User:CircuitCraft/Definition:Semi-Thue Machine
Jump to navigation
Jump to search
Definition
Let $\Sigma$ be a finite set.
Let $\Sigma^*$ be the set of finite strings over $\Sigma$.
Let $\vdash \subset \Sigma^* \times \Sigma^*$ be a relation.
Let $\vdash^* \subset \Sigma^* \times \Sigma^*$ be defined as:
- $\Gamma \vdash^* \Gamma'$
if and only if there exist $\alpha, \beta, \gamma, \gamma' \in \Sigma^*$ such that:
- $\Gamma = \alpha \gamma \beta$
- $\Gamma' = \alpha \gamma' \beta$
- $\gamma \vdash \gamma'$
Let $M = \struct {\Sigma^*, \vdash^*}$ be an abstract machine.
Then $M$ is the semi-Thue machine over $\Sigma$ generated by $\vdash$.