The temporal logic of reactive and concurrent systems
The temporal logic of reactive and concurrent systems
The formal semantics of programming languages: an introduction
The formal semantics of programming languages: an introduction
An axiomatic basis for computer programming
Communications of the ACM
Projection in Temporal Logic Programming
LPAR '94 Proceedings of the 5th International Conference on Logic Programming and Automated Reasoning
Design and Synthesis of Synchronization Skeletons Using Branching-Time Temporal Logic
Logic of Programs, Workshop
The temporal logic of programs
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Twenty Years of Theorem Proving for HOLs Past, Present and Future
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
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In Propositional Projection Temporal Logic (PPTL), a well-formed formula is generally formed by applying rules of its syntax finitely many times. However, under some circumstances, although formulas such as ones expressed by index set expressions, are constructed via applying rules of the syntax infinitely many times, they are possibly still well-formed. With this motivation, this paper investigates the relationship between formulas specified by index set expressions and concise syntax expressions by means of fixed-point induction approach. Firstly, we present two kinds of formulas, namely $\bigvee_{i\in N_0}\bigcirc^i P$ and $\bigvee_{i\in N_0}P^i$, and prove they are indeed well-formed by demonstrating their equivalence to formulas $\Diamond P$ and P+ respectively. Further, we generalize $\bigvee_{i\in N_0}\bigcirc^i Q$ to $\bigvee_{i\in N_0}P^{(i)} \wedge \bigcirc^i Q$ and explore solutions of an abstract equation $X \equiv Q \vee P \wedge \bigcirc X$. Moreover, we equivalently represent 'U' (strong until) and 'W' (weak until) constructs in Propositional Linear Temporal Logic within PPTL using the index set expression techniques.