112 research outputs found

    On the relative asymptotic expressivity of inference frameworks

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    Let Οƒ\sigma be a first-order signature and let Wn\mathbf{W}_n be the set of all Οƒ\sigma-structures with domain {1,…,n}\{1, \ldots, n\}. By an inference framework we mean a class F\mathbf{F} of pairs (P,L)(\mathbb{P}, L), where P=(Pn:n=1,2,3,…)\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots) and Pn\mathbb{P}_n is a probability distribution on Wn\mathbf{W}_n, and LL is a logic with truth values in the unit interval [0,1][0, 1]. An inference framework Fβ€²\mathbf{F}' is asymptotically at least as expressive as another inference framework F\mathbf{F} if for every (P,L)∈F(\mathbb{P}, L) \in \mathbf{F} there is (Pβ€²,Lβ€²)∈Fβ€²(\mathbb{P}', L') \in \mathbf{F}' such that P\mathbb{P} is asymptotically total-variation-equivalent to Pβ€²\mathbb{P}' and for every Ο†(xΛ‰)∈L\varphi(\bar{x}) \in L there is Ο†β€²(xΛ‰)∈Lβ€²\varphi'(\bar{x}) \in L' such that Ο†β€²(xΛ‰)\varphi'(\bar{x}) is asymptotically equivalent to Ο†(xΛ‰)\varphi(\bar{x}) with respect to P\mathbb{P}. This relation is a preorder and we describe a partial order on the equivalence classes of some inference frameworks that seem natural in the context of machine learning and artificial intelligence. Several previous results about asymptotic (or almost sure) equivalence of formulas or convergence in probability can be formulated in terms of relative asymptotic strength of inference frameworks. We incorporate these results in our classification of inference frameworks and prove two new results. Both concern sequences of probability distributions defined by directed graphical models that use ``continuous'' aggregation functions. The first considers queries expressed by a logic with truth values in [0,1][0, 1] which employs continuous aggregation functions. The second considers queries expressed by a two-valued conditional logic that can express statements about relative frequencies.Comment: 52 page

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Quantitative Hennessy-Milner Theorems via Notions of Density

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    The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g. in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance of such a general result. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstraß theorem; this allows us to cover, for instance, behavioural ultrametrics.publishe

    Topological data analysis of organoids

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    Organoids are multi-cellular structures which are cultured in vitro from stem cells to resemble specific organs (e.g., colon, liver) in their three- dimensional composition. The gene expression and the tissue composition of organoids constantly affect each other. Dynamic changes in the shape, cellular composition and transcriptomic profile of these model systems can be used to understand the effect of mutations and treatments in health and disease. In this thesis, I propose new techniques in the field of topological data analysis (TDA) to analyse the gene expression and the morphology of organoids. I use TDA methods, which are inspired by topology, to analyse and quantify the continuous structure of single-cell RNA sequencing data, which is embedded in high dimensional space, and the shape of an organoid. For single-cell RNA sequencing data, I developed the multiscale Laplacian score (MLS) and the UMAP diffusion cover, which both extend and im- prove existing topological analysis methods. I demonstrate the utility of these techniques by applying them to a published benchmark single-cell data set and a data set of mouse colon organoids. The methods validate previously identified genes and detect additional genes with known involvement cancers. To study the morphology of organoids I propose DETECT, a rotationally invariant signature of dynamically changing shapes. I demonstrate the efficacy of this method on a data set of segmented videos of mouse small intestine organoid experiments and show that it outperforms classical shape descriptors. I verify the method on a synthetic organoid data set and illustrate how it generalises to 3D to conclude that DETECT offers rigorous quantification of organoids and opens up computationally scalable methods for distinguishing different growth regimes and assessing treatment effects. Finally, I make a theoretical contribution to the statistical inference of the method underlying DETECT

    Quantitative Graded Semantics and Spectra of Behavioural Metrics

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    Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the classical linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics come in various degrees of granularity, depending on the observer's ability to interact with the system. Graded monads have been shown to provide a unifying framework for spectra of behavioural equivalences. Here, we transfer this principle to spectra of behavioural metrics, working at a coalgebraic level of generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we discuss presentations of graded monads on the category of metric spaces in terms of graded quantitative equational theories. Moreover, we obtain a canonical generic notion of invariant real-valued modal logic, and provide criteria for such logics to be expressive in the sense that logical distance coincides with the respective behavioural distance. We thus recover recent expressiveness results for coalgebraic branching-time metrics and for trace distance in metric transition systems; moreover, we obtain a new expressiveness result for trace semantics of fuzzy transition systems. We also provide a number of salient negative results. In particular, we show that trace distance on probabilistic metric transition systems does not admit a characteristic real-valued modal logic at all

    Probabilistic Guarded KAT Modulo Bisimilarity: Completeness and Complexity

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    Rethinking inconsistent mathematics

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    This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics

    Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces

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    We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation.Comment: Appendix remove

    ВСхнология комплСксной ΠΏΠΎΠ΄Π΄Π΅Ρ€ΠΆΠΊΠΈ ΠΆΠΈΠ·Π½Π΅Π½Π½ΠΎΠ³ΠΎ Ρ†ΠΈΠΊΠ»Π° сСмантичСски совмСстимых ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚ΡƒΠ°Π»ΡŒΠ½Ρ‹Ρ… ΠΊΠΎΠΌΠΏΡŒΡŽΡ‚Π΅Ρ€Π½Ρ‹Ρ… систСм Π½ΠΎΠ²ΠΎΠ³ΠΎ поколСния

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    Π’ ΠΈΠ·Π΄Π°Π½ΠΈΠΈ прСдставлСно описаниС Ρ‚Π΅ΠΊΡƒΡ‰Π΅ΠΉ вСрсии ΠΎΡ‚ΠΊΡ€Ρ‹Ρ‚ΠΎΠΉ Ρ‚Π΅Ρ…Π½ΠΎΠ»ΠΎΠ³ΠΈΠΈ онтологичСского проСктирования, производства ΠΈ эксплуатации сСмантичСски совмСстимых Π³ΠΈΠ±Ρ€ΠΈΠ΄Π½Ρ‹Ρ… ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚ΡƒΠ°Π»ΡŒΠ½Ρ‹Ρ… ΠΊΠΎΠΌΠΏΡŒΡŽΡ‚Π΅Ρ€Π½Ρ‹Ρ… систСм (Π’Π΅Ρ…Π½ΠΎΠ»ΠΎΠ³ΠΈΠΈ OSTIS). ΠŸΡ€Π΅Π΄Π»ΠΎΠΆΠ΅Π½Π° стандартизация ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚ΡƒΠ°Π»ΡŒΠ½Ρ‹Ρ… ΠΊΠΎΠΌΠΏΡŒΡŽΡ‚Π΅Ρ€Π½Ρ‹Ρ… систСм, Π° Ρ‚Π°ΠΊΠΆΠ΅ стандартизация ΠΌΠ΅Ρ‚ΠΎΠ΄ΠΎΠ² ΠΈ срСдств ΠΈΡ… проСктирования, Ρ‡Ρ‚ΠΎ являСтся ваТнСйшим Ρ„Π°ΠΊΡ‚ΠΎΡ€ΠΎΠΌ, ΠΎΠ±Π΅ΡΠΏΠ΅Ρ‡ΠΈΠ²Π°ΡŽΡ‰ΠΈΠΌ ΡΠ΅ΠΌΠ°Π½Ρ‚ΠΈΡ‡Π΅ΡΠΊΡƒΡŽ ΡΠΎΠ²ΠΌΠ΅ΡΡ‚ΠΈΠΌΠΎΡΡ‚ΡŒ ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚ΡƒΠ°Π»ΡŒΠ½Ρ‹Ρ… ΠΊΠΎΠΌΠΏΡŒΡŽΡ‚Π΅Ρ€Π½Ρ‹Ρ… систСм ΠΈ ΠΈΡ… ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½Ρ‚ΠΎΠ², Ρ‡Ρ‚ΠΎ сущСствСнноС сниТСниС трудоСмкости Ρ€Π°Π·Ρ€Π°Π±ΠΎΡ‚ΠΊΠΈ Ρ‚Π°ΠΊΠΈΡ… систСм. Книга ΠΏΡ€Π΅Π΄Π½Π°Π·Π½Π°Ρ‡Π΅Π½Π° всСм, ΠΊΡ‚ΠΎ интСрСсуСтся ΠΏΡ€ΠΎΠ±Π»Π΅ΠΌΠ°ΠΌΠΈ искусствСнного ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚Π°, Π° Ρ‚Π°ΠΊΠΆΠ΅ спСциалистам Π² области ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚ΡƒΠ°Π»ΡŒΠ½Ρ‹Ρ… ΠΊΠΎΠΌΠΏΡŒΡŽΡ‚Π΅Ρ€Π½Ρ‹Ρ… систСм ΠΈ ΠΈΠ½ΠΆΠ΅Π½Π΅Ρ€ΠΈΠΈ Π·Π½Π°Π½ΠΈΠΉ. ΠœΠΎΠΆΠ΅Ρ‚ Π±Ρ‹Ρ‚ΡŒ использована студСнтами, магистрантами ΠΈ аспирантами ΡΠΏΠ΅Ρ†ΠΈΠ°Π»ΡŒΠ½ΠΎΡΡ‚ΠΈ Β«Π˜ΡΠΊΡƒΡΡΡ‚Π²Π΅Π½Π½Ρ‹ΠΉ ΠΈΠ½Ρ‚Π΅Π»Π»Π΅ΠΊΡ‚Β». Π’Π°Π±Π». 8. Ил. 223. Π‘ΠΈΠ±Π»ΠΈΠΎΠ³Ρ€.: 665 Π½Π°Π·Π²
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