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 Hirschhorn’s 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 Ganzinger’s Ordered Resolution Prover

We present a formalization of the first half of Bachmair and Ganzinger’s 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 chapter’s 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 Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL theorem prover. We analyze sufficient conditions for the theorems’ 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 Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the S ́wierczkowski–Paulson semantics-based approach. As part of our framework’s validation, we upgrade Paulson’s 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

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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