A metric for positional games

We define an extended real-valued metric, ρ, for positional games and prove that this class of games is a topological semigroup. We then show that two games are finitely separated iff they are path-connected and iff two closely related Conway games are equivalent. If two games are at a finite distance then this distance is bounded by the maximum difference of any two atoms found in the games. We may improve on this estimate when two games have the same form, as given by a form match. Finally, we show that if ρ(G,H)=∞ then for all X we have G+X H+X, a step towards proving cancellation for positional games.     

A Rough Set Approach to Estimating the Game Value and the Shapley Value from Data

A value of a game v is a function which to each coalition S assigns the value v(S) of this coalition, meaning the expected pay–off for players in that coalition. A classical approach of von Neumann and Morgenstern [6] had set some formal requirements on v which contemporary theories of value adhere to. A Shapley value of the game with a value v [14] is a functional Φ giving for each player p the value Φ_p(v) estimating the expected pay-off of the player p in the game. Game as well as conflict theory have been given recently much attention on the part of rough and fuzzy set communities [11,8,1,4,7,2]. In particular, problems of plausible strategies [1] in conflicts as well as problems related to Shapley's value [3,2] have been addressed.We confront here the problem of estimating a value as well as Shapley's value of a game from a partial data about the game. We apply to this end the rough set ideas of approximations, defining the lower and the upper value of the game and, respectively, the lower and upper Shapley value. We also define a notion of an exact coalition, on which both values coincide giving the true value of the game; we investigate the structure of the family of exact sets showing its closeness on complements, disjoint sums, and intersections of coalitions covering the set of players. This work sets open a new area of rough set applications in mining constructs from data. The construct mined in this case are values as well as Shapley values of games.     

Interaction in Time and Space

Distributed π-calculus and ambient calculus are extended with timers which may trigger timeout recovery processes. Timers provide a useful notion of relative time with respect to the interaction in a distributed system. The rather flat notion of space in timed distributed π-calculus is improved by considering a hierarchical representation of space in timed mobile ambients. Some basic results are proven, making sound both formal approaches. An easily understood example is used for both extensions, showing how it is possible to describe a non-monotonic behaviour and use a decentralized control to coordinate the interacting components in time and space.     

The μ-basis of a planar rational curve—properties and computation

A moving line L(x,y;t)=0 is a family of lines with one parameter t in a plane. A moving line L(x,y;t)=0 is said to follow a rational curve P(t) if the point P(t₀) is on the line L(x,y;t₀)=0 for any parameter value t₀. A μ-basis of a rational curve P(t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P(t), which is the syzygy module of P(t). The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3 and 6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P(t), and has O(n²) time complexity, where n is the degree of P(t). We show that the new algorithm is more efficient than the fastest previous algorithm [7].     

Approximation by Bézier type of Meyer–König and Zeller operators

In this paper, we give direct, inverse and equivalence approximation theorems for the Bézier type of Meyer–König and Zeller operator with unified Ditzian–Totik modulus ω_φ^λ(f,t) (0≤λ≤1).     

A duality involving Borel spaces

The purpose of the paper is to show that there is a dual equivalence between sober Borel spaces and spatial Boolean σ-frames. We show that a discrete space is sober, if and only if, its cardinality is non-measurable and also show that many well known Borel spaces are sober.     

On the representation of McCarthy's in the -calculus

We study the encoding of , the call-by-name λ-calculus enriched with McCarthy's amb operator, into the π-calculus. Semantically, amb is a challenging operator, for the fairness constraints that it expresses. We prove that, under a certain interpretation of divergence in the λ-calculus (weak divergence), a faithful encoding is impossible. However, with a different interpretation of divergence (strong divergence), the encoding is possible, and for this case we derive results and coinductive proof methods to reason about that are similar to those for the encoding of pure λ-calculi. We then use these methods to derive the most important laws concerning amb. We take bisimilarity as behavioural equivalence on the π-calculus, which sheds some light on the relationship between fairness and bisimilarity.     

Probabilistic π-Calculus and Event Structures

This paper proposes two semantics of a probabilistic variant of the π-calculus: an interleaving semantics in terms of Segala automata and a true concurrent semantics, in terms of probabilistic event structures. The key technical point is a use of types to identify a good class of non-deterministic probabilistic behaviours which can preserve a compositionality of the parallel operator in the event structures and the calculus. We show an operational correspondence between the two semantics. This allows us to prove a “probabilistic confluence” result, which generalises the confluence of the linearly typed π-calculus.     

Linearity and bisimulation

Exploiting linear type structure, we introduce a new theory of weak bisimilarity for the π-calculus in which we abstract away not only τ-actions but also non-τ actions which do not affect well-typed observers. This gives a congruence far larger than the standard bisimilarity while retaining semantic soundness. The framework is smoothly extendible to other settings involving nondeterminism and state. As an application we develop a behavioural theory of secrecy in the π-calculus which ensures secure information flow for a strictly greater set of processes than the type-based approach, while still offering compositional verification techniques.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

Smooth Scheduling under Variable Rates or the Analog-Digital Confinement Game

This work considers non-terminating scheduling problems in which a system of multiple resources serves clients having variable needs. The system has m identical resources and n clients; in each time slot each resource may serve at most one client; in each such slot t each client γ has a rate, a real number ρ _γ (t), that specifies his needs in this slot. The rates satisfy the restriction ∑ _γ ρ _γ (t)≤m for any slot t. Except of this restriction, the rates can vary in arbitrary fashion. (This contrasts most prior works in this area in which the rates of the clients are constant.) The schedule is required to be smooth as follows: a schedule is Δ -smooth if for all time intervals I the absolute difference between the amount of service received by each client γ to his nominal needs of ∑ _t∈I ρ _γ (t) is less than Δ. Our objective are online schedulers that produce Δ-smooth schedules where Δ is a small constant which is independent of m and n. Our paper constructs such schedulers; these are the first online Δ-smooth schedulers, with a constant Δ, for clients with arbitrarily variable rates in a single or multiple resource system. Furthermore, the paper also considers a non-concurrent environment in which there is an additional restriction that each client is served at most once in each time slot; it presents the first online smooth schedulers for variable rates under this restriction. The above non-concurrent restriction is crucial in some applications (e.g., CPU scheduling). It has been pointed out that this restriction “adds a surprising amount of difficulty” to the scheduling problem. However, this observation was never formalized and, of course, was never proved. Our paper formalizes and proves some aspects of this observation. Another contribution of this paper is the introduction of a complete information, two player game called the analog-digital confinement game. In such a game pebbles are located on the real line; the two players, the analog player and the digital player, take alternating turns and each one, in his turn, moves some of the pebbles; the digital player moves the pebbles backwards by discrete distances while the analog player moves the pebbles forward by analog distances; the aim of the analog player is to cause one pebble (or more) to escape a pre-defined real interval while the aim of the digital player is to confine the pebbles into the interval. We demonstrate that this game is a convenient framework to study the general question of how to approximate an analog process by a digital one. All the above scheduling results are established via this game. In this derivation, the pebbles represent the clients, the analog player generates the needs of the clients and the digital player generates the schedule. Dedicated to the memory of Professor Shimon Even for his inspiration and encouragement     

Efficient Stochastic Simulation of Biological Systems with Multiple Variable Volumes

The application of concurrent calculi to the formalisation of biological systems constitutes a promising approach to the analysis in silico of biological phenomena. The Gillespie algorithm is one of the main models exploited for their stochastic simulation. While the original algorithm considers only one fixed-volume compartment, the simulation of biological systems often requires multi-compartment semantics. In this paper we present an enhanced formulation of an extended version of the algorithm which handles multiple compartments with varying volumes. The presented algorithm is used as basis for the implementation of an extension of the stochastic π-Calculus, called Sπ@, which allows an intuitive and concise formalisation of such systems. The algorithm is also efficient in presence of a high number of compartments and reactions, therefore Sπ@ represents the starting point for the development of an effective tool for the simulation of biological systems with dynamical structure even in presence of computationally expensive phenomena like diffusion.     

Robust control and differential games on a finite time horizon

It is well known that theH control problem has a state space formulation in terms of differential games. For a finite time horizon control problem, the analogous differential game is considered. The disturbance is the control for the maximizing player. In order to allow forL ² disturbances, the controls for at least one player must be allowed to be unbounded. It is shown that the value of the game is the viscosity solution of the corresponding Isaacs equation under rather general conditions.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

Differential graphical games for H∞ control of linear heterogeneous multiagent systems

Differential graphical games have been introduced in the literature to solve state synchronization problem for linear homogeneous agents. When the agents are heterogeneous, the previous notion of graphical games cannot be used anymore and a new definition is required. In this paper, we define a novel concept of differential graphical games for linear heterogeneous agents subject to external unmodeled disturbances, which contain the previously introduced graphical game for homogeneous agents as a special case. Using our new formulation, we can solve both the output regulation and H_∞ output regulation problems. Our graphical game framework yields coupled Hamilton‐Jacobi‐Bellman equations, which are, in general, impossible to solve analytically. Therefore, we propose a new actor‐critic algorithm to solve these coupled equations numerically in real time. Moreover, we find an explicit upper bound for the overall ‐gain of the output synchronization error with respect to disturbance. We demonstrate our developments by a simulation example.     

A note on two-sided stochastic control problems

We consider a class of two-sided stochastic control problems. For each continuous process π_t = π_t⁺ − π_t⁻ with bounded variation, the state process (x_t) is defined by x_t = B_t + f₀^t I(x_s - a)^d_πs+ − f₀^t I(x_s a)^dπ_s−, where a is a positive constant and (B_t) is a standard Brownian motion. We show the existence of an optimal policy so as to minimize the cost function J(π) = E [f₀^∞ e^−αsX_s² ds], with discount rate α > 0, associated with π.     

Open Bisimulation, Revisited

In the context of the π-calculus, open bisimulation is prominent and popular due to its congruence properties and its easy implementability. Motivated by the attempt to generalise it to the spi-calculus, we offer a new, more refined definition and show in how far it coincides with the original one.