Dershowitz–Manna ordering

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

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

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

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

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

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

References

• Template:Citation. (Also in Proceedings of the International Colloquium on Automata, Languages and Programming, Graz, Lecture Notes in Computer Science 71, Springer-Verlag, pp. 188–202 [July 1979].)
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