Filter Preferential Entailment for the Logic of Action in ... - CiteSeerX

Filter Preferential Entailment for the Logic of A c t i o n in Almost Continuous Worlds Erik Sandewall* Department of Computer and Information Science Linkoping University Linkoping, Sweden Abstract Mechanical systems, of the kinds which are of interest for qualitative reasoning, are characterized by a set of real-valued parameters, each of which is a piecewise continuous function of real-valued time. A temporal logic is introduced which allows the description of parameters, both in their continuous intervals and around their breakpoints, and which also allows the description of actions being performed in sequence or in parallel. If axioms are given which characterize physical laws, conditions and effects of actions, and observations or goals at specific points in time, one wishes to identify sets of actions ("plans") which account for the observations or obtain the goals. The paper proposes preference criteria which should determine the model set for such axioms. It is shown that conventional preferential entailment is not sufficient. A modified condition, filter preferential entailment is defined where preference conditions and axiom satisfaction conditions are interleaved.

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Topic

Our ultimate research goal is to find a coherent theory for temporal reasoning, knowledge based planning, and qualitative reasoning. In that context the present paper addresses the following problem. Assume that one has obtained: 1. a description of a mechanical or other physical system 2. axioms characterizing physical laws which hold in that system 3. axioms characterizing conditions and effects of actions which can be performed by an agent in the world 4. axioms characterizing the observed or desired state of the world at certain point(s) in time all expressed as logic formulas (wff) in a suitable logic. By what logical criteria can one then derive formulas characterizing a set of actions which together explain *This research was supported by the Swedish Board of Technical Development and the IT4 research program 894

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the observed state of the w o r l d resp. w h i c h obtain the desired state of the world? T h i s general problem includes b o t h " p l a n n i n g " in the A . I . sense (knowledge based planning) and " t e m p o r a l exp l a n a t i o n " , the difference being t h a t in knowledge based planning the axioms in the last group express the goals, whereas in t e m p o r a l explanation they express the given observations. For s i m p l i c i t y we shall refer to the last group of axioms as observations in the sequel, regardless of whether the application is p l a n n i n g or explanation.

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C o m b i n i n g logic and differential equations for describing the physical system

In a previous paper [San89a] we have described an approach to i n t e g r a t i n g non-monotonic logic and different i a l equations for characterizing physical systems. The solution described there was however l i m i t e d to the case in w h i c h there are no agents or actions in the world. T h e present paper extends the same approach to the case where actions may occur. T h e key ideas in the approach of the previous paper are as follows. Object systems are assumed to be characterized by a number of parameters, or fluents, which have a real-number value at each point in t i m e . T i m e is also measured as real numbers, not only as discrete time-points. A l l parameters are assumed to be piecewise continuous and d i f f e r e n t i a t e , w i t h all their derivatives. For example a scenario w i t h a bouncing ball w i l l be characterized by the p o s i t i o n , velocity, acceleration etc. of the ball as functions of t i m e . These functions are continuous except (some of t h e m ) for those moments when the ball bounces, where they are taken to be discontinuous. T h e properties of, and relationships between the parameters can then be described by logic formulas, in a language which has been suitably extended to allow time derivatives, and left and right l i m i t values of parameters at the breakpoints (i.e. the t i m e p o i n t s where discontinuities may occur). T h e following are examples of such formulas: saying t h a t at t i m e the t e m p e r a t u r e of the object b 4 is greater t h a n zero degrees Celsius,

saying t h a t the t e m p e r a t u r e of the object is always less

t h a n or equal to 100 degrees C,

saying t h a t the rate of change of the temperature of the objects is always between 0.1 degrees C per second decreasing, a n d 0.05 degrees per second increasing,

saying t h a t whenever the current coordinates of the object at h a n d , is a point where the object bounces, then the left l i m i t value of the vertical velocity has the same m a g n i t u d e , b u t the opposite sign, as the r i g h t l i m i t value of the vertical velocity. In this way the logic can characterize the behavior of the parameters around their discontinuities. For a d d i t i o n a l details and for a more systematic exposition please refer to [San89a]. Interpretations for such formulas must contain the following parts: 1. a specification of a set of breakpoints i.e. those timepoints at which discontinuities may occur 2. a m a p p i n g f r o m t i m e points x parameter symbols to parameter values (where the set of parameter symbols is closed under prefixing of 3. appropriate mappings f r o m constant symbols, such as and b 4 above, to corresponding numbers, objects, etc. Also certain general conditions must be placed on the interpretations, in particular t h a t for each particular parameter s y m b o l , the m a p p i n g in the second i t e m must be continuous, as a f u n c t i o n of t i m e , at all time points which are not break points. ( I n the break points it is allowed b u t not required to be continuous).

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Actions

For the purpose of the present paper, the formula language and the interpretations are extended w i t h actions. Formulas are allowed such as

saying t h a t an action of the action type Keepwarm(b4) is performed f r o m t i m e t 5 to time t 8 . Correspondingly interpretations are amended w i t h one more component, namely an action set which should be a set of triples; each t r i p l e consisting of the t i m e point when the action starts, the descriptor for the action, and the time point when the action ends. A c t i o n descriptors may be atomic, or may have a structure for example i.e. the tuple consisting of the symbol "Keepwarm" and the object which is being kept w a r m . A d d i t i o n a l f o r m a l details are not necessary here. The i m p o r t a n t p o i n t is t h a t each i n t e r p r e t a t i o n characterizes a possible " h i s t o r y " of the object world in two parallel ways: p a r t l y as a set of snapshots i.e. assignments of values to parameters at each point in t i m e , and p a r t l y as a set of actions each of which has a s t a r t i n g time, an ending t i m e , a n d a descriptor saying what the action is. Logic formulas refer to such interpretations, and the t r u t h value of a f o r m u l a in each i n t e r p r e t a t i o n is defined. It should be clear how one can, for example, write axioms

which specify the effects of actions i.e. the values of parameters at the t i m e when an action ends. T h e topic of the present paper is now re-phrased formally as follows. Let be a set of formulas representing observations (or goals, in a planning problem); let T be a set of formulas representing all the other given informat i o n ; we look for a f o r m u l a which characterizes the set of actions, or p l a n , t h a t accounts for or obtains the observations. Often w o u l d be a disjunction of expressions each of which characterizes an alternative explanation or

plan.

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Semantic entailment

For the purpose of the present work it was necessary to introduce a non-standard notion of semantic entailment. Let us first relate it to the t r a d i t i o n a l notions. In general a definition of whether should consist of two parts: 1. model set criterium: determine the model set for 2. formula criterium: identify which formulae wants to conclude f r o m the model set

one

where in classical logic the model set is of course simply Mod the set of all interpretations in which all members of are t r u e , and the formula c r i t e r i u m is to choose a formula which is true in all members of the model set. In non-monotonic logic the model set for is chosen differently, using model m i n i m i z a t i o n , b u t the formula c r i t e r i u m remains unchanged. For reasoning about actions, however, b o t h criteria have to be reconsidered. In preferential entailment, as defined by [Sho88], one assumes the existence of a preference relation which is a partial order on interpretations, and defines the model set for in the simplest case as

i.e. as the set of m i n i m a l members of Mod(Y). However since there may be infinite chains of successively more preferred models, whose l i m i t does not exist or is not a model for an alternative is to define the model set of as a set of paths of interpretations,

where P a t h s i s the set o f all m a x i m a l subsets o f 5 w i t h i n which is a t o t a l order. T h e formula criterium is then t h a t in each preference p a t h of the model set, there must be some member such t h a t is true in all elements in t h a t p a t h . Our previous paper proposes a definition for the preference relation for the logic t h a t was outlined in the previous section. T h e c r i t e r i u m , chronological minimization of discontinuities ( C M D ) is basically that if and are interpretations, t h e n i f f there i s some time-point such t h a t and assign the same value to all parameter for all times and the set of parameters in which have a discontinuity in t is a true subset of the set of parameters in t h a t have a discontinuity in t. We showed t h a t preferential entailment using C M D , and w i t h a reasonable set of axioms, obtains the intended set of models for a simple b u t prototypical example. From the example and f r o m general considerations we concluded t h a t C M D is a plausible choice of

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semantic entailment c o n d i t i o n for piecewise continuous systems. In the following sections we address the question whether and how the model set c r i t e r i u m has to be revised when observations and actions also occur as axioms and in the interpretations. First however a brief remark on the f o r m u l a c r i t e r i u m , how to go f r o m model set to proposed conclusions. It has sometimes been debated whether one should let "goals entail plans" [Kau88] or vice versa, i.e. schematically whether one should require

or In a separate paper [San89b] we argue t h a t neither of these alternatives constrains correctly, and t h a t b o t h of t h e m should be used together at least approximatively. T h e requirement should therefore be t h a t the model set for be approximativelu equal to the model set for r, . Since the issue is somewhat complex, it merits a paper of its o w n . We therefore o m i t a d d i t i o n a l detail f r o m the present paper, and focus on the first question of i d e n t i f y i n g the model set for Of course the c r i t e r i u m for a proposed definition of semantic entailment is not t h a t it in itself obtains the correct model set, since w h a t models are obtained depends also on the axioms. For the conventional "frame probl e m " , for example, it is perfectly possible to o b t a i n the correct model set w i t h s t a n d a r d , monotonic entailment, b u t the p r o b l e m is t h a t it may be very cumbersome to w r i t e out the axioms. T h e c r i t e r i u m for a definition of entailment is therefore whether it makes it easy to w r i t e axiomatizations which o b t a i n the right model sets. This is the claim t h a t is tentatively made for chronological m i n i m i z a t i o n of discontinuities.

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M o d e l set c r i t e r i a f o r a x i o m s w i t h observations

T h e extension of the logic to allowing actions, is closely tied to the use of observations. W i t h o u t actions there are good reasons to s t u d y the consequences of the general axioms combined w i t h observations at an i n i t i a l point in t i m e , or at no t i m e at a l l . ( T h e l a t t e r case corresponds essentially to the n o t i o n of "envisionment" in qualitative reasoning, [ d K B 8 5 ] ) . W i t h actions, observations at two or more points in t i m e are needed, for example for speci f y i n g the i n i t i a l condition and the goal for the required action-plan. We therefore first consider the case where there are observations b u t no actions. T h e set by the n o t a t i o n at the beginning of the previous section, is assumed to be non-empty, while the action set is empty. Consider the following t w o candidate model sets:

where Filter is a subset of 5 consisting of those members of S w h i c h are also models for T h e definit i o n of M 2 can therefore be w r i t t e n equivalently as

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These definitions have been made in terms of m i n i m a l models; corresponding definitions in terms of preference paths are easily constructed. T h e definition for M\ is the most straightforward one f r o m the p o i n t of view of the inference system: one just combines all the available knowledge and applies preferential entailment as usual. T h e definition for M 2 however has an i n t u i t i v e appeal: since

is the set of all possible developments in the world regardless of amy observations, it would make sense to take t h a t whole set and " f i l t e r " it w i t h the given observations. It is easily seen t h a t M2 M i , since every member of M2 satisfies all conditions for being a member of Mi. T h e opposite is however not always the case, as the foll o w i n g example shows. Consider a one-dimensional object system where an object ( w i t h zero size) is known to have coordinate 0 at t i m e 0, and coordinate 2 at time 10. It is also k n o w n t h a t the object's coordinate is continuous at all times ( i n c l u d i n g breakpoints), and t h a t the velocity of the object is always 10. Finally it is known t h a t the object's acceleration is zero at all times. This is all one knows. In the absence of any reason for a discontinuity in the object's velocity, one would expect to have one single member of the model set, namely where the velocity of the object is 0.2 at all times. We shall refer to it as the standard model. Consider however also a model where the velocity of the object is 0.1 f r o m t i m e 0 to some timepoint shortly before t i m e 10, where the velocity changes discontinously to a larger value which allows the object to be at coordinate 2 in t i m e . Such an i n t e r p r e t a t i o n satisfies b o t h the observation a x i o m (coordinate 2 at time 1 0 ) n d the other axioms, and is therefore a member of Since the i n i t i a l velocity in such a non-standard model is different f r o m a standard m o d e l , the chronological m i n i m i z a t i o n of discontinuities w i l l not prefer standard over non-standard models. On the other hand for a given i n i t i a l velocity (e.g. 0.1), C M D w i l l prefer those models where the discontinuity occurs as late as possible, or in other words the model where the object uses the maxim u m speed after the discontinuity. T h u s for every choice of i n i t i a l velocity there is exactly one preferred model in Mx. In M2 on the other h a n d , only the standard model is obtained. This is because Min w i l l not contain any model w i t h a discontinuity: for every model w i t h a discontinuity there is a corresponding model w i t h o u t the discontinuity which is preferred according to a n d which also satisfies all the axioms in T. The set Min w i l l contain one member for every possible value of the object's velocity, b u t only one of them w i l l remain after filtering w i t h the observation a x i o m . Based on this discussion and example, we suggest that the definition of M2 is the one w h i c h should be used for i d e n t i f y i n g model sets in piecewise continuous worlds, l

I t may seem strange that the velocity may have a discontinuity while at the same time the acceleration is constantly zero. It is O K , however; see [San89a] for the explanation.

a n d a s t h e f i r s t step i n t h e d e f i n i t i o n o f semantic entailment there. T h e t e r m filter preferential entailment is proposed for semantic e n t a i l m e n t using the d e f i n i t i o n of M 2 as i t s m o d e l set c r i t e r i u m . P a r e n t h e t i c a l l y , i t i s i n t e r e s t i n g t o note t h a t i f the m a x i m a l speed c o n d i t i o n is d r o p p e d (or is changed f r o m a c o n d i t i o n to a c o n d i t i o n ) , and the m o d e l m i n i m i z a t i o n i s p e r f o r m e d i n t e r m s o f m i n i m a l models r a t h e r t h a n preference p a t h s , t h e n M 1 as well as M 2 contains o n l y t h e s t a n d a r d m o d e l . T h i s is because there is an i n f i n i t e progression of n o n - s t a n d a r d models where the break p o i n t occurs l a t e r a n d l a t e r , a n d the velocity after the break p o i n t i s greater a n d greater. I n t h e i n t e r p r e t a t i o n a t t h e l i m i t o f t h a t progression, t h e o b j e c t ' s c o o r d i n a t e has a d i s c o n t i n u i t y at t i m e 10, w h i c h means t h a t the l i m i t i n t e r p r e t a t i o n is n o t a m o d e l of the given axioms in T h e r e f o r e n o n - s t a n d a r d models are de-selected. However we can n o t see this as a reason for reconside r i n g the M\ d e f i n i t i o n - it w o u l d be t o o ad as a m e t h o d for de-selecting u n i n t e n d e d models.

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M o d e l set c r i t e r i a for i n t e r p r e t a t i o n s w i t h a c t i o n sets

We proceed now to the case where i n t e r p r e t a t i o n s cont a i n n o t o n l y the " s n a p s h o t s " i.e. t h e state of the w o r l d or the p a r a m e t e r values at each p o i n t in t i m e , b u t also a set of actions each specified by s t a r t i n g t i m e , ending t i m e , a n d a c t i o n t y p e , as described in section 3. T h i s means i n p a r t i c u l a r t h a t i s going t o c o n t a i n i n t e r p r e t a t i o n s c o n t a i n i n g n o , one, or several actions in t h e i r a c t i o n sets. A n a d d i t i o n a l preference c o n d i t i o n m u s t b e involved w h e n a c t i o n sets occur in i n t e r p r e t a t i o n s , n a m e l y a preference r e l a t i o n between i n t e r p r e t a t i o n w h i c h is due to preference between t h e i r respective a c t i o n sets. Suppose for given a n d given we have a m o d e l set w i t h in particular two interpretations and where contains an a c t i o n set w h i c h is in fact necessary for e x p l a i n i n g or a c h i e v i n g the observations, a n d is essentially t h e same as except i t s a c t i o n set also contains a r e d u n d a n t act i o n w h i c h does n o t influence t h e value o b t a i n e d for the observations. In such a case one w o u l d obviously prefer over b o t h as an e x p l a n a t i o n of the observations a n d as a p l a n for t h e agent's o w n actions. T h e r e m a y however also be other preferences w h i c h are n o t so o b v i o u s , a l l the w a y to t h e c r i t e r i a based on a cost f u n c t i o n on plans. T h e question is t h e n , how shall t h e m o d e l set be m o d i f i e d to account for our preference between a c t i o n sets, or plans? T h e choice i n C M D t o a p p l y observation enforcement " a f t e r " t h e ( c h r o n o l o g i c a l ) m i n i m i z a t i o n c o n d i t i o n , generalizes n a t u r a l l y t o p l a n m i n i m i z a t i o n . W e therefore propose t h e f o l l o w i n g m o d e l set c r i t e r i u m :

where o n l y compares i n t e r p r e t a t i o n s w i t h the same action set, a n d compares i n t e r p r e t a t i o n s according to t h e preference of t h e i r respective a c t i o n sets, so t h a t at least if t h e set of actions in the i n t e r p r e t a t i o n is t h e set o f actions i n then O t h e r action-set preferences m a y of course also be a d d e d .

T h e first preference r e l a t i o n was defined in t h e previous paper a n d described above. However it now has to be s l i g h t l y revised, since one should n o t do chronological m i n i m i z a t i o n of those discontinuities w h i c h are caused by actions. O n l y d i s c o n t i n u i t i e s w h i c h occur "spontaneously" as consequences of t h e laws of n a t u r e , for example w h e n a b a l l falls over an edge, or heated water arrives to the b o i l i n g p o i n t , should be chronologically m i n i m i z e d . O t h e r w i s e the preference according to w i l l prefer a l l actions to take place as late as possible! F o r m a l l y t h i s m o d i f i c a t i o n can be achieved by i n t r o d u c i n g t w o masking relations and on timepoints

properties

w h e r e w a i v e s the C M D preference for interpretations where a n d s i m i l a r l y for (Notice if t h e n u is continuous at t i m e The relations X i a n d X T are themselves m i n i m i z e d by the relation A l s o modes ( p r o p o s i t i o n a l fluents) are dealt w i t h in t h e same way. T h e r e s u l t i n g f o r m a l semantics is as follows. Definition. An i n t e r p r e t a t i o n is a t u p l e

where H is a set of a c t i o n t y p e symbols, M is a set of mode symbols (for t r u t h valued parameters), U is a set of p a r a m e t e r symbols closed under p r e f i x i n g by , S R is a "sparse" set of t i m e - p o i n t s namely the set of breakpoints, and w i t h the same t y p e , are the masks on where modes a n d parameters are " a l l o w e d " to be discontinuous, S is the " p l a n " i.e. the set of actions, R is a m a p p i n g

w h i c h gives the ( t r u t h - ) v a l u e of a mode at each p o i n t in time, is a m a p p i n g

w h i c h s i m i l a r l y gives the (real-)value of a parameter at each p o i n t in t i m e , a n d W is a m a p p i n g f r o m t e m p o r a l constant symbols (such as ) to the d o m a i n R of real numbers u n d e r s t o o d as t i m e - p o i n t s . I n t e r p r e t a t i o n s are s u b j e c t to t h e continuity requirement t h a t for every 5, R and shall be continuous as f u n c t i o n s of t. For t n o t in S it is also required Logic f o r m u l a s w h i c h can be evaluated in such interp r e t a t i o n s are defined as desired. Definition. A p a r a m e t e r u is essentially continuous at time a n d we w r i t e ec iff

where the i n d e x a n d on Q represent the left a n d right l i m i t values, as used in section 2 above. Essential c o n t i n u i t y for modes is defined s i m i l a r l y . Definition. For every i n t e r p r e t a t i o n and timepoint R we define breaksefr as

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between 0 a n d 0.5 seconds before k bounces at coordinate a. T h e other s t r a t e g y is to catch k as it is t r a v e l l i n g left, a n d a r o u n d the t i m e t h a t the objects pass each other. T h e t i m i n g m a y b e e.g. t o i n i t i a t e the a c t i o n exactly w h e n t h e objects meet, or more generelly w h e n the x c o o r d i n a t e of A; is between 3.5 a n d 2.5. These t w o a l t e r n a t i v e strategies can be characterized by the following "strategy f o r m u l a "

It w o u l d be ideal if the m o d e l set for the axioms were e x a c t l y the set of i n t e r p r e t a t i o n s where either of these t w o strategies is p e r f o r m e d , w h i l e also the "laws of nat u r e " in axioms 1 t h r o u g h 11 are also satisfied. If this were the case t h e n t h e given a x i o m s w o u l d e n t a i l the strategy f o r m u l a (the d i s j u n c t i o n of the possible plans) using the filter p r e f e r e n t i a l m o d e l set c r i t e r i u m , a n d the conventional f o r m u l a c r i t e r i u m in the sense of section 4. U n f o r t u n a t e l y there w i l l also be other i n t e r p r e t a t i o n s w h i c h satisfy a l l t h e a x i o m s , b u t w h i c h rely o n " c o i n cidence" for o b t a i n i n g t h e i r results. For example if the openlid a c t i o n s t a r t s when is in c o o r d i n a t e 2.75 and t r a v e l l i n g i n the positive d i r e c t i o n , t h e n will certainly go i n t o the shaft, a n d m a y or m a y n o t go i n t o the shaft d e p e n d i n g o n how q u i c k l y t h e l i d opens. T h e r e w i l l b e some i n t e r p r e t a t i o n s where the l i d opens late enough for all the axioms to be satisfied. T h e question of how to deal w i t h such coincident models is discussed in [San89b], in t h e c o n t e x t of how to choose t h e f o r m u l a c r i t e r i u m . O u r concern here is only to make sure t h a t t h e filter preferential m o d e l set for axioms such as those given above, obtains exactly those i n t e r p r e t a t i o n s where the course of events in the w o r l d has been decoded correctly. In p a r t i c u l a r it must be req u i r e d t h a t a l l i n t e r p r e t a t i o n s w h i c h are i n accordance w i t h either o f the t w o p r i m a r y strategies, a n d w h i c h are i n accordance w i t h the general laws, r e m a i n i n t h e f i l t e r preferential m o d e l set. It is easily seen t h a t a l l i n t e r p r e t a t i o n s of t h a t k i n d , w h i c h have o n l y one openlid a c t i o n in their P compon e n t , satisfy a l l t h e a x i o m s . It is also clear t h a t t h e y are m i n i m a l w i t h respect t o since n o i n t e r p r e t a t i o n w i t h an e m p t y P c o m p o n e n t could satisfy all the axioms. Furt h e r m o r e if the i n t e r p r e t a t i o n s only have discontinuities where "necessary" i.e. where t h e objects bounce against the ends of the range, a n d w h e n starts f a l l i n g , they w i l l b e m i n i m a l w i t h respect t o I n t e r p r e t a t i o n s w i t h more t h a n one a c t i o n i n their P c o m p o n e n t s can n o t be members of the m o d e l set. Even i f t h e y satisfy a l l the a x i o m s , t h e y are s t i l l n o t m i n i m a l since one can remove the r e d u n d a n t actions and o b t a i n another i n t e r p r e t a t i o n w h i c h also satisfies all the axioms. I n t e r p r e t a t i o n s where the objects are allowed to bounce several t i m e s before the l i d is opened, w i l l also be members of the m o d e l set a n d r i g h t f u l l y so. O t h e r possible f o r m s of e n t a i l m e n t c o n d i t i o n s or m o d e l set c r i t e r i a e x h i b i t various k i n d s of i n t e r e s t i n g bugs. For e x a m p l e , one concern in t h e choice of c r i t e r i a is to n o t i n t r o d u c e unnecessary actions w h i c h m a y account for

n a t u r a l discontinuities. Suppose there is some type of discontinuity, for example an object falling over an edge, which may b o t h be the direct result of an action, and be the n a t u r a l effect of previous movements. One does not wish an interpretation where the discontinuity occurs as a n a t u r a l effect, to be dominated in the sense of the preference relations, by another interpretation cont a i n i n g an extra action which has the discontinuity as an effect. T h e filter preferential model set does not have t h a t bug. However if the definition of is changed by o m i t t i n g the condition P = P', then exactly this bug is obtained. Chronological m i n i m i z a t i o n of discontinuities is a natural preference c r i t e r i u m ; it captures the one-way, nonsymmetric character of time in the real world. However one must make an exception f r o m chronological minimization w i t h i n the time-span of actions. The masking UX" relations are of course the technical device for realizing those exceptions. This was illustrated also by the example above: w i t h o u t the exceptions, C M D would prefer interpretations where the lid opens and closes as late as possible. In our particular example the action involved discontinuities, at a r b i t r a r i l y chosen times, of a propositional fluent ( " m o d e " ) . Actions involving quantitative feedback, which proliferate in many real-world applications, are seen in our system as introducing discontinuities for quantitative parameters at a r b i t r a r y times.

References [dKB85] Johan de Kleer and John Seeley Brown. A qualitative physics based on confluences. In Daniel G. Bobrow, editor, Qualitative Reasoning about Physical Systems, pages 7-83, M I T press, 1985.

[Kau88]

Henry K a u t z . Formal theories of plan recogn i t i o n , position statement. In Josh Tenenberg Jay Weber and James A l l e n , editors, From Formal Systems to Practical Systems. Advance Proceedings from the Rochester Planning Workshop, pages 64-70, 1988.

[San89a] Erik Sandewall.

Combining logic and differential equations for describing real-world systems. In Proc. International Conference on Knowledge Representation, Toronto, Canada, 1989.

[San89b] Erik Sandewall. On the variety of minimizat i o n related conditions for reasoning about actions and plans. 1989. In preparation. [Sho88]

Yoav Shoham. Reasoning about Change. M I T Press. 1988.

Sandewall

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