6 research outputs found
Reasoning about transfinite sequences
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 -sequences but interact synchronously with the system in
order to restrict their behaviors. We show that the satisfiability problem for
the logics working on -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.Comment: 38 page
A logic with temporally accessible iteration
Deficiency in expressive power of the first-order logic has led to developing
its numerous extensions by fixed point operators, such as Least Fixed-Point
(LFP), inflationary fixed-point (IFP), partial fixed-point (PFP), etc. These
logics have been extensively studied in finite model theory, database theory,
descriptive complexity. In this paper we introduce unifying framework, the
logic with iteration operator, in which iteration steps may be accessed by
temporal logic formulae. We show that proposed logic FO+TAI subsumes all
mentioned fixed point extensions as well as many other fixed point logics as
natural fragments. On the other hand we show that over finite structures FO+TAI
is no more expressive than FO+PFP. Further we show that adding the same
machinery to the logic of monotone inductions (FO+LFP) does not increase its
expressive power either
Controller synthesis & Ordinal Automata
with appendixOrdinal automata are used to model physical systems with Zeno behavior. Using automata and games techniques we solve a control problem formulated and left open by Demri and Nowak in 2005. It involves partial observability and a new synchronization between the controller and the environment
The complexity of linear-time temporal logic over the class of ordinals
We consider the temporal logic with since and until modalities. This temporal
logic is expressively equivalent over the class of ordinals to first-order
logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability
problem over the class of ordinals. Among the consequences of our proof, we
show that given the code of some countable ordinal alpha and a formula, we can
decide in PSPACE whether the formula has a model over alpha. In order to show
these results, we introduce a class of simple ordinal automata, as expressive
as B\"uchi ordinal automata. The PSPACE upper bound for the satisfiability
problem of the temporal logic is obtained through a reduction to the
nonemptiness problem for the simple ordinal automata.Comment: Accepted for publication in LMC
A Hypersequent Calculus with Clusters for Data Logic over Ordinals
We study freeze tense logic over well-founded data streams. The logic features past-and future-navigating modalities along with freeze quantifiers, which store the datum of the current position and test data (in)equality later in the formula. We introduce a decidable fragment of that logic, and present a proof system that is sound for the whole logic, and complete for this fragment. Technically, this is a hy-persequent system enriched with an ordering, clusters, and annotations. The proof system is tailored for proof search, and yields an optimal coNP complexity for validity and a small model property for our fragment
Reasoning about transfinite sequences (extended abstract
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.