On Randomised Strategies in the λλ-Calculus

International audienceIn this work we study randomised reduction strategies-a notion already known in the context of abstract reduction systems-for the λ-calculus. We develop a simple framework that allows us to prove a randomised strategy to be positive almost-surely normalising. Then we propose a simple example of randomised strategy for the λ-calculus that has such a property and we show why it is non-trivial with respect to classical deterministic strategies such as leftmost-outermost or rightmost-innermost. We conclude studying this strategy for the affine λ-calculus, where duplication is syntactically forbidden

Model-independent pricing with insider information: a Skorokhod embedding approach

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider's information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox and Huesmann (2016) to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems

Tree games with regular objectives

We study tree games developed recently by Matteo Mio as a game interpretation of the probabilistic $\mu$-calculus. With expressive power comes complexity. Mio showed that tree games are able to encode Blackwell games and, consequently, are not determined under deterministic strategies. We show that non-stochastic tree games with objectives recognisable by so-called game automata are determined under deterministic, finite memory strategies. Moreover, we give an elementary algorithmic procedure which, for an arbitrary regular language L and a finite non-stochastic tree game with a winning objective L decides if the game is determined under deterministic strategies.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Permissive Controller Synthesis for Probabilistic Systems

We propose novel controller synthesis techniques for probabilistic systems modelled using stochastic two-player games: one player acts as a controller, the second represents its environment, and probability is used to capture uncertainty arising due to, for example, unreliable sensors or faulty system components. Our aim is to generate robust controllers that are resilient to unexpected system changes at runtime, and flexible enough to be adapted if additional constraints need to be imposed. We develop a permissive controller synthesis framework, which generates multi-strategies for the controller, offering a choice of control actions to take at each time step. We formalise the notion of permissivity using penalties, which are incurred each time a possible control action is disallowed by a multi-strategy. Permissive controller synthesis aims to generate a multi-strategy that minimises these penalties, whilst guaranteeing the satisfaction of a specified system property. We establish several key results about the optimality of multi-strategies and the complexity of synthesising them. Then, we develop methods to perform permissive controller synthesis using mixed integer linear programming and illustrate their effectiveness on a selection of case studies

Grid-free computation of probabilistic safety with Malliavin Calculus

This work concerns continuous-time, continuous-space stochastic dynamical systems described by stochastic differential equations (SDE). It presents a new approach to compute probabilistic safety regions, namely sets of initial conditions of the SDE associated to trajectories that are safe with a probability larger than a given threshold. The approach introduces a functional that is minimised at the border of the probabilistic safety region, then solves an optimisation problem using techniques from Malliavin Calculus, which computes such region. Unlike existing results in the literature, the new approach allows one to compute probabilistic safety regions without gridding the state space of the SDE

Pure Nash Equilibria in Concurrent Deterministic Games

We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)Comment: 72 page