# Using Temporal Logic to Analyse Temporal Logic: A Hierarchical Approach Based on Intervals

Temporal logic has been extensively utilized in academia and industry to formally specify and verify behavioural properties of numerous kinds of hardware and software. We present a novel way to apply temporal logic to the study of a version of itself, namely, propositional linear-time temporal logic (PTL). This involves a hierarchical framework for obtaining standard results for PTL, including a small model property, decision procedures and axiomatic completeness. A large number of the steps involved are expressed in a propositional version of Interval Temporal Logic (ITL) which is referred to as PITL. It is a natural generalization of PTL and includes operators for reasoning about periods of time and sequential composition. Versions of PTL with finite time and infinite time are both considered and one benefit of the framework is the ability to systematically reduce infinite-time reasoning to finite-time reasoning. The treatment of PTL with the operator Until and past time naturally reduces to that for PTL without either one. The interval-oriented methodology differs from other analyses of PTL which typically use sets of formulas and sequences of such sets for canonical models. Instead we represent models as time intervals expressible in PITL. The analysis furthermore relates larger intervals with smaller ones. Being an interval-based formalism, PITL is well suited for sequentially combining and decomposing the relevant formulas. Consequently, we can articulate issues of equal significance in more conventional analyses of PTL but normally only considered at the metalevel. A good example of this is the existence of bounded models with periodic suffixes for PTL formulas which are satisfiable in infinite time. We also describe decision procedures based on binary decision diagrams and exploit some links with finite-state automata.

Beyond the specific issues involving PTL, the research is a significant application of ITL and interval-based reasoning and illustrates a general approach to formally reasoning about sequential and parallel behaviour in discrete linear time. The work also includes some interesting representation theorems. In addition, it has relevance to hardware description and verification since the specification languages PSL/Sugar (IEEE Standard 1850) and 'temporal e' (part of IEEE Standard 1647) both contain temporal constructs concerning intervals of time as does the related SystemVerilog Assertion language contained in SystemVerilog (IEEE Standard 1800), an extension of the IEEE 1364-2001 Verilog language.