arXiv subjects

Mushfeq Khan

Publications and source records attributed to Mushfeq Khan.

4 recordsLinked to original sources

arXiv2020

From eventually different functions to pandemic numberings

A function is strongly non-recursive (SNR) if it is eventually different from each recursive function. We obtain hierarchy results for the mass problems associated with computing such functions with varying growth bounds. In particular, there is no least and no greatest Muchnik degree among those of the form SNR$_f$ consisting of SNR functions bounded by varying recursive bounds $f$. We show that the connection between SNR functions and canonically immune sets is, in a sense, as strong as that between DNR (diagonally non-recursive) functions and effectively immune sets. Finally, we introduce pandemic numberings, a set-theoretic dual to immunity.

math.LO↗

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↗

arXiv2017

Effective bi-immunity and randomness

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no ML random set. We show that the same holds when ML randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set. We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every ML random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

math.LO↗

arXiv2017

Forcing with Bushy Trees

We present several results that rely on arguments involving the combinatorics of "bushy trees". These include the fact that there are arbitrarily slow-growing diagonally noncomputable (DNC) functions that compute no Kurtz random real, as well as an extension of a result of Kumabe in which we establish that there are DNC functions relative to arbitrary oracles that are of minimal Turing degree. Along the way, we survey some of the existing instances of bushy tree arguments in the literature.

math.LO↗