Temporal Specifications with Accumulative Values

Abstract

Abstract-There is recently a significant effort to add quantitative objectives to formal verification and synthesis. We introduce and investigate the extension of temporal logics with quantitative atomic assertions, aiming for a general and flexible framework for quantitative-oriented specifications.In the heart of quantitative objectives lies the accumulation of values along a computation. It is either the accumulated summation, as with the energy objectives, or the accumulated average, as with the mean-payoff objectives. We investigate the extension of temporal logics with the prefix-accumulation assertions Sum(v) ≥ c and Avg(v) ≥ c, where v is a numeric variable of the system, c is a constant rational number, and Sum(v) and Avg(v) denote the accumulated sum and average of the values of v from the beginning of the computation up to the current point of time. We also allow the path-accumulation assertions LimInfAvg(v) ≥ c and LimSupAvg(v) ≥ c, referring to the average value along an entire computation.We study the border of decidability for extensions of various temporal logics. In particular, we show that extending the fragment of CTL that has only the EX, EF, AX, and AG temporal modalities by prefix-accumulation assertions and extending LTL with path-accumulation assertions, result in temporal logics whose model-checking problem is decidable. The extended logics allow to significantly extend the currently known energy and mean-payoff objectives. Moreover, the prefix-accumulation assertions may be refined with "controlled-accumulation", allowing, for example, to specify constraints on the average waiting time between a request and a grant. On the negative side, we show that the fragment we point to is, in a sense, the maximal logic whose extension with prefix-accumulation assertions permits a decidable model-checking procedure. Extending a temporal logic that has the EG or EU modalities, and in particular CTL and LTL, makes the problem undecidable.

Quantitative Verification of Weighted Kripke Structures

“…to constrain the average consumption of some resource x). In this context, the authors of [BCHK11] show undecidability of the logics in general, and propose several fragments for which model checking is decidable: (i) LTL with limit-average, where (roughly) a property can be rewritten as a conjunction of a qualitative logical property and a quantitative constraint; and (ii) CTL restricted to EF and EX modalities, with all kinds of numerical assertions, but where the qualitative logical part of the formula is rather poor. Notice that this decidability result is rather surprising, since reachability in two-counter machines is undecidable.…”

“…In that direction, the most relevant and advanced propositions are those of [DG09] and of [BCHK11]. In [DG09], LTL is (roughly) extended with Presburger constraints over weights and interpreted over one-counter automata (and an extension thereof).…”

“…The satisfiability and model checking problems are addressed, and it is shown that only a restriction to a single weight leads to decidability. In [BCHK11], CTL and LTL are extended with (prefix) accumulative values over finitely many variables: these logics embed numerical assertions such as Sum(x) ∼ c (e.g. to compare the accumulated amount of some resource x against some value c) and Avg(x) ∼ c (e.g.…”

“…Our contribution. We investigate further the temporal logics with prefix accumulation that has been proposed in [BCHK11], and we study the impact of restricting the logic to a single weight. The logics we consider are therefore based on CTL and LTL, and they extend the standard logics with two kinds of numerical assertions: Sum ∼ c expresses that the accumulated weight satisfies the constraint ∼ c, and Avg ∼ c expresses that the current average of the weight satisfies the constraint ∼ c. The extension of CTL is called WCTL and the extension of LTL is called WLTL.…”

“…On the contrary, allowing Avg constraints leads to undecidability for both branching and linear time. This undecidability result for our logic WLTL Avg is to be compared with the decidability of LTL with limit-average modalities of [BCHK11]: limit-average constraints can be made disjoint from the logical property expressed by the formula of LTL, whereas average modalities are really mixed with the logical property. Finally, we define a flat fragment of WLTL allowing both Sum and Avg constraints, but restricting the way they can be nested in the formula; we prove that this fragment has decidable model checking.…”