Abstract
We revisit the semantic relations between Assumption-Based Argumentation (ABA) and Logic Programming (LP) based on the recent development of model-based semantics for ABA frameworks. This effort is motivated by the close resemblance between the computation of complete ABA models and the computation of Przymuzinski’s partial stable models for logic programs. As we show these concepts coincide ipsis litteris, multiple results about the different ABA semantics (preferred, grounded, stable, semi-stable, ideal, eager) and corresponding LP semantics (regular, well-founded, stable, L-stable, ideal, eager) follow. Our approach also introduces a new translation from ABA frameworks to logic programs that has better properties than the one available in the literatue, including lower computational complexity. The combination of our new translation and model-based ABA semantics is the key to all of our results. It is also known that the more traditional assumption extension and labelling-based semantics for ABA can be obtained from ABA models using an operation called tuple projection, so it follows from our results that ABA is LP with projection.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This program is extracted from [6].
- 2.
\(\mathtt {not}\ \) denotes negation as failure [7].
- 3.
At this time, to avoid the necessity of formal concepts, only the core syntactic elements of \(\mathcal {F}\) are shown.
- 4.
In relational algebra, this is equivalent to selection, but the common name for this operation in logic programming literature is projection.
- 5.
The interpretation order \(\preceq \) is based on >.
- 6.
Logic programs whose rules may contain weak but not strong negation, and the head of each rule is an atom [10].
- 7.
\(\Psi _P(I)\) is a least fix-point of the immediate consequences operator \(\Psi \) of [11], which is guaranteed to exist and be unique for positive programs.
- 8.
The concept of deductive systems used in ABA is fully detailed in [2].
- 9.
Traditionally, \(\mathcal {A}\) is required to be non-empty. We opt to relax that condition to favor a simpler presentation of our later concepts.
- 10.
In most ABA works, the codomain of \(\bar{\ }\) is \(\mathcal {L}\), but the contraries of assumptions are implicitly assumed to be non-assumptions.
- 11.
Let S be a set and \(T = \langle S_1, S_2, \ldots , S_k \rangle \) be a tuple of sets. The projection of elements of S from T is \(\sigma _S(T) = \langle S_1 \cap S, S_2 \cap S, \ldots , S_k \cap S \rangle \).
- 12.
Given some \(\mathcal {F}\) (resp. P), a complete (resp. p-stable) model \(I = \langle T,F,U \rangle \) of \(\mathcal {F}\) is a semi-stable (resp. L-stable) model of \(\mathcal {F}\) (resp. P) iff I has minimal \(\{l \in \mathcal {L}\mid I(l) = \mathbf {u}\}\) amongst all complete (resp. p-stable) models of \(\mathcal {F}\) (resp. P).
References
Alcântara, J.F.L., Sá, S., Guadarrama, J.C.A.: On the equivalence between abstract dialectical frameworks and logic programs. Theory Pract. Log. Program. 19(5–6), 941–956 (2019)
Bondarenko, A., Dung, P., Kowalski, R., Toni, F.: An abstract, argumentation-theoretic approach to default reasoning. AI 93, 63–101 (1997)
Caminada, M., Harikrishnan, S., Sá, S.: Comparing logic programming and formal argumentation; the case of ideal and eager semantics. Argum. Comput. (Preprint), 1–28 (2020)
Caminada, M., Sá, S., Alcântara, J., Dvorák, W.: On the difference between assumption-based argumentation and abstract argumentation. IfCoLog J. Logics Appl. 2, 15–34 (2015). http://www.collegepublications.co.uk/journals/ifcolog/?00003
Caminada, M., Sá, S., Alcântara, J., Dvorák, W.: On the equivalence between logic programming semantics and argumentation semantics. Int. J. Approx. Reason. 58, 87–111 (2015)
Caminada, M., Schulz, C.: On the equivalence between assumption-based argumentation and logic programming. J. Artif. Intell. Res. 60, 779–825 (2017)
Clark, K.L.: Negation as failure. In: Logic and Data Bases, pp. 293–322. Springer, Heidelberg (1978). https://doi.org/10.1007/978-1-4684-3384-5_11
Dung, P., Kowalski, R., Toni, F.: Assumption-based argumentation. In: Simari, G., Rahwan, I. (eds.) Argumentation in Artificial Intelligence, pp. 199–218. Springer, US (2009). https://doi.org/10.1007/978-0-387-98197-0_10
Eiter, T., Leone, N., Sacca, D.: On the partial semantics for disjunctive deductive databases. Ann. Math. Artif. Intell. 19(1–2), 59–96 (1997)
Lloyd, J.W.: Foundations of Logic Programming; (2nd extended ed.). Springer-Verlag, New York Inc., New York (1987). https://doi.org/10.1007/978-3-642-83189-8
Przymusinski, T.: The well-founded semantics coincides with the three-valued stable semantics. Fundamenta Informaticae 13(4), 445–463 (1990)
Przymusinski, T.C.: Stable semantics for disjunctive programs. New Gener. Comput. 9, 401–424 (1991)
Sá, S., Alcântara, J.: Interpretations and models for assumption-based argumentation. In: Proceedings of the 34rd Annual ACM Symposium on Applied Computing. SAC ’19, ACM, New York, NY, USA (2019). https://doi.org/10.1145/3297280.329739
Schulz, C., Toni, F.: Complete assumption labellings. In: Computational Models of Argument - Proceedings of COMMA 2014, Atholl Palace Hotel, Scottish Highlands, UK, 9–12 September 2014, pp. 405–412 (2014). https://doi.org/10.3233/978-1-61499-436-7-405
Toni, F.: A tutorial on assumption-based argumentation. Argum. Comput. 5(1), 89–117 (2014)
Wu, Y., Caminada, M., Gabbay, D.: Complete extensions in argumentation coincide with 3-valued stable models in logic programming. Studia Logica 93(1–2), 383–403 (2009). Special issue: new ideas in argumentation theory
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Sá, S., Alcântara, J. (2021). Assumption-Based Argumentation Is Logic Programming with Projection. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-86772-0_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-86771-3
Online ISBN: 978-3-030-86772-0
eBook Packages: Computer ScienceComputer Science (R0)