Semigroup simulation of a Turing machine ($h{q}_{1}wh=q$)

A finite deterministic Turing machine can be represented by a finite semigroup presentation whose generators are tape symbols, state symbols, a boundary marker $h$, and an accept symbol $q$. Defining equalities replace the local scanned symbol and state according to machine instructions, with separate boundary cases for implicit blanks. Only terminal-state equalities erase tape letters and then replace $h{q}_{H}h$ by $q$. A derivation from an initial description to $q$ must encounter a terminal state before its first erasure. Simulation equalities preserve halting membership in either direction: for a deterministic transition $C\to D$, $C$ halts if and only if $D$ halts. Thus reverse use of defining equalities cannot create a false halting certificate. Boundary and erasure rules require explicit checking for every chosen encoding.