  In the last years, the adoption of active systems has increased in many
fields of computer science, such as databases, sensor networks, and software
engineering. These systems are able to automatically react to events, by
collecting information from outside and internally generating new events.
However, the collection of data is often hampered by uncertainty and vagueness
that can arise from the imprecision of the monitoring infrastructure,
unreliable data sources, and networks. The decision making mechanism used to
produce a reaction is also imprecise, and cannot be evaluated in a crisp way.
It depends on the evaluation of vague temporal constraints, which are expressed
on the collected data by humans. Despite fuzzy logic has been mainly conceived
as a mathematical abstraction to express vagueness, no attempt has been made to
fuzzify the temporal modalities. Existing fuzzy languages do not allow us to
represent temporal properties, such as "almost always" and "soon". Indeed, the
semantics of existing fuzzy temporal operators is based on the idea of
replacing classical connectives or propositions with their fuzzy counterparts.
To overcome these limitations, we propose a temporal framework, FTL (Fuzzy-time
Temporal Logic), to express vagueness on time. This framework formally defines
a set of fuzzy temporal modalities, which can be customized by choosing a
specific semantics for the connectives. The semantics of the language is sound,
and the introduced modalities respect a set of expected mutual relations. We
also prove that under the assumption that all events are crisp, FTL reduces to
LTL. Finally, for some of the possible fuzzy interpretations of the
connectives, we identify adequate sets of temporal operators, from which it is
possible to derive all the others.
