Dershowitz–Manna ordering

In mathematics, the Dershowitz–Manna ordering is a well-founded ordering on multisets named after Nachum Dershowitz and Zohar Manna. It is often used in context of termination of programs or term rewriting systems.

Suppose that ${\displaystyle (S,<_{S})}$ is a well-founded partial order and let ${\displaystyle {\mathcal {M}}(S)}$ be the set of all finite multisets on ${\displaystyle S}$. For multisets ${\displaystyle M,N\in {\mathcal {M}}(S)}$ we define the Dershowitz–Manna ordering ${\displaystyle M<_{DM}N}$ as follows:

${\displaystyle M<_{DM}N}$ whenever there exist two multisets ${\displaystyle X,Y\in {\mathcal {M}}(S)}$ with the following properties:

• ${\displaystyle X\neq \varnothing }$,
• ${\displaystyle X\subseteq N}$,
• ${\displaystyle M=(N-X)+Y}$, and
• ${\displaystyle X}$ dominates ${\displaystyle Y}$, that is, for all ${\displaystyle y\in Y}$, there is some ${\displaystyle x\in X}$ such that ${\displaystyle y<_{S}x}$.

An equivalent definition was given by Huet and Oppen as follows:

${\displaystyle M<_{DM}N}$ if and only if

• ${\displaystyle M\neq N}$, and
• for all ${\displaystyle y}$ in ${\displaystyle S}$, if ${\displaystyle M(y)>N(y)}$ then there is some ${\displaystyle x}$ in ${\displaystyle S}$ such that ${\displaystyle y<_{S}x}$ and ${\displaystyle M(x)<N(x)}$.