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Reasoning about transfinite sequences

Stéphane Demri·David Nowak
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Abstract

We introduce a family of temporal logics to specify the behavior of systems with Zeno behaviors. We extend linear-time temporal logic LTL to authorize models admitting Zeno sequences of actions and quantitative temporal operators indexed by ordinals replace the standard next-time and until future-time operators. Our aim is to control such systems by designing controllers that safely work on $ω$-sequences but interact synchronously with the system in order to restrict their behaviors. We show that the satisfiability problem for the logics working on $ω^k$-sequences is EXPSPACE-complete when the integers are represented in binary, and PSPACE-complete with a unary representation. To do so, we substantially extend standard results about LTL by introducing a new class of succinct ordinal automata that can encode the interaction between the different quantitative temporal operators.

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Stéphane Demri, David Nowak. 2006-08-17. Reasoning about transfinite sequences. https://doi.org/10.1142/s0129054107004589

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