112 research outputs found
On the relative asymptotic expressivity of inference frameworks
Let be a first-order signature and let be the set of
all -structures with domain . By an inference
framework we mean a class of pairs , where
and is a
probability distribution on , and is a logic with truth
values in the unit interval . An inference framework is
asymptotically at least as expressive as another inference framework
if for every there is
such that is asymptotically
total-variation-equivalent to and for every there is such that is
asymptotically equivalent to with respect to .
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 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
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Quantitative Hennessy-Milner Theorems via Notions of Density
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-WeierstraΓ
theorem; this allows us to cover, for instance, behavioural ultrametrics.publishe
Topological data analysis of organoids
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
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
Rethinking inconsistent mathematics
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
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
Π’Π΅Ρ Π½ΠΎΠ»ΠΎΠ³ΠΈΡ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΠΎΠΉ ΠΏΠΎΠ΄Π΄Π΅ΡΠΆΠΊΠΈ ΠΆΠΈΠ·Π½Π΅Π½Π½ΠΎΠ³ΠΎ ΡΠΈΠΊΠ»Π° ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΡΡ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ ΡΠΈΡΡΠ΅ΠΌ Π½ΠΎΠ²ΠΎΠ³ΠΎ ΠΏΠΎΠΊΠΎΠ»Π΅Π½ΠΈΡ
Π ΠΈΠ·Π΄Π°Π½ΠΈΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎ ΠΎΠΏΠΈΡΠ°Π½ΠΈΠ΅ ΡΠ΅ΠΊΡΡΠ΅ΠΉ Π²Π΅ΡΡΠΈΠΈ ΠΎΡΠΊΡΡΡΠΎΠΉ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΠΈ ΠΎΠ½ΡΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ, ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΡΡΠ²Π° ΠΈ ΡΠΊΡΠΏΠ»ΡΠ°ΡΠ°ΡΠΈΠΈ ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΡΡ
Π³ΠΈΠ±ΡΠΈΠ΄Π½ΡΡ
ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ (Π’Π΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΠΈ OSTIS). ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π° ΡΡΠ°Π½Π΄Π°ΡΡΠΈΠ·Π°ΡΠΈΡ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ, Π° ΡΠ°ΠΊΠΆΠ΅ ΡΡΠ°Π½Π΄Π°ΡΡΠΈΠ·Π°ΡΠΈΡ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΈ
ΡΡΠ΅Π΄ΡΡΠ² ΠΈΡ
ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ, ΡΡΠΎ ΡΠ²Π»ΡΠ΅ΡΡΡ Π²Π°ΠΆΠ½Π΅ΠΉΡΠΈΠΌ ΡΠ°ΠΊΡΠΎΡΠΎΠΌ, ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠΈΠ²Π°ΡΡΠΈΠΌ ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΡΡ ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΠΎΡΡΡ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΈ ΠΈΡ
ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½ΡΠΎΠ², ΡΡΠΎ
ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠ΅ ΡΠ½ΠΈΠΆΠ΅Π½ΠΈΠ΅ ΡΡΡΠ΄ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠΊΠΈ ΡΠ°ΠΊΠΈΡ
ΡΠΈΡΡΠ΅ΠΌ.
ΠΠ½ΠΈΠ³Π° ΠΏΡΠ΅Π΄Π½Π°Π·Π½Π°ΡΠ΅Π½Π° Π²ΡΠ΅ΠΌ, ΠΊΡΠΎ ΠΈΠ½ΡΠ΅ΡΠ΅ΡΡΠ΅ΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ°ΠΌΠΈ ΠΈΡΠΊΡΡΡΡΠ²Π΅Π½Π½ΠΎΠ³ΠΎ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΠ°, Π° ΡΠ°ΠΊΠΆΠ΅ ΡΠΏΠ΅ΡΠΈΠ°Π»ΠΈΡΡΠ°ΠΌ Π² ΠΎΠ±Π»Π°ΡΡΠΈ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΈ ΠΈΠ½ΠΆΠ΅Π½Π΅ΡΠΈΠΈ Π·Π½Π°Π½ΠΈΠΉ. ΠΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½Π° ΡΡΡΠ΄Π΅Π½ΡΠ°ΠΌΠΈ, ΠΌΠ°Π³ΠΈΡΡΡΠ°Π½ΡΠ°ΠΌΠΈ ΠΈ Π°ΡΠΏΠΈΡΠ°Π½ΡΠ°ΠΌΠΈ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΠΎΡΡΠΈ Β«ΠΡΠΊΡΡΡΡΠ²Π΅Π½Π½ΡΠΉ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΒ».
Π’Π°Π±Π». 8. ΠΠ». 223. ΠΠΈΠ±Π»ΠΈΠΎΠ³Ρ.: 665 Π½Π°Π·Π²
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