

![0_image_0.png](0_image_0.png)

igure Temporal dependencies in the definition of the Instant-Stack action.

Note that the Simple-Stack action subsumes the Instant-Stack action, whose temporal dependencies are depicted in   igure

## Instant-Stack

$$\begin{array}{r l}{{\textsf{f}}}&{{}\textsf{m}\ \ .\ {\left(\begin{array}{l l l}{{\textsf{BLOCK}}}&{{\nabla\textsf{UnTable}}\ \boxed{\mathbf{\nabla}}}&{\ \ \ \ }\\ {{\textsf{BLOCK}}}&{{\nabla\textsf{UnBlock}}\ \boxed{\mathbf{\nabla}}}&{\ \ \ }\end{array}\right)}}\end{array}$$

ubsumption holds because the class of intervals   obtained by homogeneity of the state OnTable as defined in the Simple-Stack action   including   and all its subintervals is a subset of the class of intervals over which the block is known to be on the table, according to the definition of Instant-Stack this latter class includes all the subintervals of If the Instant-Stack action had been defined without the V operator, then it would not specialize any more the Simple-Stack action. In fact, according to such a weaker definition of Instant-Stack, specifying that the ob ect is on the table at   does not imply that the ob ect is on the table at subintervals of - in particular, it is not possible to deduce any more that the ob ect is on the table at and its subintervals, as specified in the definition of Simple-Stack action. oreover, the weak Instant-Stack action type would not specialize the wak Simple-Stack action type  i.e., Basic-Stack  too. Thus, homogeneity helps us to define states and actions in a more accurate way, such that important inferences are captured.

As seen above, the definition of homogeneity makes use of universal temporal uantification.

emember that subsumption in a propositionally complete   escription   ogic with both existential and universal temporal  uantification is undecidable and it is still an open problem if it becomes decidable in absence of negation ettini,
. The homogeneity operator is a restricted form of universal uantification. An even more restricted form interests us here, where the concept in ∇   does not contain any other temporal operator  called simple homogeneous concept . The expressiveness of the resulting logic is enough, for example, to correctly represent the homogeneous nature of states. In  Artale, ranconi, an algorithm to compute subsumption in TL-F augmented with ettini, the homogeneity operator is proposed. Even if a formal proof is still not available, good arguments are discussed to con ecture its completeness. This would also prove decidability of this logic and of the corresponding modal logics.

## 8.2 Persistence

This ection shows how our framework can be successfully extended in a general way to cope with incrtial properties. In the basic temporal language, a property holding, say, as a post-condition of an action at a certain interval, is not guaranteed to hold anymore at other included or subse uent intervals. This is the reason why we propose an extended