Fuzzy Time in Linear Temporal Logic

Abstract

In the past years, the adoption of adaptive systems has increased in many fields of computer science, such as databases and software engineering. These systems are able to automatically react to events by collecting information from the external environment and generating new events. However, the collection of data is often hampered by uncertainty and vagueness. The decision-making mechanism used to produce a reaction is also imprecise and cannot be evaluated in a crisp way, as it depends on vague temporal constraints expressed by humans. Logic has been extensively used as an abstraction to express vagueness in the satisfaction of system properties, as well as to enrich existing modeling formalisms. However, existing attempts to fuzzify the temporal modalities still have some limitations. Existing fuzzy temporal languages are generally obtained from classical temporal logic by replacing classical connectives or propositions with their fuzzy counterparts. Hence, these languages do not allow us to represent temporal properties, such as “almost always” and “soon,” in which the notion of time is inherently fuzzy. To overcome these limitations, we propose a temporal framework, fuzzy-time temporal logic (FTL), to express vagueness on time. This framework formally defines a set of fuzzy temporal modalities that 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 mutual relations. We also prove that under the assumption that all events are crisp, FTL reduces to linear temporal logic (LTL). Moreover, 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 of the other ones.