64 research outputs found
Positive model structures for abstract symmetric spectra
We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhornâs sense, of a certain positive projective model structure on spectra, where positivity basically means the truncation of the zero level. The localization is by the set of stabilizing morphisms or their truncated version
Formalising Mathematics in Simple Type Theory
Despite the considerable interest in new dependent type theories, simple type
theory (which dates from 1940) is sufficient to formalise serious topics in
mathematics. This point is seen by examining formal proofs of a theorem about
stereographic projections. A formalisation using the HOL Light proof assistant
is contrasted with one using Isabelle/HOL. Harrison's technique for formalising
Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic
type classes. However, every formal system can be outgrown, and mathematics
should be formalised with a view that it will eventually migrate to a new
formalism
Formalizing Bachmair and Ganzingerâs Ordered Resolution Prover
We present a formalization of the first half of Bachmair and Ganzingerâs chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapterâs text, emphasizing the value of formal proofs in the field of automated reasoning
A formally verified abstract account of Gödel's incompleteness theorems
We present an abstract development of Gödelâs incompleteness theorems, performed with the help of the Isabelle/HOL theorem prover. We analyze sufficient conditions for the theoremsâ applicability to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosserâs variation of the first theorem, Jeroslowâs variation of the second theorem, and the S ÌwierczkowskiâPaulson semantics-based approach. As part of our frameworkâs validation, we upgrade Paulsonâs Isabelle proof to produce a mech- anization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation
Logarithmic topological Hochschild homology of topological K-theory spectra
Contains fulltext :
183369.pdf (preprint version ) (Open Access
Generalized Thom spectra and their topological Hochschild homology
Contains fulltext :
214009.pdf (publisher's version ) (Closed access)
Contains fulltext :
214009pre.pdf (preprint version ) (Open Access
Link Prediction in Knowledge Graphs with Concepts of Nearest Neighbours
International audienceThe open nature of Knowledge Graphs (KG) often implies that they are incomplete. Link prediction consists in infering new links between the entities of a KG based on existing links. Most existing approaches rely on the learning of latent feature vectors for the encoding of entities and relations. In general however, latent features cannot be easily interpreted. Rule-based approaches offer interpretability but a distinct ruleset must be learned for each relation, and computation time is difficult to control. We propose a new approach that does not need a training phase, and that can provide interpretable explanations for each inference. It relies on the computation of Concepts of Nearest Neighbours (CNN) to identify similar entities based on common graph patterns. Dempster-Shafer theory is then used to draw inferences from CNNs. We evaluate our approach on FB15k-237, a challenging benchmark for link prediction, where it gets competitive performance compared to existing approaches
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