A Simple Solution to the Yale Shooting Problem. Andrew B. Baker. Department of Computer Science. Stanford University

Transcription

1 A imple olution to the Yale hooting Problem Andrew B. Baker Department of Computer cience tanford University tanford, CA Abstract Most of the solutions proposed to the Yale shooting problem have either introduced new nonmonotonic reasoning methods (generally involving temporal priorities) or completely reformulated the domain axioms to represent causality explicitly. This paper presents a new solution based on the idea that since the abnormality predicate takes a situational argument, it is important for the meanings of the situations to be held constant across the various models being compared. This is accomplished by a simple change in circumscription policy: when Ab is circumscribed, Result (rather than Holds) is allowed to vary. In addition, we need an axiom ensuring that every consistent situation is included in the domain of discourse. Ordinary circumscription will then produce the intuitively correct answer. Beyond its conceptual simplicity, the solution proposed here has additional advantages over the previous approaches. Unlike the approach that uses temporal priorities, it can support reasoning backward in time as well as forward. And unlike the causal approach, it can handle rami cations in a natural manner. 1 Introduction The formalization of reasoning about change has proven to be a surprisingly dicult problem. tandard logics are inadequate for this task because of diculties such as the frame problem [McCarthy and Hayes, 1969]; nonmonotonic reasoning seems to be necessary. Unfortunately, as demonstrated by the Yale shooting problem of Hanks and McDermott [1987], the straightforward use of standard nonmonotonic logics (such as circumscription) for reasoning about action leads to counterintuitive results. There have been a large number of solutions proposed to the shooting problem [Gelfond, 1988, Haugh, 1987, Kautz, 1986, Lifschitz, 1987a, Lifschitz, 1987b, Morris, 1988, Pearl, 1988, hoham, 1988, and others], but none of them are completely satisfactory. ome of these solutions cannot handle examples that require reasoning backward in time. And others require that the domain axioms be written in a rather restrictive format. This paper presents a new approach to the shooting problem that avoids these diculties. In the next section, we describe the shooting problem. ection 3 surveys some of the previous solutions and their limitations. ection 4 presents our solution, albeit in a slightly simplied form; this solution is re ned in ection 5. In ection 6, we consider some additional temporal reasoning scenarios in order to compare the various approaches to the Yale shooting problem. Concluding remarks are contained in ection 7. 2 The shooting problem The Yale shooting problem arises regardless of which temporal formalism is used; we will use the situation calculus [McCarthy and Hayes, 1969]. A situation is the state of the world at a particular time. Given an action a and a situation s, Result(a; s) denotes the new situation after action a is performed in situation s. A truthvalued uent (the only kind of uent that will concern us) is a property that may or may not hold in a given situation. If p is a uent and s is a situation, then Holds(p; s) means that the uent p is true in situation s. ith these conventions, one might use standard rstorder logic to formalize the eects of various actions, but there are some wellknown problems with this monotonic approach. The frame problem [Mc Carthy and Hayes, 1969], to which this paper will be limited, is that we would need to write down a great many axioms specifying those properties that are unchanged by each action. And yet, intuitively, all of these frame axioms seem redundant; we would like to specify just the positive eects of an action, and then somehow say that nothing else changes. Part of the motivation behind the development of nonmonotonic reasoning was to formalize this notion, and thus to solve the frame problem; we will use circumscription [McCarthy, 1980, McCarthy, 1986]. If A is a formula, P is a predicate, and is a tuple of predicates and

2 functions, then the circumscription of P in A with varied is written as Circum(A; P; ) [Lifschitz, 1985]. This formula selects those models of A in which the extension of the predicate P is minimal (in the set inclusion sense). Besides P, only those predicates and functions in are allowed to vary during this minimization process. Consider the standard default frame axiom: 1 :Ab(p; a; s) ) (Holds(p; Result(a; s)) Holds(p; s)): (1) This says that the value of a uent persists from one situation to the next unless something is abnormal. The original intention was to circumscribe Ab with Holds varied. (e will refer to this as the standard circumscription policy.) It was hoped that this minimization of abnormality would ensure that a uent would persist unless a specic axiom forced this uent to change. Unfortunately, this approach does not work. In a sequence of events, often one can eliminate an expected abnormality at one time by introducing a totally gratuitous abnormality at another time. In this case, there will be multiple minimal models only one of which will correspond to our intuitions. The standard example, of course, is the Yale shooting problem [Hanks and McDermott, 1987]. 2 In this problem, which we are simplifying slightly from the Hanks and McDermott version, there are two uents, Loaded and, and two actions, ait and hoot. The story is that if the gun is shot while it is loaded, a person (named Fred) dies. There are no axioms about ait, so the generalpurpose frame axiom should ensure that it does not change anything. In the original situation, the gun is loaded, and Fred is alive. If the actions ait and then hoot are performed in succession, what happens? Let YP be the conjunction of the default frame axiom (1) with the following domain axioms: hat does Holds(Loaded; s) ) :Holds(; Result(hoot; s)); (2) Holds(Loaded; 0); (3) Holds(; 0): (4) Circum(YP; Ab; Holds) have to say about the truth value of Holds(; Result(hoot; Result( ait; 0)))? 1 Lower case letters represent variables. Unbound variables are implicitly universally quantied. 2 A similar scenario, involving the qualication problem rather than the frame problem, was discovered independently by Lifschitz and reported by McCarthy [1986]. e might guess that the waiting has no eect, and thus the shooting kills Fred, but circumscription is not so cooperative. Another possibility according to circumscription is that the gun mysteriously becomes unloaded while waiting, and Fred survives. This second model contains an abnormality during the waiting that was not present in the rst model, but there is no longer an abnormality during the shooting. (ince the gun is unloaded, Fred does not change from to not.) o both models are minimal, and the formalization must be altered in some way to rule out the anomalous model. 3 3 Previous approaches There have been a large number of solutions proposed to the shooting problem. This section discusses the two most popular groups of solutions. 3.1 Chronological minimization One idea, proposed in various forms by Kautz [1986], Lifschitz [1987b], and hoham [1988], is chronological minimization (the term is due to hoham). This proposal claims that we should reason forward in time; that is, apply the default assumptions in temporal order. o in the shooting scenario, we should rst conclude that the waiting action is not abnormal. Then, since the gun would remain loaded, we would conclude that Fred dies. Each of the above authors successfully constructs a nonmonotonic logic that captures this notion of chronological minimality. Kautz, for example, uses a modied version of circumscription in which abnormalities at earlier times are minimized at a higher priority than those at later times. hile this approach does in fact give the intuitively correct answer to the Yale shooting problem, it is nevertheless highly problematic. Its applicability seems to be limited to what Hanks and McDermott call temporal projection problems, or in other words, problems in which given the initial conditions, we are asked to predict what will be the case at a later time. One can also consider temporal explanation problems [Hanks and McDermott, 1987], i.e., problems requiring reasoning backward in time. For problems of this sort, chronological minimization generally does not work very well. (ee the example in ection 6.1.) For this reason, chronological minimization is not a completely satisfactory solution. 3.2 Causal minimization Another approach, developed by Haugh [1987] and by Lifschitz [1987a], is that of causal minimization. This method represents causality explicitly by stating that a uent changes its value if and only if a successful 3 The current formalization admits a third possibility: Fred might die during the waiting phase. Our solution will rule out this model also.

3 action causes it to do so. The intuition here is that there is a crucial dierence between the abnormality of a gun becoming unloaded while waiting, and the abnormality of Fred dying when shot with a loaded gun: there is a cause for the second while the rst is totally arbitrary. e will discuss the system of [Lifschitz, 1987a]. There, the eects of actions are represented with a predicate Causes(a; p; v) that indicates that if the action a is successful, then the uent p takes on the value v. (The success or failure of an action is determined by a P recond predicate that we will not discuss.) ith this formalism, one species all the known causal rules (for the current example, Causes(hoot; ; F alse)), and then circumscribes Causes with Holds varied. ince Causes does not take a situational argument, there obviously cannot be a conict in minimizing it in dierent situations. Therefore, the shooting problem cannot arise. The main drawback of this proposal is that it does not allow us to write our domain axioms in unrestricted situation calculus. Instead, we must use the Causes predicate. This is a severe restriction on our expressive power because there is simply no way to use the Causes predicate to express ramications, domain constraints, or general contextdependent eects. (ee ection 6.2 for a discussion of this issue.) In light of this diculty, it would be useful to solve the Yale shooting problem in the original formalism. 4 The solution Our approach to the Yale shooting problem consists of two innovations. First of all, when we circumscribe Ab, instead of letting the Holds predicate vary, we will let the Result function vary. That is, we will not think of Result( ait; 0) as being a xed situation, with circumscription being used to determine which uents hold in this situation. Instead, we will assume that for each combination of uents, there is some possible situation in which these uents hold. Circumscription will then be used to determine which of these situations might be the result of waiting in 0. econd, in order for this idea to succeed, we must already have every consistent situation in the domain of discourse; an axiom will be added to accomplish this. To see why this approach makes sense, let us resist the temptation of appealing to causality or temporal priorities, and instead think about the problem in its own terms. hy is it that we prefer the model in which Fred dies to the one in which waiting unloads the gun? After all, if by making the world behave more abnormally in 0, we could make it behave less abnormally somewhere else, this would seem to be a fair trade. The problem is that if waiting unloads the gun, the resultant situation is a dierent situation than it would have been, so it is not the case that the world behaves more normally somewhere else. In our preferred model, the abnormality was that changed to not when hoot was performed in a situation in which the gun was loaded. In the anomalous model, the world does not behave more normally in this situation; this situation just never comes about! But the usual circumscription policy, with Holds varied and Result xed, completely misses this subtlety. It views the Result( ait; 0)'s in the two models as the same situation, even though dierent uents hold in them. From this perspective, the shooting problem arises from the failure to index abnormality correctly; by varying Result instead of Holds, we can correct this problem. First, some details need to be discussed. e will use a manysorted language with object variables of the following sorts: for situations (s; s 1 ; s 2 ;...), for actions (a; a 1 ; a 2 ;...), for primitive uents 4 (p; p 1 ; p 2 ;...), and for times (t; t 1 ; t 2 ;...). Times are integers, and the function T ime(s):t maps a situation to its time. 5 e have the primitiveuent constants, and Loaded, the situation constant, 0, and the action constants, ait and hoot. Finally, we have the predicate constant Holds(p; s) and the function constant Result(a; s) : s. Axioms (1){(4) should be interpreted relative to these declarations. Actions always increment the time: T ime(result(a; s)) = T ime(s) + 1: (5) e also need a uniqueness of names axiom: ait 6= hoot: (6) Now, for the important part. uppose that for every point in time, and for every possible combination of uents, there is some situation at this time in which these uents hold. e will discuss how to achieve this in general in the next section, but for the Yale shooting problem we add the following existence of situations axiom: 9s (T ime(s) = t ^ Holds(; s) ^ Holds(Loaded; s)) ^ 9s (T ime(s) = t ^ Holds(; s) ^ :Holds(Loaded; s)) ^ 9s (T ime(s) = t ^ :Holds(; s) ^ Holds(Loaded; s)) ^ 9s (T ime(s) = t ^ :Holds(; s) ^ :Holds(Loaded; s)): (7) 4 ection 5 will make use of generalized uents; these are built up from the primitive uents by using And and N ot. Primitive uents are simply those that do not contain logical connectives. 5 e are using this syntax to keep explicit the special role of time; we could just as easily make time an ordinary uent. Actually, the only reason time is used in this paper is to rule out \circular" models, in which a sequence of actions performed in one situation leads back to that same situation.

4 N ot(loaded) N ot(loaded) Loaded Loaded q q q 0 ~ ~ Figure 1: A minimal model of the Yale shooting problem Let A be the conjunction of Axioms (1){(7), and circumscribe Ab in A with Result varied: B Circum(A; Ab; Result): Figures 1 and 2 are pictorial representations of models of A. Time ows horizontally, and each circle represents the set of situations at a given time in which the uents to its left either hold or do not hold as indicated. Axiom (7) ensures that each circle contains at least one situation. The arrows show the result of performing the ait action, and the arrows show the result of performing the hoot action. Diagonal arrows represent actions that change at least one uent, and hence are associated with at least one abnormality. ince the arrows from situations in which Loaded and hold lead to situations in which does not hold, these arrows must be diagonal. In the model of Figure 1, these are the only abnormalities, and therefore this is a minimal model of A and hence a model of B. Figure 1 represents the expected model of the shooting problem. Figure 2, on the other hand, represents an unexpected model: one in which waiting in 0 unloads the gun. Note, however, that this added abnormality does not reduce the abnormality anywhere else. This model is strictly inferior to the previous one, and so it is not a model of B. Therefore, our technique solves the Yale shooting problem. Proposition 1 B is consistent. Proof. A model of B corresponding to Figure 1 can be dened in a straightforward manner. 2 Proposition 2 (Fred dies) B j= :Holds(; Result(hoot; Result( ait; 0))) Proof. Consider a model of A where waiting in 0 unloads the gun; i.e., a model that satises :Holds(Loaded; Result( ait; 0)) ^ Ab(Loaded; ait; 0): By (7), we can vary Result to keep the gun loaded, and thus eliminate this abnormality without introducing any new abnormalities. (Axiom (6) ensures that Result can be varied in this limited way without changing its eect for actions other than ait.) Therefore, in all minimal models of A the gun remains loaded, and Fred dies. 2 5 Existence of situations In the last section, we violated the spirit of the nonmonotonic enterprise by adding an existence of situations axiom (7) that explicitly enumerated all possible situations. For more complicated problems, this axiom would be a bit unwieldy. In this section, we show how to write this axiom in a more general way. e add a new sort to the language: generalized uents (which are a supersort of primitive uents), with the variables f; f 1 ; f 2 ;...; and the functions And(f; f):f and N ot(f):f to build up these uents starting with primitive uents. e will also alter the declaration Holds(p; s) to Holds(f; s) so the rst argument can be any generalized uent. e have: Holds(And(f 1 ; f 2 ); s) Holds(f 1 ; s) ^ Holds(f 2 ; s); (8)

5 N ot(loaded) N ot(loaded) Loaded Loaded q q q > 0 ~ ~ Figure 2: A nonminimal model of the Yale shooting problem Holds(N ot(f); s) :Holds(f; s); (9) And(f 1 ; f 2 ) 6= p; (10) N ot(f) 6= p: (11) Axioms (8) and (9) are straightforward. Axioms (10) and (11) indicate that compound uents do not belong to the subdomain of primitive uents. ince the frame axiom only applies to primitive uents, this rules out minimal models in which compound uents persist at the expense of primitive ones. For our existence of situations axiom, we would like to introduce a function it(t; f) : s that maps a time and a uent into some situation at that time in which the uent holds: T ime(it(t; f)) = t ^ Holds(f; it(t; f)): This would guarantee that for each time, there is some situation such that And(; Loaded) holds, and one such that And(; N ot(loaded)) holds and so on. But this would also mean that even inconsistent uents like And(; ) would be true in some situation; this contradicts (8) and (9). More generally, any domain constraint will render certain uent combinations inconsistent. o instead, the existence of situations axiom will be written as a default rule: :Absit(t; f) ) (T ime(it(t; f)) = t ^ Holds(f; it(t; f))) (12) with Absit circumscribed. In order for the circumscription of Absit to have its intended eect, some uniqueness of names axioms will be necessary. e will use an abbreviation from [Lifschitz, 1987a]: UNA[f 1 ;... ; f n ], where f 1 ;... ; f n are (possibly 0ary) functions, stands for the axioms: f i (x 1 ;... ; x k ) 6= f j (y 1 ;... ; y l ) for i < j where f i has arity k and f j has arity l, and f i (x 1 ;... ; x k ) = f i (y 1 ;... ; y k ) ) (x 1 = y 1 ^... ^ x k = y k ) for f i of arity k > 0. These axioms ensure that f 1... ; f n are injections with disjoint ranges. e state that uniqueness of names applies to actions, uents, and our special it function: UNA[ ait; hoot]; (13) UNA[; Loaded; And; N ot]; (14) UNA[it]: (15) (Axiom (13) is equivalent to Axiom (6).) Let C be the conjunction of Axioms (1){(5) and (8){ (15). In addition to circumscribing Ab as before, we now circumscribe Absit with Holds, T ime, Result, and Ab allowed to vary: D Circum(C; Absit; Holds; T ime; Result; Ab) ^ Circum(C; Ab; Result): Proposition 3 D is consistent. Proposition 4 (Fred dies again) D j= :Holds(; Result(hoot; Result( ait; 0)))

6 The above approach for ensuring the existence of situations only works correctly for propositional uents. If we had some uent Interesting(x), for instance, where x could range over integers, (12) would ensure the existence of a situation in which And(Interesting(0); Interesting(1)) held, but with only nite conjunctions, it would not ensure the existence of a situation in which all integers were interesting. To do this, we would have to reify quantied formulas. Alternatively, we could use something like the following secondorder existence of situations axiom: :Absit2(t; h) ) T ime(it2(t; h)) = t ^ (Holds(p; it2(t; h)) h(p)) where h is a predicate variable. 6 Examples In order to compare the dierent approaches to the Yale shooting problem, this section will discuss three additional temporal reasoning problems. ection 6.1 contains an example that requires reasoning backward in time. ection 6.2 discusses the issue of ramications. And nally, ection 6.3 deals with changes that happen unexpectedly. 6.1 The murder mystery Consider the following temporal explanation problem, which we will call the murder mystery. In this story, Fred is alive in the initial situation, and after the actions hoot and then ait are performed in succession (the opposite of the Yale shooting order), he is dead: Holds(Loaded; s) ) :Holds(; Result(hoot; s)); Holds(; 0); 2 = Result( ait; Result(hoot; 0)); :Holds(; 2): The detective expert system is faced with the task of determining when Fred died, and whether or not the gun was originally loaded. If we used the obvious monotonic frame axioms, we would be able to conclude that the gun was originally loaded, and that Fred died during the shooting. Unfortunately, the standard circumscription policy is unable to reach this same conclusion. It has no preference for when Fred died, and even if it were told that Fred died during the shooting, it still would not conclude that the gun was originally loaded. urely, assuming that the gun was loaded is the only way to explain the Ab(; hoot; 0) that would be entailed by Fred being shot to death, but circumscription is in the business of minimizing abnormalities not explaining them. Chronological minimization only makes the situation worse. It tries to delay abnormalities as long as possible, so it avoids any abnormality during the shooting phase, by postponing Fred's death to the waiting phase. It therefore concludes that the gun must have been unloaded! 6 Causal minimization yields the intuitive answer since it is only by assuming that the gun was originally loaded that it can explain the death without introducing an additional causal rule. Our method also gives the right answer. The minimal model is the same one pictured in Figure 1, with 2 being reached by starting in 0 and following the arrow and then the arrow. There is a ne point, however. In addition to Result, the situation constants 0 and 2 also must be allowed to vary during the circumscription. In general with our approach, all situation constants and functions must be allowed to vary. This did not matter for the Yale shooting problem since in that problem, the situation 0 was fully specied. 6.2 Ramications Often, it is impractical to list explicitly all the consequences of an action. Rather, some of these consequences will be ramications; that is, they will be implied by domain constraints [Ginsberg and mith, 1988]. One of the main advantages of our method over causal minimization is that ours can handle ramications, while causal minimization cannot. A simple example of this limitation of causal minimization can be obtained from Hanks and McDermott's original version of the Yale shooting problem in which the gun was unloaded in the initial situation, and a Load action was performed before the waiting. 7 Using the notation from [Lifschitz, 1987a], we have the following causal specications: Causes(Load; Loaded; T rue); Causes(hoot; ; F alse): uppose that we added the uent Dead and a domain constraint relating Dead and : Holds(Dead; s) :Holds(; s): (16) It would be nice if by minimizing Causes we could conclude that not only does shooting make Fred not alive; it also makes him dead: Causes(hoot; Dead; T rue): (17) 6 Actually, even the standard circumscription policy says that the gun must be originally unloaded. Regardless of whether Fred dies during the shooting or the waiting, making the gun unloaded will keep Fred alive in other \possible" action sequences starting with Result( ait; 0). This, however, is just an artifact of the situation calculus. 7 The example is from Matthew Ginsberg, personal communication.

7 N ot(tolen) 0 q q q ~ tolen 2 N ot(tolen) 0 q q q ~ tolen 2 Figure 3: Two minimal models of the stolen car problem Unfortunately, there is another causally minimal model in which Load kills Fred. In this model, there is no situation in which the gun is loaded while Fred is alive; therefore, (17) does not have to be added. There have been some attempts at resolving this dif culty while remaining within the causal minimization framework, but so far none seem adequate. [Lifschitz, 1987a], for example, requires that uents be divided into two groups, primitive and nonprimitive, with the causal laws and the frame axiom limited in application to the primitive uents, and with the values of the nonprimitive uents determined by their denitions in terms of the primitive uents. o, in the above example, Dead would be a nonprimitive uent. 8 The problem with this is that the uents that change as the result of domain constraints need not be denitional in nature like Dead. uppose two objects are connected in some manner. Moving either one of them will cause the other to move as well, so each object position will have to be primitive for some actions and nonprimitive for other actions. Furthermore, if the connection between the two objects is subsequently broken, then both positions will become primitive uents. In short, the primitiveness of a uent is dependent on both the situation and the action, and to correctly formalize 8 In this paper, we also use primitive and nonprimitive uents, but for a somewhat dierent purpose. As discussed in the next paragraph, we have no diculty in letting both and Dead be primitive uents. this notion it seems that all the ramications would have to be precomputed a potentially intractable task [Ginsberg and mith, 1988]. The approach advocated in this paper handles ramications correctly. Domain constraints determine which situations can exist in the model; in the above example, there are only four types of situations that can exist at any time point: Loaded can be true or false, can be true or false, and Dead must have the opposite value of. The causal laws, like (2) further constrain the resultant situation of an action. o if the gun is loaded, and Fred is alive, then (2) will demand that Fred must be not alive after being shot, and the domain constraint (16) will ensure that Fred is dead in this resulting situation. Finally, the minimization of abnormality will keep the gun loaded since we have not axiomatized the notion of running out of bullets. The strange behavior of the causal approach cannot occur here, because we require all possible situations to exist in the model. 6.3 The stolen car problem e consider one nal example, Kautz's stolen car problem [Kautz, 1986]. In the initial situation, your car is not stolen. After you wait two times, it is stolen: :Holds(tolen; 0); 2 = Result( ait; Result( ait; 0)); Holds(tolen; 2):

8 tolen N ot(tolen) N ot(tolen) tolen 2 > q q q > 0 Figure 4: A peculiar minimal model of the modied stolen car problem as the car stolen during the rst waiting phase or the second? This problem diers in character from those discussed previously: if we formalized this using the monotonic frame axioms, we would have a contradictory theory. Nevertheless, the problem seems simple enough; the car could have been stolen during either of the two waiting phases, and there is no reason to prefer one over the other. And indeed, the standard circumscription policy will yield models corresponding to both of these alternatives. But as pointed out in [Kautz, 1986], chronological minimization will claim that the car must have been stolen during the second waiting action. This is clearly unreasonable. The basic causalitybased formalism cannot deal correctly with this problem either. ince it only allows changes that are caused, it must assume that waiting causes the car to be stolen. o not only is the car stolen in the rst waiting phase, it will always be stolen whenever you wait. everal authors have augmented the causal formalism to better address problems of this sort. Lifschitz and Rabinov [1988], for example, allow \miracles," i.e., changes that cannot be explained. hen their new predicate M iracle is minimized at a lower priority than Causes, they are able to conclude that a miracle occurred during either the rst waiting action or during the second. Another approach, proposed by Morgenstern and tein [1988], explains unexpected changes by assuming that additional actions (selected from a list of known action types) have been performed, possibly in parallel with the actions that we know about. o in this case, if they had an appropriate causal rule for the action teal, Morgenstern and tein would be able to conclude that a teal action had been performed during one of the waiting phases. Finally, the solution proposed in this paper also runs into some diculties with the stolen car problem. It does correctly handle the simple version. Figure 3 shows that, as desired, both possibilities (the car being stolen during either the rst or the second waiting phase) are minimal models. Unfortunately, if we have another uent, say, such that Holds(; 0); besides the two expected minimal models, there will be an additional one (shown in Figure 4) that satises: Ab(; ait; 0) ^ Ab(tolen; ait; Result( ait; 0)): Not only is the car stolen during the second waiting action, poor Fred dies during the rst waiting action! 9 In this model, waiting steals the car in a situation in which Fred is dead; this abnormality is dierent from the car being stolen while Fred is still alive. ome words of explanation are in order. First of all, this third model may be warranted in some cases. If Fred were the security guard, then it is plausible that the thief killed him before stealing the car. Of course, for two arbitrary uents, it is a bit silly to posit an 9 This was pointed out by Matthew Ginsberg.

9 abnormality of one uent not to get rid of an abnormality of the second uent but merely to change this abnormality into another abnormality that is \almost the same." But since circumscription's denition of a minimal model is based on set inclusion, circumscription does not recognize the concept of \almost the same." econdly, the Ab facts may be thought of as highly specic causal rules. Thus, for example, Ab(; ait; 0) might be interpreted as saying that ait causes the value of to change if it is performed in a particular situation one that is at time 1 in which Fred is alive and the car is not stolen. o in order to explain unexpected changes, our approach, rather than assuming the occurrence of additional actions (as in [Morgenstern and tein, 1988]) or miracles (as in [Lifschitz and Rabinov, 1988]), instead extends the causal theory in a minimal (and admittedly peculiar) fashion. This is obviously not the right thing to do. There is no reason, however, why the improvements to causal minimization cannot also be integrated with the current approach. 7 Conclusion This paper has presented a new approach to nonmonotonic temporal reasoning. The approach correctly formalizes problems requiring reasoning both forward and backward in time, and it allows for some of an action's eects to be specied indirectly using domain constraints. It should be noted that, in a certain sense, our solution works for the same reason that causal minimization does. Causal minimization is not tempted to unload the gun because it is minimizing the extent of the Causes predicate rather than actual changes in the world; unloading the gun would prevent the Causes(hoot; ; F alse) fact from being used, but it would not eliminate the fact itself. imilarly, our solution minimizes even those abnormality facts associated with situations that do not really happen. But since we stick with the standard axioms (rather than introducing a special Causes predicate), our approach appears to be the more robust of the two: for us, even those abnormalities that arise as ramications will not be eliminated by the assumption of gratuitous changes. There are several possible directions for future work. One project would be to extend this approach to handle more expressive models of time, of action, and of causality. Another question is how to augment the formalism with default rules concerning the typical values of uents; we might, for instance, wish to assume that guns are loaded by default. But if we add defaults of this kind carelessly, the Yale shooting problem is likely to reappear. Lastly and most importantly, there is the issue of implementation. It would be interesting if there were a formulation of this paper's theory that, when given to an \o the shelf" circumscriptive theorem prover, would lead to chains of reasoning that were both intuitive and ecient. There has been some preliminary work on these topics, but much more remains to be done. Acknowledgements I would like to acknowledge helpful discussions with Matthew Ginsberg, Vladimir Lifschitz, Karen Myers, Arkady Rabinov, Yoav hoham, and David mith. I especially thank Matthew Ginsberg for his comments on several drafts of this paper and for his encouragement. The author is supported by an AFOR graduate fellowship. References [Gelfond, 1988] Michael Gelfond. Autoepistemic logic and formalization of commonsense reasoning. In Proceedings of the econd International orkshop on NonMonotonic Reasoning, Munich, [Ginsberg and mith, 1988] Matthew L. Ginsberg and David E. mith. Reasoning about action I: A possible worlds approach. Articial Intelligence, 35:165{195, [Hanks and McDermott, 1987] teve Hanks and Drew McDermott. Nonmonotonic logics and temporal projection. Articial Intelligence, 33:379{412, [Haugh, 1987] Brian A. Haugh. imple causal minimizations for temporal persistence and projection. In Proceedings of the ixth National Conference on Articial Intelligence, pages 218{223, [Kautz, 1986] Henry A. Kautz. The logic of persistence. In Proceedings of the Fifth National Conference on Articial Intelligence, pages 401{405, [Lifschitz and Rabinov, 1988] Vladimir Lifschitz and Arkady Rabinov. Miracles in formal theories of action. Draft, To appear in Articial Intelligence. [Lifschitz, 1985] Vladimir Lifschitz. Computing circumscription. In Proceedings of the Ninth International Joint Conference on Articial Intelligence, pages 121{127, [Lifschitz, 1987a] Vladimir Lifschitz. Formal theories of action. In Matthew L. Ginsberg, editor, Readings in Nonmonotonic Reasoning. Morgan Kaufmann, Los Altos, CA, [Lifschitz, 1987b] Vladimir Lifschitz. Pointwise circumscription. In Matthew L. Ginsberg, editor, Readings in Nonmonotonic Reasoning. Morgan Kaufmann, Los Altos, CA, 1987.

10 [McCarthy and Hayes, 1969] John McCarthy and Patrick J. Hayes. ome philosophical problems from the standpoint of articial intelligence. In B. Meltzer and D. Mitchie, editors, Machine Intelligence 4, pages 463{502. Edinburgh University Press, [McCarthy, 1980] John McCarthy. Circumscription a form of nonmonotonic reasoning. Articial Intelligence, 13:27{39, [McCarthy, 1986] John McCarthy. Applications of circumscription to formalizing commonsense knowledge. Articial Intelligence, 28:89{116, [Morgenstern and tein, 1988] Leora Morgenstern and Lynn Andrea tein. hy things go wrong: A formal theory of causal reasoning. In Proceedings of the eventh National Conference on Articial Intelligence, pages 518{523, [Morris, 1988] Paul H. Morris. The anomalous extension problem in default reasoning. Articial Intelligence, 35:383{399, [Pearl, 1988] Judea Pearl. On logic and probability. Computational Intelligence, 4:99{103, [hoham, 1988] Yoav hoham. Chronological ignorance: Experiments in nonmonotonic temporal reasoning. Articial Intelligence, 36:279{331, 1988.