18,254 research outputs found
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Ordinal Measures of the Set of Finite Multisets
Well-partial orders, and the ordinal invariants used to measure them, are relevant in set theory, program verification, proof theory and many other areas of computer science and mathematics. In this article we focus on a common data structure in programming, finite multisets of some well partial order. There are two natural orders one can define on the set of finite multisets of a partial order: the multiset embedding and the multiset ordering. Though the maximal order type of these orders is already known, other ordinal invariants remain mostly unknown. Our main contributions are expressions to compute compositionally the width of the multiset embedding and the height of the multiset ordering. Furthermore, we provide a new ordinal invariant useful for characterizing the width of the multiset ordering
Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem
Injective multiset functions have a key role in the theoretical study of
machine learning on multisets and graphs. Yet, there remains a gap between the
provably injective multiset functions considered in theory, which typically
rely on polynomial moments, and the multiset functions used in practice, which
rely on \unicode{x2014} whose injectivity on
multisets has not been studied to date.
In this paper, we bridge this gap by showing that moments of neural networks
do define injective multiset functions, provided that an analytic
non-polynomial activation is used. The number of moments required by our theory
is optimal essentially up to a multiplicative factor of two. To prove this
result, we state and prove a , which is of
independent interest.
As a corollary to our main theorem, we derive new approximation results for
functions on multisets and measures, and new separation results for graph
neural networks. We also provide two negative results: (1) moments of
piecewise-linear neural networks cannot be injective multiset functions; and
(2) even when moment-based multiset functions are injective, they can never be
bi-Lipschitz.Comment: NeurIPS 2023 camera-read
Information Theory over Multisets
Starting from Shannon theory of information, this paper presents the case of producing information in the form of multisets, and encoding information using multisets. We review the entropy rate of a multiset information source and derive a formula for the information content of a multiset. Then we study the encoder and channel part of the system, obtaining some results about multiset encoding length and channel capacity
The Multiset Partition Algebra: Diagram-Like Bases and Representations
There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur--Weyl Duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. My thesis begins with a construction of an analogous basis for the multiset partition algebra. It continues with applications of this basis to constructing the irreducible representations of the multiset partition algebra and analysis of subalgebras analogous to the Tanabe algebras and the Brauer algebra. Finally, I address connections between the multiset partition algebra and longstanding questions in the representation theory of the symmetric group including the Kronecker problem and the restriction problem from GL_n to S_n
Standard monomial theory for wonderful varieties
A general setting for a standard monomial theory on a multiset is introduced
and applied to the Cox ring of a wonderful variety. This gives a degeneration
result of the Cox ring to a multicone over a partial flag variety. Further, we
deduce that the Cox ring has rational singularities.Comment: v3: 20 pages, final version to appear on Algebras and Representation
Theory. The final publication is available at Springer via
http://dx.doi.org/10.1007/s10468-015-9586-z. v2: 20 pages, examples added in
Section 3 and in Section
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