The Complete Extensions do not form a Complete Semilattice
This is an incremental correction to a foundational result in abstract argumentation theory, clarifying a potential misconception for researchers in the field.
The paper identifies an error in Dung's seminal proof that the set of complete extensions forms a complete semilattice, providing counterexamples and tracing the mistake, but later retracts the claim as the counterexample was based on a misunderstanding of the definition of 'glb'.
In his seminal paper that inaugurated abstract argumentation, Dung proved that the set of complete extensions forms a complete semilattice with respect to set inclusion. In this note we demonstrate that this proof is incorrect with counterexamples. We then trace the error in the proof and explain why it arose. We then examine the implications for the grounded extension. [Reason for withdrawal continued] Page 4, Example 2 is not a counterexample to Dung 1995 Theorem 25(3). It was believed to be a counter-example because the author misunderstood ``glb'' to be set-theoretic intersection. But in this case, ``glb'' is defined to be other than set-theoretic intersection such that Theorem 25(3) is true. The author was motivated to fully understand the lattice-theoretic claims of Dung 1995 in writing this note and was not aware that this issue is probably folklore; the author bears full responsibility for this error.