Andreas Blass
Yuri Gurevich
Abstract
The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many well-founded relations U1,…,Un then R is well-founded. A question arises how to bound the ordinal height |R| of the relation R in terms of the ordinals αi = |Ui|. We introduce the notion of the stature ∥P∥ of a well partial ordering P and show that |R| ≤ ∥α1 × … × αn∥ and that this bound is tight. The notion of stature is of considerable independent interest. We define ∥ P ∥ as the ordinal height of the forest of nonempty bad sequences of P, but it has many other natural and equivalent definitions. In particular, ∥ P ∥ is the supremum, and in fact the maximum, of the lengths of linearizations of P. And ∥α1 × … × αn∥ is equal to the natural product α1 ⊗ … ⊗ αn.
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Index Terms
- Program termination and well partial orderings
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