An Efficient Normalisation Procedure for Linear Temporal Logic: Isabelle/HOL Formalisation

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\underset{i=1}{\overset{n}{\bigwedge }}\mathbf{G}\mathbf{F}{\phi }_i\vee \mathbf{F}\mathbf{G}{\psi }_i$, where ${\phi }_i$ and ${\psi }_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present an executable formalisation of a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up.