Related papers: On The Composition Lemma
We provide simple proofs of analogues for coverings numbers of lattices of several recently studied basic statements on the ranks of tensors. We highlight the differences and analogies between the proofs in both settings.
The main results in this note concern the characterization of the length of continua 1 (Theorems 2.5) and the parametrization of continua with finite length (Theorem 4.4). Using these results we give two independent and relatively…
We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This…
These notes are a self-contained short proof of the stability of persistence diagrams.
We develop a theory for describing composite objects in physics. These can be static objects, such as tables, or things that happen in spacetime (such as a region of spacetime with fields on it regarded as being composed of smaller such…
The purpose of this article is to present my new proof of the the construction and the convergence theorem of spectral sequences of filtered complexes, which is much shorter and cleaner than the "standard" proof.
Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates…
We argue for a compositional semantics grounded in a strongly typed ontology that reflects our commonsense view of the world and the way we talk about it in ordinary language. Assuming the existence of such a structure, we show that the…
We present a non-vacuous definition of compositionality. It is based on the idea of combining the minimum description length principle with the original definition of compositionality (that is, that the meaning of the whole is a function of…
We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2].
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
We provide a simple proof of Kamp's theorem.
Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary. In fact, all we need is the theorem on the dimensions…
This short note present a "proof" of $P\neq NP$. The "proof" with double quotation marks is to indicate that we do not know whether the proof is correct or not (We're confused because we do know in which we make the mistakes).
In this short \'etude, we observe that the full structure of a recollement on a stable infinity-category can be reconstructed from minimal data: that of a reflective and coreflective full subcategory. The situation has more symmetry than…
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas' lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas' lemma from it using…
We state a construction theorem for specifications starting from single-site conditional probabilities (singleton part). We consider general single-site spaces and kernels that are absolutely continuous with respect to a chosen product…
Compositional verification algorithms are well-studied in the context of model checking. Properly selecting components for verification is important for efficiency, yet has received comparatively less attention. In this paper, we address…
Automatic security protocol analysis is currently feasible only for small protocols. Since larger protocols quite often are composed of many small protocols, compositional analysis is an attractive, but non-trivial approach. We have…
Lexicographic composition is a natural way to build an aggregate choice function from component choice functions. As the name suggests, the components are ordered and choose sequentially. The sets that subsequent components select from are…