Semantic Correspondence between Stable Argument Extensions and

In Chapter 5, we present a justification approach for literals with respect to an answer set of a logic program. This approach relies on the correspondence between answer sets and the stable semantics of the translated AA framework. Importantly, we will base these justifications on stable argument extensions instead of labellings, since for stable

A4 : {not q} ⊢ p

A5 : {not p} ⊢ q

A7 : {not p, not r} ⊢ r

A6: {not r} ⊢ r A1: {not p} ⊢ not p

A2 : {not q} ⊢ not q A3 : {not r} ⊢ not r

Figure 4.2: The translated AA framework AAP7 from Example 4.15.

argument labellings the labels of all arguments are directly characterised by the respective stable argument extension, i.e. all arguments contained in the stable argument extension are labelled in and all arguments not in the stable argument extension are labelled out3. Therefore, we reformulate the correspondence results between (2-valued) stable models and stable argument labellings from the previous section to state correspondence between answer sets and stable argument extensions.

Usually, answer sets only contain classical literals. However, if l /∈ S for an answer set S of P and some classical literal l ∈ LitP, then not l is considered satisfied with respect to S. Thus, we introduce the notion of Answer Sets with NAF literals, i.e. answer sets which also comprise all true NAF literals.

Definition 4.6(Answer Set with NAF Literals). Let S ⊆ LitP be a set of classical literals. ∆S = {not l ∈ NAFLitP | l /∈ S} consists of all NAF literals not l whose corresponding

classical literal l is not contained in S. If S is an answer set of P, then SNAF = S ∪ ∆S is an answer set with NAF literals of P.

Since for consistent logic programs the answer set semantics coincides with the (2- valued) stable semantics [GL91, Prz90], the correspondence results from Theorems 4.18 and 4.19 concerning (2-valued) stable models also hold between answer sets and stable argument labellings and can be reformulated in terms of stable argument extensions.

We reformulate the “consistency” condition for argument labellings, which is stated in terms of assumption-arguments labelled out, namely ∀a ∈ HBP : {not a} ⊢ not a /∈ out(LabArg) ∨ {not ¬a} ⊢ not ¬a /∈ out(LabArg), as a condition on arguments in the stable extension, namely ∀a ∈ HBP :∄Asms ⊢ a ∈ E ∨ ∄Asms ⊢ ¬a ∈ E, where E is a stable extension.

3_{This is in general not the case for, e.g., complete extensions. In order to determine the labels of}

all arguments according to a complete extension, the attacks between arguments need to be taken into account.

Corollary 4.20. Let P be a consistent logic program and hArP, AttPi the translated AA framework of P. If E ⊆ ArP is a stable argument extension of hArP, AttPi such that

∀a ∈ HBP :∄Asms ⊢ a ∈ E ∨ ∄Asms ⊢ ¬a ∈ E,

then SNAF = {k | ∃Asms ⊢ k ∈ E} is an answer set with NAF literals of P.

This follows from Theorem 4.18 and the correspondence between LabArg2Mod and LabArg2Mod_Wu from Proposition 4.12. Note that this Corollary extends the correspon- dence result of Dung [Dun95b] (see Section 4.4) to answer sets, i.e. to logic programs that may comprise explicitly negated atoms.

Corollary 4.21. Let P be a consistent logic program and hArP, AttPi the translated AA framework of P. S ⊆ LitP is an answer set of P if and only if E = {Asms ⊢ k | Asms ⊆ ∆S} is a stable argument extension of hArP, AttPi such that ∀a ∈ HBP :∄Asms ⊢ a ∈ E ∨∄Asms ⊢ ¬a ∈ E.

This follows from Theorem 4.19. Note that this Corollary also extends the correspon- dence result of Dung [Dun95b] (see Section 4.4) to answer sets, i.e. to logic programs that may comprise explicitly negated atoms.

Given an answer set S, we call E = {Asms ⊢ k | Asms ⊆ ∆S} the corresponding stable argument extension of S.

4.5

Related Work

Throughout this chapter, we mentioned various closely related works: Bondarenko et al. [BTK93, BDKT97] present some correspondences between the semantics of logic pro- grams and their translated ABA frameworks (see Section 4.3.1), which we extended using our new assumption labellings. Concerning the correspondence between the semantics of logic programs and AA frameworks, both Dung [Dun95b] and Wu et al. [WCG09] present various results, which we extended too.

We focussed on the translation of logic programs into ABA and AA frameworks, whereas other authors have investigated the opposite direction. Dung [Dun95b] gives a translation of AA frameworks into logic programs and proves some semantic corre- spondence, which is (among others) extended by Osorio et al. [OZNC05] and Wu et al. [WCG09]. Furthermore, Caminada and Schulz [CS15] present a translation of ABA frameworks into logic programs and show semantic correspondence.

More generally, the question whether different non-monotonic reasoning formalism can be translated into one another and how their semantics relate has received considerable attention. Early work focussed on formalisms such as default logic, circumscription, and autoepistemic logic [Imi87, Got95, Jan99] and the formulation of the answer set semantics in other logical formalisms, such as equilibrium logic [Pea96]. More recently, work has been

done regarding translations between argumentation frameworks and other non-monotonic logics.

Thimm and Kern-Isberner [TKI08] investigate the correspondence between defeasible logic programming (DeLP), which is commonly classified as an argumentation framework, and answer set programming. Lam et al. [LGR16] study the relation between the AS- PIC+ argumentation framework and defeasible logic, and Young et al. [YMR16] between ASPIC+ and prioritised default logic. Heyninck and Straßer [HS16] give translations and correspondence results between ASPIC+, ABA frameworks, and adaptive logics. Further- more, Bochman [Boc16] studies a translation from abstract dialectical frameworks (ADFs) into causal calculus.

Furthermore, various authors investigate mappings between different argumentation frameworks. Oren et al. [ORL10] present a mapping between AA frameworks and eviden- tial argumentation frameworks (EAFs) and prove semantic correspondence. Polberg and Oren [PO14] investigate mappings between EAFs and argumentation frameworks with ne- cessities and show that there exists no natural translation between the two which preserves the semantics. Polberg [Pol17] extends that work and additionally investigates mappings with ADFs.

4.6

Summary

In this chapter, we reviewed and extended existing correspondence results between the semantics of a logic program and its translated ABA and AA frameworks.

Concerning the translated ABA framework, existing results only showed how to derive a corresponding 3-valued interpretation from an assumption extension, but not vice versa. Furthermore, the corresponding 3-valued interpretations were defined based on the argu- ments supported by the assumption extension, rather than on the assumption extension itself. We introduced direct mappings between 3-valued interpretations of a logic program and assumption labellings of the translated ABA framework, which do not require to con- struct arguments. We then proved that the mapping of complete assumption labellings yields 3-valued stable models (analogous to existing results), and that the mapping of 3-valued stable models yields complete assumption labellings. These results can be ex- tended to the correspondence between grounded, preferred, ideal, semi-stable, and stable assumption labellings and, respectively, well-founded, 3-valued M-stable, ideal, 3-valued L-stable, and (2-valued) stable models.

With regards to the translated AA framework of a logic program, various mappings and correspondence results exist, both regarding argument extensions and labellings. We compared these mappings with new mappings obtained by concatenating our mappings between 3-valued interpretations and assumption labellings and between assumption la- bellings and argument labellings. We showed that for complete assumption labellings and 3-valued stable models, our mappings yield the same outcome as existing mappings. Thus, existing correspondence results between complete, grounded, preferred, ideal, and stable

argument labellings and, respectively, 3-valued stable, well-founded, 3-valued M-stable, ideal, and (2-valued) stable models also hold for our mappings. However, in the general case the outcome of our mappings and existing mappings may not be the same. We also show that in contrast to existing mappings, our mappings always preserve the labels/truth values of certain assumptions/literals when translating back and forth between 3-valued interpretations and assumption labellings.

In the next chapter, we introduce a justification approach for logic programs under the answer set semantics, which is based upon the correspondence results presented in this chapter.

Chapter 5

Justifying Answer Sets using

Argumentation

5.1

Introduction

If ASP is used for applications in real-world scenarios involving non-experts, it is useful to have an explanation as to why a literal does or does not belong to an answer set. Answer set justification has thus been identified as an important but not yet sufficiently studied research area [LD04, BD08]. In this chapter, we present two methods for justifying literals with respect to an answer set of a consistent logic program by applying the notions of arguments and attacks of the translated ABA and AA framework of a logic program. Our approach is based upon the semantic correspondence results between logic programs and their translated ABA and AA frameworks presented in Chapter 4. Of particular impor- tance for this chapter is the result that every answer set of a logic program corresponds to a stable argument extension of the translated AA framework (Corollaries 4.20 and 4.21). Our first justification approach, an Attack Tree, expresses how to construct an argu- ment for a literal in question (the supporting argument) as well as which arguments attack the argument for the literal in question (the attacking arguments); the same information is provided for all arguments attacking the attacking arguments, and so on. The second justification approach, an ABA-Based Answer Set (ABAS) Justification of a literal, rep- resents the same information as an Attack Tree, but expressed in terms of literals rather than arguments. An ABAS Justification comprises facts and NAF literals necessary to derive the literal in question (the “supporting literals”) as well as information about lit- erals that are in conflict with the literal in question (the “attacking literals”). The same information is provided for all supporting and attacking literals of the literal in question, for all their supporting and attacking literals, and so on.

The chapter is organised as follows. In Section 5.2, we give some definitions specific to this chapter and in Section 5.3 we introduce a motivating (medical) example and a technical example, which will serve as the running examples throughout this chapter. In Section 5.4, we introduce Attack Trees as our first justification method, show their relationship with abstract dispute trees of the translated AA framework, and characterise the explanations they provide as admissible fragments of the answer set in question. Based on Attack Trees, we define two forms of ABAS Justifications: Basic ABA-Based Answer Set Justifications, introduced in Section 5.5, illustrate how to flatten Attack Trees, yielding a justification in terms of literals and their relations. Labelled ABA-Based Answer Set Justifications, introduced in Section 5.6, constitute a more elaborate version of Basic ABA- Based Answer Set Justifications, following the same flattening strategy, but additionally using labels to solve some deficiencies of the basic variant. In Section 5.7, we present a web- platform implementing Attack Trees and Labelled ABA-Based Answer Set Justifications. In Section 5.8, we compare ABAS Justifications to related work and in Section 5.9 we summarise the contributions of this chapter.