Axiomatic system

In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system. Properties An axiomatic system is said to be consistent if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion). In an axiomatic syste

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (11)

Related publications

Related people (2)

Related people

Related units (3)

Related units

Related concepts (64)

Related concepts

Related courses (8)

Related courses

Related lectures (5)

Related lectures

Regularity in metric spaces

Serguei Timochine

Using arguments developed by De Giorgi in the 1950's, it is possible to prove the regularity of the solutions to a vast class of variational problems in the Euclidean space. The main goal of the present thesis is to extend these results to the more abstract context of metric spaces with a measure. In particular, working in the axiomatic framework of Gol'dshtein – Troyanov, we establish both the interior and the boundary regularity of quasi-minimizers of the p-Dirichlet energy. Our proof works for quite general domains, assuming some natural hypotheses on the (axiomatic) D-structure. Furthermore, we prove analogous results for extremal functions lying in the class of Sobolev functions in the sense of Hajłasz – Koskela, i.e. functions characterized by the single condition that a Poincaré inequality be satisfied. Our strategy to prove these regularity results is first to show that, in a very general setting, the (Hölder) continuity of a function is a consequence of three specific technical hypotheses. This part of the argument is the essence of the De Giorgi method. Then, we verify that for a function u which is a quasi-minimizer in an axiomatic Sobolev space or an extremal Sobolev function in the sense of Hajłasz – Koskela, these technical hypotheses are indeed satisfied and u is thus (Hölder) continuous. In addition to that, we establish the Harnack's inequality for these extremal functions, and we show that the Dirichlet semi-norm of a piecewise-extremal function is equivalent to the sum of the Dirichlet semi-norms of its components.

EPFL2006

Analytic twists of modular forms and applications

Alexandre François Peyrot

We are interested in the study of non-correlation of Fourier coefficients of Maass forms against a wide class of real analytic functions. In particular, the class of functions we are interested in should be thought of as some archimedean analogs of Frobenius trace functions. In the first part of the thesis, we give an axiomatic definition for this class, and prove that these functions satisfy properties similar to that of Frobenius trace functions. In particular, we prove non-correlation statements analogous to those given by Fouvry, Kowalski and Michel for algebraic trace functions. In the second part of the thesis, we establish the existence of large values of Hecke-Maass L-functions with prescribed argument. In studying these problems, one encounters sums of Fourier coefficients of Maass forms against real oscillatory functions. In some cases, one can prove that these functions satisfy the axioms discussed previously.

EPFL2017

Never Say Never Probabilistic & Temporal Failure Detectors (Extended)

Rachid Guerraoui, David Kozhaya, Yvonne Anne Pignolet-Oswald

The failure detector approach for solving distributed computing problems has been celebrated for its modularity. This approach allows the construction of algorithms using abstract failure detection mechanisms, defined by axiomatic properties, as building blocks. The minimal synchrony assumptions on communication, which enable to implement the failure detection mechanism, are studied separately. Such synchrony assumptions are typically expressed as eventual guarantees that need to hold, after some point in time, forever and deterministically. But in practice, they never do. Synchrony assumptions may hold only probabilistically and temporarily. In this paper, we study failure detectors in a realistic distributed system N, with asynchrony inflicted by probabilistic synchronous communication. We address the following paradox about the weakest failure detector to solve the consensus problem (and many equivalent problems), i.e., S: an implementation of “consensus with probability 1” is possible in N without using randomness in the algorithm itself, while an implementation of “S with probability 1” is impossible to achieve in N. We circumvent this paradox by introducing a new failure detector S*, a variant of S with probabilistic and temporal accuracy. We prove that S* is implementable in N and we provide an optimal S* implementation. Interestingly, we show that S* can replace S , in several existing deterministic consensus algorithms using S, to yield an algorithm that solves “consensus with probability 1”. In fact, we show that such result holds for all decisive problems (not only consensus) and also for failure detector P (not only S). The resulting algorithms combine the modularity of distributed computing practices with the practicality of networking ones.

IPDPS2016

Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These top

Axiom

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξ

First-order logic

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

MATH-101(e): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.