Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium. Daniele Theseider Dupre

Transcription

1 Ramications in an Event-based Language Kristof Van Belleghem 1 Marc Denecker 2 Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium. Daniele Theseider Dupre Department of Computer Science, University of Torino, Corso Svizzera 185, Torino, Italy. fkristof, marcdg@cs.kuleuven.ac.be, dtd@di.unito.it Abstract In the last couple of years, several high-level languages have been proposed for modeling actions and change, following the example of Gelfond and Lifschitz's A language and tackling increasingly complex issues. In this paper we present a narrative-based language ER with a linear time structure, which is designed to deal correctly with simultaneous actions as well as with general uent and change dependencies, in particular ramications. We argue that a combination of state constraints, causal laws and action preconditions is necessary to correctly represent all such dependencies. In particular, weintroduce causal laws stating that a change is triggered by achange in truth value of a complex uent formula weshowsuchlaws to be useful for a compact representation of ramications, in particular those of simultaneous actions. A semantics for such complex causal laws is provided based on the principle of inductive denitions. Moreover ER is able to deal in a very exible way with complete and incomplete knowledge on action occurrences, action ordering and the initial state of the world. 1 Introduction Recently a number of high-level languages have been proposed for modeling actions and change, as a tool for studying the principles underlying time and causality. The original A language ([GL92]) models inertia and direct eects of actions in a branching time topology, with possible uncertainty on the initial state of the world. Extensions of A tackle gradually more complex issues. The E language ([KM97]) uses a narrativebased ontology modeled after the Event Calculus ([KS96]). It allows for modeling uncertainty on the initial state of the world and includes an initial idea on dealing with some ramications. The authors have also devised extensions of E to represent scenarios with incomplete knowledge on action ordering or action occurrences. The goal of this paper is to design a narrative-based language ER with a core somewhat similar to E and able to correctly represent simultaneous actions and a very general set of indirect eects, both of simple actions and simultaneous actions. Moreover the language is intended to deal in a exible way with complete and/or incomplete knowledge on action occurrences, the action ordering or the initial state. We argue that to represent all ramications and qualications of actions, it is necessary to include in the language state constraints as well as explicit action pre- 1 Supported by Vlaams Instituut voor de Bevordering van het Wetenschappelijk-Technologisch Onderzoek in de Industrie (IWT) 2 Supported by GOA 93/97-03

2 conditions and derived eect rules 1 : these three types of formulae are at least in part independent, i.e. there are derived eect rules that do not correspond to any state constraint and vice versa, and there are action preconditions not related to a state constraint. We motivate this and indicate dierences between our approach and those in the recent literature. We also argue that in many applications a clear and natural representation of indirect eects of actions in general and of the eects of simultaneous actions in particular, can be obtained by using complex causal laws, i.e. eect rules stating that a change is triggered by the change in truth value of a complex uent formula. ER includes such complex derived eect rules. A complete treatment of simple and complex derived eect rules requires relying on a strong semantics. We base the semantics of ER on the principle of inductive denitions (see section 3.1). This yields at the same time sucient expressiveness to deal with the frame and ramication problems in the presence of negative and possibly cyclic dependencies, while it has the advantage that the intuitions underlying the formal semantics (i.e. inductive denitions) are generally well-understood. We assume in the main part of this paper that actions are deterministic, have no duration, and can be simultaneous, and that all changes are discrete. In this setting we intend to deal correctly with all immediate ramications, i.e. all ramications occurring at the time of the action(s) they are ramications of. We further indicate how ER can be extended to deal with delayed ramications and with nondeterministic actions. We do not handle default qualications, as we consider dealing with defaults an entirely dierent problem than the inertia and ramication problems. In the next section, we present the syntax of ER and motivate the design of the language. Section 3 discusses and denes the semantics of ER. The fourth section sketches the more interesting contributions of the language and several extensions. Due to space considerations, most issues in this paper have not been discussed in as much depth as they deserve. For more details we refer the reader to [VDT98], where we add more examples, motivations and in-depth discussions, several results concerning the semantics, detailed comparisons with related work, a mapping to open logic programming, extensions for dealing with nondeterminism and delayed eects, and discussions of various other related issues. 2 The syntax of ER Basically, aner theory consists of a set of eect rules determining direct and indirect eects of actions (a theory of causation), combined with a general rst order theory describing the truth of uents at certain times, the occurrence and order of actions, and general state constraints and action preconditions. Formally : Denition 2.1 An ER-signature is a tuple < Sorts Functors Vars > with Sorts = ft A F Pg,representing the sorts time, action, uent and predicate. Functors consists of { a set Tof constants of sort T, denoted t t 1 ::, which includes all real numbers { a set Aof constants of sort A, denoted a a 1 :: 1 similar to \causal laws" or \causal rules" in the literature we will use these terms as synonyms

3 { a set Fof constants of sort F, denoted f f 1 :: { four typed predicate symbols 2 : T T! P Happens : AT! P Initially : F!P Holds : FT!P. Vars = Vars A [ Vars T, disjoint innite sets of variables of sort A resp. T, denoted asa A 1 :: resp. T T 1 ::. Terms (i.e. constants or variables) of sort T will be denoted by, action terms by. A uent literal l f is either a uent constant f or its negation :f. We dene bf as the set of uent literals. A uent formula F f is any expression that can be constructed using uent constants and the operators : ^ _ (! $ can also be used for convenience). For any F f, and, the atoms Holds(F f ), Happens( ), Initially(F f ) are general atoms. Given, the formulae of the ER language are: direct eect rules of the form: a causes l f if F f derived eect rules of the form: initiating F f causes l f if F f 0 any sentence (i.e. formula without free variables) constructed in the usual way ofholds Happens Initially atoms and the connectives and quanti- ers : ^ _ 8 9 (and! $). Some classes of sentences are of particular importance: state constraints of the form 8T:Holds(F f T) action preconditions of the form 8T:Happens(a T )! Holds(F f T) Other sentences may state complete or incomplete information about the truth of uents at certain times, the occurrence and order of actions, or the initial state. Denition 2.2 An ER-theory is a tuple < e p > such that is an ERsignature, e is a set of direct or derived eect rules based on, p is a set of sentences based on. In the remainder of this section, wemotivate the types of formulae proposed above. The most important issue in ER is addressing the ramication problem ([HM87]), i.e. dealing with indirect eects of actions. In the literature, ramications have generally been considered strongly related to state constraints. If an action's direct eect results in a violation of a state constraint, there will either be indirect eects restoring the validity of the constraint, or the action is impossible. It has been argued convincingly in for example [MT95] that state constraints are insucient to convey all of the information required to determine possible state changes: it is unclear in general if an action violating a state constraint will give rise to indirect eects or if it is simply impossible. Also, as argued in [Pin94], there can be multiple sets of indirect eects able to restore the state constraint, and the intended set can not be determined without additional information. For this reason, causal laws have been proposed ([MT95, KM97, 1]) to represent ramications. Causal laws are explicit change propagation rules describing that certain changes in uents cause certain other changes. Interestingly, in all existing approaches incorporating them, these causal laws are still tightly coupled with state constraints: in all aforementioned approaches they are used as a way of restoring integrity of some explicit or implicit state constraint. 2 In addition, we assume an equality predicate for A and T and the predicates true false :!P

4 Consider the example of an alarm system that detects if somehow people enter a building. We assume the building has many possible entrances (doors, windows, possibly unexpected ways of getting in). In other words, there are many actions able to bring someone in the building and these actions may notevenallbeknown. 3 We formalise the system using the uents in (stating that there is someone inside), active (the alarm system is active) and ring (the alarm bell is ringing). While the system is active, anyone entering the building triggers the alarm: if in becomes true when active is already true, ring becomes true. In ER this reads initiating in causes ring if active However, the corresponding state constraint 8T:Holds(in ^ active! ring T) is not valid, since activating the alarm system when someone is already in the building is not supposed to cause the bell to ring. Moreover we can assume that the proposed constraint may also be violated by shutting down the bell, without deactivating the alarm system. So there is no state constraint related to this indirect eect, it is simply the result of a change propagation. In ER, derived eect rules are independent change propagation rules. Their semantics is independent of the state constraints in the theory (as opposed to [1]), and does not include an implicit state constraint itself (as opposed to [MT95, KM97]). Thanks to this uncoupling, a ner-grained representation of temporal domains can be achieved, as illustrated above. In ER then the resulting state after an action occurrence is uniquely determined by the previous state and the direct and derived eect rules in the theory. The role of state constraints is then reduced to ltering out unwanted models, which boils down to functioning as an implicit action precondition: if a state constraint is violated by the combined direct and indirect eects of an action executed in a particular state, that action is impossible in that state. Obviously this depends on the presence or absence of causal rules related to the constraint which can restore its validity, which in turn can be dependent onhow the constraint was violated. 4 ER also includes explicit action preconditions. This is necessary because not all action preconditions are related to state constraints. For example, in a chess game amove is only possible if the moved piece is initially on the starting position of the move. This can not be represented by a state constraint. For concisely representing indirect eects of actions we found it very convenient to use derived eect rules incorporating complex uent formulae. Moreover this approach proves to be successful, without any modication, in modeling the eects of simultaneous actions, as will be discussed in section 4. As an example we present the suitcase domain from [Lin95]. A suitcase is equipped with a spring mechanism which opens the suitcase when its two latches are open at the same time. In ER this is concisely formulated by the rule initiating l 1 ^ l 2 causes open if true 3 This implies that the use of an indirect eect rule relating the alarm bell to someone's presence in the building can not be circumvented by adding new direct eect rules for each action able to bring someone inside (which would clearly be an undesirable approach in a decent knowledge representation system in any case). 4 [1] describes an automatic way of deriving the intended causal rules related to a state constraint, using additional inuence information. For causal laws actually corresponding to state constraints, we can use a variant ofthismethodiner,butwelack the space to discuss it here.

5 where l 1, l 2 represent that latch 1 resp. 2 are open and open that the suitcase is open. The intended reading of this rule is that when l 1 ^ l 2 is strongly initiated (i.e. changes from false to true), open is weakly initiated (i.e. becomes true if not already true). With action preconditions, (complex) eect rules, and state constraints, we are able to characterise temporal domains. Apart from that, we need to represent scenario information in such a domain, like actual action occurrences, an initial state, known uent values at certain time points. We choose to represent these as a standard rst order theory, as a part of p. This allows us to deal easily with complete and incomplete information, as we discuss in section 4. 3 The Semantics of ER In this section we discuss and dene the semantics of ER. Evidently the most important concern is solving the frame and ramication problems. This is usually done by means of an inertia axiom stating that uents persist unless they are changed, in combination with a form of minimisation of change. This minimisation can be achieved by using a circumscription policy or techniques like Clark completion ([Cla78]). It is unclear to us if a variant of circumscriptive minimisation can yield a general solution to the frame and ramication problems. The many increasingly complex variants proposed to date suggest that such a general solution is not evident, even though distinct variants yield solutions for particular classes of theories. Clark completion in turn is a simple and intuitive technique, but only applicable to a very restricted class of theories. However, more powerful extensions of Clark completion can be devised which are intuitively not much more complex: examples are more advanced logic programming semantics likestableorwell-founded semantics. We will apply such an extension of Clark completion to ER's eect rules. Intuitively, the semantics we propose for a set of eect rules is to read them as an inductive denition for a predicate initiates. Provided there are no cycles in the eect rules, this coincides with their completion. However, we argue that cyclic dependencies naturally occur in eect rules. As an example, consider two connected gear wheels. Any action which makes one gear turn, makes the other one turn as well, and any action which stops one gear, stops the other one. This can be represented by the following rules: initiating turning 1 causes turning 2 if true initiating :turning 1 causes :turning 2 if true initiating turning 2 causes turning 1 if true initiating :turning 2 causes :turning 1 if true which introduce a cyclic dependency. Evidently, in cases with mutually dependent eects, no eect should take place unless some exterior eect causes it (e.g. in the example a motor is started). In other words we comply with Shoham ([Sho90]) who insists that causation is anti-reexive, i.e. that causes for a fact should never include the fact itself. However, we claim that this should not be enforced by forbidding cycles in the causal rules on a syntactical level : the example shows that such cycles sometimes arise naturally. Rather, cyclic dependencies should be given their intuitive meaning, which is that if and only if one of the mutually dependent eects has an \external cause", both of them occur.

6 Negative dependencies (in the sense that the absence of a particular eect is a precondition for another eect to occur) do at rst sight not occur in eect rules, but a closer look at the complex eect rules reveals that this is a false impression. Take the suitcase domain presented earlier, which contains the rule initiating l 1 ^ l 2 causes open if true: The intended reading of this rule is that open is initiated if the conjunction l 1 ^ l 2 becomes true. This can happen in three ways, intuitively if l 1 is initiated and l 2 is initiated if l 1 is initiated, l 2 holds and :l 2 is not initiated if l 2 is initiated, l 1 holds and :l 1 is not initiated We will dene the semantics of complex eect rules by mapping them to an equivalent set of primitive rules like those above. These rules contain negative dependencies, yet giving them a natural semantics needs not be problematic: in the above example it is clear that when l 1 is initiated while l 2 holds, open is expected to be initiated rather than l 2 terminated 5. In general, the intended semantics is clear if the eects can be ordered in layers (stratied) such that each eect only depends negatively on more primitive eects, i.e. eects in lower layers. 6 Consider now the addition of a rule \initiating l 1 ^:open causes :l 2 if true" to the example. This rule implies that l 2 is terminated if l 1 is initiated while open is false and not initiated. Then the initiations of open and :l 2 depend negatively on each other. The intended eect of opening the rst latch is no longer clear: according to one rule the suitcase should open, according to the other rule latch 2 should close. In this case one can at best argue that the eect is nondeterministic, but in our view nondeterminism, if intended, should be modeled explicitly and not follow from tricky combinations of rules. Moreover, one can also write rules in which an eect depends negatively on itself, in which case there is no intuitive interpretation at all. Hence, we consider denitions containing cycles over negation as erroneous. We do not disallow them, as this would require non-trivial restrictions on the syntax. Rather, we handle them by assigning an \undened" truth value to eects that are only justied by cycles over negation. We interpret this truth value as an error indication. The above intuitions are formalised by the principle of inductive denition. 3.1 Principle of Inductive Denition The semantics and expressivity of inductive denitions are studied in a subarea of mathematical logic, the area of Iterated Inductive Denitions (IID) [BFPS81, Mos74]. Here we are only interested in the semantics. We need the following concepts. Given an atom set P, the set of (3-valued) valuations on P is dened as VP, the set of all functions P!ft u fg. On VP, a partial order F is dened as the pointwise extension of the order u F t u F f more precisely, 8I I 0 2VP : I F I 0 i 8l 2 P : I(l) F I(l 0 ). VP F is a chain complete poset with least element?, thevaluation which assigns u to each atom. Given an atom set P, we dene P = P [f:ljl 2 Pg[ft u fg. Valuations can be naturally extended to P. Adenition rule in P is an object l B where l 2 P and 5 Wesayauent f is terminated i :f is initiated. 6 In the example, open is in a higher layer than l1 and l2.

7 B P. l is called the head, B the body of the rule. A denition on P is any set D of rules in P which contains at least one rule for each atom p 2 P (possibly p ffg). Given P and a denition D on P, we need to characterise a valuation I D which denes the truth values of all atoms according to D. In IID this involves stratifying the denition, but it has been argued in [Den96] that techniques inspired by logic programming semantics formalise the same intuitions in a more general and syntax independent way. We present this technique. A proof tree T for an atom p 2 P is a tree of elements of P such that the root of T is p for each non-leaf node n of T with immediate descendants B, n B 2D T is maximal, i.e. atoms occur only in non-leaf nodes (since for each atom there is at least one rule in D). Leaf nodes then contain only t,u,f or negative literals. T is nite, i.e. contains no innite branches Given some valuation I 2VP of P, for each l 2 P, we dene its supported value w.r.t. I, denoted SV I (l), as the truth value proven by its \best" proof tree. Formally: SV I (l) =t if l has a proof tree with all leaves containing true facts w.r.t. I SV I (l) =f if each proof tree of l has a false fact w.r.t. I in a leaf SV I (l) =u otherwise i.e. if each proof tree of l contains a non-true leaf, and some proof tree contains only non-false leaves. For a denite (i.e. without negative literals in the body of rules) denition D, I D is the valuation mapping each p 2 P to SV? (p), i.e. each atom is mapped to its supported value (w.r.t.?). For non-denite denitions, I D is obtained as a xpoint of this operation: Denition 3.1 The positive induction operator PR D : VP!VP : I! I 0 is dened such that 8p 2 P : I 0 (p) = SV I (p). This operator can be proven to have a least xpoint. Hence, we can dene: given < P D >, I D = PR D ". Below, we apply this semantics to eect rules. 3.2 Formal semantics of ER Denition 3.2 Given an ER-signature as dened before, a temporal interpretation of is a structure I =< P Fun H > with: P =ft 1 t 2 jt 1 t 2 2 Tg[ finitially(l f )jl f 2 b Fg[ fhappens(a t)ja 2 A t2 Tg[ fholds(l f t)jl f 2 b F t2 Tg[ finitiates(t l f )jt 2 T l f 2 b Fg Fun : T! IR, a mapping of time to the real numbers, mapping each real to itself H : P!ft fg, a truth assignment function. H denes relations interpreting Happens Holds Initially and Initiates we denote them Ha Ho Initially Initiates respectively. A temporal interpretation needs to satisfy the following conditions is the classical total order on IR. Well-founded event topology: the set E = ftj9a : Ha(a t)g has a least element, denoted e start and 8t t 0 :[t t 0 ] \ E is a nite set. 7 7 This condition plays the same role as the induction axiom in for example [LR94]

8 8t 8f : :Initiates(t f) _:Initiates(t :f) Consistency: 8f : Initially(f) $:Initially(:f) Denition of initial state: 8t e start : Ho(l f t) $Initially(l f ) Inertia: 8t 1 t 2 8l f : t 1 t 2 ^ (:9t 3 : t 1 t 3 t 2 ^Initiates(t 3 :l f ))!Ho(l f t 2 ) $Initiates(t 1 l f ) _Ho(l f t 1 ) Initiates denotes strong initiation: 8t 2 T 8l f 2 F b : Initiates(t l f )!Ho(:l f t) An ER-interpretation I is a model of a ER-theory < e p > if it is a model of both e and p.to dene whether I satises p,we extend the truth function H to all closed formulae F, in the classical way. For complex Holds and Initially atoms, the interpretation of Holds(F f ) is dened as the interpretation of the formula F 0 obtained from F f by substituting each uent atom f by Holds(f Fun()). Likewise for the interpretation of Initially(F f ). An interpretation I satises p if all formulae in p are true in I. Next we focus on the semantics of the eect theory e.we rst reduce direct and derived eect rules to an inductive denition of Initiates. Tothisendweintroduce the concept of a supporting set: Denition 3.3 Let F f be a uent formula and F 0 1 f =(l ^ f 1 :: ^ l f 1 m n 1 ) _ ::: _ (l ^ f 1 :: ^ m l f n m ) its three-valued disjunctive normal form. A supporting set S of F f is any set i fl f 1 :: l f i n i g, 81 i m. A formula is true i all literals of some supporting set are true. It follows that F f is initiated i F f is not already true, and for some supporting set S of F f, all literals of some subset S i S are initiated and all literals in S p = S n S i are true and not terminated. Assuming a temporal interpretation I, we dene the grounding of an eect rule as the set of primitive (ground) rules it corresponds to. Below, given a set of uent literals S, :S denotes the set f:fjf 2 S ^ f is positiveg [ffj:f 2 Sg, and Initiates(t S) denotes the set finitiates(t l f )jl f 2 Sg. Ho(S t) denotes the truth value Ho(^lf 2Sl f t). For readibility reasons we write rule bodies as sequences rather than sets. Then Denition 3.4 The grounding of a direct eect rule \a causes l f if F f " is the set: finitiates(t l f ) Ha(a t) Ho(F f t) jt 2 Tg. The grounding of a derived eect rule \initiating F f causes l f if F f 0 " is the set: finitiates(t l f ) Initiates(t S i ) :Initiates(t :S p ) Ho(S p t) :Ho(F f t) Ho(F f 0 t) j t 2 T and S i [ S p is a supporting set of F f g. The grounding D init of e is the union of the groundings of all eect rules of e. D init is an inductive denition on the atom domain P 0 = finitiates(t l f )jt 2 T l f 2 Fg, b for which I D init is dened as in section 3.1. Initiates(t l f )istrueinthis valuation if l f is weakly initiated by t. Hence: Denition 3.5 Given an ER-theory < e p > and a temporal interpretation I of : I j= p i 8F 2 p : H(F )=t. I j= e i 8t 2 T l f 2 b F : Initiates(t l f ) $ I D init(initiates(t l f )) ^:Ho(l f t). I j= ER i I j= p and I j= e. Note that when I D init contains any truth value u, the second condition is unsatisable.

9 4 Interesting Contributions of ER We can only briey discuss some of the more interesting contributions of ER. As mentioned before, ER deals naturally with simultaneous actions. A nice example is from [BG93]: a glass on a table spills its contents as soon as the table is in a non-horizontal position : initiating :(up l $ up r ) causes wet if true where up l, up r represent that the left resp. right side of the table are lifted from the oor and wet that the table is wet. Consider then two actions lift l and lift r: lift l causes up l if true lift r causes up r if true Assuming the table is initially on the oor, executing either of the actions will cause the water to spill, except if they are executed at the same time. It can be checked that the given causal law leads to the intended conclusion in all cases. This result extends to the wide range of similar examples where simultaneous actions are problematic. On the basis of these observations, we conjecture that a key element in concisely modeling simultaneous actions is the use of complex causal rules. Another issue in ER is the exible representation of both complete and incomplete knowledge on the occurrence and order of actions and on the initial situation. Since this information is represented as a rst order theory, it is in general incomplete (it is complete if and only if the rst order theory has only one model). However it is possible to explicitly state that knowledge about any part of the scenario is complete by using explicit Clark completion style axioms. 8 This oers a lot of exibility : it allows one to specify complete knowledge about very precise parts of the scenario (for example, about all occurrences of a certain action type, or about all action occurrences in a particular time interval), while leaving other parts incomplete. As an example consider the well-known stolen car problem, formalised as follows: park causes parked if true steal causes :parked if true 8T:Happens(steal T )! Holds(parked T ) Happens(park t 1 ) ^ t 1 <t 2 ^:Holds(parked t 2 ) This specication represents incomplete information on action occurrences, and entails 9T:t 1 <T <t 2 ^ Happens(steal T ). We can assert complete knowledge on actions, i.e. that park is the only action, by adding: 8A T:Happens(A T ) $ A = park ^ T = t 1 In that case the specication is inconsistent. At the time of submission of this paper, we have developed extensions of ER for dealing with delayed causation and with nondeterministic actions and ramications. Moreover we have determined a mapping from ER theories to Open Logic Programming under justication semantics ([Den96]), a semantics based on inductive denitions, and proven the correctness of this mapping. The mapping deals with basic ER theories as well as with the aforementioned extensions of ER. Finally, using nondeterministic ramications, wehave improved and adapted to ER the method presented in [1] for using inuence information to automatically generate derived eect rules. 8 Extensions of ER, in particular for dealing with delayed ramications, will incorporate stronger principles than explicit completion in particular an inductive denition semantics will be applied to the then arising theory of actions.

10 References [BG93] C. Baral and M. Gelfond. Representing concurrent actions in extended logic programming. In Proc. 13th IJCAI, pages 866{871, Chambery, [BFPS81] W. Buchholz, S. Feferman, W. Pohlers, and W. Sieg. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Springer- Verlag, Lecture Notes in Mathematics 897, [Cla78] K. Clark. Negation as failure. In H. Gallaire and J. Minker, editors, Logic and databases, pages 293{322. Plenum Press, [Den96] M. Denecker. Inductive Denitions, Logic Programming, Knowledge Representation. Technical report, K.U. Leuven, [GL92] M. Gelfond and V. Lifschitz. Describing Action and Change by Logic Programs. In Proc. of the 9th Int. Joint Conf. and Symp. on Logic Programming, [HM87] S. Hanks and D. McDermott. Nonmonotonic logic and temporal projection. Articial Intelligence, 33:379{412, [KM97] A. Kakas and R. Miller. A simple declarative language for describing narratives with actions. to appear in : Journal of Logic Programming, Special Issue on Reasoning about Actions, [KS96] R. A. Kowalski and M. Sergot. A logic-based calculus of events. New Generation Computing, 4(4):319{340, [Lin95] F. Lin. Embracing causality in specifying the indirect eects of actions. In C. Mellish, editor, Proceedings of the International Joint Conference on Articial Intelligence, pages 1985{1991, [LR94] F. Lin and R. Reiter. State constraints revisited. J. of Logic and computation, special issue on actions and processes, 4:655{678, [MT95] N. McCain and H. Turner. A causal theory of ramications and qualications. In C. Mellish, editor, Proceedings of the International Joint Conference on Articial Intelligence, pages 1978{1984, [Mos74] Y. N. Moschovakis. Elementary Induction on Abstract Structures. North- Holland Publishing Company, Amsterdam- New York, [Pin94] J. A. Pinto. Temporal reasoning in the situation calculus. Technical Report KRR-TR-94-1, Computer Science Dept., University of Toronto, [Sho90] Y. Shoham. Nonmonotonic reasoning and causation. Cognitive Science, 214:213{252, [1] M. Thielscher. Ramication and causality. Articial Intelligence, 89(1-2):317{ 364, [VDT98] K. Van Belleghem, M. Denecker and D. Theseider Dupre. Ramications and dependencies in an event-based language. Preliminary version of long paper, K.U.Leuven.