Abstract
The concept of optimal fixpoint was introduced by Manna and Shamir [6, 7] in order to extract the maximum amount of “useful” information from a recursive definition. In this paper, we extend the concept of optimal fixpoint to arbitrary posets and investigate conditions which guarantee their existence. We prove that if a poset is chain-complete and has bounded joins, then every monotonic function has an optimal fixpoint. We also provide a sort of converse which generalizes a Theorem of A. Davis [2]. If a lower semilattice has bounded joins and every monotonic function has a fixpoint, then it is chain-complete.
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Gallier, J.H. On the existence of optimal fixpoints. Math. Systems Theory 13, 209–217 (1979). https://doi.org/10.1007/BF01744296
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DOI: https://doi.org/10.1007/BF01744296