Abstract
Linear Temporal Logic (LTL) is one of the most commonly used formalisms for representing and reasoning about temporal properties of computations. Its application domains range from formal verification to artificial intelligence. Many real-time extensions of LTL have been proposed over the years, including Timed Propositional Temporal Logic (TPTL), that makes it possible to constrain the temporal ordering of pairs of events as well as the exact time elapsed between them.The paper focuses on TPTL and BoundedTPTLwith Past (▪), a bounded variant of TPTL enriched with past operators, which has been recently introduced to formalise a meaningful class of timeline-based planning problems. ▪ allows one to refer to the past while keeping the computational complexity under control: in contrast to the full TPTLwith Past (TPTL+P), whose satisfiability problem is non-elementary, the satisfiability problem for ▪ is ▪-complete.The paper deals with the satisfiability problem for TPTL and ▪ by providing an original tableau system for each of them that suitably generalises Reynolds' one-pass and tree-shaped tableau for LTL. First, we show how to handle past operators, by devising a one-pass and tree-shaped tableau system for LTLwith Past (LTL+P). Then, we adapt it to TPTL and ▪, providing full proofs of the soundness and completeness of the resulting systems. In particular, completeness is proved by exploiting a novel model-theoretic argument that, compared to the one originally employed for the LTL system, provides a deeper understanding of the crucial role of the prune rule of the system.
Highlights
Among the reasoning methods used to decide the satisfiability of logical formulae, tableau methods are among the earliest proposed and most studied solutions [4]
Classic tableau methods for logics of the linear time, such as, for instance, Linear Temporal Logic (LTL) [8, 9], build a graph structure which is traversed to look for possible models of the formula
We can apply the general tableau rules to the G(TPTL+P) translation of any TPTLb+P formula, provided that, similar to the Timed Propositional Temporal Logic (TPTL) case, a proper temporal shift operator can be defined. This can be done by exploiting the following observation: thanks to the bounds applied to the TPTLb+P temporal operators, whose semantics is implemented in G(TPTL+P) formulae by means of the guards, when interpreting a timing constraint like x ≤ y + c, the distance between variables x and y cannot be greater than an upper bound W that depends on the bounds applied to the temporal operators of the formula
Summary
Among the reasoning methods used to decide the satisfiability of logical formulae, tableau methods are among the earliest proposed and most studied solutions [4]. Proposed as a formal tool for the verification of real-time systems, it recently found interesting applications in the area of artificial intelligence, to encode a meaningful class of timeline-based planning problems [5]. This and other application scenarios benefit from/require the use of past operators, which allow the logic to compactly predicate about events in the past of the current time point. In contrast to the case of LTL, where past operators can be supported without harm, adding them to TPTL greatly increases the complexity of its satisfiability problem, which becomes non-elementary [1].
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