۱ Introduction

۲ Classic Modal Logic

۳ Fuzzy Modal Logic

۴ Simulation and bisimulation

۵ Bisimilarity

۶ Modal Invariance

۷ Conclusion

References

This paper intends to contribute with a new fuzzy modal logic to model and reason about transition systems involving uncertainty in behaviours. Our formalism supports fuzziness at transitions and on the proposition symbols assignment levels. Against of other approaches in the literature, our bisimulation and bisimilarity notions generalise the analogous standard notions of classic modal logic and of process algebras. Moreover, the outcome of our logic is also fuzzy, with the semantic interpretation of connectives supported by the G¨odel algebra.

Introduction

For 50 years, fuzzy sets and fuzzy logic have been an area of active research (cf. [13]). Fuzzy automata [17], fuzzy Markov processes [1], fuzzy petri nets [25,16], fuzzy reactive frames [24] and fuzzy discrete event systems [22] are some of the formalisms that have been considering to model computational systems that deal with uncertainty and fuzzy sets. In this work, we will focus on fuzzy transition systems or fuzzy labelled transition systems, which are a generalisation of transition systems or labelled transition systems (widely used in computer science) with [0, 1]- weights on the transitions [26,27,15]. It is well known that bisimulations and simulations are a worth way of comparing two transition systems. They have been considered in several frameworks, such as fuzzy automata [7,8] fuzzy Markov process [9], fuzzy discrete systems [22], weighted labelled transition system [26,27,15]. All of them have special motivations and, consequently, different formulations. For example, in [26,6,5] bisimulations are defined as equivalence relations. There are other approaches that focus on horizontal and vertical bisimulations [15] and some other which define bisimulations as fuzzy relations [7].