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## Igure

Temporal dependencies in the definition of the Boil-Spaghetti plan.

1. An interpretation I satisfies the terminological axiom i 
, for every A concept is subsumed by a concept if and every interval . An interpretation I is a model for a concept    if   t  / 0 for some If a concept has a model, then it is satisfiable, otherwise it is unsatisfiable.

Each TL-F concept expression is always satisfiable, with the proviso that the temporal constraints introduced by the existential uantifiers are consistent. This latter condition can be easily checked during the reduction of the concept into a normal form when the minimal temporal network  see   ection    , definition  .    is computed.

It is interesting to note that only the relations s, f, mi  are really necessary, because it is possible to express any temporal relationship between two distinct intervals using only these three relations and their transpositions si, fi, m    alpern hoham,
. The following e uivalences hold

$$\begin{array}{r l r l}{{\mathbf{a}}}&{{}\quad\cdot\quad{\color{red}{\quad\boxed{\mathbf{a}}}}}&{{}\quad\equiv}\\ {\mathbf{d}}&{{}\quad\cdot\quad{\color{red}{\quad\boxed{\mathbf{a}}}}}&{{}\quad\equiv}\\ {\mathbf{0}}&{{}\quad\cdot\quad{\color{red}{\quad\boxed{\mathbf{a}}}}}&{{}\quad\equiv}\end{array}$$
$$\begin{array}{c c c c}{{\mathrm{mi}}}&{{}}&{{\mathrm{mi}}}&{{}}&{{\mathrm{\Phi}}}\\ {{\mathrm{s}}}&{{}}&{{\mathrm{f}}}&{{}}&{{\mathrm{\Phi}}}\\ {{\mathrm{s}}}&{{}}&{{\mathrm{fi}}}&{{}}&{{\mathrm{\Phi}}}\end{array}$$

To assign a meaning to A ox axioms, the temporal interpretation function is extended to temporal intervals so that is an element of T* for each T. The semantics of assertions is the following is satisfied by an interpretation I i is satisfied by I i is satisfied by I i and is
;
satisfied by I i iven a knowledge base , an individual   in    is said to be an  instance of a concept      at the interval    if Now we are able to give a semantic definition for the reasoning task we already called specific plan recognition with respect to a plan description. This is an inference service that computes if an individual action/plan is an instance of an action/plan type at a certain interval, i.e., the task known as instance recognition in the escription ogic community.

iven a knowledge base   , an interval  , an individual   and a concept   , the instance recognition problem is to test whether

## 5. Action And Plan Representation: Two Examples

An action description represents how the world state may evolve in relation with the possible occurrence of the action itself. A plan is a complex action  it is described by means of temporally related world states and simpler actions. The following introduces examples of action and plan representations from two well known domains, the cooking domain autz,