In this paper, we propose to extend First-Order Linear-time Temporal Logic with Past adding two operators “at next” and “at last”, which take in input a term and a formula and return the value of the term at the next state in the future or last state in the past in which the formula holds. The new logic, named LTL-EF, can be interpreted with different models of time (including discrete, dense, and super-dense time) and with different first-order theories (à la Satisfiability Modulo Theories (SMT)). We show that the “at next” and “at last” can encode (first-order) ${\mathit{MTL}}_{0,\infty }$ with counting. We provide rewriting procedures to reduce the satisfiability problem to the discrete-time case (to leverage on the mature state-of-the-art corresponding verification techniques) and to remove the extra functional symbols. We implemented these techniques in the nuXmv model checker enabling the analysis of LTL-EF and ${\mathit{MTL}}_{0,\infty }$ based on SMT-based model checking. We show the feasibility of the approach experimenting with several non-trivial valid and satisfiable formulas.

在本文中，我们建议扩展“一阶线性时间时间逻辑”，使其具有“过去”和“最后”两个运算符，这两个运算符输入一个术语和一个公式，并在下一个返回值该公式适用的将来状态或过去的最后状态。可以使用不同的时间模型（包括离散时间，密集时间和超密集时间）和不同的一阶理论（可满足性模数理论（SMT））来解释名为LTL-EF的新逻辑。我们证明“在下一个”和“在最后”可以编码（一阶）${\mathit{MTL}}_{0，\infty }$与计数。我们提供了重写程序，以将可满足性问题减少到离散时间情况（以利用成熟的最新技术，相应的验证技术）并删除多余的功能符号。我们在nuXmv模型检查器中实施了这些技术，从而可以分析LTL-EF和${\mathit{MTL}}_{0，\infty }$基于基于SMT的模型检查。我们展示了使用几种非平凡有效且可满足的公式进行实验的方法的可行性。