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href="/search/advanced?terms-0-term=Arsiwalla%2C+X+D&terms-0-field=author&size=50&order=-announced_date_first">Advanced Search</a> </div> </div> <input type="hidden" name="order" value="-announced_date_first"> <input type="hidden" name="size" value="50"> </form> <div class="level breathe-horizontal"> <div class="level-left"> <form method="GET" action="/search/"> <div style="display: none;"> <select id="searchtype" name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Arsiwalla, X D"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.01148">arXiv:2405.01148</a> <span> [<a href="https://arxiv.org/pdf/2405.01148">pdf</a>, <a href="https://arxiv.org/format/2405.01148">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Neurons and Cognition">q-bio.NC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="History and Philosophy of Physics">physics.hist-ph</span> </div> </div> <p class="title is-5 mathjax"> Qualia and the Formal Structure of Meaning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.01148v1-abstract-short" style="display: inline;"> This work explores the hypothesis that subjectively attributed meaning constitutes the phenomenal content of conscious experience. That is, phenomenal content is semantic. This form of subjective meaning manifests as an intrinsic and non-representational character of qualia. Empirically, subjective meaning is ubiquitous in conscious experiences. We point to phenomenological studies that lend evide… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.01148v1-abstract-full').style.display = 'inline'; document.getElementById('2405.01148v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.01148v1-abstract-full" style="display: none;"> This work explores the hypothesis that subjectively attributed meaning constitutes the phenomenal content of conscious experience. That is, phenomenal content is semantic. This form of subjective meaning manifests as an intrinsic and non-representational character of qualia. Empirically, subjective meaning is ubiquitous in conscious experiences. We point to phenomenological studies that lend evidence to support this. Furthermore, this notion of meaning closely relates to what Frege refers to as "sense", in metaphysics and philosophy of language. It also aligns with Peirce's "interpretant", in semiotics. We discuss how Frege's sense can also be extended to the raw feels of consciousness. Sense and reference both play a role in phenomenal experience. Moreover, within the context of the mind-matter relation, we provide a formalization of subjective meaning associated to one's mental representations. Identifying the precise maps between the physical and mental domains, we argue that syntactic and semantic structures transcend language, and are realized within each of these domains. Formally, meaning is a relational attribute, realized via a map that interprets syntactic structures of a formal system within an appropriate semantic space. The image of this map within the mental domain is what is relevant for experience, and thus comprises the phenomenal content of qualia. We conclude with possible implications this may have for experience-based theories of consciousness. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.01148v1-abstract-full').style.display = 'none'; document.getElementById('2405.01148v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.04406">arXiv:2212.04406</a> <span> [<a href="https://arxiv.org/pdf/2212.04406">pdf</a>, <a href="https://arxiv.org/format/2212.04406">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1093/comnet/cnad022">10.1093/comnet/cnad022 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Cosine Rule-Based Discrete Sectional Curvature for Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Plessis%2C+J+F+D">J. F. Du Plessis</a>, <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.04406v1-abstract-short" style="display: inline;"> How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete spacetime in quantum gravity; inferring network geometry in network science; and manifold learning in data science. The key contribution of this paper is to introduc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.04406v1-abstract-full').style.display = 'inline'; document.getElementById('2212.04406v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.04406v1-abstract-full" style="display: none;"> How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete spacetime in quantum gravity; inferring network geometry in network science; and manifold learning in data science. The key contribution of this paper is to introduce and validate a new estimator of discrete sectional curvature for random graphs with low metric-distortion. The latter are constructed via a specific graph sprinkling method on different manifolds with constant sectional curvature. We define a notion of metric distortion, which quantifies how well the graph metric approximates the metric of the underlying manifold. We show how graph sprinkling algorithms can be refined to produce hard annulus random geometric graphs with minimal metric distortion. We construct random geometric graphs for spheres, hyperbolic and euclidean planes; upon which we validate our curvature estimator. Numerical analysis reveals that the error of the estimated curvature diminishes as the mean metric distortion goes to zero, thus demonstrating convergence of the estimate. We also perform comparisons to other existing discrete curvature measures. Finally, we demonstrate two practical applications: (i) estimation of the earth's radius using geographical data; and (ii) sectional curvature distributions of self-similar fractals. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.04406v1-abstract-full').style.display = 'none'; document.getElementById('2212.04406v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.01838">arXiv:2211.01838</a> <span> [<a href="https://arxiv.org/pdf/2211.01838">pdf</a>, <a href="https://arxiv.org/ps/2211.01838">ps</a>, <a href="https://arxiv.org/format/2211.01838">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> </div> <p class="title is-5 mathjax"> On the Operator Origins of Classical and Quantum Wave Functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a>, <a href="/search/math?searchtype=author&query=Chester%2C+D">David Chester</a>, <a href="/search/math?searchtype=author&query=Kauffman%2C+L+H">Louis H. Kauffman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2211.01838v2-abstract-short" style="display: inline;"> We investigate operator algebraic origins of the classical Koopman-von Neumann wave function $蠄_{KvN}$ as well as the quantum mechanical one $蠄_{QM}$. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and qua… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.01838v2-abstract-full').style.display = 'inline'; document.getElementById('2211.01838v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.01838v2-abstract-full" style="display: none;"> We investigate operator algebraic origins of the classical Koopman-von Neumann wave function $蠄_{KvN}$ as well as the quantum mechanical one $蠄_{QM}$. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $蠄_{KvN}$ and $蠄_{QM}$ are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle ${\cal E}$ follows from this. $蠄_{KvN}$ and $蠄_{QM}$ are shown to be sections in ${\cal E}$. The difference between $蠄_{KvN}$ and $蠄_{QM}$ originates from a quantization map interpreted as "twisting" of sections over ${\cal E}$. We also show that the Schr枚dinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schr枚dinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.01838v2-abstract-full').style.display = 'none'; document.getElementById('2211.01838v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.05456">arXiv:2205.05456</a> <span> [<a href="https://arxiv.org/pdf/2205.05456">pdf</a>, <a href="https://arxiv.org/format/2205.05456">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Heaps of Fish: arrays, generalized associativity and heapoids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zapata-Carratala%2C+C">Carlos Zapata-Carratala</a>, <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a>, <a href="/search/math?searchtype=author&query=Beynon%2C+T">Taliesin Beynon</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2205.05456v1-abstract-short" style="display: inline;"> In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. In doing so we rediscover the ternary structures known as heaps and are able to give a more comprehensive treatment of their mergence in the context of dagger categories and their generalizat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.05456v1-abstract-full').style.display = 'inline'; document.getElementById('2205.05456v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.05456v1-abstract-full" style="display: none;"> In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. In doing so we rediscover the ternary structures known as heaps and are able to give a more comprehensive treatment of their mergence in the context of dagger categories and their generalizations. Our key insight is to approach associativity as a confluence property of hypergraph rewrite systems. This approach allows us to define a notion of ternary category and heapoid, where morphisms bind three objects simultaneously, and suggests a systematic study of higher arity forms of associativity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.05456v1-abstract-full').style.display = 'none'; document.getElementById('2205.05456v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.09738">arXiv:2201.09738</a> <span> [<a href="https://arxiv.org/pdf/2201.09738">pdf</a>, <a href="https://arxiv.org/format/2201.09738">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Mathematics">math.GM</span> </div> </div> <p class="title is-5 mathjax"> An Invitation to Higher Arity Science </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zapata-Carratala%2C+C">Carlos Zapata-Carratala</a>, <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.09738v1-abstract-short" style="display: inline;"> Analytical thinking is dominated by binary ideas. From pair-wise interactions, to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this article we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.09738v1-abstract-full').style.display = 'inline'; document.getElementById('2201.09738v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.09738v1-abstract-full" style="display: none;"> Analytical thinking is dominated by binary ideas. From pair-wise interactions, to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this article we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this we discuss several higher order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.09738v1-abstract-full').style.display = 'none'; document.getElementById('2201.09738v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Feedback and communications welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2111.03460">arXiv:2111.03460</a> <span> [<a href="https://arxiv.org/pdf/2111.03460">pdf</a>, <a href="https://arxiv.org/format/2111.03460">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> </div> <p class="title is-5 mathjax"> Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a>, <a href="/search/math?searchtype=author&query=Gorard%2C+J">Jonathan Gorard</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2111.03460v2-abstract-short" style="display: inline;"> How do spaces emerge from pregeometric discrete building blocks governed by computational rules? To address this, we investigate non-deterministic rewriting systems (multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation, we outline how spatial structures can be functorially inherited from pregeometric type-theoretic constructions.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.03460v2-abstract-full').style.display = 'inline'; document.getElementById('2111.03460v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2111.03460v2-abstract-full" style="display: none;"> How do spaces emerge from pregeometric discrete building blocks governed by computational rules? To address this, we investigate non-deterministic rewriting systems (multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation, we outline how spatial structures can be functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopy types are constructed from rewriting rules. These correspond to morphisms of an $n$-fold category. Subsequently, the $n \to \infty$ limit of the Wolfram model rulial multiway system is identified as an $\infty$-groupoid, with the latter being relevant given Grothendieck's homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an ${\left(\infty, 1\right)}$-topos. This correspondence to higher categorical structures offers a new way to understand how spaces relevant to physics may arise from pregeometric combinatorial models. A key issue we have addressed here is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it eliminates ad hoc assumptions about geometric attributes of a model such as an a priori background or pre-assigned geometric data. Instead, geometry is inherited functorially by higher structures. This is relevant for formally justifying different choices of underlying spacetime discretization adopted by models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions, corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.03460v2-abstract-full').style.display = 'none'; document.getElementById('2111.03460v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">52 pages, 18 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.10822">arXiv:2105.10822</a> <span> [<a href="https://arxiv.org/pdf/2105.10822">pdf</a>, <a href="https://arxiv.org/format/2105.10822">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a>, <a href="/search/math?searchtype=author&query=Gorard%2C+J">Jonathan Gorard</a>, <a href="/search/math?searchtype=author&query=Elshatlawy%2C+H">Hatem Elshatlawy</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.10822v2-abstract-short" style="display: inline;"> We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.10822v2-abstract-full').style.display = 'inline'; document.getElementById('2105.10822v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.10822v2-abstract-full" style="display: none;"> We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order $n$ may naturally be formalized as an $n$-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ${\infty}$-groupoid. Via Grothendieck's homotopy hypothesis, this ${\infty}$-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of homotopical multiway systems may potentially be used for making formal connections to homotopy spaces upon which models relevant to physics may be instantiated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.10822v2-abstract-full').style.display = 'none'; document.getElementById('2105.10822v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 5 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.07775">arXiv:1701.07775</a> <span> [<a href="https://arxiv.org/pdf/1701.07775">pdf</a>, <a href="https://arxiv.org/format/1701.07775">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Neurons and Cognition">q-bio.NC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A Forward Model at Purkinje Cell Synapses Facilitates Cerebellar Anticipatory Control </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Herreros-Alonso%2C+I">Ivan Herreros-Alonso</a>, <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a>, <a href="/search/math?searchtype=author&query=Verschure%2C+P+F+M+J">Paul F. M. J. Verschure</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1701.07775v1-abstract-short" style="display: inline;"> How does our motor system solve the problem of anticipatory control in spite of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous system? To a great extent, our highly-skilled motor responses are a result of a reactive feedback system, originating in the brain-stem and spinal cord, combined wi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.07775v1-abstract-full').style.display = 'inline'; document.getElementById('1701.07775v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.07775v1-abstract-full" style="display: none;"> How does our motor system solve the problem of anticipatory control in spite of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous system? To a great extent, our highly-skilled motor responses are a result of a reactive feedback system, originating in the brain-stem and spinal cord, combined with a feed-forward anticipatory system, that is adaptively fine-tuned by sensory experience and originates in the cerebellum. Based on that interaction we design the counterfactual predictive control (CFPC) architecture, an anticipatory adaptive motor control scheme in which a feed-forward module, based on the cerebellum, steers an error feedback controller with counterfactual error signals. Those are signals that trigger reactions as actual errors would, but that do not code for any current or forthcoming errors. In order to determine the optimal learning strategy, we derive a novel learning rule for the feed-forward module that involves an eligibility trace and operates at the synaptic level. In particular, our eligibility trace provides a mechanism beyond co-incidence detection in that it convolves a history of prior synaptic inputs with error signals. In the context of cerebellar physiology, this solution implies that Purkinje cell synapses should generate eligibility traces using a forward model of the system being controlled. From an engineering perspective, CFPC provides a general-purpose anticipatory control architecture equipped with a learning rule that exploits the full dynamics of the closed-loop system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.07775v1-abstract-full').style.display = 'none'; document.getElementById('1701.07775v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">NIPS 2016</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Neural Information Processing Systems, 29: 3828-3836, (2016) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.07106">arXiv:1612.07106</a> <span> [<a href="https://arxiv.org/pdf/1612.07106">pdf</a>, <a href="https://arxiv.org/format/1612.07106">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Neurons and Cognition">q-bio.NC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Biological Physics">physics.bio-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s41109-016-0018-8">10.1007/s41109-016-0018-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Global Dynamical Complexity of the Human Brain Network </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arsiwalla%2C+X+D">Xerxes D. Arsiwalla</a>, <a href="/search/math?searchtype=author&query=Verschure%2C+P">Paul Verschure</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.07106v1-abstract-short" style="display: inline;"> How much information do large brain networks integrate as a whole over the sum of their parts? Can the dynamical complexity of such networks be globally quantified in an information-theoretic way and be meaningfully coupled to brain function? Recently, measures of dynamical complexity such as integrated information have been proposed. However, problems related to the normalization and Bell number… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.07106v1-abstract-full').style.display = 'inline'; document.getElementById('1612.07106v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.07106v1-abstract-full" style="display: none;"> How much information do large brain networks integrate as a whole over the sum of their parts? Can the dynamical complexity of such networks be globally quantified in an information-theoretic way and be meaningfully coupled to brain function? Recently, measures of dynamical complexity such as integrated information have been proposed. However, problems related to the normalization and Bell number of partitions associated to these measures make these approaches computationally infeasible for large-scale brain networks. Our goal in this work is to address this problem. Our formulation of network integrated information is based on the Kullback-Leibler divergence between the multivariate distribution on the set of network states versus the corresponding factorized distribution over its parts. We find that implementing the maximum information partition optimizes computations. These methods are well-suited for large networks with linear stochastic dynamics. We compute the integrated information for both, the system's attractor states, as well as non-stationary dynamical states of the network. We then apply this formalism to brain networks to compute the integrated information for the human brain's connectome. Compared to a randomly re-wired network, we find that the specific topology of the brain generates greater information complexity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.07106v1-abstract-full').style.display = 'none'; document.getElementById('1612.07106v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 6 figures</span> </p> </li> </ol> <div class="is-hidden-tablet"> <!-- feedback for mobile only --> <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a> 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