In this chapter, we presented the background on AA and ABA frameworks as well as on logic programs used throughout this thesis.
Chapter 6 relies only on the background on AA frameworks from Section 2.2.1 and Chapter 3 only makes use of concepts regarding ABA and AA frameworks, presented in Section 2.2. All other chapters apply both concepts from argumentation and logic programming presented in this chapter.
Chapter 3
Labellings for Assumption-Based
and Abstract Argumentation
3.1
Introduction
As introduced in Section 2.2.1, the semantics of AA frameworks can be expressed in terms of either argument extensions or labellings. Argument labellings have the advan- tage over argument extensions that they do not only distinguish between accepted and non-accepted arguments, but further divide the non-accepted arguments into rejected and undecided ones. Since argument labellings and extensions correspond [CG09, BCG11], ar- gument labellings can also be used to characterise the semantics of a flat ABA framework in terms of its corresponding AA framework. In this chapter, we transfer the idea of argu- ment labellings to assumptions, yielding a new characterisation of the semantics of ABA frameworks. In contrast to argument labellings, which label whole arguments, assumption labellings label each assumption as in (accepted), out (rejected), or undec (undecided). Assumption labellings have the advantage over assumption extensions that rejected (out) assumptions and assumptions that are neither accepted nor rejected (undec) are dis- tinguished. This distinction can be important in applications such as decision making. Undecided assumptions can for example provide an indication that further information from an expert is required in order to make a definite decision about their acceptability.
We propose assumption labellings for all semantics defined for flat ABA frameworks, i.e. admissible, grounded, complete, preferred, ideal, semi-stable, and stable semantics, and prove that there is a one-to-one correspondence between the respective assumption labellings and extensions. We also investigate the relation between assumption labellings of flat ABA frameworks and argument labellings of the corresponding AA frameworks, showing a one-to-one correspondence for the grounded, complete, preferred, ideal, and stable semantics. These results extend existing work on the correspondence between the semantics of flat ABA frameworks and AA frameworks, as illustrated in Figure 3.1. Since semi-stable argument and assumption extensions do not correspond [CSAD15a], it is un- surprising that the respective labellings do not correspond either, as shown in Figure 3.2. Concerning the admissible semantics we prove a one-to-many correspondence between as- sumption and argument labellings. Based on this dissimilarity, we introduce a variant of admissible argument labellings for AA frameworks, called committed admissible argu- ment labellings, which correspond more closely to admissible assumption labellings than the original admissible argument labellings, as illustrated in Figure 3.3. We furthermore introduce labellings for possibly non-flat ABA frameworks, and prove correspondence with the extension semantics for possibly non-flat ABA frameworks, as shown in Figure 3.4.
The chapter is organised as follows. In Section 3.2, we introduce assumption labellings for the different semantics of flat ABA frameworks and prove their correspondence with as- sumption extensions of flat ABA frameworks. In Section 3.3, we simplify the definition of assumption labellings for flat ABA frameworks by considering only certain sets of assump- tions as attackers of assumptions. We furthermore introduce a graphical representation of flat ABA frameworks and illustrate how assumption labellings can be easily determined and represented using these graphs. In Section 3.4, we investigate the correspondence
complete/grounded/ preferred/stable/
ideal arg. ext.
[Dun95b,DMT07]
complete/grounded/ preferred/stable/
ideal arg. lab.
[CG09,BCG11]
complete asm. lab.
Sec. 3.2.2
grounded/preferred/ stable/ideal asm. lab.
Sec. 3.2.3 complete/grounded/
preferred/stable/ ideal asm. ext.
[BDKT97,DMT07] [CG09,BCG11] [DMT07, Ton12, CSAD15a] complete Sec. 3.2.2 grounded/ preferred/ stable/ideal Sec. 3.2.3 complete Sec. 3.4.2 grounded/ preferred/ stable/ideal Sec. 3.4.3 ABA as AA ABA as AA AA
ABA complete/grounded/preferred/stable/
ideal asm. lab. wrt. arg.-supp. sets Sec. 3.3.1 ABA complete/ grounded/ preferred/ stable/ideal Sec. 3.3.1
Figure 3.1: A summary of results concerning the complete, grounded, preferred, stable, and ideal semantics in this chapter, where applicable, in the context of previous work. Bidirectional arrows indicate semantic correspondence, and bold indicates novel work pre- sented in this chapter.
semi-stable arg. ext.
[Cam06b]
semi-stable arg. lab.
[CG09,BCG11]
semi-stable asm. lab. Sec. 3.2.3
semi-stable asm. ext.
[CSAD15a]
[CG09,BCG11]
[CSAD15a]
Sec.3.2.3
semi-stable asm. lab. wrt. arg.-supp. sets Sec. 3.3.1 ABA Sec. 3.3.1 Sec. 3.4.4 ABA AA ABA as AA ABA as AA
Figure 3.2: Results concerning the semi-stable semantics in this chapter, where applicable, in the context of previous work. Bidirectional arrows indicate semantic correspondence, crossed out arrows denote non-correspondence, and bold indicates novel work in this chap- ter.
admissible arg. ext. [Dun95b] committed admissible arg. lab. Sec. 3.4.5 admissible asm. lab. Sec. 3.4.5 admissible asm. ext. [BDKT97] [CG09] [DMT07] admissible arg. lab. [CG09] Sec. 3.4.5 Sec. 3.4.5 Sec. 3.4.5 Sec. 3.2.1 Sec. 3.4.5
admissible asm. lab. wrt. arg.-supp. sets Sec. 3.3.1 ABA Sec. 3.3.1 AA AA ABA ABA as AA ABA as AA ABA as AA … … … …
Figure 3.3: Results concerning the admissible semantics in this chapter, where applicable, in the context of previous work. Bidirectional arrows indicate semantic correspondence (arrows with the same starting point but different end points indicate one-to-many corre- spondence), and bold indicates novel work in this chapter.
For possibly non-flat ABA: admissible/complete/ grounded/preferred/ stable/ideal/semi-stable
asm. lab. Sec. 3.5
For possibly non-flat ABA: admissible/complete/
grounded/preferred/
stable asm. ext. [BDKT97]
ideal/semi-stable asm. ext. Sec. 3.5.3
possibly non-flat ABA
Sec. 3.5
For flat ABA: admissible/complete/ grounded/preferred/ stable/ideal/semi-stable asm. ext. [BDKT97,DMT07,CSAD15a] flat ABA flat ABA
flat ABA Sec. 3.5
admissible/complete/ grounded/preferred/stable
[BDKT97]
ideal/semi-stable Sec. 3.5.3
For flat ABA:
complete asm. lab. Sec. 3.2.2 grounded/preferred/ stable/ideal/semi-stable
asm. lab. Sec. 3.2.3 admissible asm. lab.
Sec. 3.2.1 complete Sec. 3.2 semi-stable Sec.3.3 grounded/ preferred/ stable/ideal Sec. 3.3
Figure 3.4: Results for any (possibly non-flat) ABA framework in this chapter, where applicable, in the context of previous work. Bidirectional arrows indicate semantic corre- spondence, and bold indicates novel work in this chapter.
between assumption labellings of flat ABA frameworks and argument labellings of their corresponding AA frameworks, and introduce committed admissible argument labellings as a variant of admissible argument labellings for AA frameworks. In Section 3.5, we extend the definition of assumption labellings to possibly non-flat ABA frameworks. We discuss related work in Section 3.6 and summarise the contributions of this chapter in Section 3.7.