The Complete Extensions do not form a Complete Semilattice

TL;DR

This paper identifies and corrects an error in Dung's proof that the set of complete extensions forms a complete semilattice, showing the proof is invalid and discussing implications for the grounded extension.

Contribution

It demonstrates that Dung's proof is incorrect through counterexamples and clarifies the misunderstanding regarding the lattice structure of complete extensions.

Findings

Dung's proof of the complete semilattice property is invalid

Counterexamples show the set of complete extensions does not always form a complete semilattice

Implications for the grounded extension are discussed

Abstract

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.