PMA changes the truth valuations of the ground atoms in each model as little as ... the program will run forever if no system error occurs. However, this is ...
R e a s o n i n g a b o u t Persistence: A T h e o r y o f A c t i o n s Y a n Z h a n g a n d N o r m a n Y . Foo Knowledge Systems Group Basser Department of Computer Science University of Sydney N S W 2006, Australia
Abstract Winslett proposed a method for reasoning about action called the possible models approach ( P M A ) . The PMA successfully removed the major difficulty manifested by Ginsberg and Smith's possible worlds approach (PWA). In this paper, we show that Winslett's PMA fails to solve the frame and ramification problems for some actions, as does the PWA. From this observation, we classify actions as definite and indefinite, and find that, in general, the P M A is not appropriate for both definite and indefinite actions. We propose a new approach to formalize actions based on persistence. We compare our approach with the PMA in detail, and show that our new formalization can avoid the problems in the PMA and PWA in most cases, and give more intuitive results for reasoning about action, regardless of whether the action is definite or indefinite.
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Introduction
Winslett proposed a method for reasoning about action, which she called a possible models approach ( P M A ) [Winslett, 1988]. The P M A was devised to remove some difficulties in Ginsberg and Smith's possible worlds approach (PWA) for reasoning about action [Ginsberg and Smith, 1987a; Ginsberg and Smith, 1987b]. As argued by Winslett, although the PWA is an elegant, simple and powerful technique for reasoning about action, it fails to solve the frame problem (What facts about the world remain true when an action is performed?), ramification problem (What facts about the world must change when an action is performed?) and qualification problem (When is it reasonable to assume that an action will succeed?), if the PWA is forced to operate with incomplete information [Winslett, 1988]. The basic idea of the PWA is to take a logical theory as the description of the world. The effect of an action is modeled by incorporating a set of formulas F (that specify the effects of the action) into the world description T. The description of the possible worlds after the action are the maximal subsets S of that are consistent and [Ginsberg and Smith, 1987a].
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The P M A , on the other hand, distinguishes the state of the world from a theory that describes the world. The state of the world is identified with a Herbrand model. This distinction leads to different effects in representing the frame principle of actions between the PWA and PMA. In particular, the frame principle of actions says that there are minimal changes in the world when an action is performed. The PWA translates this into "minimal changes in the description of the world when an action is performed." However, as has been shown by Winslett [Winslett, 1988], since a minimal change in the state of the world does not necessarily correspond to a minimal change in the description of the world, the PWA may not give a correct handling of incomplete information. On the other hand, the P M A represents the minimal change on the state of the world rather than on the description of the world when an action is performed, and thus can avoid the problems with the PWA. Does the PMA always give satisfactory results for reasoning about action? In this paper, we show that the PMA is sometimes incorrect in that it may give unintuitive results relative to the frame problem, and for its dual, the ramification problem. In developing our alternative approach we found it helpful to classify actions into definite and indefinite actions to systematize a comparison between the P M A and our new proposal. The P M A is shown to be inadequate for both definite and indefinite actions in the general case. Our proposal for reasoning about actions is based on persistence. We show that our approach can avoid the problems with the P M A and PWA in most cases, and yield more intuitive results, regardless of whether the action is definite or indefinite. The paper is organized as follows. In the following section we briefly review the P M A , and illustrate the problem with the P M A by an intuitive example. We then start to develop our new approach. In section 3 we introduce the concepts of definite and indefinite actions, which are helpful for our discussion. We define the persistence set in section 4, and based on this definition we give a precise representation of the effect of action. In section 5 we present more examples to show how our approach works for reasoning about action. We then compare our approach with the P M A in detail from a logical viewpoint in section 6. Finally, we discuss the related work with our approach presented in this paper.
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T h e P r o b l e m w i t h the P M A
We first briefly review Winslett's PMA. Let T be a first order theory, which is treated as a description of the world, and C be a subset of formulas in T representing the domain constraints about the world called protected formulas. To reason about the effects of performing an action with post-condition F (a first order formula), the P M A considers the effect of the action on each possible state of the world, that is, on each model of T. The PMA changes the truth valuations of the ground atoms in each model as little as possible in order to make both F and the protected formulas of T true in that model. The possible states of the world after the performance of the action are all those models thus produced. Winslett made a Herbrand universe assumption so that models are simply subset of the Herbrand base.
->terminated(p, a) and -ab-terminated(p,a) are omitted. Now consider the action continue. After this action, what is the status of program p? Intuitively, after a while, the status of p has two possibilities — one is still running, the other is terminated (since there is no any explicit information to infer that system errors may have occurred after continue, p's status will not be abnormally terminated). So the post-condition of continue is simply running(p,a) V terminated(p,a). Now what can we say for the result of continue? From the previous definition, it is easy to show that Incorporate(Post(continue), M) = SoThat is, the only possible state of the world resulting from action continue is that nothing has changed, i.e., the program will run forever if no system error occurs. However, this is unreasonable from our intuition. It is not difficult to show that the PWA also gives the same result: the only possible world to add running(p} a) V terminated(p, a) in T is T itself. D The reason for this unintuitive result is that the PMA represents the frame principle of actions by the minimal change on models when an action is performed, and this is not always correct. This can be easily seen if we consider the persistence of facts in the state of the world when an action is performed. Informally, we say that a fact persists if given that no relevant action has occurred over a stretch of time, the fact does not change its truth value over that time [Shoham, 1988]. In our example, the effect of action continue is running(p, a) V terminated(p,a), and the initial state of the world is {system-ok(a), running(p, a)}. The only fact that persists is system-ok(a), and since running(p,a) V terminated(p, a) may or may not cause the truth value of running(p, a) to change, we have no any reason to say that running(p, a) must persist, where the PMA does just because the state {system-ok(a), running(p, a)} satisfies the formula running(p, a)\/terminated(p, a) before continue is performed. The assumptions underlying the PMA can therefore be sometimes too strong.
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T h i s p r o p o s i t i o n s i m p l y says t h a t each m u t a b l e fact o b t a i n e d by the PSA is based on some logical relevance, i.e., influence or indefiniteness. However, in the P M A , some fact m a y be m u t a b l e for no reason. T h i s is w h y the P M A m a y have some u n i n t e n d e d effects in reasoning a b o u t an a c t i o n .
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Related W o r k
In t h i s paper, we presented a f o r m a l i z a t i o n of actions based on persistence, w h i c h we call the Persistence Set Approach ( P S A ) , and compared i t w i t h the P M A i n d e t a i l . We showed t h a t the P S A provides a unified f r a m e work for representing effects of definite and indefinite actions, and argued t h a t it is conceptually simple and plausible for reasoning a b o u t a c t i o n . We notice t h a t the c o m p u t a t i o n of persistence sets seems d i f f i c u l t generally. In the f u l l version of t h i s paper ( [ Z h a n g a n d Foo, 1992J) we present an a l g o r i t h m i c description and a c o m p u t a t i o n a l analysis of the P S A necessary for i t s i m p l e m e n t a t i o n . We argue t h a t it is possible to get a c o m p u t a t i o n ally t r a c t a b l e m e t h o d for some special cases, i.e., o n l y definite actions occur or all constraints are H o r n clauses. M o r e details a b o u t the c o m p u t a t i o n of persistence sets w i l l be considered in our f u r t h e r w o r k . We t h i n k t h a t the PSA can be a p p l i e d as a general m e t h o d o l o g y for m o d e l i n g state change. T h e r e are a n u m b e r of issues related to the P S A t h a t we are considering. Here we list some of t h e m : • the p r o b l e m of m u l t i p l e extensions • persistence u p d a t i n g in knowledge bases • representing t e m p o r a l persistence T h e P S A , at least in general, induces m u l t i p l e extensions f r o m a single description of the w o r l d (i.e., E x a m p l e 1). In order to avoid c o n f l i c t i n g extensions (or u n i n t u i t i v e resulting states) it is necessary to provide m o r e d o m a i n i n f o r m a t i o n i n the P S A for some a p p l i cations. It seems feasible to i n t r o d u c e some perference p o l i c y i n t o the P S A t o handle the p r o b l e m o f m u l t i p l e extensions (i.e., the idea of prioritized circumscription [Lifschitz, 1985] or eptistemic entrenchment [Gardenfors and M a k i n s o n , 1988J). T h e other related w o r k is knowledge base updates. W h i l e there are m a n y approaches in t h i s area, most o f t h e m have some l i m i t a t i o n s [ K a t s u n o a n d M e n d e l zon, 1991]. One of the difficulties is to u p d a t e k n o w l edge base w i t h indefinite i n f o r m a t i o n (i.e., s o m e t h i n g like As the P S A provides a unified v i e w p o i n t for processing definite a n d i n d e f i n i t e i n f o r m a t i o n , we argue t h a t the idea of persistence m a y p r o v i d e new insights i n t o knowledge base updates. F u r t h e r m o r e , it seems t h a t the P S A also provides a unified representat i o n for upadtes and actions. In [ Z h a n g a n d Foo, 1993b], we show t h a t under the persistent semantics, given a specific u p d a t e o p e r a t o r some related a c t i o n (or a sequence of actions) can be generated. T h e P S A presents a p r i n c i p l e of persistence for reasoning a b o u t change, w h i c h is q u i t e different f r o m the p r i n c i p l e of minimality t h a t is w i d e l y used in current approaches. I t w o u l d n o t b e d i f f i c u l t t o c o m b i n e the
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persistence idea i n t o other f o r m a l i s m s . In [Zhang and Foo, 1993a] we presented a persistence-based f o r m a l i z a t i o n w i t h i n the s i t u a t i o n calculus f r a m e w o r k [ M c C a r t h y a n d Hayes, 1969] to represent t e m p o r a l persistence, and showed t h a t the Yale S h o o l t i n g P r o b l e m [Hanks and McD e r m o t t , 1987] is solved in a n a t u r a l way, t h a t is quite different f r o m those m i n i m a l i t y - b a s e d methods.
Acknowledgements T h i s research is s u p p o r t e d in p a r t by a grant f r o m the A u s t r a l i a n Research C o u n c i l . T h e first author is supp o r t e d by a S O P F scholarship f r o m the A u s t r a l i a n Gove r n m e n t . Discussions w i t h members of the Knowledge Systems G r o u p of the University of Sydney have i m proved the presentation.
References
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