Discounting in LTL

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 \emph{how well} the system satisfies the specification. One direction in this effort is to refine the "eventually" operators of temporal logic to {\em 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

Near-Optimal Scheduling for LTL with Future Discounting

We study the search problem for optimal schedulers for the linear temporal logic (LTL) with future discounting. The logic, introduced by Almagor, Boker and Kupferman, is a quantitative variant of LTL in which an event in the far future has only discounted contribution to a truth value (that is a real number in the unit interval [0, 1]). The precise problem we study---it naturally arises e.g. in search for a scheduler that recovers from an internal error state as soon as possible---is the following: given a Kripke frame, a formula and a number in [0, 1] called a margin, find a path of the Kripke frame that is optimal with respect to the formula up to the prescribed margin (a truly optimal path may not exist). We present an algorithm for the problem; it works even in the extended setting with propositional quality operators, a setting where (threshold) model-checking is known to be undecidable

Linear Time Logics - A Coalgebraic Perspective

We describe a general approach to deriving linear time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching behaviour and linear behaviour. Concretely, we define logics whose syntax is determined by the choice of linear behaviour and whose domain of truth values is determined by the choice of branching, and we provide two equivalent semantics for them: a step-wise semantics amenable to automata-based verification, and a path-based semantics akin to those of standard linear time logics. We also provide a semantic characterisation of the associated notion of logical equivalence, and relate it to previously-defined maximal trace semantics for such systems. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear time property. We conclude with a generalisation of the logics, dual in spirit to logics with discounting, which increases their practical appeal in the context of resource-aware computation by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the results in the previous version, with new proofs; Section 6 contains new result

Determination of the fair value of a multifunctional family farm: a case study

The article analyses the problems of the determination of the fair value of a multifunctional family farm using the method of discounted cash flow, presents a model of determination of the fair value of a multifunctional family farm and tests it for a selected family farm. The specificity of the cash flows in a multifunctional family farm is related to the cash flows from financial support, different value drivers of the earnings before interests and tax and their calculation methodology, and the value of created public goods and externalities. Two types of discount rates are used to determine the value of a family farm: marketbased and social discount rate (SDR). It is appropriate to use the SDR to discount the cash flow of investment, the economic and social benefits whereof are distributed among present and future generations. The stages of the determination of the fair value of a farmer’s farm include: value drivers’ decomposition; differentiation of cash flow and discount rates and measuring value drivers; forecasting cash flow and discount rate value drivers; value drivers’ composition; determination of the terminal value; and cash flow discounting

Discounting in Strategy Logic

Discounting is an important dimension in multi-agent systems as long as we want to reason about strategies and time. It is a key aspect in economics as it captures the intuition that the far-away future is not as important as the near future. Traditional verification techniques allow to check whether there is a winning strategy for a group of agents but they do not take into account the fact that satisfying a goal sooner is different from satisfying it after a long wait. In this paper, we augment Strategy Logic with future discounting over a set of discounted functions D, denoted SLdisc[D]. We consider "until" operators with discounting functions: the satisfaction value of a specification in SLdisc[D] is a value in [0, 1], where the longer it takes to fulfill requirements, the smaller the satisfaction value is. We motivate our approach with classical examples from Game Theory and study the complexity of model-checking SLdisc[D]-formulas.Comment: Extended version of the paper accepted at IJCAI 202