Gödel semantics of fuzzy argumentation frameworks with consistency degrees

Jiachao Wu; Lingqiang Li; Weihua Sun; Jiachao Wu; Lingqiang Li; Weihua Sun

doi:10.3934/math.2020260

AIMS Mathematics

2020, Volume 5, Issue 4: 4045-4064. doi: 10.3934/math.2020260

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Research article

Gödel semantics of fuzzy argumentation frameworks with consistency degrees

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2.
Department of Mathematics, Liaocheng University, Liaocheng 252059, China
3.
School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China

Received: 13 January 2020 Accepted: 23 April 2020 Published: 27 April 2020
MSC : 03E72, 03E75

Argumentation frameworks (AF) play important roles in artificial intelligence. This paper is an exploration in establishing semantics of fuzzy AFs by fuzzy sets. There are many ways to characterize the semantics of fuzzy AFs. In this paper, our work is based on the assumption that some inconsistency of the system is permitted. Firstly, we formalize the conflict-freeness with a consistency degree x and the acceptability with a consistency degree y. Various types of extensions are then defined in a way similar to Dung's approach. The conflict-freeness and acceptability can be seen as an interpretation of the corresponding notion in Janssen's work. Formally, we add the conflict-freeness into the admissible extensions and the preferred extensions. We also introduce the complete extensions and the grounded extensions. Moreover, some basic properties are proven, such as the Fundamental Lemma, the algorithm of the grounded extension, etc. At last, it is proven to be consistent with Dung's original semantics in crisp AFs.
- argumentation frameworks,
- fuzzy arguments,
- fundamental lemma,
- characteristic function,
- fuzzy argumentation frameworks
Citation: Jiachao Wu, Lingqiang Li, Weihua Sun. Gödel semantics of fuzzy argumentation frameworks with consistency degrees[J]. AIMS Mathematics, 2020, 5(4): 4045-4064. doi: 10.3934/math.2020260

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Abstract

Argumentation frameworks (AF) play important roles in artificial intelligence. This paper is an exploration in establishing semantics of fuzzy AFs by fuzzy sets. There are many ways to characterize the semantics of fuzzy AFs. In this paper, our work is based on the assumption that some inconsistency of the system is permitted. Firstly, we formalize the conflict-freeness with a consistency degree x and the acceptability with a consistency degree y. Various types of extensions are then defined in a way similar to Dung's approach. The conflict-freeness and acceptability can be seen as an interpretation of the corresponding notion in Janssen's work. Formally, we add the conflict-freeness into the admissible extensions and the preferred extensions. We also introduce the complete extensions and the grounded extensions. Moreover, some basic properties are proven, such as the Fundamental Lemma, the algorithm of the grounded extension, etc. At last, it is proven to be consistent with Dung's original semantics in crisp AFs.

References

[1]	L. Amgoud and C. Cayrol, Inferring from inconsistency in preference-based argumentation frameworks, J. Autom. Reasoning, 29 (2002), 125-169. doi: 10.1023/A:1021603608656
[2]	L. Amgoud, C. Cayrol, M. C. Lagasquie-Schiex, et al. On bipolarity in argumentation frameworks, Int. J. Int. Syst., 23 (2008), 1062-1093. doi: 10.1002/int.20307
[3]	P. Baroni, M. Caminada and M. Giacomin, An introduction to argumentation semantics, The knowledge engineering review, 26 (2011), 365-410. doi: 10.1017/S0269888911000166
[4]	P. Baroni, F. Cerutti, M. Giacomin, et al. AFRA: Argumentation framework with recursive attacks, Int. J. Appr. Reas., 52 (2011), 19-37. doi: 10.1016/j.ijar.2010.05.004
[5]	S. Bistarelli, F. Rossi, F. Santini, A novel weighted defence and its relaxation in abstract argumentation, Int. J. Appr. Reas., 92 (2018), 66-86. doi: 10.1016/j.ijar.2017.10.006
[6]	C. Caginalp and G. Caginalp, Establishing cryptocurrency equilibria through game theory, AIMS Math., 4 (2019), 420-436. doi: 10.3934/math.2019.3.420
[7]	M. Caminada, On the issue of reinstatement in argumentation, European Workshop on Logics in Artificial Intelligence, (2006), 111-123.
[8]	C. da Costa Pereira, A. G. Tettamanzi, S. Villata, Changing one's mind: Erase or rewind? possibilistic belief revision with fuzzy argumentation based on trust, Proc. 22nd IJCAI, (2011), 164-171.
[9]	P. Dondio, Multi-value and probabilistic argumentation frameworks, Proc. 5th COMMA, (2014), 253-260.
[10]	P. M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games, Artif. Intell., 77 (1995), 321-357. doi: 10.1016/0004-3702(94)00041-X
[11]	P. E. Dunne, A. Hunter, P. McBurney, et al. Weighted argument systems: Basic definitions, algorithms, and complexity results, Artif. Intel., 175 (2011), 457-486. doi: 10.1016/j.artint.2010.09.005
[12]	D. M. Gabbay, Equational approach to argumentation networks, Argument & Computation, 3 (2012), 87-142.
[13]	D. M. Gabbay and O. Rodrigues, Equilibrium states in numerical argumentation networks, Logica Universalis, 9 (2015), 411-473. doi: 10.1007/s11787-015-0119-7
[14]	P. McBurney, S. Parsons, I. Rahwan, (editors) Argumentation in Multi-Agent Systems, Lecture Notes in Computer Science, 7543 (2012), 1-8. Springer.
[15]	A. Hunter, A probabilistic approach to modelling uncertain logical arguments, Int. J. Approx. Reason., 54 (2013), 47-81. doi: 10.1016/j.ijar.2012.08.003
[16]	J. Janssen, M. De Cock, D. Vermeir, Fuzzy argumentation frameworks, Proc. 12th IPMU, (2008), 513-520.
[17]	S. Kaci and C. Labreuche, Argumentation framework with fuzzy preference relations, Proc. 13th IPMU, (2010), 554-563.
[18]	H. Li, N. Oren, T. J. Norman, Probabilistic argumentation frameworks, Proc. TAFA-11, (2011), 1-16. Springer.
[19]	X. Li and M. Bohner, An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875-1881. doi: 10.1016/j.camwa.2012.03.013
[20]	X. Li, J. Shen, R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.
[21]	X. Li, X. Yang, T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.
[22]	X. Lv, A degree condition for fractional (g, f, n)-critical covered graphs, AIMS Math., 5 (2020), 872-878. doi: 10.3934/math.2020059
[23]	S. Modgil, Reasoning about preferences in argumentation frameworks, Artif. Intel., 173 (2009), 901-934. doi: 10.1016/j.artint.2009.02.001
[24]	S. H. Nielsen and S. Parsons, A generalization of Dung's abstract framework for argumentation: Arguing with sets of attacking arguments, Proc. 3rd ArgMAS, (2006), 7-19.
[25]	S. Polberg and N. Oren, Revisiting support in abstract argumentation systems, Proc. 5th COMMA, (2014), 369-376.
[26]	H. Prakken, An abstract framework for argumentation with structured arguments, Argument and Computation, 1 (2010), 93-124. doi: 10.1080/19462160903564592
[27]	R. Stranders, M. de Weerdt, C. Witteveen, Fuzzy argumentation for trust, Proc. 5th CLIMA, (2007), 214-230. Springer.
[28]	N. Tamani and M. Croitoru, Fuzzy argumentation system for decision support, Proc. 15th IPMU, (2014), 77-86. Springer.
[29]	J. Wu, H. Li, N. Oren, et al. Gödel fuzzy argumentation frameworks, Proc. 6th COMMA, (2016), 447-458.
[30]	D. Yang, X. Li, J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal-Hybri., 32 (2019), 294-305. doi: 10.1016/j.nahs.2019.01.006
[31]	X. Yang, X. Li, Q. Xi, et al. Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng., 15 (2018), 1495-1515. doi: 10.3934/mbe.2018069
[32]	F. F. Zhao, L. Q. Li, S. B. Sun, et al. Rough approximation operators based on quantale-valued fuzzy generalized neighborhood systems, Iran. J. Fuzzy Syst., 16 (2019), 53-63.

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