11,437 research outputs found

    Self-Supervised Learning to Prove Equivalence Between Straight-Line Programs via Rewrite Rules

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    We target the problem of automatically synthesizing proofs of semantic equivalence between two programs made of sequences of statements. We represent programs using abstract syntax trees (AST), where a given set of semantics-preserving rewrite rules can be applied on a specific AST pattern to generate a transformed and semantically equivalent program. In our system, two programs are equivalent if there exists a sequence of application of these rewrite rules that leads to rewriting one program into the other. We propose a neural network architecture based on a transformer model to generate proofs of equivalence between program pairs. The system outputs a sequence of rewrites, and the validity of the sequence is simply checked by verifying it can be applied. If no valid sequence is produced by the neural network, the system reports the programs as non-equivalent, ensuring by design no programs may be incorrectly reported as equivalent. Our system is fully implemented for a given grammar which can represent straight-line programs with function calls and multiple types. To efficiently train the system to generate such sequences, we develop an original incremental training technique, named self-supervised sample selection. We extensively study the effectiveness of this novel training approach on proofs of increasing complexity and length. Our system, S4Eq, achieves 97% proof success on a curated dataset of 10,000 pairs of equivalent programsComment: 30 pages including appendi

    Boosting the Cycle Counting Power of Graph Neural Networks with I2^2-GNNs

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    Message Passing Neural Networks (MPNNs) are a widely used class of Graph Neural Networks (GNNs). The limited representational power of MPNNs inspires the study of provably powerful GNN architectures. However, knowing one model is more powerful than another gives little insight about what functions they can or cannot express. It is still unclear whether these models are able to approximate specific functions such as counting certain graph substructures, which is essential for applications in biology, chemistry and social network analysis. Motivated by this, we propose to study the counting power of Subgraph MPNNs, a recent and popular class of powerful GNN models that extract rooted subgraphs for each node, assign the root node a unique identifier and encode the root node's representation within its rooted subgraph. Specifically, we prove that Subgraph MPNNs fail to count more-than-4-cycles at node level, implying that node representations cannot correctly encode the surrounding substructures like ring systems with more than four atoms. To overcome this limitation, we propose I2^2-GNNs to extend Subgraph MPNNs by assigning different identifiers for the root node and its neighbors in each subgraph. I2^2-GNNs' discriminative power is shown to be strictly stronger than Subgraph MPNNs and partially stronger than the 3-WL test. More importantly, I2^2-GNNs are proven capable of counting all 3, 4, 5 and 6-cycles, covering common substructures like benzene rings in organic chemistry, while still keeping linear complexity. To the best of our knowledge, it is the first linear-time GNN model that can count 6-cycles with theoretical guarantees. We validate its counting power in cycle counting tasks and demonstrate its competitive performance in molecular prediction benchmarks

    Model Diagnostics meets Forecast Evaluation: Goodness-of-Fit, Calibration, and Related Topics

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    Principled forecast evaluation and model diagnostics are vital in fitting probabilistic models and forecasting outcomes of interest. A common principle is that fitted or predicted distributions ought to be calibrated, ideally in the sense that the outcome is indistinguishable from a random draw from the posited distribution. Much of this thesis is centered on calibration properties of various types of forecasts. In the first part of the thesis, a simple algorithm for exact multinomial goodness-of-fit tests is proposed. The algorithm computes exact pp-values based on various test statistics, such as the log-likelihood ratio and Pearson\u27s chi-square. A thorough analysis shows improvement on extant methods. However, the runtime of the algorithm grows exponentially in the number of categories and hence its use is limited. In the second part, a framework rooted in probability theory is developed, which gives rise to hierarchies of calibration, and applies to both predictive distributions and stand-alone point forecasts. Based on a general notion of conditional T-calibration, the thesis introduces population versions of T-reliability diagrams and revisits a score decomposition into measures of miscalibration, discrimination, and uncertainty. Stable and efficient estimators of T-reliability diagrams and score components arise via nonparametric isotonic regression and the pool-adjacent-violators algorithm. For in-sample model diagnostics, a universal coefficient of determination is introduced that nests and reinterprets the classical R2R^2 in least squares regression. In the third part, probabilistic top lists are proposed as a novel type of prediction in classification, which bridges the gap between single-class predictions and predictive distributions. The probabilistic top list functional is elicited by strictly consistent evaluation metrics, based on symmetric proper scoring rules, which admit comparison of various types of predictions

    Limit theorems for non-Markovian and fractional processes

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    This thesis examines various non-Markovian and fractional processes---rough volatility models, stochastic Volterra equations, Wiener chaos expansions---through the prism of asymptotic analysis. Stochastic Volterra systems serve as a conducive framework encompassing most rough volatility models used in mathematical finance. In Chapter 2, we provide a unified treatment of pathwise large and moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhiraja, Dupuis and Ellis. This powerful approach also enables us to investigate the pathwise large deviations of families of white noise functionals characterised by their Wiener chaos expansion as~XΔ=∑n=0∞ΔnIn(fnΔ).X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). In Chapter 3, we provide sufficient conditions for the large deviations principle to hold in path space, thereby refreshing a problem left open By PĂ©rez-Abreu (1993). Hinging on analysis on Wiener space, the proof involves describing, controlling and identifying the limit of perturbed multiple stochastic integrals. In Chapter 4, we come back to mathematical finance via the route of Malliavin calculus. We present explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models, including rough volatility models and their multi-factor versions. Our general setup encompasses both European options on a stock and VIX options. In particular, we develop a detailed analysis of the two-factor rough Bergomi model. Finally, in Chapter 5, we consider the large-time behaviour of affine stochastic Volterra equations, an under-developed area in the absence of Markovianity. We leverage on a measure-valued Markovian lift introduced by Cuchiero and Teichmann and the associated notion of generalised Feller property. This setting allows us to prove the existence of an invariant measure for the lift and hence of a stationary distribution for the affine Volterra process, featuring in the rough Heston model.Open Acces

    Foundations for programming and implementing effect handlers

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    First-class control operators provide programmers with an expressive and efficient means for manipulating control through reification of the current control state as a first-class object, enabling programmers to implement their own computational effects and control idioms as shareable libraries. Effect handlers provide a particularly structured approach to programming with first-class control by naming control reifying operations and separating from their handling. This thesis is composed of three strands of work in which I develop operational foundations for programming and implementing effect handlers as well as exploring the expressive power of effect handlers. The first strand develops a fine-grain call-by-value core calculus of a statically typed programming language with a structural notion of effect types, as opposed to the nominal notion of effect types that dominates the literature. With the structural approach, effects need not be declared before use. The usual safety properties of statically typed programming are retained by making crucial use of row polymorphism to build and track effect signatures. The calculus features three forms of handlers: deep, shallow, and parameterised. They each offer a different approach to manipulate the control state of programs. Traditional deep handlers are defined by folds over computation trees, and are the original con-struct proposed by Plotkin and Pretnar. Shallow handlers are defined by case splits (rather than folds) over computation trees. Parameterised handlers are deep handlers extended with a state value that is threaded through the folds over computation trees. To demonstrate the usefulness of effects and handlers as a practical programming abstraction I implement the essence of a small UNIX-style operating system complete with multi-user environment, time-sharing, and file I/O. The second strand studies continuation passing style (CPS) and abstract machine semantics, which are foundational techniques that admit a unified basis for implementing deep, shallow, and parameterised effect handlers in the same environment. The CPS translation is obtained through a series of refinements of a basic first-order CPS translation for a fine-grain call-by-value language into an untyped language. Each refinement moves toward a more intensional representation of continuations eventually arriving at the notion of generalised continuation, which admit simultaneous support for deep, shallow, and parameterised handlers. The initial refinement adds support for deep handlers by representing stacks of continuations and handlers as a curried sequence of arguments. The image of the resulting translation is not properly tail-recursive, meaning some function application terms do not appear in tail position. To rectify this the CPS translation is refined once more to obtain an uncurried representation of stacks of continuations and handlers. Finally, the translation is made higher-order in order to contract administrative redexes at translation time. The generalised continuation representation is used to construct an abstract machine that provide simultaneous support for deep, shallow, and parameterised effect handlers. kinds of effect handlers. The third strand explores the expressiveness of effect handlers. First, I show that deep, shallow, and parameterised notions of handlers are interdefinable by way of typed macro-expressiveness, which provides a syntactic notion of expressiveness that affirms the existence of encodings between handlers, but it provides no information about the computational content of the encodings. Second, using the semantic notion of expressiveness I show that for a class of programs a programming language with first-class control (e.g. effect handlers) admits asymptotically faster implementations than possible in a language without first-class control

    Analysis of spatial point patterns on surfaces

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    With the advent of improved data acquisition technologies more complex spatial datasets can be collected at scale meaning theoretical and methodological developments in spatial statistics are imperative in order to analyse and generate meaningful conclusions. Spatial statistics has seen a plethora of applications in life sciences with particular emphasis on ecology, epidemiology and cell microscopy. Applications of these techniques provides researchers with insight on how the locations of objects of interest can be influenced by their neighbours and the environment. Examples include understanding the spatial distribution of trees observed within some window, and understanding how neighbouring trees and potentially soil contents can influence this. Whilst the literature for spatial statistics is rich the common assumption is that point processes are usually restricted to some d-dimensional Euclidean space, for example cell locations in a rectangular window of 2-dimensional Euclidean space. As such current theory is not capable of handling patterns which lie on more complex spaces, for example cubes and ellipsoids. Recent efforts have successfully extended methodology from Euclidean space to spheres by using the chordal distance (the shortest distance between any two points on a sphere) in place of the Euclidean distance. In this thesis we build on this work by considering point processes lying on more complex surfaces. Our first significant contribution discusses the construction of functional summary statistics for Poisson processes which lie on compact subsets of Rd which are off lower dimension. We map the process from its original space to the sphere where it is possible to take advantage of rotational symmetries which allow for well-defined summary statistics. These in turn can be used to determine whether an observed point patterns exhibits clustered or regular behaviour. Partnering this work we also provide a hypothesis testing procedure based on these functional summary statistics to determine whether an observed point pattern is complete spatially random. Two test statistics are proposed, one based on the commonly used L-function for planar processes and the other a standardisation of the K-function. These test statistics are compared in an extensive simulation study across ellipsoids of varying dimensions and processes which display differing levels of aggregation or regularity. Estimates of first order properties of a point process are extremely important. They can provide a graphical illustration of inhomogeneity and are useful in second order analysis. We demonstrate how kernel estimation can be extended from a Euclidean space to a Riemannian manifold where the Euclidean metric is now substituted for a Riemannian one. Many of the desirable properties for Euclidean kernel estimates carry over to the Riemannian setting. The issue of edge correction is also discussed and two criteria for bandwidth selection are proposed. These two selection criteria are explored through a simulation study. Finally, an important area of research in spatial statistics is exploring the interaction between different processes, for example how different species of plant spatially interact within some window. Under the framework of marked point processes we show that functional summary statistics for multivariate point patterns can be constructed on the sphere. This is extended to more general convex shapes through an appropriate mapping from the original shape to the sphere. A number of examples highlight that these summary statistics can capture independence, aggregation and repulsion between components of a multivariate process on both the sphere and more general surfaces.Open Acces

    Walking with the Earth: Intercultural Perspectives on Ethics of Ecological Caring

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    It is commonly believed that considering nature different from us, human beings (qua rational, cultural, religious and social actors), is detrimental to our engagement for the preservation of nature. An obvious example is animal rights, a deep concern for all living beings, including non-human living creatures, which is understandable only if we approach nature, without fearing it, as something which should remain outside of our true home. “Walking with the earth” aims at questioning any similar preconceptions in the wide sense, including allegoric-poetic contributions. We invited 14 authors from 4 continents to express all sorts of ways of saying why caring is so important, why togetherness, being-with each others, as a spiritual but also embodied ethics is important in a divided world

    How to Be a God

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    When it comes to questions concerning the nature of Reality, Philosophers and Theologians have the answers. Philosophers have the answers that can’t be proven right. Theologians have the answers that can’t be proven wrong. Today’s designers of Massively-Multiplayer Online Role-Playing Games create realities for a living. They can’t spend centuries mulling over the issues: they have to face them head-on. Their practical experiences can indicate which theoretical proposals actually work in practice. That’s today’s designers. Tomorrow’s will have a whole new set of questions to answer. The designers of virtual worlds are the literal gods of those realities. Suppose Artificial Intelligence comes through and allows us to create non-player characters as smart as us. What are our responsibilities as gods? How should we, as gods, conduct ourselves? How should we be gods

    BECOMEBECOME - A TRANSDISCIPLINARY METHODOLOGY BASED ON INFORMATION ABOUT THE OBSERVER

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    ABSTRACT Andrea T. R. Traldi BECOMEBECOME A Transdisciplinary Methodology Based on Information about the Observer The present research dissertation has been developed with the intention to provide practical strategies and discover new intellectual operations which can be used to generate Transdisciplinary insight. For this reason, this thesis creates access to new knowledge at different scales. Firstly, as it pertains to the scale of new knowledge generated by those who attend Becomebecome events. The open-source nature of the Becomebecome methodology makes it possible for participants in Becomebecome workshops, training programmes and residencies to generate new insight about the specific project they are working on, which then reinforce and expand the foundational principles of the theoretical background. Secondly, as it pertains to the scale of the Becomebecome framework, which remains independent of location and moment in time. The method proposed to access Transdisciplinary knowledge constitutes new knowledge in itself because the sequence of activities, described as physical and mental procedures and listed as essential criteria, have never been found organised 6 in such a specific order before. It is indeed the order in time, i.e. the sequence of the ideas and activities proposed, which allows one to transform Disciplinary knowledge via a new Transdisciplinary frame of reference. Lastly, new knowledge about Transdisciplinarity as a field of study is created as a consequence of the heretofore listed two processes. The first part of the thesis is designated ‘Becomebecome Theory’ and focuses on the theoretical background and the intellectual operations necessary to support the creation of new Transdisciplinary knowledge. The second part of the thesis is designated ‘Becomebecome Practice’ and provides practical examples of the application of such operations. Crucially, the theoretical model described as the foundation for the Becomebecome methodology (Becomebecome Theory) is process-based and constantly checked against the insight generated through Becomebecome Practice. To this effect, ‘information about the observer’ is proposed as a key notion which binds together Transdisciplinary resources from several studies in the hard sciences and humanities. It is a concept that enables understanding about why and how information that is generated through Becomebecome Practice is considered of paramount importance for establishing the reference parameters necessary to access Transdisciplinary insight which is meaningful to a specific project, a specific person, or a specific moment in time
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