## Publications

105 results found

Edalat A, Farjudian A, Mohammadian M, et al., 2020, Domain theoretic second-order Euler’s method for solving initial valueproblems, Mathematical Foundations of Programming Semantics, Publisher: Elsevier, Pages: 105-128, ISSN: 1571-0661

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.

Davari MJ, Edalat A, Lieutier A, 2020, The convex hull of finitely generable subsets and its predicate transformer, Thirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Publisher: ACM/IEEE

We consider the domain of non-empty convex andcompact subsets of a finite dimensional Euclidean space torepresent partial or imprecise points in Computational Geom-etry. The convex hull map on such imprecise points is givendomain-theoretically by an inner and an outer convex hull. Weprovide a practical algorithm to compute the inner convex hullwhen there are a finite number of convex polytopes as partialpoints. A notion of pre-inner support function is introduced,whose convex hull gives the support function of the innerconvex hull in a general setting. We then show that the convexhull map is Scott continuous and can be extended to finitelygenerable subsets, represented by the Plotkin power domain ofthe underlying domain. This in particular allows us to computethe convex hull of attractors of iterated function systems infractal geometry. Finally, we derive a program logic for theconvex hull map in the sense of the weakest pre-condition fora given post-condition.

Edalat A, Ghorban S, Ghoroghi A, 2018, Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments, *Games*, Vol: 9, ISSN: 2073-4336

We show that a Bayesian game where the type space of each agent is a bounded set of m-dimensional vectors with non-negative components and the utility of each agent depends linearly on its own type only is equivalent to a simultaneous competition in m basic games which is called a uniform multigame. The type space of each agent can be normalised to be given by the ( m - 1 ) -dimensional simplex. This class of m-dimensional Bayesian games, via their equivalence with uniform multigames, can model decision making in multi-environments in a variety of circumstances, including decision making in multi-markets and decision making when there are both material and social utilities for agents as in the Prisoner’s Dilemma and the Trust Game. We show that, if a uniform multigame in which the action set of each agent consists of one Nash equilibrium inducing action per basic game has a pure ex post Nash equilibrium on the boundary of its type profile space, then it has a pure ex post Nash equilibrium on the whole type profile space. We then develop an algorithm, linear in the number of types of the agents in such a multigame, which tests if a pure ex post Nash equilibrium on the vertices of the type profile space can be extended to a pure ex post Nash equilibrium on the boundary of its type profile space in which case we obtain a pure ex post Nash equilibrium for the multigame.

Edalat A, Maleki M, 2018, Differential calculus with imprecise input and its logical framework, International Conference on Foundations of Software Science and Computation Structures 2018, Publisher: Springer International Publishing, Pages: 459-475

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.

Cittern D, Nolte T, Friston K,
et al., 2018, Intrinsic and extrinsic motivators of attachment under active inference, *PLoS ONE*, Vol: 13, ISSN: 1932-6203

This paper addresses the formation of infant attachment types within the context ofactive inference: a holistic account of action, perception and learning in the brain. Weshow how the organised forms of attachment (secure, avoidant and ambivalent) mightarise in (Bayesian) infants. Specifically, we show that these distinct forms of attachmentemerge from a minimisation of free energy – over interoceptive states relating to internalstress levels – when seeking proximity to caregivers who have a varying impact on theseinteroceptive states. In line with empirical findings in disrupted patterns of affectivecommunication, we then demonstrate how exteroceptive cues (in the form ofcaregiver-mediated AMBIANCE affective communication errors, ACE) can result indisorganised forms of attachment in infants of caregivers who consistently increase stresswhen the infant seeks proximity, but can have an organising (towards ambivalence)effect in infants of inconsistent caregivers. In particular, we differentiate disorganisedattachment from avoidance in terms of the high epistemic value of proximity seekingbehaviours (resulting from the caregiver’s misleading exteroceptive cues) that precludethe emergence of coherent and organised behavioural policies. Our work, the first toformulate infant attachment in terms of active inference, makes a new testableprediction with regards to the types of affective communication errors that engenderambivalent attachment.

Edalat A, Maleki M, 2017, Differentiation in logical form, Logic in Computer Science (LICS 2017), Publisher: ACM / IEEE

We introduce a logical theory of differentiation for areal-valued function on a finite dimensional real Euclidean space.A real-valued continuous function is represented by a localic ap-proximable mapping between two semi-strong proximity lattices,representing the two stably locally compact Euclidean spaces forthe domain and the range of the function. Similarly, the Clarkesubgradient, equivalently the L-derivative, of a locally Lipschitzmap, which is non-empty, compact and convex valued, is repre-sented by an approximable mapping. Approximable mappings ofthe latter type form a bounded complete domain isomorphic withthe function space of Scott continuous functions of a real variableinto the domain of non-empty compact and convex subsets ofthe finite dimensional Euclidean space partially ordered withreverse inclusion. Corresponding to the notion of a single-tie ofa locally Lipschitz function, used to derive the domain-theoreticL-derivative of the function, we introduce the dual notion ofa single-knot of approximable mappings which gives rise toLipschitzian approximable mappings. We then develop the notionof a strong single-tie and that of a strong knot leading to aStone duality result for locally Lipschitz maps and Lipschitzianapproximable mappings. The strong single-knots, in which aLipschitzian approximable mapping belongs, are employed todefine the Lipschitzian derivative of the approximable mapping.The latter is dual to the Clarke subgradient of the correspondinglocally Lipschitz map defined domain-theoretically using strongsingle-ties. A stricter notion of strong single-knots is subsequentlydeveloped which captures approximable mappings of continu-ously differentiable maps providing a gradient Stone dualityfor these maps. Finally, we derive a calculus for Lipschitzianderivative of approximable mapping for some basic constructorsand show that it is dual to

Bilokon P, Edalat A, 2017, A domain-theoretic approach to Brownian motion and general continuous stochastic processes, *Theoretical Computer Science*, Vol: 691, Pages: 10-26, ISSN: 0304-3975

We introduce a domain-theoretic framework for continuous-time, continuous-statestochastic processes. The laws of stochastic processes are embedded into the spaceof maximal elements of the normalised probabilistic power domain on the space ofcontinuous interval-valued functions endowed with the relative Scott topology. We usethe resultingω-continuous bounded complete dcpo to obtain partially defined stochas-tic processes and characterise their computability. For a given continuous stochasticprocess, we show how its domain-theoretic, i.e., finitary, approximations can be con-structed, whose least upper bound is the law of the stochastic process. As a mainresult, we apply our methodology to Brownian motion. We construct a partially de-fined Wiener measure and show that the Wiener measure is computable within thedomain-theoretic framework.

Cittern D, Edalat A, 2017, A neural model of empathic states in attachment-based psychotherapy, *Computational Psychiatry*, Vol: 1, Pages: 132-167, ISSN: 2379-6227

We build on a neuroanatomical model of how empathic states can motivatecaregiving behaviour, via empathy circuit-driven activation of regions in thehypothalamus and amygdala which in turn stimulate a mesolimbic-ventral pal-lidum pathway, by integrating findings related to the perception of pain in selfand others. Based on this we propose a network to capture states of personaldistress and empathic concern, which are particularly relevant for psychothera-pists conducting attachment-based interventions. This model is then extendedfor the case of Self-Attachment therapy in which conceptualised components ofthe self serve as both the source of and target for empathic resonance, and weconsider how states of empathic concern involving an other that is perceived asbeing closely related to the self might enhance the motivation for self-directedbonding. We simulate our model computationally, and discuss the interplaybetween the bonding and empathy protocols of the therapy.

Cittern D, Edalat A, Ghaznavi I, 2017, An immersive virtual reality mobile platform for self-attachment, Artificial Intelligence and Simulation of Behaviour (AISB) 2017, Publisher: AISB

Psychotherapy is among the most effective techniques forcombating mental health issues, and virtual reality is beginning to beexplored as a way to enhance the efficacy of various psychotherapeu-tic treatments. In this paper we propose an immersive virtual realitymobile platform for Self-Attachment psychotherapy. Under the Self-Attachment therapeutic framework, the causes of disorders such aschronic anxiety and depression are traced back to the quality of theindividual’s attachment with their primary caregiver during child-hood. Our proposed platform aims to assist the user in enhancingtheir capacities for self-regulation of emotion, by means of earningsecure attachment through the experience of positive attachment in-teractions, missed in their childhood. In the virtual environment pro-vided by the platform, the adult-self of the user learns to create andstrengthen an affectional and supportive bond with the inner-child.It is hypothesised that by long term potentiation and neuroplasticity,the user gradually develops new neural pathways and matures intoan effective secure attachment object for the inner-child, thereby en-abling the self-regulation of emotions.

Edalat A, 2017, Self attachment: A holistic approach to computational psychiatry, Computational Neurology and Psychiatry, Editors: Peter, Bhattacharya, Cochran, Publisher: Springer, Pages: 273-314, ISBN: 9783319499581

There has been increasing evidence to suggest that the root cause of muchmental illness lies in a sub-optimal capacity for affect regulation. Cognition andemotion are intricately linked and cognitive deficits, which are characteristic ofmany psychiatric conditions, are often driven by affect dysregulation, which itselfcan usually be traced back to sub-optimal childhood development. This view is supported by Attachment Theory, a scientific paradigm in developmental psychology,that classifies the type of relationship a child has with a primary care-giver to one offour types of insecure or secure attachments. Individuals with insecure attachment intheir childhoods are prone to a variety of mental illness, whereas a secure attachmentin childhood provides a secure base in life. We therefore propose, based on previouswork, a holistic approach to Computational Psychiatry, which is informed by thedevelopment of the brain during infancy in social interaction with its primary care-givers. We identify the protocols governing the interaction of a securely attachedchild with its primary care-givers that produce the capacity for affect regulation inthe child. We contend that these protocols can be self-administered to construct,by neuroplasticity and long term potentiation, new “optimal” neural pathways inthe brains of adults with insecure attachment history. This procedure is called Self-attachment and aims to help individuals create their own attachment objects whichhas many parallels with Winnicott’s notion of transitional object, Bowlby’s comfort objects, Kohut’s empathetic self-object as well as religion as an attachment object. We describe some mathematical models for Self-attachment: a game-theoreticmodel, a model based on the notion of a strong pattern in an energy based associativeneural network and several neural models of the human brain.

White J, Edalat A, 2016, Iran is ready to thrive, *NEW SCIENTIST*, Vol: 229, Pages: 29-29, ISSN: 0262-4079

Edalat A, 2015, Introduction to self-attachment and its neural basis, The 2015 International Joint Conference on Neural Networks (IJCNN), Publisher: IEEE, Pages: 1-8

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, ethology and psychology of religion 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 ognitive neural pathways for decision making.

Cittern D, Edalat A, 2015, Towards a Neural Model of Bonding in Self-Attachment, International Joint Conference on Neural Networks (IJCNN), Publisher: IEEE, ISSN: 2161-4393

Cittern D, Edalat A, 2015, Reinforcement Learning for Nash Equilibrium Generation, Autonomous Agents and Multiagent Systems (AAMAS), Publisher: International Foundation for Autonomous Agents and Multiagent Systems, Pages: 1727-1728

Edalat A, 2015, Extensions of domain maps in differential and integral calculus, Logic in Computer Science (LICS) 2015, Publisher: IEEE, Pages: 426-437, ISSN: 1043-6871

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 Ltopology. 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.

Edalat A, 2014, A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations, *Theoretical Computer Science*, Vol: 564, Pages: 89-106, ISSN: 0304-3975

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.

Bilokon P, Edalat A, 2014, A domain-theoretic approach to Brownian motion and general continuous stochastic processes, Twenty-Ninth Annual ACM/IEEE Symposium on LOGIC IN COMPUTER SCIENCE (LICS), Publisher: ACM

We introduce a domain-theoretic framework for continuous-time,continuous-state stochastic processes. The laws of stochastic processesare embedded into the space of maximal elements of thenormalised probabilistic power domain on the space of continuousinterval-valued functions endowed with the relative Scott topology.We use the resulting !-continuous bounded complete dcpo to definepartial stochastic processes and characterise their computability.For a given continuous stochastic process, we show how itsdomain-theoretic, i.e., finitary, approximations can be constructed,whose least upper bound is the law of the stochastic process. As amain result, we apply our methodology to Brownian motion. Weconstruct a partial Wiener measure and show that the Wiener measureis computable within the domain-theoretic framework.

Edalat A, Lin Z, 2014, A Neural Model of Mentalization/Mindfulness based Psychotherapy, The 2014 International Joint Conference on Neural Networks (IJCNN 2014)

We introduce and implement a neural model for mentalization/mindfulness basedpsychotherapy. It uses Dan Levine’s neural model of pathways foremotional-cognitive decision making, which is integratedwith a competitive Hopfield network built up from the new concept of strong patternsfor the six basic emotions and for mentalization ormindfulness. We adopt a particular form of Q-learning toreinforce the mentalizing/mindful pattern in the network,which represents the process of psychotherapy. In a successfulcourse of therapy, the mentalizing/mindful pattern becomes themore dominant pattern compared to negative emotions and thebrain makes decisions that are more deliberate and thoughtfulthan heuristic and automatic.

Cittern D, Edalat A, 2014, An Arousal-Based Neural Model of Infant Attachment, IEEE Symposium on Computational Intelligence, Cognitive Algorithms, Mind, and Brain, Publisher: IEEE, Pages: 57-64

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Edalat A, 2013, Capacity of strong attractor patterns to modelbehavioural and cognitive prototypes, Neural Information Processing Systems (NIPS) 2013

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 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.

Edalat A, Mancinelli F, 2013, Strong Attractors of Hopfield Neural Networks to ModelAttachment Types and Behavioural Patterns, Piscataway, New Jersey, USA, International Joint Conference on Neural Networks (IJCNN 2013), Publisher: IEEE

We study the notion of a strong attractor ofa Hopfield neural model as a pattern that has been storedmultiple times in the network, and examine its propertiesusing basic mathematical techniques as well as a variety ofsimulations. It is proposed that strong attractors can be usedto model attachment types in developmental psychology as wellas behavioural patterns in psychology and psychotherapy. Westudy the stability and basins of attraction of strong attractorsin the presence of other simple attractors and show that they areindeed more stable with a larger basin of attraction comparedwith simple attractors. We also show that the perturbationof a strong attractor by random noise results in a cluster ofattractors near the original strong attractor measured by theHamming distance. We investigate the stability and basins ofattraction of such clusters as the noise increases and establishthat the unfolding of the strong attractor, leading to its break-up, goes through three different stages. Finally the relationbetween strong attractors of different multiplicity and theirinfluence on each other are studied and we show how the impactof a strong attractor can be replaced with that of a new strongattractor. This retraining of the network is proposed as a modelof how attachment types and behavioural patterns can undergochange.

Di Gianantonio P, Edalat A, 2013, A Language for Differentiable Functions, 16th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS), Publisher: Springer, Pages: 337-352

We introduce a typed lambda calculus in which real numbers, realfunctions, and in particular continuously differentiable and more generally Lipschitz functions canbe defined. Given an expression representing a real-valued functionof a real variable in this calculus, we are able to evaluate the expression on anargument but also evaluate the generalised derivative, i.e., theL-derivative, equivalently the Clarke gradient, of the expression on anargument. Moreover, the whole hierarchy of higher order functions onreal numbers can be defined. The language is an extension of PCF witha real number data-type, similar to Real PCF and RL, but is equippedwith primitives for min and max and weighted average to capturecomputable continuously differentiable or Lipschitz functions on real numbers. Wepresent an operational semantics and a denotational semantics based oncontinuous Scott domains and several logical relations on thesedomains. We then prove an adequacy result for the two semantics. Thedenotational semantics is closely linked with AutomaticDifferentiation also called Algorithmic Differentiation, which hasbeen an active area of research in numerical analysis for decades, andour framework can also be considered as providing denotationalsemantics for Automatic Differentiation. We derive a definabilityresult showing that for any computable Lipschitz function there is aclosed term in the language whose evaluation on any real numbercoincides with the value of the function and whose derivativeexpression also evaluates on the argument to the value of thegeneralised derivative of the function.

Edalat A, Ghoroghi A, Sakellariou G, 2012, Multi-games and a double game extension of the Prisoner’s Dilemma, 10th Conference on Logic and the Foundations of Game and Decision Theory

We propose a new class of games, called Multi-Games (MG), in which a given number of players play a fixed number of basic games simultaneously. Each player can have different sets of strategies for the different basic games. In each round of the MG, each player will have a specific set of weights, one for each basic game, which add up to one and represent the fraction of the player's investment in each basic game. The total payoff for each player is then the convex combination, with the corresponding weights, of the payoffs it obtains in the basic games. The basic games in a MG can be regarded as different environments for the players, and, in particular, we submit that MG can be used to model investment in a global economy with different national or continental markets. When the players' weights for the different games in MG are private information or types with given conditional probability distributions, we obtain a particular class of Bayesian games. We show that for the class of so-called completely pure regular Double Game (DG) with finite sets of types, the Nash equilibria (NE) of the basic games can be used to compute a Bayesian Nash equilibrium of the DG in linear time with respect to the number of types of the players. We study a DG for the Prisoner's Dilemma (PD) by extending the PD with a second so-called Social Game (SG), generalising the notion of altruistic extension of a game in which players have different altruistic levels (or social coefficients). We show that, with respect to the SG we choose, the payoffs for PD give rise to two different types of DG's. We study two different examples of Bayesian games in this context in which the social coefficients have a finite set of values and each player only knows the probability distribution of the opponent's social coefficient. In the first case we have a completely pure regular DG for which we deduce a Bayesian NE. Finally, we use the second example to compare various strategies in a round-robin tou

Edalat A, Lieutier A, Pattinson D, 2012, A Computational Model for Multi-Variable Differential Calculus, *Informarion and Computation*, Vol: 224, Pages: 22-45, ISSN: 0890-5401

We develop a domain-theoretic computational model for multi-variabledifferential calculus, which for the first time gives rise to datatypes for piecewise differentiable or more generally Lipschitz functions, by constructing an effectively given continuous Scott domain for real-valued Lipschitz functions on finite dimensional Euclidean spaces. The model for real-valued Lipschitz functions of $n$ variables, is built as asub-domain of the product of two domains by tupling together consistent information aboutlocally Lipschitz functions and theirdifferential properties as given by their L-derivative or equivalently Clarke gradient, which has values given by non-empty, convex and compact subsets of $\R^n$. To obtain a computationally practical framework, the derivative information is approximated by the best fit compact hyper-rectangles in $\R^n$. In this case, we show that consistency of the function and derivative information can be decided by reducing it to a linear programming problem. This provides an algorithm to check consistency on the rationalbasis elements of the domain, implying that the domain can be equipped with aneffective structure and giving a computable framework formulti-variable differential calculus.We also develop a domain-theoretic, interval-valued, notion of line integraland show that if a Scott continuous function, representing a non-empty, convex and compact valued vector field, is integrable, then its interval-valued integral over any closed piecewise $C^1$ path contains zero. In the case that the derivative information is given interms of compact hyper-rectangles, we use techniques from the theory of minimal surfaces to deduce the converse result: a hyper-rectangular valued vector field is integrable if its interval-valued line integral over any piecewise $C^1$ path contains zero. This gives a domain-theoretic extension of the fundamental theorem of path integration. Finally, weconstruct the least and the greatest piecewise linear functionst

He P, Edalat A, 2011, Visual hull from imprecise polyhedral scene, *Proceedings - 2011 International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission, 3DIMPVT 2011*, Pages: 164-171

We present a framework to compute the visual hull of a polyhedral scene, in which the vertices of the polyhe-dra are given with some imprecision. Two kinds of visual event surfaces, namely VE and EEE surfaces are modelled under the geometric framework to derive their counterpart object, namely partial VE and partial EEE surfaces, which contain the exact information of all possible visual event surfaces given the imprecision in the input. Correspondingly, a new definition of visual number is proposed to label the cells of Euclidean space partitioned by partial VE and partial EEE surfaces. The overall algorithm maintains the same computational complexity as the classical method and generates a partial visual hull which converges to the classical visual hull as the input converges to an exact value. © 2011 IEEE.

Edalat A, 2010, A differential operator and weak topology for Lipschitz maps, *TOPOLOGY AND ITS APPLICATIONS*, Vol: 157, Pages: 1629-1650, ISSN: 0166-8641

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Edalat A, 2009, A computable approach to measure and integration theory, *INFORMATION AND COMPUTATION*, Vol: 207, Pages: 642-659, ISSN: 0890-5401

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Longo G, Asarin E, Barr M,
et al., 2009, Editors' note: bibliometrics and the curators of orthodoxy, *MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE*, Vol: 19, Pages: 1-4, ISSN: 0960-1295

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Edalat A, 2008, Weak topology and a differentiable operator for Lipschitz maps, 23rd Annual IEEE Symposium on Logic in Computer Science, Publisher: IEEE COMPUTER SOC, Pages: 364-375, ISSN: 1043-6871

Edalat A, Pattinson D, 2007, Denotational semantics of hybrid automata, *Journal of Logic and Algebraic Programming*, Vol: 73, Pages: 3-21, ISSN: 1567-8326

We introduce a denotational semantics for non-linear hybrid automata and relate it to the operational semantics given in terms of hybrid trajectories. The semantics is defined as least fixpoint of an operator on the continuous domain of functions of time that take values in the lattice of compact subsets of n-dimensional Euclidean space. The semantic function assigns to every point in time the set of states the automaton can visit at that time, starting from one of its initial states. Our main results are the correctness and computational adequacy of the denotational semantics with respect to the operational semantics given in terms of hybrid trajectories. Moreover, we show that our denotational semantics can be effectively computed, which allows for the effective analysis of a large class of non-linear hybrid automata.

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