Reasoning About Transfinite Sequences

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 and model-checking 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.

The Complexity of Temporal Logic with Until and Since over Ordinals

“…As a consequence, we are also able to improve some results from [Roh97,DN07]. For instance the uniform satisfiability problem is pspace-complete and we obtain alternative proofs for results in [DN07].…”

“…In this section, we show that the main results from [DN07] are subsumed by the current paper. We also solve an open problem from [Cac06,DN07].…”

“…The proofs in [Rey03,Rey] are non trivial and difficult to grasp and it is therefore difficult to compare our proof technique with those of [Rey03,Rey] even though we believe cross-fertilization would be fruitful. We provide uniform proofs and we improve upper bounds for decision problems considered in [Cac06,DN07,Roh97]. We also compare our results and techniques with Rohde's thesis [Roh97].…”

“…We also solve an open problem from [Cac06,DN07]. For every fixed countable ordinal α ≤ ω, let us introduce the logic LTL(O α ) where the set of temporal operators O α is defined as follows:…”

“…The main results of [Cac06,DN07] are the following: for every k ≥ 1, the satisfiability problem for LTL(O k ) restricted to models of length ω k is pspace-complete and LTL(O ω ) restricted to models of length ω ω is decidable. Observe that LTL(O k ) cannot express the temporal operator U over the class of countable ordinals but it can do it on models of length ω k .…”