Abbas Edalat

Bisimulation for labelled Markov processes

In this paper we introduce a new class of labelled transition systems-Labelled Markov Processes-and define bisimulation for them. Labelled Markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete, it could be the reals, for example. We assume that it is a Polish space (the underlying topological space for a complete separable metric space). The mathematical theory of such systems is completely new from the point of view of the extant literature on probabilistic process algebra; of course, it uses classical ideas from measure theory and Markov process theory. The notion of bisimulation builds on the ideas of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result that we prove is that a notion of bisimulation for Markov processes on Polish spaces, which extends the Larsen-Skou definition for discrete systems, is indeed an equivalence relation. This turns our to be a rather hard mathematical result which, as far as we know, embodies a new result in pure probability theory. This work heavily uses continuous mathematics which is becoming an important part of work on hybrid systems.

A computational model for metric spaces

For every metric space X, we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and ω-continuous iff X is separable. The computational model BX is used to give domain-theoretic proofs of Banachs fixed point theorem and of two classical results of Hutchinson: on a complete metric space, every hyperbolic iterated function system has a unique non-empty compact attractor, and every iterated function system with probabilities has a unique invariant measure with bounded support. We also show that the probabilistic power domain of BX provides an ω-continuous computational model for measure theory on a separable complete metric space X.

Dynamical systems, measures, and fractals via domain theory

We introduce domain theory in the computation of dynamical systems, iterated function systems (fractals) and measures. For a discrete dynamical system (X, f), given by the action of a continuous map f: X → X on a metric space X, we study the extended dynamical systems (VX, Vf) and (UX, Uf) where V is the Vietoris functor and U is the upper space functor. In fact, from the point of view of computing the attractors of (X, f), it is natural to study the other two systems: A compact attractor of (X, f) is a fixed point of (VX, Vf) and a fixed point of (UX, Uf). We show that if (X, f) is chaotic, then so is (UX, Uf). When X is locally compact UX is a continuous bounded complete dcpo. If X is second countable as well, then UX will be ω-continuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of non-deterministic functions on CUX where C is the convex (Plotkin) power domain. We also establish an interesting link between measure theory and domain theory. We show that the set, M(X), of Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective way of obtaining measures on X. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.

Domain theory and integration

We present a domain-theoretic framework for measure theory and integration of bounded read-valued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilistic power domain of its upper space. Any bounded Borel measure on the compact metric space can then be obtained as the least upper bound of an /spl omega/-chain of linear combinations of point valuations (simple valuations) on the zipper space, thus providing a constructive setup for these measures. We use this setting to develop a theory of integration based on a new notion of integral which generalises and shares all the basic properties of the Riemann integral. The theory provides a new technique for computing the Lebesgue integral. It also leads to a new algorithm for integration over fractals of iterated function systems. >

DOMAINS FOR COMPUTATION IN MATHEMATICS, PHYSICS AND EXACT REAL ARITHMETIC

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic.

A logical characterization of bisimulation for labeled Markov processes

This paper gives a logical characterization of probabilistic bisimulation for Markov processes. Bisimulation can be characterized by a very weak modal logic. The most striking feature is that one has no negation or any kind of negative proposition. Bisimulation can be characterized by several inequivalent logics; we report five in this paper and there are surely many more. We do not need any finite branching assumption yet there is no need of infinitely conjunction. We give an algorithm for deciding bisimilarity of finite state systems which constructs a formula that witnesses the failure of bisimulation.

A New Representation for Exact Real Numbers

Abstract We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infinite product of matrices, with non-negative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S1, is expressed as the image of the base interval [0, ∞] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first so-called sign matrix determines an interval on which the real number lies. The subsequent so-called digit matrices have non-negative integer coefficients and successively refine that interval. Based on the classification of lfts according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S1 by π/4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.

A domain-theoretic approach to computability on the real line

In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of Stoltenberg-Hansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turing-machine based approach which dates back to Grzegorczyk and Lacombe, is used by Pour-El & Richards in their foundational work on computable analysis, and, moreover, is the standard notion of computability among physicists as in the work of Penrose. Our framework makes it possible to capture partial functions in an elegant way and it extends to the complex numbers and the n-dimensional Euclidean space.

Power Domains and Iterated Function Systems

We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domain-theoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.

Semantics of exact real arithmetic

In this paper, we incorporate a representation of the non-negative extended real numbers based on the composition of linear fractional transformations with non-negative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.