Editing Least fixed point (section)

==Greatest fixed points==
The greatest fixed point of a function can be defined analogously to the least fixed point, as the fixed point which is greater than any other fixed point, according to the order of the poset. In [[computer science]], greatest fixed points are much less commonly used than least fixed points. Specifically, the posets found in [[domain theory]] usually do not have a greatest element, hence for a given function, there may be multiple, mutually incomparable [[Maximal element|maximal]] fixed points, and the greatest fixed point of that function may not exist. To address this issue, the ''optimal fixed point'' has been defined as the most-defined fixed point compatible with all other fixed points. The optimal fixed point always exists, and is the greatest fixed point if the greatest fixed point exists. The optimal fixed point allows formal study of [[recursive]] and [[corecursive]] functions that do not converge with the least fixed point.<ref>{{cite book |last1=Charguéraud |first1=Arthur |chapter=The Optimal Fixed Point Combinator |title=Interactive Theorem Proving |series=Lecture Notes in Computer Science |date=2010 |volume=6172 |pages=195–210 |doi=10.1007/978-3-642-14052-5_15 |isbn=978-3-642-14051-8 |chapter-url=https://www.chargueraud.org/research/2010/fix/fix.pdf |access-date=30 October 2021}}</ref> Unfortunately, whereas [[Kleene's recursion theorem]] shows that the least fixed point is effectively computable, the optimal fixed point of a [[computable function]] may be a non-computable function.<ref>{{cite thesis |type=Ph.D. thesis |last1=Shamir |first1=Adi |title=The fixedpoints of recursive definitions |date=October 1976 |publisher=Weizmann Institute of Science |url=https://weizmann.primo.exlibrisgroup.com/permalink/972WIS_INST/1d4esio/alma990002185270203596 |oclc=884951223 |language=en}} Here: Example 12.1, pp. 12.2–3</ref>