This paper examines modifications of the definition of lowness for prefix-free complexity \(K\). An oracle \(x\) is low for \(K\) when for every string \(\sigma\), \[ K(\sigma)\le K^X(\sigma)\text{ up to an additive constant.}\tag{\(\ast\)} \] The authors introduce an ostensibly weaker property: \(X\) is low for \(K\) along an infinite sequence \(A\) when for every \(\sigma\) that is an initial segment of \(A\), \((\ast)\) holds. They then prove that, in fact, \(X\) is low for \(K\) along some sequence iff \(X\) is low for \(K\) along all sequences iff \(x\) is low for \(K\).
Likewise, the authors consider an apparent strengthening of “weakly low for \(K\)” to “weakly low for \(K\) along \(A\)” – by requiring that \((\ast)\) hold not just for infinitely many \(\sigma\), but for infinitely many initial segments of \(A\) – and establish a similar equivalence result. The paper also discusses consequences of requiring \((\ast)\) to hold on other sets of strings.