Vol. 2(1997): nr 12. Reasoning about Actions, Antonis Kakas. University of Cyprus, 75 Kallipoleos Street, P.O. Box 537, CY-1678 Nicosia, CYPRUS

Transcription

1 Linkoping Electronic Articles in Computer and Information Science Vol. 2(1997): nr 12 Reasoning about Actions, Narratives and Ramications Antonis Kakas Department of Computer Science, University of Cyprus, 75 Kallipoleos Street, P.O. Box 537, CY-1678 Nicosia, CYPRUS Rob Miller School of Library Archive and Information Systems, University College London, University of London, Gower Street, London EC1E 6BT, U.K. Linkoping University Electronic Press Linkoping, Sweden http: /

2 Republished on February 16, 1998 by Linkoping University Electronic Press Linkoping, Sweden Linkoping Electronic Articles in Computer and Information Science ISSN Series editor: Erik Sandewall c1997 Antonis Kakas and Rob Miller Typeset by the authors using LATEX Formatted using etendu style Recommended citation: <Authors>. <Title>. Linkoping Electronic Articles in Computer and Information Science, Vol. 2(1997): nr 12. http: / October 16, This URL will also contain a link to the authors' home pages. The publishers will keep this article on-line on the Internet (or its possible replacement network in the future) for a period of 25 years from the date of publication, barring exceptional circumstances as described separately. The on-line availability of the article implies a permanent permission for anyone to read the article on-line, to print out single copies of it, and to use it unchanged for any non-commercial research and educational purpose, including making copies for classroom use. This permission can not be revoked by subsequent transfers of copyright. All other uses of the article are conditional on the consent of the copyright owners. The publication of the article on the date stated above included also the production of a limited number of copies on paper, which were archived in Swedish university libraries like all other written works published in Sweden. The publisher has taken technical and administrative measures to assure that the on-line version of the article will be permanently accessible using the URL stated above, unchanged, and permanently equal to the archived printed copies at least until the expiration of the publication period. For additional information about the Linkoping University Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: http: / or by conventional mail to the address stated above.

3 Abstract The Language E is a simple declarative language for describing the eects of action occurrences within a given narrative, using an ontology of actions, time points and uents (i.e. properties which can change their truth values over time). This paper shows how E may be extended to deal with ramications. More precisely, we show how Language E domain descriptions can include statements describing permanent relationships or constraints between uents, and how the model theoretic semantics of E can be extended in an intuitive way to ensure that the eects of actions are appropriately propagated via such statements, whilst retaining E's simple approach to the frame problem. We also show how Event Calculus style logic programs may be used to compute consequences of such domain descriptions using standard SLDNF, even when only incomplete information is given about some initial state of aairs. Because of E's generality, these techniques are easily adaptable to other formalisms for reasoning about actions, such as the Language A and the Situation Calculus. Posted and under public review in the News Journal of Electronic Transactions on Articial Intelligence (ETAI)

4 1 1 Introduction The idea of action description languages was rst introduced by Gelfond and Lifschitz [7, 8], with their proposal for the Language A. The intention was that such languages would have a specialised syntax and semantics, and would be at a high enough level of abstraction so as to be easily readable, understandable and intuitive. They could then be used as specications for theories of action and change written in more general purpose or computation oriented formalisms. Since then, many variants and extensions of A have been proposed. The majority of these, like A itself, are based on an ontology inherited from the Situation Calculus. The Language E [10] was introduced partly to illustrate that Gelfond and Lifschitz's methodology can be applied using ontologies other than that of the Situation Calculus. In particular, it showed that for narrative reasoning (that is, reasoning about actions which actually occur at various times, and reasoning about the properties that hold or do not hold at dierent times as a consequence of such occurrences), it is appropriate to use an ontology where the ow of time is independent of the action occurrences embedded within it. E's ontology of actions, time points and uents facilitates a particularly simple and intuitive syntax and semantics for describing the eects of such occurrences. Our approach has been inspired by Kowalski and Sergot's Event Calculus [13] and its extensions (see [27] for a comprehensive overview). A key characteristic of the E's semantics is that it is modular, and thus easily extendible. To illustrate what is meant by this more precisely, two extensions of the \core" semantics are briey described in an appendix of [10], to deal with qualications and ramications respectively. The purpose of the present paper is to expand on the theme of ramications, and in particular to show how the translation method described in [10], from basic E domain descriptions to Event Calculus style logic programs, may be extended to deal with domains containing ramication statements. For generality, the semantics of E described both in this paper and in [10] is in terms of an arbitrary structure of time. All denitions and results apply whether time is linear, branching, continuous or discrete. Indeed, as shown in [10], as a special case time may be dened as a forward branching structure of situations, each characterised by a unique sequence of actions, as in various versions of the situation calculus. In this case, as shown in [10], the semantics of E is equivalent to that of A (and thus, by the results of Kartha [11], equivalent to the core Situation Calculi of Reiter [23], Pednault [22] and Baker [1]). Hence both the representational and the computational mechanisms reported here to deal with ramications are, like those in [10], readily adaptable for use in the context of the Language A and the Situation Calculus. The paper is organised as follows. In Section 2, a review is given of the syntax and semantics of the basic Language E (without ramication statements). In Section 3, we describe E's extension to deal with ramications, illustrating its utility using two benchmark problems from the literature. In Section 4, a translation method into logic programs is given, followed by a proof of soundness with respect to the standard SLDNF (i.e. Prolog) proof procedure. As in [10], this proof holds even when complete information about some initial state of aairs is unavailable. Finally, in Section 5 we discuss the advantages and limitations of our approach, and compare it with other recent proposals for dealing with ramications within logic based approaches to reasoning about the eects of actions.

5 2 2 A Review of the Basic Language E 2.1 Syntax Like the Language A, the Language E is really a collection of languages. The particular vocabulary of each language depends on the domain being represented, but always includes a set of uent constants, a set of action constants, and a partially ordered set of time-points. A uent literal may either be a uent constant or its negation, as shown in the following denitions. Denition 1 [Domain Language] A domain language is a tuple h; ; ; i, where is a partial (possibly total) ordering dened over the non-empty set of time points, is a non-empty set of action constants, and is a non-empty set of uent constants. 2 Denition 2 [Fluent literal] A uent literal of E is an expression either of the form F or of the form :F, where F 2. 2 As in [10], for the remainder of this paper we assume a particular domain language E = h; ; ; i. We will often write T 1 T 2 to mean T 1 T 2 and T 1 6= T 2. Three types of statements are used to describe domains without ramications; h-propositions (\h" for \happens"), t-propositions (\t" for \time point") and c-propositions (\c" for \causes"). Their intended meanings are clear from their denitions: Denition 3 [h-proposition] An h-proposition in E is an expression of the form A happens-at T where A 2 and T 2. 2 Denition 4 [t-proposition] A t-proposition in E is an expression of the form L holds-at T where L is a uent literal of E and T 2. 2 Denition 5 [c-proposition] A c-proposition in E is an expression either of the form or of the form A initiates F when C A terminates F when C where F 2, A 2, and C is a set of uent literals of E. 2 \A initiates F when C" should be regarded as meaning \C is a minimally sucient set of conditions for an occurrence of A to have an initiating eect on F ". C-propositions of the form \A initiates F when ;" and \A terminates F when ;" can be written more simply as \A initiates F " and \A terminates F " respectively. For domains without indirect eects, a domain description in E is a triple h; ; i, where is a set of c-propositions, is a set of h-propositions and is a set of t-propositions. As a simple example domain, consider taking a photograph. Suppose that we need to refer to the actions of loading the camera with lm, looking

6 3 at the view and taking a picture, so that = fload; Look; T akeg, and refer to the properties of the camera having lm in it and a picture having been taken, so that = floaded; P ictureg. For and we will pick the real numbers with the usual ordering relation. Given this vocabulary, an example of a t-proposition is :P icture holds-at 1 (tsp1) Examples of h-propositions are Load happens-at 2 Look happens-at 5 T ake happens-at 8 (tsp2) (tsp3) (tsp4) and examples of c-propositions are Load initiates Loaded T ake initiates P icture when floadedg (tsp5) (tsp6) 2.2 Semantics The semantics of E is based on simple denitions of interpretations and models. Since we are primarily interested in inferences about what holds at particular time-points in, it is sucient to dene an interpretation as a mapping of uent/time-point pairs to true or false (i.e. a \holds" relation). An interpretation satises a uent literal or set of uent literals at a particular time-point if it assigns the relevant truth values to each of the corresponding uent constants: Denition 6 [Interpretation] An interpretation of E is a mapping H : 7! ftrue; falseg 2 Denition 7 [Point satisfaction] Given a set of uent literals C of E and a time point T 2, an interpretation H satises C at T i for each uent constant F 2 C, H(F; T ) = true, and for each uent constant F 0 such that :F 0 2 C, H(F 0 ; T ) = false. 2 The denition of a model in E is parametric on the denitions of an initiation point and a termination point. For the basic E (without ramication statements), these denitions are particularly straightforward. Initiation and termination points are simply time-points where a c-proposition and an h-proposition combine to describe a direct eect on a particular uent: Denition 8 [Initiation/termination point (without ramications)] Let H be an interpretation of E, let D = h; ; i be a domain description, let F 2 and let T 2. T is an initiation-point (respectively terminationpoint) for F in H relative to D i there is an A 2 such that (i) there is both an h-proposition in of the form \A happens-at T " and a c-proposition in of the form \A initiates F when C" (respectively \A terminates F when C") and (ii) H satises C at T. 2 Having dened these preliminary notions, it is straightforward to describe which interpretations are models of a particular domain description.

7 4 Three basic properties need to be satised; (1) uents change their truth values only via occurrences of initiating or terminating actions, (2) initiating a uent establishes its truth value as true, and (3) terminating a uent establishes its truth value as false. In addition, (4) every model must match with each of the t-propositions in the domain description: Denition 9 [Model (without ramications)] Given a domain description D = h; ; i in E, an interpretation H of E is a model of D i, for every F 2 and T; T 0 ; T 1 ; T 3 2 such that T 1 T 3, the following properties hold: 1. If there is no initiation-point or termination-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 1 ) = H(F; T 3 ). 2. If T 1 is an initiation-point for F in H relative to D, and there is no termination-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 3 ) = true. 3. If T 1 is a termination-point for F in H relative to D, and there is no initiation-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 3 ) = false. 4. For all t-propositions in of the form \F holds-at T ", H(F; T ) = true, and for all t-propositions of the form \:F holds-at T 0 ", H(F; T 0 ) = false. Denition 10 [Consistency] A domain description is consistent i it has a model. 2 Denition 11 [Entailment] A domain description D entails the t- proposition \F holds-at T ", written \D j= F holds-at T ", i for every model H of D, H(F; T ) = true. D entails the t-proposition \:F holds-at T " i for every model H of D, H(F; T ) = false. 2 It is easy to see from the above denitions that, for example, the collection of t-, h- and c-propositions (tsp1){(tsp6) entails the t-proposition P icture holds-at 10 Hence the Language E solves this (non-violent) version of the Yale Shooting Problem. The framework described above allows for various modes of reasoning within various structures of time. As another example, we can construct a set of time-points out of the set = fload; Look; T akeg of action constants used above. includes all sequences of these constants, e.g. [Load; Look; T ake] and [Load; Load]. We denote the empty sequence by S 0. It also includes auxiliary \start" points, Start([Load; Look; T ake]), Start([Load; Load]), etc, for each such sequence other than S 0. The partial ordering on this set is the obvious one. Given any sequence [ 1 ; : : : ; m ], where each i 2, it is such that S 0 Start([ 1 ; : : :; m?1 ]) [ 1 ; : : :; m?1 ] Start([ 1 ; : : :; m ]) Clearly, this gives a branching structure of time analogous to that used in many versions of the Situation Calculus. Now consider the domain description which includes the set of all h-propositions of the form 2

8 5 n happens-at Start([ 1 ; : : : ; n ]) and, in addition, the following c-propositions and t-propositions T ake initiates P icture when floadedg :P icture holds-at S 0 P icture holds-at [Look; T ake] Again, it is easy to see that this domain entails the t-proposition Loaded holds-at S 0 Thus the semantics of E facilitates reasoning both forwards and backwards within various (branching or linear) structures of time. A general result in [10] shows that for domain descriptions with time structures such as h ; i above and the set of h-propositions, E's semantics is equivalent to that of the Language A, so that A can be regarded as a special case of E. 1 Note that proponents of the Situation Calculus generally regard reasoning with this type of branching structure of time as \hypothetical" reasoning about (possible future) actions (as opposed to reasoning about the \actual" time-line). Hence, viewed in this light, the h-propositions in refer to hypothetical or possible future action occurrences. (This is in contrast to the very dierent use of the syntactically similar \occurrence facts" in [3], which refer to \actual" action occurrences.) More generally, [10] also discusses how E can be used to formulate an intuitive notion of planning, and how various notions of explanation can be dened to deal with inconsistency within E domain descriptions. It also describes how Event Calculus style logic programs can be used to compute consequences (t-propositions) of E domain descriptions of the basic type described above, even in the absence of complete information about which uents hold at some initial time-point. 3 Ramications in E The ramication problem arises in domains whose description most naturally includes permanent constraints or relationships between uents. In formalisms which allow for such statements, the eects of actions may sometimes be propagated via groups of these constraints. The problem is to adequately describe these propagations of eects, whilst retaining a solution to the frame problem { that is, the problem of succinctly expressing that most actions leave most uents unchanged. As briey described in the appendix of [10], E domain descriptions can include ramication statements, called r-propositions, to describe constraints between uents: Denition 12 [r-proposition] An r-proposition in E is an expression of the form L whenever C where L is a uent literal and C is a set of uent literals of E. 2 For the remainder of the paper, we will assume that domain descriptions are of the form h; ; ; i, where, and are as before, and is a set of r-propositions. The intended meaning of \L whenever C" is as follows: \at 1 A slightly dierent notation is used in [10] to refer to the time-points within.

9 6 every time-point that C holds, L holds, and hence every action occurrence that brings about C also brings about L". As a simple illustration of the use of r-propositions, consider the canonical \stuy room" example. Imagine a room with a window and a vent. We can open and close both the window and the vent, with the obvious eects: CloseWindow initiates WindowClosed CloseVent initiates VentClosed OpenWindow terminates WindowClosed OpenVent terminates VentClosed (sr1) (sr2) (sr3) (sr4) Now suppose that we also want to assert that the room becomes stuy whenever both the vent and the window are closed. We could of course attempt to describe this fact using extra c-propositions: CloseWindow initiates Stuffy when fventclosedg CloseVent initiates Stuffy when fwindowclosedg OpenWindow terminates Stuffy OpenVent terminates Stuffy However, there are problems with this approach. First, given the semantics of the previous section, these c-propositions fail to capture the \static" aspect of the intended constraint. In a domain description with no h- propositions or t-propositions at all, it would be possible to construct a model where, for example, WindowClosed and VentClosed were true at all time-points, but Stuffy was false. Second, this representation would fail to deal with simultaneous action occurrences. If both the vent and the window were initially open, closing both of them simultaneously would fail to make the room stuy. If the vent was initially closed and the window initially open, simultaneously opening the vent and closing the window would cause an inconsistency. The alternative is to use three r-propositions: Stuffy whenever fventclosed; WindowClosedg :Stuffy whenever f:ventclosedg :Stuffy whenever f:windowclosedg (sr5) (sr6) (sr7) and to extend the semantics of E appropriately. As we shall see, the extension described below deals with the two problems described above, as well as with other problems raised by dierent examples. It is also easy to see that this approach has the additional advantage of being more succinct { imagine a slightly larger example where the room has a small window, a large window, a vent and a door, or where a number of dierent actions have the same eect of closing the vent. In fact, we need make only very few modications to the semantics of E in order to deal with r-propositions. The rst modication is to extend condition (4) in Denition 9 of a model in the obvious way to cover the static aspect of r-propositions. Conditions (1), (2) and (3) stay the same (since they are parametric on the denitions of an initiation- and a terminationpoint, which are extended below), so that the new denition is: Denition 13 [Model] Given a domain description D = h; ; ; i in E, an interpretation H of E is a model of D i, for every F 2 and T; T 0 ; T 1 ; T 3 2 such that T 1 T 3, the following properties hold: 1. If there is no initiation-point or termination-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 1 ) = H(F; T 3 ).

10 7 2. If T 1 is an initiation-point for F in H relative to D, and there is no termination-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 3 ) = true. 3. If T 1 is a termination-point for F in H relative to D, and there is no initiation-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 3 ) = false. 4. For all t-propositions in of the form \F holds-at T ", H(F; T ) = true, for all t-propositions of the form \:F holds-at T 0 ", H(F; T 0 ) = false, and for all r-propositions in of the form \L whenever C", if H satises C at T then H satises flg at T. Using this new denition of a model, and given an arbitrary T 2, we can, for example, now infer the t-proposition from the two t-propositions :WindowClosed holds-at T :Stuffy holds-at T VentClosed holds-at T by considering the \contrapositive", static eect of the r-proposition (sr5). It remains only to extend the denitions of an initiation point and a termination point to accommodate the possibility of uents being indirectly initiated and terminated via r-propositions. Again the modication is straightforward. Intuitively, in order to nd time-points at which the uent literal L is established via the r-proposition \L whenever C", we need to look for time-points at which one or more of the conditions in C become established, and at which the remaining conditions are already and continue to be satised (up to some time-point beyond the point in question). This intuition translates directly into the following xed-point denition 2 : Denition 14 [Initiation/termination point] Let H be an interpretation of E = h; ; ; i, and D = h; ; ; i be a domain description. Let W be the set 2 2 and let the operator F : W 7! W be dened as follows. For each, hin; Tei 2 W denote F(hIn; Tei) by hin 0 ; Te 0 i. Then for any F 2 and T 2, (F; T ) is in In 0 (respectively in Te 0 ) i one of the following two conditions holds. 1. There is an A 2 such that (i) there is both an h-proposition in of the form \A happens-at T " and a c-proposition in of the form \A initiates F when C" (respectively \A terminates F when C") and (ii) H satises C at T. 2. There is an r-proposition in of the form \F whenever C" (respectively \:F whenever C") and a partition fc 1 ; C 2 g of C such that (i) C 1 is non-empty, for each uent constant F 0 2 C 1, (F 0 ; T ) 2 In, and for each uent literal :F 0 2 C 1, (F 0 ; T ) 2 Te, and (ii) there is some T 2 2, T T 2, such that for all T 1, T T 1 T 2, H satises C 2 at T 1. 2 Note that Michael Thielscher's online ETAI discussion question, at refers to a slightly earlier version of this paper with a more informal wording of this denition, in which the least xed point construction is not described explicitly. 2

11 8 Let hin f ; Te f i be the least xed point of the (monotonic) operator F starting from the empty tuple h;; ;i. T is an initiation-point (respectively terminationpoint) for F in H relative to D i (F; T ) 2 In f (respectively (F; T ) 2 Te f ). 2 It is useful to note that any initiation or termination point at some time T relative to D, dened in this way, must refer to at least one known h- proposition at T in the domain D. Therefore a domain which contains a nite number of h-propositions will necessarily contain only a nite number of initiation and termination points. We can see how the new denition of initiation and termination points works by extending the stuy room example with some t- and h-propositions. For the sake of variety, suppose that the set of time-points is the set of natural numbers with the usual ordering relation, and that in our domain description D sr = h sr ; sr ; sr ; sr i, sr consists of c-propositions (sr1){(sr4), sr consists of r-propositions (sr5){(sr7), sr contains the h-propositions CloseWindow happens-at 2 CloseVent happens-at 2 (sr8) (sr9) and sr contains the t-propositions :WindowClosed holds-at 0 :VentClosed holds-at 0 (sr10) (sr11) Intuitively, we can see that at any given time-point after 2, say 4, the room should be stuy. To show that the semantics matches with this intuition, we need to show that D sr j= Stuffy holds-at 4 and that D sr is consistent. It is sucient to observe that the following interpretation H sr is the only model of D sr. For all time-points (naturals) 2 and 2 fstuffy; WindowClosed; VentClosedg, H sr (; ) = false if 2 H sr (; ) = true if > 2 According to Denition 14, the only possible initiation point or termination point for any of the uents in H sr relative to D sr is 2, since this is the only point to which an h-proposition refers. The combination of (sr1) with (sr8) and the combination of (sr2) with (sr9) ensure that (in any model) 2 is an initiation point for WindowClosed and VentClosed, by condition (1) of Denition 14. Hence, since ffventclosed; WindowClosedg; ;g is a partition of fventclosed; WindowClosedg, (sr5) ensures that (in any model) 2 is an initiation point for Stuffy by condition (2) of Denition 14. It is now straightforward to check that H sr satises all four conditions of Denition 13, and easy to see that there can be no other model of this domain. (Variations of this stuy room example also behave correctly when represented in E. In particular, if we replace (sr9) with \CloseVent happens-at 3" our semantics does not give rise to the type of anomalous model which is problematic for some other approaches, e.g. [1], in which a change at 3 from :Stuffy to Stuffy is avoided by incorporating an intuitively unjustied change from WindowClosed to :WindowClosed.) Our second example is taken from [28], and is problematic for what the author terms as \categorization-based" approaches to ramication such as [14] and [25] (see [28] for a detailed discussion of why this is so). An

12 9 electric circuit is arranged as in Figure 1, so that, when activated, the relay instantaneously pulls switch Switch2 open (i.e. disconnects it). We will use the uent constants Switch1, Switch2 and Switch3 to represent that the respective switches are closed (i.e. making a connection), and the symbols Relay and Light respectively to indicate that these components are activated. :Switch1 Switch2 sp PP s s s s s Switch3 :Relay :Light Figure 1 The representation of and reasoning about this domain is unproblematical for E. The various permanent constraints on the state of the circuit can be represented with 7 r-propositions: Light whenever fswitch1; Switch2g :Light whenever f:switch1g :Light whenever f:switch2g Relay whenever fswitch1; Switch3g :Relay whenever f:switch1g :Relay whenever f:switch3g :Switch2 whenever frelayg (c1) (c2) (c3) (c4) (c5) (c6) (c7) The direct eects of opening (disconnecting) and closing (connecting) each of the three switches are captured in 6 c-propositions: CloseSwitch1 initiates Switch1 OpenSwitch1 terminates Switch1 CloseSwitch2 initiates Switch2 when f:relayg OpenSwitch2 terminates Switch2 CloseSwitch3 initiates Switch3 OpenSwitch3 terminates Switch3 (c8) (c9) (c10) (c11) (c12) (c13) Taking the set of time-points to be the non-negative reals, we can describe the initial conguration of the circuit, pictured in Figure 1, as follows: :Switch1 holds-at 0 Switch2 holds-at 0 Switch3 holds-at 0 :Light holds-at 0 :Relay holds-at 0 (c14) (c15) (c16) (c17) (c18)

13 10 (In fact, given condition (4) of Denition 13, only (c15) and (c16) are needed to adequately describe this initial conguration.) If we add an occurrence of the action CloseSwitch1, say at 0:2, to this domain description: CloseSwitch1 happens-at 0:2 (c19) it should follow from E's semantics that, at points after 0:2, the relay is activated and Switch2 is disconnected, so that the light remains o. So, for example, taking D c to consist of propositions (c1){(c19), it should be the case that D c j= :Light holds-at 0:4 and that D c is consistent. As in the previous example, it is sucient to show that there is a single model H c for D c, where H c is dened as follows. For all 2 fswitch1; Switch2; Switch3; Relay; Lightg and time-points (nonnegative reals) 2, H c (; ) = true H c (; ) = true H c (; ) = false if 0:2 and 2 fswitch2; Switch3g if > 0:2 and 2 fswitch1; Switch3; Relayg otherwise Clearly, the only possible initiation point or termination point for any of the uents in any interpretation relative to D c is 0:2. The combination of (c8) with (c19) ensures that (in any model) 0:2 is an initiation point for Switch1, by condition (1) of Denition 14. Hence, by condition (2) of Denition 14, 0:2 is an initiation point for Relay in H c, because ffswitch1g; fswitch3gg is a partition of the set fswitch1; Switch3g which appears on the righthand side of (c4), and any time-point T 2 > 0:2 is such that, for all T 1 in the closed interval [0:2; T 2 ], H c (Switch3; T 1 ) = true. Therefore, again by condition (2) of Denition 14, 0:2 is a termination point for Switch2 in H c, since ffrelayg; ;g is a partition for the set frelayg which appears on the righthand side of (c7). Notice that 0:2 is not a termination point for Switch3 in H c, because Switch3 does not appear in the left-hand side of any r-proposition in D c, and there is no h-proposition mentioning the action OpenSwitch3 in D c. Neither is 0:2 an initiation point for Light in H c { although ffswitch1g; fswitch2gg is a partition of the set fswitch1; Switch2g which appears on the righthand side of (c1), condition (2) of Denition 14 is not applicable because there is no T 2 > 0:2 such that H c (Switch2; T 2 ) = true. From this argument it is clear that in every model of D c, 0:2 must be an initiation point for Switch1 and Relay and a termination point for Switch2, and that Switch3 and Light cannot have any initiation or termination points. Hence H c is the only possible model of D c. Note that variations of D c involving simultaneous action occurrences also give the intuitively expected results. For example, consider the domain description D c +, which is the same as D c except that it also includes the h-proposition OpenSwitch3 happens-at 0:2 (c20) It is straightforward to check that the only model of D c + is H c +, where for all 2 fswitch1; Switch2; Switch3; Relay; Lightg and time-points (nonnegative reals) 2,

14 11 H c + (; ) = true H c + (; ) = true H c + (; ) = false Thus, for example, if 0:2 and 2 fswitch2; Switch3g if > 0:2 and 2 fswitch1; Switch2; Lightg otherwise D c + j= Light holds-at 0:4 4 Logic Programs for E Domains with Ramications In this section we present a generalisation of the translation given in [10] (from E domains to logic programs) which can also be applied to domains containing r-propositions. We then show that the derivation of HoldsAt literals (translations of t-propositions) from these programs via SLDNF is sound with respect to E's notion of semantic entailment. Before presenting the translation itself, we give some preliminary propositions and denitions which will be useful in proving soundness. The rst proposition follows from a straightforward observation that in Denition 13 of a model we could replace condition (1) with two more specic conditions relating to the truth values true and false: Proposition 1 Let H be a model of the domain description D and let T 1 T 3. Then for every F 2 1. If H(F; T 1 ) = true and there is no termination-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 3 ) = true, and 2. If H(F; T 1 ) = false and there is no initiation-point T 2 for F in H relative to D such that T 1 T 2 T 3, then H(F; T 3 ) = false. Proof: For the rst part of the proposition, there are two cases to consider. (i) There is an initiation-point T 0, T 1 T 0 T 3, for F in H relative to D. In this case this half of the proposition follows from condition (2) of Denition 13. (ii) There is no initiation-point T 0, T 1 T 0 T 3, for F in H relative to D. In this case this part of the proposition follows from condition (1) of Denition 13. The proof of the second part of the proposition is analogous. 2 Aside from the generalisation to deal with r-propositions, we take the opportunity here to dene a translation which, unlike that in [10], can deal with a limited class of domain descriptions containing an innite number of h-propositions. This allows us, for example, to deal with domains containing a situation-calculus-like structure of time, such as the second example of Section 2.2, which contains an innite number of h-propositions of the form n happens-at Start([ 1 ; : : : ; n ]). However, our domain descriptions must be occurrence-sparse, i.e. cannot contain nite intervals of time in which an innite number of actions occur. The formal denition of occurrence-sparsity is as in [10]: Denition 15 [Occurrence Sparsity] Let D = h; ; ; i be a domain description written in a language E = h; ; ; i. D and are occurrencesparse i for any two points T 1 ; T 2 2 there are only a nite number of h-propositions in of the form \A happens-at T " such that T 1 T T 2. 2

15 12 To avoid technical complications, we only consider domains with the following property; for all time-points at which some action occurs, there exists some future time-point. We will call such domains end-free. We could modify (and thereby complicate) the denitions to deal with occurrences at the \end(s) of time", but there seems little point, especially since no such points exist in most common representations of (both branching and linear) time, and since in any case we would not want to make predictions beyond such points. Denition 16 [End-free domain description] A domain description D is end-free i for every h-proposition \A happens-at T " in D, there is a T 0 2 such that T T 0. 2 Finally, translatable domain descriptions must contain only a nite number of t-, c- and r-propositions: Denition 17 [T-c-r-nite domain] The domain description h; ; ; i is t-c-r-nite i, and are all nite, and for each c-proposition in either of the form \A initiates F when C" or of the form \A terminates F when C", and for each r-proposition in of the form \L whenever C", C is also nite. 2 Our translation will also utilise an observation about Denition 14, the least xed point denition of an initiation and a termination point. When the interpretation to which it refers is a model, and in the case that the domain description is occurrence-sparse and end-free, this denition can be re-cast as the following proposition. Proposition 2 below mirrors the statement of Denition 14, except that H is now a model, and condition (2)(ii) is re-stated in terms of the absence of initiation and termination points. Proposition 2 Let H be a model of the occurrence-sparse, end-free domain description D = h; ; ; i, let F 2 and let T 2. Then T is an initiation-point (respectively termination-point) for F in H relative to D i one of the following two conditions holds. 1. There is an A 2 such that (i) there is both an h-proposition in of the form \A happens-at T " and a c-proposition in of the form \A initiates F when C" (respectively \A terminates F when C") and (ii) H satises C at T. 2. There is an r-proposition in of the form \F whenever C" (respectively \:F whenever C") and a partition fc 1 ; C 2 g of C such that (i) C 1 is non-empty, for each uent constant F 0 2 C 1, T is an initiation point for F 0, and for each uent literal :F 0 2 C 1, T is a termination point for F 0, and (ii) H satises C 2 at T, and for each uent constant F 0 2 C 2, T is not a termination point for F 0, and for each uent literal :F 0 2 C 2, T is not an initiation point for F 0. Proof: It is sucient to show the equivalence of condition 2(ii) of the proposition statement and condition 2(ii) of Denition 14, under the conditions of the proposition that H is a model and D is occurrence-sparse and end-free, and in the particular case where hin; Tei = hin f ; Te f i (the least xed point of the operator F). \Only if" part: Let T 2 be as dened in condition 2(ii) of Denition 14. Since D is occurrence-sparse there are a nite number (possibly zero) of h- propositions in D which refer to a time-point T, where T T T 2. Hence

16 13 there is a (minimal) time-point T 00 such that there are no h-propositions in D which refer to a time-point T 1 such that T T 1 T 00 T 2. Condition 2(ii) of the proposition, that we need to show, must hold since otherwise, by conditions (2) and (3) of Denition 13 of a model, H would not satisfy C 2 at T 00. \If" part: By Denition 16 of an end-free domain description, there exists a time-point greater than T. Let T 0 be such a time-point, so that T T 0. As above, since D is occurrence-sparse, there are a nite number (possibly zero) of h-propositions in D which refer to a time-point T, where T T T 0. Hence there is a (minimal) time-point T 00 such that there are no h-propositions in D which refer to a time-point T 1 such that T T 1 T 00 T 0. Hence, by condition (1) of Denition 13, condition 2(ii) of Denition 14 is satised with T 2 = T We are now in a position to describe the form of our logic program translations. In [10], this translation assumed a background program module able to correctly compute the order relation. Here we assume that, in addition, this background program is also able to correctly compute or generate the h-propositions of the domain in question. Clearly, where the domain contains only a nite number of h-propositions, this extra requirement can be satised simply by listing the (translations of) the h-propositions as individual facts. The background program module, or basis program, is dened as follows: Denition 18 [Basis program] Given the domain description D = h; ; ; i written in the language E = h; ; ; i, the program B(E; D) is a basis program for E and D i for all T; T 0 2, B(E; D) succeeds on the query T T 0 if and only if T T 0, and nitely fails otherwise. for all T; T 0 2, B(E; D) succeeds on the query T T 0 if and only if T T 0, and nitely fails otherwise. for all T 2 and A 2, B(E; D) succeeds on the query Happens- At(A; T ) if and only if the h-proposition \A happens-at T " is in D, and nitely fails otherwise. the set of answer substitutions for the variable pair ha; ti in the (unground) query HappensAt(a; t) is fha; T i j \A happens-at T " is in Dg. None of the following predicate symbols appear in B(E; D): HoldsAt, Given, ClippedBetween, Whenever, PossiblyInitiates, Initiates, PossiblyTerminates, Terminates, Starts, PossiblyStarts, SetStarts, PossiblySetStarts, ContinuesToHold, PossiblyContinuesToHold, DisjunctiveForm, Partition, Resolve, NothingHoldsIn, ConverseList. An example basis program, for the situation-calculus-like domain in Section 2.2, is given in Appendix A. As in [10], the programs described here are sound even when domain descriptions are `incomplete', i.e. even when, for some uent constant F and time T, neither \F holds-at T " nor \:F holds-at T " is entailed by D. The 2

17 14 implicit completion often associated with logic programs is avoided by representing t-propositions containing negative uent literals as positive program literals; \:F holds-at T " is represented by the literal HoldsAt(Neg(F ); T ). \F holds-at T " is represented by the literal HoldsAt(Pos(F ); T ). The negative literal not HoldsAt(F; T ) is simply interpreted as `the t-proposition \F holds-at T " is not provable'. (Similar techniques are also used in [20], [7], [6] and [2].) It is convenient to dene translation operators and which convert E uent literals to program terms such as Pos(F ) and Neg(F ): Denition 19 [lp-term and lp-complement] Given a uent literal L of E = h; ; ; i, the lp-term of L, written (L), is dened to be Pos(F ) if L = F for some F 2 Neg(F ) if L = :F for some F 2 and the lp-complement of L, written (L), is dened to be Neg(F ) if L = F for some F 2 Pos(F ) if L = :F for some F 2 2 The full logic program translation of a domain D is dened as an extension of some basis program B(E; D) for D: Denition 20 [LP [B(E; D)]] Given an occurrence sparse, end-free, t-c-r- nite domain description D = h; ; ; i written in the language E = h; ; ; i, and a basis program B(E; D), the logic program LP [B(E; D)] is de- ned as the program B(E; D) augmented with the following domain-specic clauses For each t-proposition \L holds-at T " in, the clause Given((L); T ): For each r-proposition \L whenever fl 1 ; : : : ; L n g" in, the clause Whenever((L); [(L 1 ); : : : ; (L n )]): For each c-proposition \A initiates F when fl 1 ; : : :; L n g" in, the clause Initiates(A; F; t) HoldsAt((L 1 ); t); : : :; HoldsAt((L n ); t): and the clause PossiblyInitiates(A; F; t) not HoldsAt((L 1 ); t); : : : ; not HoldsAt((L n ); t): For each c-proposition \A terminates F when fl 1 ; : : : ; L n g" in, the clause

18 15 Terminates(A; F; t) HoldsAt((L 1 ); t); : : :; HoldsAt((L n ); t): and the clause PossiblyTerminates(A; F; t) not HoldsAt((L 1 ); t); : : : ; not HoldsAt((L n ); t): and the following general clauses HoldsAt(l; t) Given(l; t): (LP1) HoldsAt(Pos(f); t 3 ) Given(Pos(f); t 1 ); t 1 t 3 ; not ClippedBetween(t 1 ; Pos(f); t 3 ): HoldsAt(Pos(f); t 1 ) Given(Pos(f); t 3 ); t 1 t 3 ; not ClippedBetween(t 1 ; Neg(f); t 3 ): HoldsAt(Neg(f); t 3 ) Given(Neg(f); t 1 ); t 1 t 3 ; not ClippedBetween(t 1 ; Neg(f); t 3 ): HoldsAt(Neg(f); t 1 ) Given(Neg(f); t 3 ); t 1 t 3 ; not ClippedBetween(t 1 ; Pos(f); t 3 ): HoldsAt(l; t 3 ) Starts(l; t 1 ); t 1 t 3 ; not ClippedBetween(t 1 ; l; t 3 ): HoldsAt(l; t) Whenever(l 1 ; c); Resolve(l 1 ; c; l; t): Resolve(l 1 ; c; l; t) DisjunctiveForm(l 1 ; c; d); Partition(d; [l]; c 1 ); NothingHoldsIn(c 1 ; t): NothingHoldsIn([ ]; t): NothingHoldsIn([Pos(f)jc]; t) HoldsAt(Neg(f); t); NothingHoldsIn(c; t): NothingHoldsIn([Neg(f)jc]; t) HoldsAt(Pos(f); t); NothingHoldsIn(c; t): ClippedBetween(t 1 ; Pos(f); t 3 ) t 2 t 3 ; t 1 t 2 ; PossiblyStarts(Neg(f); t 2 ): ClippedBetween(t 1 ; Neg(f); t 3 ) t 2 t 3 ; t 1 t 2 ; PossiblyStarts(Pos(f); t 2 ): (LP2a) (LP2b) (LP2c) (LP2d) (LP3) (LP4) (LP5) (LP6) (LP7) (LP8) (LP9) (LP10)

19 16 Starts(Pos(f); t) HappensAt(a; t); Initiates(a; f; t): Starts(Neg(f); t) HappensAt(a; t); Terminates(a; f; t): Starts(l; t) Whenever(l; c); Partition(c; [l 1 jc 1 ]; c 2 ); SetStarts([l 1 jc 1 ]; t); ContinuesToHold(c 2 ; t): SetStarts([ ]; t): SetStarts([l 1 jc 1 ]; t) Starts(l 1 ; t); SetStarts(c 1 ; t): ContinuesToHold([ ]; t): ContinuesToHold([Pos(f)jc]; t) HoldsAt(Pos(f); t); not PossiblyStarts(Neg(f); t); ContinuesToHold(c; t): ContinuesToHold([Neg(f)jc]; t) HoldsAt(Neg(f); t); not PossiblyStarts(Pos(f); t); ContinuesToHold(c; t): PossiblyStarts(Pos(f); t) HappensAt(a; t); PossiblyInitiates(a; f; t): PossiblyStarts(Neg(f); t) HappensAt(a; t); PossiblyTerminates(a; f; t): PossiblyStarts(l; t) Whenever(l; c); Partition(c; [l 1 jc 1 ]; c 2 ); PossiblySetStarts([l 1 jc 1 ]; t); PossiblyContinuesToHold(c 2 ; t): PossiblySetStarts([ ]; t): PossiblySetStarts([l 1 jc 1 ]; t) PossiblyStarts(l 1 ; t); PossiblySetStarts(c 1 ; t): PossiblyContinuesToHold([ ]; t): PossiblyContinuesToHold([Pos(f)jc]; t) not HoldsAt(Neg(f); t); not Starts(Neg(f); t); PossiblyContinuesToHold(c; t): PossiblyContinuesToHold([Neg(f)jc]; t) not HoldsAt(Pos(f); t); not Starts(Pos(f); t); PossiblyContinuesToHold(c; t): (LP11) (LP12) (LP13) (LP14) (LP15) (LP16) (LP17) (LP18) (LP19) (LP20) (LP21) (LP22) (LP23) (LP24) (LP25) (LP26) and the following auxiliary denitions (or equivalent):

20 17 Partition([ ]; [ ]; [ ]): Partition([hjr]; [hjr 1 ]; r 2 ) Partition(r; r 1 ; r 2 ): Partition([hjr]; r 1 ; [hjr 2 ]) Partition(r; r 1 ; r 2 ): DisjunctiveForm(l; c; [ljc 1 ]) ConverseList(c; c 1 ): ConverseList([ ]; [ ]): ConverseList([Pos(f)jc 1 ]; [Neg(f)jc 2 ]) ConverseList(c 1 ; c 2 ): ConverseList([Neg(f)jc 1 ]; [Pos(f)jc 2 ]) ConverseList(c 1 ; c 2 ): 2 (Note that Resolve dened in clause (LP5) above is just a simple and naive implementation of a propositional, resolution-based theorem prover for positive or negative literals. Resolve(l 1 ; c; l; t) means that it is possible to show that l holds by resolution starting from the clause corresponding to Whenever(l 1 ; c) applied at the instant t. The program rst transforms the \implication" of the r-proposition into disjunctive normal form, using the predicate DisjunctiveForm, then Partition picks out the literal l of interest, and nally shows through the predicate NothingHoldsIn that the rest of the disjunction is false, by showing that for each of its literals its negation holds.) For domain descriptions without r-propositions and with only a nite number of h-propositions, the translation above essentially reduces to the translation given in [10]. However, even in this special case there are some improvements. In particular, the replacement of [10]'s clauses (LP1a) and (LP1b) with clauses (LP2a){(LP2d) sometimes allows for extra derivations of t-propositions when domains are incompletely specied 3. Intuitively, it is easy to see that clauses (LP2a){(LP2d) are justied by Proposition 1. (LP2a) reects statement (1) of the proposition, (LP2b) reects the contrapositive of statement (2), (LP2c) reects statement (2) and (LP2d) reects the contrapositive of statement (1). The soundness property for the translation to logic programs given in Denition 20 above is stated as follows. Proposition 3 Let D be an occurrence sparse, end-free, t-c-r-nite domain description and let B(E; D) be a basis program for E and D. Then for any uent literal L of E and any T 2, if then LP [B(E; D)] `SLDNF HoldsAt((L); T ) D j= L holds-at T 3 Thanks to Pascal Walheim for suggesting this improvement. He pointed out that, for example, the program in [10] could not derive (the translation of) \F1 holds-at 3" from (the translation of) the domain description f\f1 holds-at 1"; \A happens-at 2"; \A initiates F1 when ff2g"g. 2

21 18 The proof of this proposition given below is similar to that of the soundness result in [10] (for the translation of the basic E language into logic programs). It uses induction on the `length' length() of SLDNF derivations starting from goals of the form HoldsAt((L); T ) or Starts((L); T ), where length() is dened in Denition 21 below in terms of successful calls to HappensAt and Whenever. It is dened so that each SLDNF sub-derivation of a HoldsAt or Starts sub-goal within has `length' less than the top-level derivation. Denition 21 [length()] Let be a successful SLDNF derivation of the goal HoldsAt((L); T ) or Starts((L); T ) in LP [B(E; D)]. length() is de- ned inductively as follows: X length() = S + size() where 2B S is the number of successful calls to HappensAt and Whenever at the top level of, i.e. not called within a (negation-as-failure) nitelyfailed subsidiary derivation of. B is the set of all nitely-failed subsidiary derivations from negative calls (to ClippedBetween or PossiblyStarts) appearing at the top level of. size() = 0 if there is no successful call to HappensAt or Whenever in any branch of. size() = 1 + max(flength( 0 ) j 0 2 A g) if there is a successful call to HappensAt or Whenever inside, where A is the set of all successful SLDNF derivations of a HoldsAt or Starts goal, called as a negated sub-goal in one of the branches of 4. Proof of Proposition 3 Let be a successful SLDNF derivation of a goal of the form HoldsAt((L); T ) or Starts((L); T ) in LP [B(E; D)]. We will use induction on length() to show that, for any given model H of D, if is a derivation of HoldsAt((L); T ) then when L = F for some F 2 then H(F; T ) = true, and when L = :F 0 for some F 0 2 then H(F 0 ; T ) = false. if is a derivation of Starts((L); T ) then when L = F for some F 2 then T is an initiation-point for F relative to H, and when L = :F 0 for some F 0 2 then T is a termination-point for F relative to H. In the sequel of this proof we will use the term \sound derivation" of a HoldsAt or Starts goal to mean that it satises the appropriate property above. It is clear that the program LP [B(E; D)] is entirely symmetrical as regards treatment of literals of the form Pos(f) and Neg(f). This reects 4 We assume the convention that max(;) = 0. 2

22 19 the symmetric nature of E's semantics. Therefore it is sucient to consider only the case where L = F for some F 2. BASE CASE (length() = 0): Case 1: is a derivation of Starts(Pos(F ); T ). Clearly, the form of (LP11) and (LP13) requires that a successful derivation of Starts has length at least 1. Hence in this case the base case is trivially satised. Case 2: is a derivation of HoldsAt(Pos(F ); T ). The goal can succeed only starting from clauses (LP1), (LP2a) or (LP2b), as success on (LP3) or (LP4) would require length() 1. We consider each of these possibilities in turn: (i) Success on (LP1): Trivially, H(F; T ) = true by condition (4) of Denition 13 of a model. (ii) Success on (LP2a): Clearly, the success of Given(Pos(F ); T 1 ) for some T 1 T means that \F holds-at T 1 " 2 D and so H(F; T 1 ) = true by condition (4) of Denition 13 of a model. Also, since the subsidiary derivation for ClippedBetween fails with size() = 0, the unground sub-calls to HappensAt(a; t 2 ) and Whenever(Neg(F ); c) (inside the sub-call to PossiblyStarts(Neg(F ); t 2 )) fail. So there are no h-propositions in D, and hence no initiation or termination-points for F relative to H. Hence H(F; T ) = true by condition (1) of Denition 13 of a model. (iii) Success on (LP2b): The proof is exactly analogous to possibility (ii). INDUCTIVE STEP (length() = n): Suppose all successful SLDNF derivations 0 of any HoldsAt or Starts goals which are less than length n are sound in the sense described at the beginning of this proof. To show that it then follows that is also sound, it is useful to prove some intermediate results as follows. Intermediate Result 3.1. Suppose for some set of literals C (represented as a list) and some time-point T, PossiblyContinuesToHold(C; T ) nitely fails, and that all of its subsidiary derivations of HoldsAt and Starts subgoals are sound. Then either H (the model of D xed in this proof) does not satisfy C at T, or for some uent constant F 2 C, T is a terminationpoint for F, or for some uent literal :F 2 C, T is an initiation-point for F. (In other words, the condition 2(ii) in Proposition 2 is not satised for this specic set C of uent literals.) Proof of Intermediate Result 3.1. Clearly, the form of clauses (LP24){ (LP26) ensures that the nitely failed call to PossiblyContinuesToHold must contain a successful call to HoldsAt or Starts, which are sound by assumption. (End of proof of Intermediate Result 3.1.) Intermediate Result 3.2. Suppose for some uent literal L and some time-point T, PossiblyStarts((L); T ) nitely fails and that all of its subsidiary derivations of HoldsAt and Starts sub-goals are sound. Then if (L) = Pos(F ) for some F then T is not an initiation-point for F, and if (L) = Neg(F ) for some F then T is not a termination-point for F.