Propositional Temporal Linear Logic and its Application to Concurrent Systems

In computer science, concepts of resource such as data consumption and of time such as execution time are very important. Logical systems which can treat them have been applied in that field. Linear logic has been called a resource conscious logic. The expressive power is enough to describe a dynamic change in process environments. However, linear logic is not enough to treat a dynamic change in environments with the passage of time since it does not include a concept of time directly. A typical example is the relation between linear logic and Petri nets. It is well known that the reachability problem for Petri nets is equivalent to the provability for the corresponding sequent of linear logic. But linear logic cannot naturally represent timed Petri nets which are extensions of ordinary Petri nets with respect to time concept. So we extend linear logic with respect to time concept in order to introduce a resource-conscious and time-dependent logical system, that is, temporal linear logic. This system has some temporal operators “◦” which means a resource usable only once at the next time, “2” which means a resource usable only once at anytime, and a modal storage operator “!” which means a resource usable any times at anytime. We can show that the reachability problem for timed Petri nets is equivalent to the provability for the corresponding sequent of temporal linear logic. In this paper, we also represent the description of synchronous communication model by temporal linear logic. The expressive power of temporal linear logic will be applicable to various fields of computer science. key words: temporal linear logic, concurrent systems, synchronous communication, timed Petri nets, reachability, decidability.