arXiv subjects

Paul Kim Long V. Nguyen

Publications and source records attributed to Paul Kim Long V. Nguyen.

2 recordsLinked to original sources

arXiv2019

The Strength of the Grätzer-Schmidt Theorem

The Grätzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable lattices that are complete is $Π^1_1$-complete; the set of indices of computable lattices that are algebraic is $Π^1_1$-complete; the set of compact elements of a computable lattice is $Π^{1}_{1}$ and can be $Π^1_1$-complete; and the set of compact elements of a distributive computable lattice is $Π^{0}_{3}$, and there is an algebraic distributive computable lattice such that the set of its compact elements is $Π^0_3$-complete.

math.LO↗

arXiv2014

Algorithmic randomness for Doob's martingale convergence theorem in continuous time

We study Doob's martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, Doob randomness is strictly weaker than computable randomness and is incomparable with Schnorr randomness.

cs.LO↗