Dershowitz–Manna ordering

"Multiset ordering" redirects here. For the multiset variant of the recursive path ordering, see multiset path 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 $(S,<_{S})$ is a well-founded partial order and let ${\mathcal {M}}(S)$ be the set of all finite multisets on $S$. For multisets $M,N\in {\mathcal {M}}(S)$ we define the Dershowitz–Manna ordering $M<_{DM}N$ as follows:

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

• $X\neq \varnothing$,
• $X\subseteq N$,
• $M=(N-X)+Y$, 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

• $M\neq N$, 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

• Dershowitz, Nachum; Manna, Zohar (1979), "Proving termination with multiset orderings", Communications of the ACM, 22 (8): 465–476, CiteSeerX 10.1.1.1013.432, doi:10.1145/359138.359142, MR 0540043, S2CID 17906810. (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. (ed.), 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.