{'fraction_non_alphanumeric': 0.02147239263803681, 'fraction_numerical': 0.0, 'mean_word_length': 5.046296296296297, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Satisfiability over the Integers

We investigate the satisfiability problem for metric temporal logic (MTL) with both past and future operators over linear discrete bi-infinite time models — where time is unbounded both in the future and in the past — isomorphic to the integer numbers. We provide a technique to reduce satisfiability over the integers to satisfiability over the well-known mono-infinite time model of natural numbers, and we show how to implement the technique through an automata-theoretic approach. We also prove that MTL satisfiability over the integers is EXPSPACE-complete, hence the given algorithm is optimal in the worst case.