A deterministic machine has finitely many states and tape symbols, and a partial transition table. Each transition writes a symbol, changes state, and moves exactly one cell left or right. There is no distinguished initial state: mortality concerns every configuration.

We use finite tape segments, halting when a transition is absent or the head would leave the segment. Mortality means that one bound works for every segment length, every tape content, every head position and every state. This is the bounded finite-configuration formulation in Section 6.2. It is equivalent to uniform mortality on unbounded tapes: every finite run uses a finite segment, and every segment run embeds in an unbounded tape. The connection to the infinite-run formulation of Hooper's theorem additionally uses compactness, and belongs to the proof obligation, not the definition.

States are 0,…,states−1 and symbols 0,…,symbols−1, where the stored counts are increased by one to keep them nonempty. Table rows are indexed by states, columns by symbols. Missing entries halt; output numbers are reduced modulo the respective sizes. $true$ means move right. These conventions describe all deterministic machines and give an effective interpretation of every table.