Equational approach to argumentation networks

Abstract

This paper provides equational semantics for Dung's argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the “extensions” of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, and much more. The equational approach has its conceptual roots in the nineteenth century following the algebraic equational approach to logic by George Boole, Louis Couturat, and Ernst Schroeder.

Keywords:

• argumentation networks
• equational approach
• support in argumentation
• time-dependent networks
• higher level attacks
• numerical argumentation

Acknowledgements

I am grateful to Martin Caminada, Nachum Dershowitz, Phan Minh Dung, David Makinson, Alex Rabinowich, and Serena Villata for helpful discussions. I am also indebted to the referees for their penetrating comments and methodological criticisms. Research done under ISF project: Integrating Logic and Network Reasoning.

Notes

The other solutions are

See http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem (Sobolev 2001).

We have in these papers not only weighted nodes (arguments) and weighted arrows (attacks), but also higher level attacks, temporal dependence, and a lot more. These notions are being reproduced now by many authors. See, for example, Baroni, Cerutti, Giacomin, and Guida (2009) and Dung (1995) for higher level attacks originating in Barringer et al. (2005), and our discussions and further development in Gabbay (2009a,b,c). These papers are part of a general methodological approach to applied logic and connect many areas together.

Hanh et al. point out that their approach is sceptical generalisation of the standard argumentation framework, while others such as Sanjay's, Gabbay's, or Baroni et al.’s are rather credulous. Hence, in each extension of these approaches, there is a sceptical part that is one of Hanh et al.’s extensions. Hanh et al. also point out that Sanjay's generalisation of grounded semantics is rather more liberal than theirs, and hence his characteristic function is not monotonic.

This is a neater condition than the one mentioned in Remark 2.19, which was more explanatory than efficient.