, when behaviors begin to proliferate, new landmarks prevent an easy comparison of similar behaviors, because they are attached to a specific behavior . This is the reason why we extended QSIM so it can now produce envisionments as well as directly generate behaviors . The remainder of this paper presents two algorithms to aggregate similar states and behaviors in an envisionment graph. 2 . Chatter The graph in figure 1 contains many short cycles . A filled circle represents a state at some time point, an empty circle a state at some time interval . Analyzing why such cycles are produced shows that many states differ only by the direction of change of one or several variables . These distinctions are very often useless and the graph as it is in figure 1 is not easy to interpret. In the following sections, we explain how to group states which, ignoring the directions of change of the variables, are identical . We introduce the concept of gmag-equivalence and describe how to aggregate states in a same equivalence class . 2 .1 . Building Equivalence Classes The algorithm to eliminate the proliferation of such cycles is based on partitioning the graph into equivalence classes, which gather states for which all the variables have the same qualitative magnitudes . The precise definition of the equivalence relation gmag-equivalent is the following: Definition: Two states S1 and S2 are gmag-equivalent if and only if : b' v, gmag(v, Sl) = gmag(v, S2) It is easy to check that this relation is an equivalence relation over the set of possible states X From this definition and the set of possible states X9, we build a partition P ofX9, P = (Ci), eac~ Ci being an equivalence class. The next step is to identify the differences between all the states in Ci and to aggregate them to build a new state Ti which generalizes all the states in Ci . 2 .2 . Generalizing Equivalent States Since all the states in Ci are gmag-equivalent, they differ only by the directions of change of the variables and it makes sense to define the magnitude of a variable v in a class Ci, gmag(v, Ci). The magnitude of v in Ti, qmag(v, Ti) is set to gmag(v, Ci). To determine the direction of change of v in Ti, we build the set D (v, Ci) of its possible directions of change in Ci: D (v, CI) = U qdir(v, S) . If D(v, Ci) has only one element, then qdir(v, Ti) = D(v, Ci). If it has more than SCCi one element, then we set qdir(v, Ti) to ignt . Note that we can lose information in building Ti : having qdir(v, Ti) = ign does not necessarily imply that v can take on any direction of change in the states that Ti generalizes (but, for instance, inc or std only). We also lose information on how states in Ci are connected together . tign means that the direction of change is ignored. Figure 1 : Envisionment graph of A -~ B -> C -> D 2 .3 . Setting Time Label, Predecessors and Successors In building generalizations, we also want to keep the distinction between P-states and I-states, and connect the more general state to other states in the graph . We begin with defining the set of successors and predecessors of a class : succ(Ci) = U succ(S) pred(Ci) = U pred(S) SCCi SCCi We also make a distinction between Pand I-successors and predecessors, and split these sets into P-succ and I-succ : P-succ(Ci) = {S : S E succ(Ci) and time(S) =point} P-pred(Ci) = {S : S E pred(Ci) and time(S) =point} I-succ(Ci) = IS : S E succ(Ci) and time(S) = interval) I-pred(Ci) = {S : S E pred(Ci) and time(S) = interval} The successors and predecessors of Ti are defined straight-forwardly : P-succ(Ti) = IS : S E P-succ(Ci) and S o Ci} P-pred(Ti) = {S : S E P-pred(Ci) and S e Cl} 1-succ(Ti) = {S : S E 1-succ(Ci) and S o Ci) Ipred(Ti) = IS : S E Ipred(Ci) and S z CiI Defining whether Ti is a Por I-state depends on Ti's successors and predecessors : " If P-succ = P-pred = 0 then Ti must be a P-state and we set : time(Ti) = point. pred(Ti) = Ipred(Ti) succ(Ti) = I-succ(Ti) The successors of Ti's predecessors and the predecessors of Ti's successors must be updated as well : b' S e pred(Ti), succ(S) = (Tj U (succ(S) Ci) VS e succ(Ti), pred(S) = (Ti} U (pred(S) Ci) If I-succ =1-pred = empty then Ti must be an I-state and we set : time(Ti) = interval . pred(Ti) = Ppred(Ti) succ(Ti) = P-succ(Ti) Successors of Ti's predecessors and predecessors of Ti's successors are updated in exactly the same way . Otherwise, if we want to maintain the semantics of P-states and I-states, it is not possible to give a unique time label to Ti and Ti must be split into two identical states, TiP and Tii, one being a P-state and the other an I-state : b' v, gval(v, TiP) = qval(v, Tii) = qval(v, Ti) time(TiP) = point and time(Tii) = interval pred(TiP) = I-pred(Ti) succ(Tit) = P-succ(Ti) pred(Tii) = P-pred(Ti) U (TiP) succ(TiP) = I-succ(Ti) U (Ti) Predecessors and successors of Tii and TiP are updated slightly differently : b' S e pred(TiP), succ(S) _ (TiP) U (succ(S) Ci) b' S e succ(Tii), pred(S) _ (Tii) U (pred(S) Ci) d S e pred(Tii), S;TiP, succ(S) = (Tii) U (succ(S) Ci) b S e succ(TiP), S;~fii, pred(S) = (TiP) U (pred(S) Ci) Thus, if P is an element of I-pred(Ti) and Q an element of P-succ(Ti), the sequence (P TiP Tj Q) is a possible behavior, which can be interpreted like this : the system is in state P for a certain amount of time, and then evolves to be in state TiP . This state is not instantaneous (system in state Tii ) and the system finally reaches state Q . A transition between T~ and TiP is invalid because an I-state can only lead to a P-state in which at least one variable has a different qualitative value [12] . 2 .4 . Example Figure 1 shows the envisionment graph2 of a system modeling an isothermal batch reactor where three reactions in series occur: A->B->C->D . Cxdenotes the concentration of species x . The modeling equations are : dCA = -ki CA dCB dt = ki CAk2 CB d dt = k2 CBk3 CC dCD = k3 CC dt The simulation started from an initial state with species A at a concentration Co, and was constrained to produce analytic functions only (analytic functions are infinitely differentiable and 2We redrew the graph by hand, because it was not easy to highlight equivalence classes on the graph drawn by QSIM.