be on one surface, in its analysis of the Blocks world domain cited in the paper. Tim can infer these invariants from its sub-space analysis.

Gerevini and Schubert (1996a, 1996b) have also examined the potential for inferring parameter domains that are similar to the operator parameter types inferred by tim. Their domains are inferred by an iterative process of accretion which is similar to the attribute space extension process of TIM . However, the accretion process they describe is synthetic, in that the parameter domains are synthesised directly from the operator descriptions and initial state. Tim is an analytic system that constructs its types from an analysis of the functional properties of the domain objects. This analytic approach provides a rich information source from which other structures, including the domain invariants, can be derived.

SOME OF the implicative constraints inferred by DISCOPLAN correspond to an implicit type assignment and would arise in the type structure built by TIM. A further implicative constraint generated by DISCOPLAN refers to the separation of functional roles of objects. In particular, the irreflexivity of on, as in:

## V Y (On(X, Y) (X = Y))

can be captured using this kind of constraint. Tim cannot currently infer these invariants. Because TIM uses an analysis based on the state view of objects in the domain it is able to generate a broader collection of invariants, including state membership and unique state invariants currently not produced by discoplan.

Although DISCOPLAN can deal with negative preconditions and TIM cannot yet manage them, the invariants they produce overall are currently less powerful than those inferred by T I M .

Apart from the work of Gerevini and Schubert, there is some older work on the inference of invariants which also relies on the generation of candidate invariants which are then confirmed by an inductive process against the domain operators. Two examples are the work of Kelleher and Cohn (1992) and Morris and Feldman (1989). The former work concentrates on identifying directed mutual persistence relations, which hold between pairs of facts in a domain when, once both are established, the second continues to hold while the first does. The use of these relations leads to the inference of a collection of constraints which fall into the uniqueness invanants inferred by TIM. In the work described in (Morris
& Feldman, 1989) the authors build invariants by using truth counts which are counts of the nmber of propositions from particular identified sets which must be true in any state of the domain. Sets for which this count is 1 can then be used to build invariants which are a s ubset of the state membership and uniqueness invariants. The authors describe methods for attempting to identify the sets of facts from which to work. This work, in common with that of Kelleher and Cohn and of Gerevini and Schubert, builds invariants by first hypothesising a possible seed for the invariants and then determining their validity by analysing the effects of the operators on these seeds. In contrast to this generale-and-lest strategy, TIM produces only correct invariants which it infers from a deep, structural analysis of the domain. The inference of invariants does not exhaust the possibilities of this analysis. For example, the type structure is inferred automatically during this analysis, which has been shown to have dramatic potential for the efficiency of planning. The relationship between enablers, and the state transitions they enable, determines an ordering on the satisfaction of goals, which also has significance for efficiency. Further, the state-based view of the behaviour of