Computing with time and events
2. MINIMAL MODELS REVISITED
¬A(x) → A(x)is a counterexample to this. The usual way out is to move to (Kleene’s) three-valued logic. However, the natural language examples that we are interested in do not require clauses of the offending type, and we therefore decided to omit a discussion of the three-valued semantics for logic programming. In the one case where it would be necessary, nested uses of the HoldsAt predicate arising from iterated nominalization, there exists a trick, due to Robert Stärk, allowing one to use classical models only. This will be explained in Chapter 6.
For the following discussion, fix a domain of the models to be consid- ered. In our case, the domain must contain the reals together with a finite number of objects, event types and fluents. The general theory of logic pro- gramming gives us that the set of models (on the given domain) of the com- pletion has a nice ordering in which there is aleastmodel. Strictly speaking we would have to formulate this ordering using three-valued logic, but with the simplification we have made, this ordering is given by the relation ‘sub- structure of’ or⊆, whereM⊆N if the interpretation of a relation symbol R on M is a subset of the interpretation of R on N (recall that M and
N have the same domain). In the ordering⊆there exists a minimal struc- ture which is a model of comp(P). In fact, in most cases of interest to us, comp(P)has a unique model. The structure of this model is of particular importance.
It can be shown for instance that the example scenario in Section 3.1 determines a unique modelM, in which the fluentscrossing,one-sideand other-side are represented by finite sets of halfopen intervals with rational endpoints. To describe the structure of the parametrized fluentdistance(x)
inM, one needs the more general notion of a semialgebraic set:
DEFINITION15. A subset ofRnissemialgebraicif it is a finite union of
sets of the form{x∈ Rn |f
1 =. . .=fk = 0, g1 > 0, . . . , gl >0}, where
thefi, gj are polynomials.
The reader may check that a finite union of intervals with rational end- points is indeed semialgebraic; each interval (p, q] can be brought in the form{x|x−p >0, q−x >0}∪{x|x=q}, which is semialgebraic.
InM, the fluentdistance(x)determines a semialgebraic subset ofR×
R. Observe also that the events (start and reach) mark the beginning and end of fluents. The structure of the modelMis thus very similar to that of the canonical model of the event calculus given in Section 2.1.
The question is how far this generalizes. Intuitively at least, fluents cor- responding to natural language expressions (e.g. verbs) are semialgebraic, and we would like this to fall out of the setup, without further stipulations. Again speaking intuitively, there seems to be some connection between the property of inertia as formalized in the event calculus, and the simplicity of the sets described by fluents. A fluent which holds at rational points and is false at irrational points is somehow incompatible with inertia, because it would seem to need a great many events to turn it on and off.
64 5. COMPUTING WITH TIME AND EVENTS
As has been remarked above, in natural language semantics it is a con- tested issue whether the fundamental temporal entities are points or inter- vals. The event calculus neatly sidesteps this issue, by taking the basic entities to be events and fluents, which are not explicit functions of time and which can be interpreted on structures with very different ontologies for time. Even if we take the structure underlying time to be R, that does not constitute an ontological commitment to points. Ontological commit- ment is generated rather by representation theorems, which correlate the events and fluents with point sets in a given structure. It may then very well turn out that, even when time is taken to be R, fluents and events can be represented as sets of intervals, so that points have no role of their own to play. The situation is slightly more complicated in the case of fluents ad- mitting real parameters, for example fluents representing possibly changing partial objects. One would expect change to be piecewise continuous, with at most a finite number of jumps, corresponding to events explicitly men- tioned in the scenario. The kind of change it is possible to program depends on the one hand on the constraint language chosen, on the other hand on the constraint logic program. Now in the structure {0,1,+,·, <} only semial- gebraic sets can be defined; but it may well be that more complicated sets are definable in the theory consisting of a normal programP together with the constraint theoryT. The next few theorems give some pertinent results. This material leans heavily on [117]. The reader for whom this is all new is advised to read only the statements of the theorems and skip the proofs. We will provide informal glosses of the main results whenever possible.
The first definition isolates the kind of programs and queries we are interested in, namely those which make computations finite.
DEFINITION 16. A query ?c, G is finitely evaluable with respect to a
programP, if its derivation tree is finite, i.e. if all branches in a derivation tree starting from ?c, G end either in success or in finite failure. A normal programP isfinitely evaluableif every query is finitely evaluable w.r.t.P.
A query ?c, G may contain both variables over the reals and over ob- jects, events and fluents. For definiteness, we call the former x and the latter y. We are now interested in the structure of the real part (t and x) when the remaining variables (the y) are held fixed. The next theorem and its corollary form the technical backbone of this book.
THEOREM 4. Let T be the constraint theory describing the structure
A. Let P be a normal program consisting of the axioms of the event cal- culus together with a scenario. Let ?Gbe a finitely evaluable query in the language of the event calculus. Let b be an assignment to the variablesy. Then there exists a semialgebraic set defined by a constraintc(x)such that
T+comp(P)|=∀(G(xb) ↔ c(x)).
Informally, this theorem says that inertia indeed constrains the sets de- finable by a logic program to be of a very simple kind.
2. MINIMAL MODELS REVISITED 65