The algorithm is an algorithm in the sense of $Lax185574.Algorithms$, a word RAM program together with the exponent of the word length it needs. It comes first, before the class, the sentence and ε, and it knows none of them. Only the constant $c$ of the O-notation depends on them.

$A.Decides c (G, φ) (G ⊨ φ) T$ is a statement about the encoding of the pair $(G, φ)$ of $Lax185574.Encodings$, which is the encoding of the graph followed by the encoding of the sentence. At every word length that is admissible for $c$, every halting run of $A$ on this input writes $1$ if $G ⊨ φ$ and $0$ otherwise, and $A$ halts within $c · T + c$ steps once this number fits into a word. Here $T = n^(3+ε)$ is the real power. The additive $c$ is part of the O-notation, and it covers small graphs, such as the graph without vertices, where $T$ vanishes. An n-vertex graph with m edges is encoded by a word of length n + 4m + 1 ≤ 2n² + 1, so reading the input fits into the time bound.

Every graph lies in a monadically dependent class, for instance the class consisting of it alone. So the algorithm decides every instance correctly at every word length that is admissible for some such class. The time bound is claimed for members of a monadically dependent class only.

Monadic dependence is the transduction-based definition of $Lax185574.MonadicDependence$: the class does not transduce the class of all graphs. It agrees with the archive's definition $Lax710763.MonadicDependence$ and with the definition of the paper, see $Lax185574.MonadicDependence$.