Doctoral Dissertation Temporal Linear Logic and Its Applications

Linear logic, introduced by Girard in 1987, has been called a resource conscious logic. In order to express a dynamic change in process environment, it is useful to consider a concept of resource such as data consumption. The expressive power of linear logic is evidenced by some very natural encodings of computational models such as Petri nets, counter machines, Turing machines, and others. For example, in Petri nets, tokens are considered as resources that are consumed and transitions are considered as reusable resources. It is well known that the reachability problem for ordinary Petri nets is equivalent to the provability for the corresponding sequent of linear logic. Also, as a formal logical system, linear logic satisfies some basic theorems. In it the cut elimination theorem and the soundness and completeness theorems for phase semantics which is a standard semantics of linear logic hold true. In particular, the cut elimination theorem can be applied to logic programming, uniform proof and proof search, and so on. We think that linear logic has been given various applications in computer science through its resource consciousness and usefulness as a formal system. However, since linear logic does not include a concept of time directly, it is not enough to treat a dynamic change in environments with the passage of time such as execution time and waiting time. A typical example is the encoding of timed Petri nets. Although ordinary Petri nets can be encoded into linear logic naturally as stated above, the encoding of timed Petri nets into the corresponding sequent is too complex for linear logic since the reachability problem for timed Petri nets includes a time concept. Thus, it can be considered to extend linear logic with respect to the time concept. The aim of this thesis is to construct a resource-conscious and time-dependent logical system by means of extending linear logic and to provide an application to computer science. We think that such a logic can treat a dynamic change in environments with the passage of time. We call it temporal linear logic. The basic idea is to introduce temporal operators. We assume discrete linear time in this thesis and introduce “◦”, which means next, and “2”, which means anytime. We make the interpretation of a formula include a time concept. For inference rules, we refer to modal logic S4. S4 can thus be regarded as temporal logic, which also has a “2” operator meaning “always”, which is similar to ours. In this way, temporal linear logic includes linear logic as its subsystem. It can succeed to the resource-consciousness in linear logic. Also, S4 can be embedded into temporal linear logic. We can say that the time concept works in our logical system. In addition, the cut elimination theorem and the subformula property hold in this logic as in the form of linear logic. The full propositional fragment of temporal linear logic has a complete semantics in terms of temporal phase spaces, which are an extension of phase spaces in the semantics of linear logic. Using our temporal linear logic, several cases of a dynamic change in environments with the passage of time can be expressed. Timed Petri nets can be encoded naturally into it, that is, the reachability problem for timed Petri nets is equivalent to the provability for the corresponding sequent. This is connected to the decidability of temporal linear logic fragments. A logic programming language based on temporal linear logic is designed by using the idea of Miller’s uniform proof. We also represent the description of a communication model, which is our own model, by temporal linear logic. It can distinguish a synchronous calculus from an asynchronous calculus.