An optimal pastification algorithm for LTL[X,F] and LTL[X,G]

We investigate a fragment of Linear Temporal Logic (LTL) comprising the tomorrow (X) and eventually (F) modalities, and present a singly exponential time algorithm for the pastification problem within this fragment. The pastification problem consists of constructing, for a given LTL formula, an equivalent formula that exclusively employs past temporal operators. While the best known algorithms for this task in full LTL–and in the fragment under consideration–exhibit triply exponential time complexity, our approach achieves optimal complexity for this fragment. The proposed algorithm proceeds in two main stages: (i) the input formula is first translated into a tailored normal form, and then (ii) a pure past formula is synthesized from a tree-like structure derived from the normalized formula. With minor adaptations, the algorithm extends to handle the fragment of LTL featuring the tomorrow and globally modalities. We provide an implementation of the algorithm in the temporal reasoning tool BLACK, and report on an experimental evaluation of its performance.1