Discounting in LTL

Abstract

In recent years, there is growing need and interest in formalizing and reasoning about the quality of software and hardware systems. As opposed to traditional verification, where one handles the question of whether a system satisfies, or not, a given specification, reasoning about quality addresses the question of how well the system satisfies the specification. One direction in this effort is to refine the "eventually" operators of temporal logic to discounting operators: the satisfaction value of a specification is a value in [0, 1], where the longer it takes to fulfill eventuality requirements, the smaller the satisfaction value is. In this paper we introduce an augmentation by discounting of Linear Temporal Logic (LTL), and study it, as well as its combination with propositional quality operators. We show that one can augment LTL with an arbitrary set of discounting functions, while preserving the decidability of the model-checking problem. Further augmenting the logic with unary propositional quality operators preserves decidability, whereas adding an average-operator makes some problems undecidable. We also discuss the complexity of the problem, as well as various extensions.LTL disc [D] is actually a family of logics, each parameterized by a set D of discounting functions -strictly decreasing functions from AE to [0, 1] that tend to 0 (e.g., linear decaying, exponential decaying, etc.). LTL disc [D] includes a discounting-"until" (U η ) operator, parameterized by a function η ∈ D. We solve the model-checking threshold problem for LTL disc [D]: given a Kripke structure K, an LTL disc [D] formula ϕ and a threshold t ∈ [0, 1], the algorithm decides whether the satisfaction value of ϕ in K is at least t.In the Boolean setting, the automata-theoretic approach has proven to be very useful in reasoning about LTL specifications. The approach is based on translating LTL formulas to nondeterministic Büchi automata on infinite words [28]. Applying this approach to the discounted setting, which gives rise to infinitely many satisfaction values, poses a big algorithmic challenge: model-checking algorithms, and in particular those that follow the automata-theoretic approach, are based on an exhaustive search, which cannot be simply applied when the domain becomes infinite. A natural relevant extension to the automata-theoretic approach is to translate formulas to weighted automata [22]. Unfortunately, these extensively-studied models are complicated and many problems become undecidable for them [15]. We show that for threshold problems, we can translate LTL disc [D] formulas into (Boolean) nondeterministic Büchi automata, with the property that the automaton accepts a lasso computation iff the formula attains a value above the threshold on that computation. Our algorithm relies on the fact that the language of an automaton is non-empty iff there is a lasso witness for the non-emptiness. We cope with the infinitely many possible satisfaction values by using the discounting behavior of the eventualities and the given thre...

Synthesis with Rational Environments

“…In recent years, researchers have developed more refined quantitative temporal logics, which enable a formal reasoning about the quality of systems. In particular, we plan to study rational synthesis for the multi-valued logics LTL[F] [1], which enables a prioritization of different satisfaction possibilities, and LTL[D] [2], in which discounting is used in order to reduce the satisfaction value of specifications whose eventualities are delayed. The rational synthesis problem for these logics induce a game with much richer, possibly infinitely many, profiles, making the search for a stable solution much more challenging.…”

Metamathematics for Systems Design

“…One example of such semantics is temporal logics with discounting [4,5,70], where events in the farther future have the smaller significance. For example, for the LTL specification Fp (''p eventually becomes true'') we have the following difference: -in the usual Boolean semantics the formula is true no matter how long it takes before p is true for the first time; -in contrast, in the quantitative discounted semantics of [5], the formula's truth value is 1=2 i in case p is true for the first time at time i.…”

Automatic Generation of Quality Specifications

“…For all computations π and LTL queries ϕ, the function µ π,ϕ is continuous. That is, for every infinite sequence (a n ) ∞ n=1 of points in [0, 1] k such that lim n→∞ a n = a, it holds that lim n→∞ (µ π,ϕ (a n )) = µ π,ϕ (a).…”

“…This includes, for example, probabilistic [16], fuzzy [18], and accumulative [9] settings. An exciting direction in this effort is the development of formalisms and methods for reasoning about the quality of systems [2,1]. The working assumption in these works is that satisfying a specification is not a yes/no matter.…”

“…The logic LTL extends LTL by quality operators. The satisfaction value of an LTL formula in a computation refines the 0/1 value of LTL formulas to a real value in [0,1]. The higher the value is, the better is the quality of the computation.…”