Temporal Reasoning and Constraint Programming: a Survey
Rosella Gennari

Writing this thesis meant, first of all, reading a great amount of literature 
about Temporal Reasoning and Constraint Programming, dealing with the different
formalizations of Temporal Constraint Programming and, above all, trying to get
a homogeneous work.
We could have divided the literature about Temporal Constraint Programming into
two main streams: in the first one, we would have classified the research on 
algorithms for solving TCPs; the other one is characterized by the study of 
algebraic or logical properties of the underlying temporal constraint 
frameworks.
However, this classification turned out to be too simplistic, since these two
fields are not so clearly separate; as we have seen, most of the investigation 
on Allen's subalgebras, which are the main tools for reasoning in the 
qualitative approach to Temporal Constraint Programming, has been motivated by 
the fact that the problem of satisfiability for this Allen's algebra (IA) 
turned out to be NP-hard. Therefore, we have preferred to classify the 
different approaches to Temporal Constraint Programming into three main 
branches, as we did in chapter 2.
The main two approaches we have identified are the quantitative (metric), 
first introduced by Dechter, Meiri and Pearl, and the qualitative based on 
Allen's interval algebra.
The metric characterization shows its efficiency when dealing with temporal
problems involving metric information, like "Paulo leaves home at 9:10 a.m."
or "Eyal goes to the park either at 5:00 or at 5:15 p.m.". Allen's approach is
instead useful if the temporal information focuses on relations between 
intervals, like ``Alessandra wakes up either before or after Raffaella does''; 
for instance, the relation of disjointness between intervals (the example just 
proposed) cannot be characterized in the metric approach with binary 
constraints.
In section 2.1 we have discussed the seminal work of Dechter, Meiri and Pearl 
on Temporal Constraint Programming with metric information; since the satis-
fiability problem for their general framework is NP-hard, authors have 
researched on what we have called simple temporal problems. We have studied 
the literature concerning this approach and presented it as a useful tool to 
enhance the search for solutions of general temporal constraint problems or to 
approximate them.
At the end of this section (cfr. p. 27), we tried to summarize the main
techniques one can use when dealing with this kind of constraint problems.
In section 2.2, we have presented the qualitative approach, focusing on 
Allen's interval algebra. As the satisfiability problem for constraint 
problems based on IA is NP--hard, we directed our attention to subalgebras of 
IA which are tractable and expressive enough for many applications; for 
instance, we introduced the PA algebra and its properties as a tool for 
studying IA problems. This idea has guided us for the rest of our exposition; 
whenever we introduced metric temporal problems in this section, we used them 
to study interesting subalgebras of IA. In fact, one of the most promising 
direction of research aims at classifying the maximal tractable subalgebras of 
IA with different expressive power; for instance, we have seen that the 
algebra of Nebel and B\¨urckert is the maximal tractable subalgebra of IA that 
contains all of the Allen atoms, but it cannot express the relation of 
sequentiality, that is given by {before, afterg} however, this relation can be 
expressed by the 8 maximal tractable subalgebras A(r, b) of Drakengren and 
Jonsson.
As in the metric approach, at the end of section 2.2 we have reviewed and
summarized the main techniques that Constraint Programming offers to solve
IA problems; the main one is given by path--consistency, for historical reason
and because it is enough to guarantee consistency of the tractable subalgebras
we have presented.
At the end, in section 2.3, we have briefly introduced a new line of investi­
gation, which we called "the mixed approach". Research is open in this field;
local consistency procedures and new classes of constraint relations are 
currently under investigation.
Temporal Constraint Programming could grow further, taking into consideration 
event calculus and the systems adopted to reason about actions and changes. In 
our frameworks, time is the only ontological object, as events, fluents, 
states or actions are identified with their period of occurrence; this choice
provides a computational and logical system easy to manage and efficient. 
However, the creation of a new constraint-based logical system, in which 
events, fluents or actions are considered as new objects of different type, 
represents an attractive challenge.