Execution of TILCO Specifications Technical Report

For the specification of real-time system behaviour, the definition of temporal constraints amongst events and actions is needed [1], [2], [3]: properties of invariance, precedence amongst events, periodicity, liveness and safety conditions, etc. To this end, several different temporal logics with different degrees of expressiveness have been proposed. Some of them are based on propositional logic — e.g., PTL, TPTL, RTTL, ITL ([3]). Others are based on first or higher order logic — e.g., TRIO [4], MTL [5], interval temporal logic [6], TILCO [7]. According to [7], in which TILCO (Temporal Interval Logic with Compositional Operators) temporal logic has been presented, only a few examples of quantitative temporal logics exist. In these cases, an operator expressing the distance between time points is usually defined. First-order temporal logics that provide a metric for time usually allow quantification over the temporal domain — e.g., RTL [8], MTL [9], TRIO [4] — whereas a prohibition of this kind of quantification has been shown to be a necessary condition for the existence of feasible automated verification mechanisms [10]. All these temporal logics are based on time points rather than on intervals and provide a sharp distinction between past and future (e.g., MTL: G, H; TRIO: Past() and Futr()). In contrast, TILCO does not allow the quantification over time, presents a uniform model for time from past to future and unique operators for stating facts and events along the time axis [7], [11], [12]. The problem of executability of specifications given by means of temporal logics has often been misunderstood with that of validation. The validation of a logic specification usually consists in proving high-level properties, which are also given in the form of logical expressions, by means of theorem provers [13]. Other techniques are based on the so-called history-checking [14]. In these cases, the time ordering is not satisfied, history-checking is mainly oriented to validating specifications against off-line generated histories of system inputs and outputs. There are at least three different definitions of executability [15]. According to concept of execution, which is closer to that used in software engineering, the execution of a specification leads to the on-line production of outputs on the basis of inputs and system status. Thus the time ordering of events must be satisfied with real-time constraints. This concept of execution is quite far from that of validation since it is supposed that the specification under execution is a verified and validated specification. In this report, the TILCO-Executor algorithm for executing TILCO is presented. The meaning adopted by TILCO-Executor for executability consists in using the specification itself as a prototype of the real-time system, thus allowing, in each time instant, for the on-line generation of system outputs on the basis of current inputs and internal state and inputs and outputs history. The algorithm is capable of executing a fragment of TILCO specifications, and can be used for execution other first order temporal logics under similar restrictions. In order to simplify the execution of TILCO specifications a Basic Temporal Logic (BTL) with a lower number of operators has been introduced, this is capable of expressing all complex operators of TILCO. In the following it is briefly described how to translate TILCO to BTL. Inference rules for BTL are presented and it is described how to graphically represent the BTL specifications as Temporal Inference Networks, TIN. An algorithm for executing temporal inference network is presented and the possibility of real-time execution is highlighted.