Planning under Uncertainty in Dynamic Domains - Carnegie Mellon ...

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28 Chapter 3. Planning under UncertaintyActions and exogenous eventsWeaver operators are very similar to operators in prodigy 4.0, but include a durationfunction and generalise from a single outcome to multiple possible outcomes, oneof which will take place when an instantiated operator is executed. Thus each operatorhas an eect distribution function, edf() that takes a state as input and returnsa probability distribution of eects, that is, a nite set edf(s) =f(p i ; i )j1 i ngfor some n, where each i is a list of add and delete statements denoting an eect setas in the operators of prodigy 4.0, each p i is a positive number and P np i=1 i =1.The eect distribution function is represented as a binary tree whose internalnodes are labelled with literals mentioning variables from the domain and whose leafnodes are labelled with probability distributions of outcomes. Each internal node hasone out-arc labelled false and one labelled true. To nd the appropriate eect distributionfor a given state, the tree is traversed from the root node, taking the branchlabelled true or false respectively depending on whether the literal at the node istrue or false in the state. Figure 3.5 showsaversion of the Move-Barge operator fromthe oil-spill domain used in the previous sections that has a simple branching eectdistribution function with one internal node labelled with (barge-ready ).In this case the two possible eect distributions have the same set of outcomes withdierent probabilities, but this need not be true in general. The probability valuesare part of the domain denition, assigned externally to Weaver. Deriving themusing machine learning or knowledge acquisition techniques would make an feasibleextension but is not covered in this thesis.Informally, when an instantiated operator is applied in some state s in a domainwith no exogenous events, the resulting state is determined probabilistically as follows.First the eect distribution function is called to nd the eect distribution for s,edf(s). Then an eect set is chosen at random, with the probabilities given in theeect distribution. Finally this eect set is applied to the state in the same way thatthe single outcome of an instantiated operator in prodigy 4.0 is applied to thestate. For example, if the instantiated-operator(Move-Barge =barge1 =Richmond =west-coast) is applied inany state in which (at barge1 Richmond) and (operational barge1) are true but(barge-ready barge1) is false, (operational barge1) will be true in the resultingstate with probability 0.1 and will be false with probability 0.9, while (at barge1Richmond) will be false and (at barge1 west-coast) will be true with certainty.The situation is complicated signicantly by the presence of exogenous events inthe planning domain. Since events specify ways the world can change independentlyof an action taken in a plan, the resulting state when an action is taken depends onwhat events may take eect between the start and end of the action. Since some ofthese events may have begun before the action is taken, at least some aspects of thehistory of the world are needed, and in order to model such events it is necessaryto introduce a notion of time. A variety of theoretical models in the AI literaturecan handle exogenous events and are in the same spirit as the situation calculus on
3.2. A representation for planning under uncertainty 29Move-Barge(Operator(duration (/ (distance ) (speed )))(preconds(( Barge)( Place)( (and Place ( di ))))(at ))(effects(branch (barge-ready )((0.667 ()((add (at ))(del (at ))))(0.333 ()((add (at ))(del (at ))(del (operational )))))((0.1 ()((add (at ))(del (at ))))(0.9 ()((add (at ))(del (at ))(del (operational ))))))))Figure 3.5: The Move-Barge action from the oil-spill domain used in Weaver hasa duration and a probability distribution of possible outcomes.which the prodigy 4.0 representation rests e.g. [Allen et al. 1991; Kartha 1995;Shanahan 1995; Baral 1995].In this thesis I have chosen a simple and restricted language for events that is stilladequate for planning with some interesting domains. While some approaches, basedon circumscription, are independent of an underlying model of the language, my approachis based on the adoption of a Markov decision process as an underlying model.This approach was chosen partly because the existing techniques did not handle probabilisticreasoning well, although the random-worlds approach is promising [Bacchuset al. 1997], and partly to make it easier to compare the planning system developedin this thesis with other approaches to planning under uncertainty, many of whichare based on Markov decision processes [Boutilier & Dearden 1994; Dean & Lin 1995;Littman 1996].Exogenous events haveavery similar representation to operators. Figure 3.6 shows
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