Acta Informatica (2018) 55:129–152
Parameterized linear temporal logics meet costs: still not
costlier than LTL
Received: 22 February 2016 / Accepted: 10 September 2016 / Published online: 13 October 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract We continue the investigation of parameterized extensions of linear temporal logic
(LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model
checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al.
and Kupferman et al. showed that this is the case for parametric LTL (PLTL) and PROMPT-
LTL respectively, which have temporal operators equipped with variables that bound their
scope in time. Later, this was also shown to be true for parametric LDL (PLDL), which
extends PLTL to be able to express all ω-regular properties. Here, we generalize PLTL to
systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in
terms of the cost accumulated during time. Again, we show that model checking and solving
games for speciﬁcations in PLTL with costs is not harder than the corresponding problems
for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.
Parameterized linear temporal logics address a serious shortcoming of linear-temporal logic
(LTL) : LTL is not able to express timing constraints, e.g., while G(q → F p) expresses
that every request q is eventually answered by a response p, the waiting time between requests
and responses might diverge. This is typically not the desired behavior, but cannot be ruled
out by LTL.
To overcome this shortcoming, parameterized LTL (PLTL) was introduced by Alur et
al. , which extends LTL with parameterized operators of the form F
and y are variables. The formula G(q → F
p) expresses that every request q is answered by
p within an arbitrary, but ﬁxed number of steps α(x). Here, α is a variable valuation, a mapping
Supported by the project “TriCS” (ZI 1516/1-1) of the German Research Foundation (DFG).
A preliminary version of this work appeared in the proceedings of GandALF 2015 .
Reactive Systems Group, Saarland University, 66123 Saarbrücken, Germany