# Dershowitz–Manna ordering

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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 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).

## References

• Dershowitz, Nachum; Manna, Zohar (1979), "Proving termination with multiset orderings", Communications of the ACM, 22 (8): 465–476, doi:10.1145/359138.359142, MR 0540043. (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].)
• Huet, G.; Oppen, D. C. (1980), "Equations and rewrite rules: A survey", in Book, R., Formal Language Theory: Perspectives and Open Problems, New York: Academic Press, pp. 349–405.
• Jouannaud, Jean-Pierre; Lescanne, Pierre (1982), "On multiset orderings", Information Processing Letters, 15 (2): 57–63, doi:10.1016/0020-0190(82)90107-7, MR 0675869.