Table 1.

Table of mathematical notation and acronyms

${\mathbf{\text{I}}}_{\mathbf{\text{ac}}}(x)$	the size of the algorithmic information content of an encoded object $x$	Section 2
${\mathbf{\text{I}}}_{\mathbf{\text{ac}}}(y|x)$	the size of the (conditional) algorithmic information content of an encoded object $y$ given an encoded object $x$	Section 2
FDDDS	acronym for finite discrete deterministic dynamical system	Section 2
$c_{\mathbf{\text{ I }}}$	constant that determines the equivalence class ${\mathbf{\text{I}}}_{\mathbf{\text{ac}}}(\cdot )$	Section 2
${\mathcal{S}}_t$	a single state of a FDDDS $\mathcal{S}$ at time instant $t$	Section 2
$\mathcal{S}{\upharpoonright }_t^{t^{\prime }}$	a state space trajectory $({\mathcal{S}}_t,\dots ,{\mathcal{S}}_{t^{\prime }})$ of $\mathcal{S}$ from the state ${\mathcal{S}}_t$ until state ${\mathcal{S}}_{t^{\prime }}$	Section 2
$\mathcal{P}$	an algorithmic perturbation	Section 3
$O$	an observer Turing machine, which is a particular type of two-tape Turing machine whose first tape receives input and the second tape stores formal knowledge	([1], Section 3.2)
$\mathcal{O}$	a formal observer system, which is a particular type of FDDDS that simulates $O$	Section 3
${\mathcal{S}}_t^{\prime }$	a single state of $\mathcal{S}$ at time instant $t$ that occurs after $\mathcal{S}$ being perturbed/observed at any time instant $<t$	Section 3
$c_{\mathcal{O}}$	constant that determines the minimum conditions for the observation to take place	Section 3
ODE	acronym for observer-dependent emergence	Section 3a
$c_e$	constant that determines the minimum conditions for observer-dependent emergence	Section 3a
AOIE	acronym for asymptotically observer-independent emergence	Section 3b
$\mathbf{\text{P}}$	a sequence (or collection) of Turing machines	Section 3b(i)
$\mathbf{\text{S}}$	a discrete deterministic dynamical system whose each contiguous subsequence of its entire state space trajectory is a state space trajectory of a particular FDDDS ${\mathcal{S}}^{(k)}$	Section 3b(ii)
$\mathfrak{S}$	a macro-level discrete deterministic dynamical system whose micro-level systems (or parts) cannot interact with each other	Section 3b(ii)
${\mathfrak{S}}^{\prime }$	a macro-level discrete deterministic dynamical system whose micro-level systems (or parts) do interact with each other	Section 3b(ii)