Computational recognition of verbal humour re-mains a challenging task, requiring an understanding of lan-guage, delivery style, emotions, and cultural context. Most existing approaches focus on binary classification and lack datasets that capture psychological dimensions of humour alongside variations in expression. We introduce MultiHuSE, a multi-modal dataset comprising 2,407 high-definition videos of 50 demographically diverse actors performing 1,463 text samples across four psychological humour styles (affiliative, aggressive, self-enhancing, and self-deprecating), as well as neutral content. A subset is additionally annotated for underlying emotions. The dataset uniquely captures multiple actor interpretations of the same texts, enabling systematic analysis of expressive diversity. Baseline experiments show that multimodal fusion outperforms unimodal approaches (80.1 % vs. 77.4% accuracy) in humour style classification, with particularly strong gains for affiliative humour (66 % to 74 %). While text provides the strongest individual signal, fusion models deliver meaningful im-provements. We hope that MultiHuSE provides empirical support for psychological theories linking humour and emotion, while also opening new avenues for research in human communication, well-being, and AI-driven interaction. The dataset is available for academic use under an End-User Licence Agreement.

We present a novel, yet rather simple construction within the traditional framework of Scott domains to provide semantics to probabilistic programming, thus obtaining a solution to a long-standing open problem in this area. We work with the Scott domain of random variables from a standard and fixed probability space---the unit interval or the Cantor space---to any given Scott domain. The map taking any such random variable to its corresponding probability distribution provides a Scott continuous surjection onto the probabilistic power domain of the underlying Scott domain, which preserving canonical basis elements, establishing a new basic result in classical domain theory. If the underlying Scott domain is effectively given, then this map is also computable. We obtain a Cartesian closed category by enriching the category of Scott domains by a partial equivalence relation to capture the equivalence of random variables on these domains. The constructor of the domain of random variables on this category, with the two standard probability spaces, leads to four basic strong commutative monads, suitable for defining the semantics of probabilistic programming.

We construct a continuous domain, as a model of interval analysis, for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlying machine architecture, domain-theoretic methods may be complete, i.e., the result may be obtained to any degree of accuracy. Furthermore, unlike methods based on interval arithmetic which require access to the syntactic representation of the vector field, domain-theoretic methods only deal with the semantics of the field, in the sense that the field is assumed to be given via finitely-representable approximations, to within any required accuracy. In contrast to the domain-theoretic first-order Euler method, the second-order method uses the local Lipschitz properties of the field. This is achieved by using a domain for Lipschitz functions, whose elements are consistent pairs that provide approximations of the field and its local Lipschitz properties. In the special case where the field is differentiable, the local Lipschitz properties are exactly the local differential properties of the field. In solving IVPs, Lipschitz continuity of the field is a common assumption, as a sufficient condition for uniqueness of the solution. While the validated methods for solving IVPs commonly impose further restrictions on the vector field, the second-order Euler method requires no further condition. In this sense, the method may be seen as the most general of its kind. To avoid complicated notations and lengthy arguments, the results of the paper are stated for the second-order Euler method. Nonetheless, the framework, and the results, may be extended to any higher-order Euler method, in a straightforward way.

We consider the domain of non-empty convex and compact subsets of a finite dimensional Euclidean space to represent partial or imprecise points in computational geometry. The convex hull map on such imprecise points is given domain-theoretically by an inner and an outer convex hull. We provide a practical algorithm to compute the inner convex hull when there are a finite number of convex polytopes as partial points. A notion of pre-inner support function is introduced, whose convex hull gives the support function of the inner convex hull in a general setting. We then show that the convex hull map is Scott continuous and can be extended to finitely generable subsets, represented by the Plotkin power domain of the underlying domain. This in particular allows us to compute, for the first time, the convex hull of attractors of iterated function systems in fractal geometry. Finally, we derive a program logic for the convex hull map in the sense of the weakest pre-condition for a given post-condition and show that the convex hull predicate transformer is computable.

We develop a domain-theoretic Differential Calculus for locally Lipschitz functions on finite dimensional real spaces with imprecise input/output. The inputs to these functions are hyper-rectangles and the outputs are compact real intervals. This extends the domain of application of Interval Analysis and exact arithmetic to the derivative. A new notion of a tie for these functions is introduced, which in one dimension represents a modification of the notion previously used in the one-dimensional framework. A Scott continuous sub-differential for these functions is then constructed, which satisfies a weaker form of calculus compared to that of the Clarke sub-gradient. We then adopt a Program Logic viewpoint using the equivalence of the category of stably locally compact spaces with that of semi-strong proximity lattices. We show that given a localic approximable mapping representing a locally Lipschitz map with imprecise input/output, a localic approximable mapping for its sub-differential can be constructed, which provides a logical formulation of the sub-differential operator.

We introduce a logical theory of differentiation for a real-valued function on a finite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two stably locally compact Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the L-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Approximable mappings of the latter type form a bounded complete domain isomorphic with the function space of Scott continuous functions of a real variable into the domain of non-empty compact and convex subsets of the finite dimensional Euclidean space partially ordered with reverse inclusion. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic L-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map defined domain-theoretically using strong single-ties. A stricter notion of strong single-knots is subsequently developed which captures approximable mappings of continuously differentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satisfied by the Clarke subgradient.

We build on a previous neural model of bonding circuitry, in which the orbitofrontal cortex mediates between facilitative and stress reactivity to social stimuli, via the dorsomedial and paraventricular nucleus of the hypothalamus. We integrate recent neuroscientific findings, and consider how the introduction of additional reward could drive a further, counter-conditioning based re-balancing mechanism between activation of these networks, via increasing prefrontal-driven inhibition of the central nucleus of the amygdala. We simulate our model computationally, and hypothesise that such a process may be involved in a particular phase of self-bonding in the newly introduced self-attachment psychotherapy.

We introduce the notion of self-attachment which, based on an interdisciplinary set of concepts, proposes a new psychotherapeutic technique. The underlying ideas include findings and paradigms in developmental psychology and neuroscience, neuroplasticity and long term term potentiation, fMRI studies on human bond making and religious experience, and experiments in energy based artificial neural networks. The proposed self-attachment therapeutic technique is distinguished by its intervention to create an internal and passionate affectional bond within the individual between the “adult self”, representing the logical and cognitive faculty, and the “inner child”, representing the unregulated and undeveloped emotional circuits. The aim is to create more optimal circuits for emotional regulation. The proposed self-attachment protocols internally emulate within the individual the interactions of a good enough primary care-giver and child in order to moderate the child's arousal level, minimise its negative affects and maximize its positive affects. These interactions are assumed, in developmental neuroscience and in developmental psychology, to be the basis of secure attachment of children with their parents, which leads to an optimal regulation of neurotransmitters, hormones, and the emotional dynamics of the individual. We report on several case studies of this technique in recent years. Finally, we propose a simple mathematical model to capture the impact of self-attachment protocols using the notion of strong patterns in energy based neural networks and employ a recently developed mathematical model to examine the impact of self-attachment using emotional and cognitive neural pathways for decision making.

We introduce in the context of differential and integral calculus several key extensions of higher order maps from a dense subset of a topological space into a continuous Scott domain. These higher order maps include the classical derivative operator and the Riemann integration operator. Using a sequence of test functions, we prove that the subspace of real-valued continuously differentiable functions on a finite dimensional Euclidean space is dense in the space of Lipschitz maps equipped with the L-topology. This provides a new result in basic mathematical analysis, which characterises the L-topology in terms of the limsup of the sequence of derivatives of a sequence of C1 maps that converges to a Lipschitz map. Using this result, it is also shown that the generalised (Clarke) gradient on Lipschitz maps is the extension of the derivative operator on C1 maps. We show that the generalised Riemann integral (R-integral) of a real-valued continuous function on a compact metric space with respect to a Borel measure can be extended to the integral of interval-valued functions on the metric space with respect to valuations on the probabilistic power domain of the space of non-empty and compact sets of the metric space. We also prove that the Lebesgue integral operator on integrable functions is the extension of the R-integral operator on continuous functions. We finally illustrate an application of these results by deriving a simple proof of Green's theorem for interval-valued vector fields.

We introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions.

We introduce and implement a neural model for mentalization/mindfulness based psychotherapy. It uses Dan Levine's neural model of pathways for emotional-cognitive decision making, which is integrated with a competitive Hopfield network built up from the new concept of strong patterns for the six basic emotions and for mentalization or mindfulness. We adopt a particular form of Q-learning to reinforce the mentalizing/mindful pattern in the network, which represents the process of psychotherapy. In a successful course of therapy, the mentalizing/mindful pattern becomes the more dominant pattern compared to negative emotions and the brain makes decisions that are more deliberate and thoughtful than heuristic and automatic.

We study the notion of a strong attractor of a Hopfield neural model as a pattern that has been stored multiple times in the network, and examine its properties using basic mathematical techniques as well as a variety of simulations. It is proposed that strong attractors can be used to model attachment types in developmental psychology as well as behavioural patterns in psychology and psychotherapy. We study the stability and basins of attraction of strong attractors in the presence of other simple attractors and show that they are indeed more stable with a larger basin of attraction compared with simple attractors. We also show that the perturbation of a strong attractor by random noise results in a cluster of attractors near the original strong attractor measured by the Hamming distance. We investigate the stability and basins of attraction of such clusters as the noise increases and establish that the unfolding of the strong attractor, leading to its breakup, goes through three different stages. Finally the relation between strong attractors of different multiplicity and their influence on each other are studied and we show how the impact of a strong attractor can be replaced with that of a new strong attractor. This retraining of the network is proposed as a model of how attachment types and behavioural patterns can undergo change.

We solve the mean field equations for a stochastic Hopfield network with temperature (noise) in the presence of strong, i.e., multiply stored patterns, and use this solution to obtain the storage capacity of such a network. Our result provides for the first time a rigorous solution of the mean field equations for the standard Hopfield model and is in contrast to the mathematically unjustifiable replica technique that has been hitherto used for this derivation. We show that the critical temperature for stability of a strong pattern is equal to its degree or multiplicity, when sum of the cubes of degrees of all stored patterns is negligible compared to the network size. In the case of a single strong pattern in the presence of simple patterns, when the ratio of the number of all stored patterns and the network size is a positive constant, we obtain the distribution of the overlaps of the patterns with the mean field and deduce that the storage capacity for retrieving a strong pattern exceeds that for retrieving a simple pattern by a multiplicative factor equal to the square of the degree of the strong pattern. This square law property provides justification for using strong patterns to model attachment types and behavioural prototypes in psychology and psychotherapy.

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterisation of the basic open subsets of the L-topology in terms of ties or primitive maps of functions. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative.

We introduce a computable framework for Lebesgue's measure and integration theory in the spirit of domain theory. For an effectively given locally compact second countable Hausdorff space and an effectively given locally finite Borel measure on the space, we define the notion of a computable measurable set with respect to the given measure, which is stronger than Sanin's recursive measurable set. The set of computable measurable subsets is closed under complementation, finite unions and finite intersections. We then introduce interval-valued measurable functions and develop the notion of computable measurable functions using interval-valued simple functions. This leads us to the interval versions of the main results of the theory of Lebesgue integration which provide a computable framework for measure and integration theory. The Lebesgue integral of a computable integrable function with respect to an effectively given (sigma-)finite Borel measure on an effectively given (locally) compact second countable Hausdorff space can be computed up to any required accuracy. We show that, with respect to the metric induced from the L1norm, the set of Scott continuous interval-valued functions is dense in the set of interval-valued integrable functions.

We construct a domain-theoretic calculus for Lipschitz and differentiate functions, which includes addition, subtraction and composition. We then develop a domain-theoretic version of the inverse function theorem for a Lipschitz function, in which the inverse function is obtained as a fixed point of a Scott continuous functional and is approximated by step functions. In the case of a C/sup 1/ function, the inverse and its derivative are obtained as the least fixed point of a single Scott continuous functional on the domain of differentiable functions and are approximated by two sequences of step functions, which are effectively computed from two increasing sequences of step functions respectively converging to the original function and its derivative. In this case, we also effectively obtain an increasing sequence of polynomial step functions whose lower and upper bounds converge in the C/sup 1/ norm to the inverse function. A similar result holds for implicit functions, which combined with the domain-theoretic model for computational geometry, provides a robust technique for construction of curves and surfaces.

Domain theory, which was introduced by Dana Scott in the early 1970s, works on the principle of incorporating partial elements into denotational domains. The resulting structures are ordered by a natural notion of refinement or approximation, and they carry a topology that expresses the process of passing to a limit.

We introduce a domain-theoretic framework for differential calculus. We define the set of primitive maps as well as the derivative of an interval-valued Scott continuous function on the domain of intervals, and show that they are dually related, providing an extension of the classical duality of differentiation and integration as in the fundamental theorem of calculus. It is shown that, for locally Lipschitz functions of a real variable, the domain-theoretic derivative coincides with the Clarke's derivative. We then construct a domain for differentiable real-valued functions of a real variable by pairing consistent information about the function and information about its derivative. The set of classical $C^1$ functions, equipped with its $C^1$ norm, is embedded into the set of maximal elements of this countably based, bounded complete continuous domain. This domain also provides a model for the differential properties of piecewise $C^1$ functions, locally Lipschitz functions and more generally of all continuous functions. We prove that consistency of function information and derivative information is decidable on rational step functions, which shows that our domain can be given an effective structure. We thus obtain a data type for differential calculus. As an immediate application, we present a domain-theoretic formulation of Picard's theorem, which provides a data type for solving differential equations.

We provide an algorithmic formalization of ordinary differential equations in the framework of domain theory. Given a Scott continuous, interval-valued and time-dependent scalar field and a Scott continuous initial function consistent with the scalar field, the domain-theoretic analogue of the classical Picard operator, whose fix-points give the solutions of the differential equation, acts on the domain of continuously differentiable functions by successively updating the information about the solution and the information about its derivative. We present a linear and a quadratic algorithm respectively for updating the function information and the derivative information on the basis elements of the domain. In the generic case of a classical initial value problem with a continuous scalar field, which is Lipschitz in the space component, this provides a novel technique for computing the unique solution of the differential equation up to any desired accuracy, such that at each stage of computation one obtains two continuous piecewise linear maps which bound the solution from below and above, thus giving the precise error. When the scalar field is continuous and computable but not Lipschitz, it is known that no computable classical solution may exist. We show that in this case the interval-valued domain-theoretic solution is computable and contains all classical solutions. This framework also allows us to compute an interval-valued solution to a differential equation when the initial value and/or the scalar field are interval-valued, i.e. imprecise.

A data-type for differential calculus is introduced, which is based on domain theory. We define the integral and also the derivative of a Scott continuous function on the domain of intervals, and present a domain-theoretic generalization of the fundamental theorem of calculus. We then construct a domain for differentiable real valued functions of a real variable. The set of classical C/sup 1/ functions, equipped with its C/sup 1/ norm, is embedded into the set of maximal elements of this domain, which is a countably based bounded complete continuous domain. This gives a data type for differential calculus. The construction can be generalized to C/sup k/ and C/sup /spl infin// functions. As an immediate application, we present a domain-theoretic generalization of Picard's theorem, which provides a data type for solving differential equations.

We provide a semantical framework for exact real arithmetic using linear fractional transformations on the extended real line. We present an extension of PCF with a real type which introduces an eventually breadth-first strategy for lazy evaluation of exact real numbers. In this language, we present the constant redundant if, rif, for defining functions by cases which, in contrast to parallel if (pif), overcomes the problem of undecidability of comparison of real numbers in finite time. We use the upper space of the one-point compactification of the real line to develop a denotational semantics for the lazy evaluation of real programs. Finally two adequacy results are proved, one for programs containing rif and one for those not containing it. Our adequacy results in particular provide the proof of correctness of algorithms for computation of single-valued elementary functions. 1 Introduction It is well known that the accumulation of round-off errors in floating point programs can l...

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.

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.

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.

We construct an approximating chain of simple valuations on the upper space of a compact metric space whose lub is a given probability measure on the metric space. We show that whenever a separable metric space is homeomorphic to a G δ subset of an ω-continuous dcpo equipped with its Scott topology, the space of probability measures of the metric space equipped with the weak topology is homeomorphic with a subset of the maximal elements of the probabilistic power domain of the ω-continuous dcpo. Given an effective approximation of a probability measure by an increasing chain of normalised valuations on the upper space of a compact metric space, we show that the expected value of any Hölder continuous function on the space can be obtained up to any given accuracy. We present a novel application in computing integrals in dynamical systems. We obtain an algorithm to compute the expected value of any Hölder continuous function with respect to the unique invariant measure of the Feigenbaum map in the periodic doubling route to chaos.

Given a probability measure on a compact metric space, we construct an increasing chain of valuations on the upper space of the metric space whose least upper bound is the measure. We then obtain the expected value of any Holder continuous function with respect to the measure up to any precision. We prove that the Scott topology induces the weak topology of the space of probability measures in the following general setting: Whenever a separable metric space is embedded into a subset of the maximal elements of an /spl omega/-continuous dcpo, which is a G/sub /spl delta// subset of the dcpo equipped with the Scott topology, we show that the space of probability measures of the metric space equipped with the weak topology is then embedded into a subspace of the maximal elements of the probabilistic power domain of the dcpo. We present a novel application in the theory of periodic doubling route to chaos.

Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number of steps. Based on a domain-theoretic approach to integration, we show how to define integration in Real PCF. We propose two approaches to integration in Real PCF. One consists in adding integration as primitive. The other consists in adding a primitive for maximization of functions and then recursively defining integration from maximization. In both cases we have an adequacy theorem for the corresponding extension of Real PCF. Moreover based on previous work on Real PCF definability, we show that Real PCF extended with the maximization operator is universal, which implies that it is also fully abstract.

Fractal images defined by an iterated function system (IFS) are specified by a finite number of contractive affine transformations. In order to plot the image specified by the transformations on the screen of a digital computer, it is necessary to determine a bounding area for the image. This paper derives a formula that expresses the dimensions of this bounding area in terms of the transformations.

We establish domain-theoretic models of finite-state discrete stochastic processes, Markov processes and vector recurrent iterated function systems. In each case, we show that the distribution of the stochastic process is canonically obtained as the least upper bound of an increasing chain of simple valuations in a probabilistic power domain associated to the process. This leads to various formulas and algorithms to compute the expected values of functions which are continuous almost everywhere with respect to Me distribution of the stochastic process. We prove the existence and uniqueness of the invariant distribution of a vector recurrent iterated function system which is used in fractal image compression. We also present a finite algorithm to decode the image.

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.>