Situation Calculus Specifications for Event Calculus Logic Programs From: AAAI Technical Report SS-95-07. Compilation copyright © 1995, AAAI (www.aaai.org). All rights reserved. Rob Miller Departmentof Computing, Imperial College of Science, Technology& Medicine, 180, Queen’s Gate, London SW72BZ, ENGLAND email: [email protected] Abstract A version of the Situation Calculus is presented whichis Many-sortedfirst order predicate calculus together with able to deal with informationabout the actual occurrenceof parallel and prioritised circumscription is used to describe actions in time. Baker’s solution to the frame problem the Narrative Situation Calculus. Variable names begin using circumscriptionis adaptedto enable default reasoning with a lower case letter. All variables in formulas are about action occurrences, as well as about the effects of universally quantified with maximumscope unless actions. A translation of Situation Calculus style theories into Event Calculus style logic programsis defined, and otherwise indicated. To simplify descriptions of the results are given on its soundnessand completeness. implementations, logic programsare written in a subset of the same language, supplemented with the symbol "not" 1. Introduction (negation-as-failure). Meta-variablesare often italicised, that, for example, "F" might represent an arbitrary ground This paper compares two formalisms and two associated term of a particular sort. The parallel circumscription of default reasoning techniques for reasoning about action m predicates P1 ..... V the Situation Calculus using a variant of Baker’s Pn in a sentence T with V1 ..... k circumscriptive solution to the frame problem [Baker, allowedto vary is written as 1991], and the logic-programming based Event Calculus [Kowalski & Sergot, 1986], in which default reasoning is CIRC[T; Pl ..... Pn; V1..... Vk] realised through negation-as-failure. The version of the Situation Calculus used enables information about the If RI ..... Rmare also circumscribed, at a higher priority occurrences of actions along a time line to be represented. thanPI ..... Pn, this is written as A course of actions identified as actually occurring is referred to as a narrative, and this formalismis referred to CIRC[T; R1 ..... Rm; P1 ..... Pn,VI..... Vk] as the Narrative Situation Calculus. Information about a A CIRC[T; PI ..... Pn ; VI ..... Vk] narrative might be incomplete, so that default assumptions might be required. The circumscription policy incorporated Justification for this notation can be found, for example, in the Narrative Situation Calculus minimises action in [Lifschitz, 1995]. One other piece of notation for occurrences along the time-line. The original Event specifying uniqueness-of-names axioms will be useful. Calculus incorporates an analogous default assumption UNA[F1,..,Fm]represents the set of axioms necessary to that the only action occurrences are those provable from ensure inequality betweendifferent terms built up from the the theory. (possibly 0-ary) function symbols1 . ... Fm. It stands for the axioms The present paper shows that under certain circumstances the Narrative Situation Calculus may be regarded as a Fi(xl........ Xk)# Fj(Yl........ Yn) specification for Event Calculus style logic programs. The programs presented here are described as "Event Calculus for i<j where Fi has arity k and Fj has arity n, together style" because of their use of "Initiates" and "Terminates" with the following axiomfor each F of arity k>0. predicates to describe the effects of actions, becauseof the i form of their persistence axioms, and because of the use of Xk)=Fi(yI ........ A xk=Yk] a time-line rather than the notion of a sequenceor structure Fi(x1 ........ Yk)--"> [x 1 =Y1 A...... of situations. They differ from someother variants of the Event Calculus in that they do not assume complete 2. A Narrative Situation Calculus knowledgeof an initial state, and in that properties can In this section an overview is given of the Narrative hold (and persist) even if they have not been explicitly Situation Calculus employed here as a specification initiated by an action. The programsdescribed are "sound" language. This work is presented more fully in [Miller & for a wide class of domains in that they only allow Shanahan, 1994]. A class of many sorted first order derivation of "Holds" information which is semantically languages is defined, and the types of sentence which can entailed by their circumscriptive specifications. Where appear in particular domaindescriptions are then described. total information is available about the initial state of Finally, the circumscription policy is discussed. affairs, the programs are also "complete" in this same sense. Definition {Narrative domain language }. A Narrative Twomore domain independent axioms are included in the domainlanguage is a first order languagewith equality of Narrative Situation Calculus, concerning properties of four sorts; a sort ct of actions with sort variables narratives and time-points: {a,al,a2,... }, a sort 0g of generalisedfluents I with sort variables {g,gl,g2.... }, a sort o of situations with sort State(t)=S0 6-- --,3al,tl [Happens(al,tl) ^ tl<t] variables{s,sl,s2 .... }, and a sort R of time-points with State(t)--Result(a1,State(t 1)) (N2) sort variables{ t, t 1, t2 ....... }. Thesort Sghas three sub- [Happens(al,tl) ^ tl<t sorts; the sub-sort 0+ of positive fluents with sort --,3a2,t2[Happens(a2,t2)^ [al;~a2 v tlc:t2] variables {f+,f~,f~,...1 2 }, the sub-sort 0- of negativefluents ^tl<t2^t2<t]] withsort variables{ f.,f_l,f.2,... }, andthe sub-sortSf of Axiom(N1) relates all time points before the first action fluents with sort variables{f, fl,f2,... }, suchthat occurrence to the initial situation SO, and Axiom(N2) says that if action A1 happens at TI, T1 is before T and 0+ n 0- = O 0+ L; 0- = 0f Of c 0g no other action happens between TI and T, then the situation at is equal to Result(A1,State(Tl)). It has time-point constant symbolscorresponding to the real numbers,a finite numberof action and positive fluent Several types of axioms are either required or ~illowed in constants, a single situation constant SO, and no negative Narrative Situation Calculus theories 3. The following or generalised fluent constants. It has five functions: definitions specify the form of such sentences. Result: tx x c~ ~ o, And: 0g x Sg --~ 0g, Neg: $+ ---) 0-, Sit: Sg ~ t~, and State: R --~ ~, and six predicates (other Definition {Initial conditions description}. A formula is than equality): Holds ranging over 0g x ~, Ab ranging an initial conditions description if over a x Sf x o, Absit ranging over Sg, < (infix) it is of the form ranging over R x R, < (infix) ranging over R x R, and Happens ranging over ct x R. [] Holds(F, S0) or Holds(Neg(F),S0) Only models are considered in which the predicates < and < whereF is a positive fluent constant. [] are interpreted in the usual way as the "less-than" and "less-than-or-equal-to" relationships betweenreal numbers. Definition {Action description}. A formula is an action "Happens(A,T)"represents that an action A occurs at time description if it is of the form T2, and "State(T)" represents the situation at time Holds(F, Result(A,s)) Several domain independent axioms will always appear in or Holds(Neg(F),Result(A,s)) Narrative Situation Calculus theories. The following five axioms are taken from [Baker, 1991]. or Holds(F,Result(A,s)) [Holds(Fl,s) ^ ... ^ Holds(Fn,s)] Holds(And(gl,g2),s) --- [Holds(gl,s) AHolds(g2,s)] or Holds(Neg(F),Result(A,s)) Holds(Neg(f+),s)-- ~Holds(f+,s) 032) [Holds(Fl,s) ^ ... ^ Holds(Fn,s)] Holds(g,Sit(g)) 4-- -,Absit(g) 033) whereA is an action constant, F, FI .... Fnare groundfluent Sit(g l)=Sit(g2) -o g 1 034) terms, and for each i and j, l.g.i,j<n, Fi~Neg(Fj). [Holds(f, Result(a,s)) -- Holds(f,s)] 6-- ---,Ab(a,f,s) Definition { Occurrencedescription }. A formula is an Axioms(BI)-(B4) are Baker’s "existence-of-situations" occurrencedescription if it is of the form axioms. Along with the minimisation of Absit, these ensure that all consistent combinations of fluents are Happens(A,T) accounted for in the overall theory. To solve the frame problem, Ab is also minimised (at a lower priority) where A is an action constant and T is a real number.[] allowing Result to vary. For details the reader should consult [Baker, 1991] or [Miller, 1994]. Definition {Narrative domain description}. Given a narrative domainlanguage with positive fluent constants F1 ...... Fn and action constants A1 ...... Am, a formulaN is a narrative domaindescription if it is a conjunctionof I Generalised fluents supply names to conjunctions of action descriptions, initial conditions descriptions, primitive fluents. For example the generalised fluent occurrence descriptions, the frame axiom(F1), existence- "And(Loaded,Neg(Alive))"represents the joint property beingloaded and dead. 2This approach is modified to represent (possibly 3Unlike in [Miller & Shanhan, 1994], domain constraints overlapping) actions with a duration in [Miller & Shanhan, betweenfluents are not consideredhere, since no translation 1994]. of these into logic programswill be given. 1/46 of-situations axioms (B I)-(B4), axioms (NI) and (N2), Example 1 {The Yale Shooting Problem, Nysp]. uniqueness-of-names axiom Holds(Neg(Alive),Result(Shoot,s)) (Y1) UNA[FI...... Fn,And,Neg] ^ UNA[AI...... Am] Holds(Loaded,s) Holds(Loaded,S0) (Y2) and a domainclosure axiomfor fluents Holds(Alive,S0) (Y3) f=F1 v .... v f=Fn v f=Neg(Fl) v .... v f=Neg(Fn ) UNA[Alive,Loaded,And,Neg] CY4) UNA[Sneeze,Shoot] Although domain constraints have not been explicitly (Y5) included in narrative domain descriptions, care must be f=Alive v f=Loaded v CY6) taken, since domain constraints might be derived from f=Neg(Alive) v f=Neg(Loaded) pairs of action descriptions, together with Axiom(B2).