1,635 research outputs found

    Complete and easy type Inference for first-class polymorphism

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    The Hindley-Milner (HM) typing discipline is remarkable in that it allows statically typing programs without requiring the programmer to annotate programs with types themselves. This is due to the HM system offering complete type inference, meaning that if a program is well typed, the inference algorithm is able to determine all the necessary typing information. Let bindings implicitly perform generalisation, allowing a let-bound variable to receive the most general possible type, which in turn may be instantiated appropriately at each of the variable’s use sites. As a result, the HM type system has since become the foundation for type inference in programming languages such as Haskell as well as the ML family of languages and has been extended in a multitude of ways. The original HM system only supports prenex polymorphism, where type variables are universally quantified only at the outermost level. This precludes many useful programs, such as passing a data structure to a function in the form of a fold function, which would need to be polymorphic in the type of the accumulator. However, this would require a nested quantifier in the type of the overall function. As a result, one direction of extending the HM system is to add support for first-class polymorphism, allowing arbitrarily nested quantifiers and instantiating type variables with polymorphic types. In such systems, restrictions are necessary to retain decidability of type inference. This work presents FreezeML, a novel approach for integrating first-class polymorphism into the HM system, focused on simplicity. It eschews sophisticated yet hard to grasp heuristics in the type systems or extending the language of types, while still requiring only modest amounts of annotations. In particular, FreezeML leverages the mechanisms for generalisation and instantiation that are already at the heart of ML. Generalisation and instantiation are performed by let bindings and variables, respectively, but extended to types beyond prenex polymorphism. The defining feature of FreezeML is the ability to freeze variables, which prevents the usual instantiation of their types, allowing them instead to keep their original, fully polymorphic types. We demonstrate that FreezeML is as expressive as System F by providing a translation from the latter to the former; the reverse direction is also shown. Further, we prove that FreezeML is indeed a conservative extension of ML: When considering only ML programs, FreezeML accepts exactly the same programs as ML itself. # We show that type inference for FreezeML can easily be integrated into HM-like type systems by presenting a sound and complete inference algorithm for FreezeML that extends Algorithm W, the original inference algorithm for the HM system. Since the inception of Algorithm W in the 1970s, type inference for the HM system and its descendants has been modernised by approaches that involve constraint solving, which proved to be more modular and extensible. In such systems, a term is translated to a logical constraint, whose solutions correspond to the types of the original term. A solver for such constraints may then be defined independently. To this end, we demonstrate such a constraint-based inference approach for FreezeML. We also discuss the effects of integrating the value restriction into FreezeML and provide detailed comparisons with other approaches towards first-class polymorphism in ML alongside a collection of examples found in the literature

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Apartment classes of integral symplectic groups

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    In this note we present an alternative proof of a theorem of Gunnells, which states that the Steinberg module of Sp2n(Q)\operatorname{Sp_{2n}}(\mathbb{Q}) is a cyclic Sp2n(Z)\operatorname{Sp_{2n}}(\mathbb{Z})-module, generated by integral apartment classes.Comment: 16 pages. Comments welcome

    2023-2024 Catalog

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    The 2023-2024 Governors State University Undergraduate and Graduate Catalog is a comprehensive listing of current information regarding:Degree RequirementsCourse OfferingsUndergraduate and Graduate Rules and Regulation

    Due Tomorrow, Do Tomorrow: Measuring and Reducing Procrastination Behavior Among Introductory Physics Students in an Online Environment

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    This work is focused on the measurement and prevention of procrastination behavior among college level introductory physics students completing online assignments in the form of mastery-based online learning modules. The research is conducted in two studies. The first study evaluates the effectiveness of offering students the opportunity to earn a small amount of extra credit for completing portions of their homework early. Unsupervised machine learning is used to identify an optimum cutoff duration which differentiates taking a short break during a continuous study session from a long break between two different study sessions. Using this cutoff, the study shows that the extra credit encouraged students to complete assignments earlier. The second study examines the impact of adding a planning-prompt survey prior to a string of assignments. In the survey, students were asked to write a plan for when and where they would work on their online homework assignments. Using a difference in differences method, a multilinear modeling technique adopted from economics research, the study shows that the survey led to students completing their homework on average 18 hours earlier and spreading their efforts on the homework over time significantly more. On the other hand, behaviors associated with disengagement, such as guessing or answer-copying, were not impacted by the introduction of the planning prompt. These studies showcase novel methods for measurement of procrastination behavior, as well as evaluating the effectiveness of the designed interventions to help students avoid waiting until the last minute to make progress on assigned tasks

    Universality and Examples in the Context of Functorial Semi-Norms

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    Let F: C → Vect be a functor from a category C to vector spaces over a normed field. A functorial semi-norm on F is a factorization of F over the forgetful functor snVect → Vect, where snVect denotes the corresponding category of semi-normed vector spaces. Functorial semi-norms, in particular the ℓ¹-semi-norm, on singular homology were introduced by Gromov in his study of simplicial volume of manifolds. The latter is a homotopy invariant that, roughly speaking, measures the complexity of the fundamental class of an oriented closed connected manifold. In the present thesis, we investigate three aspects of functorial semi-norms: Universal functorial semi-norms (joint work with Clara Löh). For a fixed functor, we define a relation among all its functorial semi-norms, whose minimal elements we call universal. We then prove certain existence results of such universal functorial semi-norms. Inflexibility from a computational perspective. Crowley and Löh established a bidirectional correspondence between functorial semi-norms on singular homology and so-called inflexible manifolds. The construction of such manifolds is based on the construction and study of certain differential graded algebras, which are purely algebraic objects. As such they are very amenable to computations, not only by humans but also via computer programs. We present fragments of a software that facilitates such computations, some new examples that we found in this way, and two results about algorithmic decidability. Excisive approximation of ℓ¹-homology. It is a well-known fact, that ℓ¹-homology does not satisfy the excision axiom in the sense of Eilenberg and Steenrod. On the other hand, the fact that singular homology satisfies excision is already visible at the level of the singular chain complex functor, namely the latter is excisive in the sense of Goodwillie calculus. The latter, however, also provides the framework for constructing a universal (or best) excisive approximation to a given functor. We apply this theory to the ℓ¹-chain complex functor and show that its excisive approximation vanishes. In appendices, we include a proof of the fact that the singular chain complex functor is excisive in the sense of Goodwillie calculus, and we relate this abstract form of excision to the classical one in the form of Mayer-Vietoris sequences

    Riemannian statistical techniques with applications in fMRI

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    Over the past 30 years functional magnetic resonance imaging (fMRI) has become a fundamental tool in cognitive neuroimaging studies. In particular, the emergence of restingstate fMRI has gained popularity in determining biomarkers of mental health disorders (Woodward & Cascio, 2015). Resting-state fMRI can be analysed using the functional connectivity matrix, an object that encodes the temporal correlation of blood activity within the brain. Functional connectivity matrices are symmetric positive definite (SPD) matrices, but common analysis methods either reduce the functional connectivity matrices to summary statistics or fail to account for the positive definite criteria. However, through the lens of Riemannian geometry functional connectivity matrices have an intrinsic non-linear shape that respects the positive definite criteria (the affine-invariant geometry (Pennec, Fillard, & Ayache, 2006)). With methods from Riemannian geometric statistics, we can begin to explore the shape of the functional brain to understand this non-linear structure and reduce data-loss in our analyses. This thesis o↵ers two novel methodological developments to the field of Riemannian geometric statistics inspired by methods used in fMRI research. First we propose geometric- MDMR, a generalisation of multivariate distance matrix regression (MDMR) (McArdle & Anderson, 2001) to Riemannian manifolds. Our second development is Riemannian partial least squares (R-PLS), the generalisation of the predictive modelling technique partial least squares (PLS) (H. Wold, 1975) to Riemannian manifolds. R-PLS extends geodesic regression (Fletcher, 2013) to manifold-valued response and predictor variables, similar to how PLS extends multiple linear regression. We also generalise the NIPALS algorithm to Riemannian manifolds and suggest a tangent space approximation as a proposed method to fit R-PLS. In addition to our methodological developments, this thesis o↵ers three more contributions to the literature. Firstly, we develop a novel simulation procedure to simulate realistic functional connectivity matrices through a combination of bootstrapping and the Wishart distribution. Second, we propose the R2S statistic for measuring subspace similarity using the theory of principal angles between subspaces. Finally, we propose an extension of the VIP statistic from PLS (S. Wold, Johansson, & Cocchi, 1993) to describe the relationship between individual predictors and response variables when predicting a multivariate response with PLS. All methods in this thesis are applied to two fMRI datasets: the COBRE dataset relating to schizophrenia, and the ABIDE dataset relating to Autism Spectrum Disorder (ASD). We show that geometric-MDMR can detect group-based di↵erences between ASD and neurotypical controls (NTC), unlike its Euclidean counterparts. We also demonstrate the efficacy of R-PLS through the detection of functional connections related to schizophrenia and ASD. These results are encouraging for the role of Riemannian geometric statistics in the future of neuroscientific research.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 202

    CSI-Otter: Isogeny-based (Partially) Blind Signatures from the Class Group Action with a Twist

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    In this paper, we construct the first provably-secure isogeny-based (partially) blind signature scheme. While at a high level the scheme resembles the Schnorr blind signature, our work does not directly follow from that construction, since isogenies do not offer as rich an algebraic structure. Specifically, our protocol does not fit into the linear identification protocol abstraction introduced by Hauck, Kiltz, and Loss (EUROCYRPT\u2719), which was used to generically construct Schnorr-like blind signatures based on modules such as classical groups and lattices. Consequently, our scheme is provably-secure in the poly-logarithmic (in the number of security parameter) concurrent execution and does not seem susceptible to the recent efficient ROS attack exploiting the linear nature of the underlying mathematical tool. In more detail, our blind signature exploits the quadratic twist of an elliptic curve in an essential way to endow isogenies with a strictly richer structure than abstract group actions (but still more restrictive than modules). The basic scheme has public key size 128128~B and signature size 88~KB under the CSIDH-512 parameter sets---these are the smallest among all provably secure post-quantum secure blind signatures. Relying on a new ring variant of the group action inverse problem rGAIP, we can halve the signature size to 4~KB while increasing the public key size to 512~B. We provide preliminary cryptanalysis of rGAIP and show that for certain parameter settings, it is essentially as secure as the standard GAIP. Finally, we show a novel way to turn our blind signature into a partially blind signature, where we deviate from prior methods since they require hashing into the set of public keys while hiding the corresponding secret key---constructing such a hash function in the isogeny setting remains an open problem
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