On the Semantics of Simple Contrapositive Assumption-Based Argumentation Frameworks

Transcription

1 On the Semantics of Simple Contrapositive Assumption-Based Argumentation Frameworks Jesse Heyninck 1 and Ofer Arieli 2 Institute of Philosophy II, Ruhr University Bochum, Germany School of Computer Science, The Academic College of Tel-Aviv, Israel CLAR Supported by the Sofja Kovalevskaja award of the Alexander von Humboldt-Foundation, funded by the German Ministry for Education and Research 2 Supported by the Israel Science Foundation (grant number 817/15). 1 / 34

2 The Plan Simple Contrapositive Assumption-Based Frameworks Preferential and Stable Semantics The Grounded Semantics A More Plausible Case Preferential Entailments Outlook 2 / 34

3 Assumption-Based Argumentation Assumption-Based Framework: ABF = L, Γ, Ab,. Premises Γ = {p q} Assumptions Ab = {p, q, s} Logic L Argumentation Framework Contrariness Operator p = { p} q = { q} {p, s} {q, s} 3 / 34

4 The Argumentation Pipeline Assumption-Based Framework Argumentation Framework Acceptable Assumptions Accepted Conclusions 4 / 34

5 Logic Definition A (propositional) logic for a language L is a pair L = L,, where is a consequence relation for L satisfying the following conditions: Reflexivity: if ψ Γ then Γ ψ. Monotonicity: if Γ ψ and Γ Γ, then Γ ψ. Transitivity: if Γ ψ and Γ, ψ φ then Γ, Γ φ. 5 / 34

6 Connectives Definition We shall assume that the language L contains at least the following connectives: a -negation, satisfying: p p and p p (for every atomic p) a -conjunction, satisfying: Γ ψ φ iff Γ ψ and Γ φ a -disjunction, satisfying: Γ, φ ψ σ iff Γ, φ σ and Γ, ψ σ a -implication, satisfying: Γ, φ ψ iff Γ φ ψ. a -falsity F, satisfying: F ψ for every formula ψ. 6 / 34

7 Some Conditions on Logics Definition Θ is -inconsistent if Θ F. A logic L = L, is explosive, if for every ψ L, the set {ψ, ψ} is -inconsistent. We say that L is contrapositive, if for every Γ {ψ} L it holds that: Γ ψ iff: ψ = F, or for every φ Γ we have that Γ \ {φ}, ψ φ. 7 / 34

8 Assumption-based framework Definition An assumption-based framework is a tuple ABF = L, Γ, Ab, where: L = L, is a propositional Tarskian logic Γ (the strict assumptions) is a -consistent set of L-formulas, and Ab (the candidate/defeasible assumptions)(assumed to be nonempty), : Ab (L) is a contrariness operator (such that for every ψ Ab \ {F} it holds that ψ ψ and ψ ψ). 8 / 34

9 Simple Contrapositive ABF Definition A simple contrapositive ABF is an assumption-based framework ABF = L, Γ, Ab,, where: L is an explosive and contrapositive logic, and, ψ = { ψ}. 9 / 34

10 Attacks Definition Let ABF = L, Γ, Ab, be an assumption-based framework,, Θ Ab, and ψ Ab. attacks ψ iff Γ, φ for some φ ψ. attacks Θ if attacks some ψ Θ. 10 / 34

11 Example Example Let L = CL, Γ = {p s}, and Ab = {p, s, t}. {s} {p} {t} 11 / 34

12 Example Example Let L = CL, Γ = {p s}, and Ab = {p, s, t}. {s} {p} {t} 11 / 34

13 Argumentation Semantics Definition ([3]) Let ABF = L, Γ, Ab, and Ab. We say that: is closed if = Ab Cn (Γ ). is conflict-free iff there is no that attacks. is naive iff it is closed and maximally conflict-free. defends a set Ab iff for every closed set Θ that attacks there is that attacks Θ. is admissible iff it is closed, conflict-free, and defends every. is complete iff it is admissible and contains every Ab that it defends. is grounded iff it is minimally complete. is preferred iff it is maximally admissible. is stable iff it is closed, conflict-free, and attacks every ψ Ab \. 12 / 34

14 Argumentation Semantics Definition ([3]) Let ABF = L, Γ, Ab, and Ab. We say that: is closed if = Ab Cn (Γ ). is conflict-free iff there is no that attacks. is naive iff it is closed and maximally conflict-free. defends a set Ab iff for every closed set Θ that attacks there is that attacks Θ. is admissible iff it is closed, conflict-free, and defends every. is complete iff it is admissible and contains every Ab that it defends. is grounded iff it is minimally complete. is preferred) iff it is maximally admissible. is stable iff it is closed, conflict-free, and attacks every ψ Ab \. 12 / 34

15 Example Example Let L = CL, Γ = {p s; p t}, and Ab = {p, s, t}. {s} {p} {t}, {s}, {t}, {p, t} and {s, t} are admissible. {p} is not admissible since it is not closed. 13 / 34

16 Example Example Let L = CL, Γ = {p s; p t}, and Ab = {p, s, t}. {s} {p} {t} {t}, {p, t} and {s, t} are complete. 13 / 34

17 Example Example Let L = CL, Γ = {p s; p t}, and Ab = {p, s, t}. {s} {p} {t} {p, t} and {s, t} are preferred (and stable). 13 / 34

18 Example Example Let L = CL, Γ = {p s; p t}, and Ab = {p, s, t}. {s} {p} {t} {t} is grounded. 13 / 34

19 Entailment Definition Given an assumption-based framework ABF = L, Γ, Ab,. For Sem {Grd, Prf, Stb}, we denote: ABF Semψ iff Γ, ψ for every Sem(ABF). ABF Semψ iff Γ, ψ for some Sem(ABF). Where ABF = L, Γ, Ab,, we will also sometimes say that Γ, Ab Sem ψ if ABF Sem ψ (for some {, }). 14 / 34

20 Preferential and Stable Semantics 15 / 34

21 Preferential and Stable Semantics Proposition Let ABF = L, Γ, Ab, be a simple contrapositive ABF and Ab. Then: is naive iff is stable iff is preferred. 16 / 34

22 But why? We [...] note that in every semi-stable labelling of an AF without stable labellings there exists an odd-length cycle whose arguments are all labelled undec [7] In other words, odd length cycles are in most cases responsible for preferred extensions not being stable. Example Suppose that L = CL, Γ = {p s, s q, q p}, and Ab = {p, q, s}. {s} {p} {q} 17 / 34

23 But why? We [...] note that in every semi-stable labelling of an AF without stable labellings there exists an odd-length cycle whose arguments are all labelled undec [7] In other words, odd length cycles are in most cases responsible for preferred extensions not being stable. Example Suppose that L = CL, Γ = {p s, s q, q p}, and Ab = {p, q, s}. {s} {p} {q} 17 / 34

24 Relation with MCS Definition Let ABF = L, Γ, Ab,. A set Ab is maximally consistent in ABF, if Γ, F and Γ, F for every Ab. The set of the maximally consistent sets in ABF is denoted MCS(ABF). Proposition Let ABF = L, Γ, Ab, be a simple contrapositive ABF and Ab. Then: is naive iff is stable iff is preferred iff MCS(ABF) 18 / 34

25 The Grounded Semantics 19 / 34

26 A Problematic Example Example Let L = CL, Γ =, and Ab = {p, p, s}. {p, s} {p} {p, p, s} {s} { p, s} { p} s is an innocent bystander that is not derivable using the grounded extension. 20 / 34

27 A Problematic Example Example Let L = CL, Γ =, and Ab = {p, p, s}. {p, s} {p} {p, p, s} {s} { p, s} { p} s is an innocent bystander that is not derivable using the grounded extension. Contamination problems. 20 / 34

28 A Problematic Example Example Let L = CL, Γ =, and Ab = {p, p, s}. {p, s} {p} {p, p, s} {s} { p, s} { p} s is an innocent bystander that is not derivable using the grounded extension. Contamination problems. No correspondence with maximal consistent subsets. 20 / 34

29 A simple solution Add F to Ab. Example (Example 4 continued) Let L = CL, Γ =, and Ab = {p, p, s, F}. {p, s} {p} {p, p, s, F} {s} { p, s} { p} 21 / 34

30 A simple solution Add F to Ab. Example (Example 4 continued) Let L = CL, Γ =, and Ab = {p, p, s, F}. {p, s} {p} {p, p, s, F} {s} { p, s} { p} 21 / 34

31 In Fact: Theorem Let ABF = L, Γ, Ab, be a simple contrapositive assumption-based framework in which F Ab. Then Grd(ABF) = MCS(ABF). 22 / 34

32 Interference Definition Given a logic L = L,, let Γ i (i = 1, 2) be two sets of L-formulas, and let ABF i = L, Γ i, Ab i, i (i = 1, 2) be two ABFs based on L. We denote by Atoms(Γ i ) (i = 1, 2) the set of all atoms occurring in Γ i. We say that Γ 1 and Γ 2 are syntactically disjoint if Atoms(Γ 1 ) Atoms(Γ 2 ) =. ABF 1 and ABF 2 are syntactically disjoint if so are Γ 1 Ab 1 and Γ 2 Ab 2. We denote: ABF 1 ABF 2 = L, Γ 1 Γ 2, Ab 1 Ab 2, 1 2. An entailment satisfies non-interference, if for every two syntactically disjoint frameworks ABF 1 = L, Γ 1, Ab 1, 1 and ABF 2 = L, Γ 2, Ab 2, 2 where Γ 1 Γ 2 is consistent, it holds that: ABF 1 ψ iff ABF 1 ABF 2 ψ for every L-formula ψ s.t. Atoms(ψ) Atoms(Γ Ab ). 23 / 34

33 Non-Interference Theorem For Sem {Naive, Prf, Stb}, both Sem and Sem satisfy non-interference with respect to simple contrapositive assumption-based frameworks. Theorem Grd satisfies non-interference for any simple contrapositive ABF in which F Ab. 24 / 34

34 Preferential Entailments 25 / 34

35 KLM properties Definition ([6]) A relation between ABFs and formulas is cumulative, if the following conditions are satisfied: Cautious Reflexivity (CR): For every -consistent ψ it holds that ψ ψ Cautious Monotonicity (CM): If Γ, Ab φ and Γ, Ab ψ then Γ, Ab, φ ψ Cautious Cut (CC): If Γ, Ab φ and Γ, Ab, φ ψ then Γ, Ab ψ. Left Logical Equivalence (LLE): If φ ψ and ψ φ then Γ, Ab, φ ρ iff Γ, Ab, ψ ρ. Right Weakening (RW): If φ ψ and Γ, Ab φ then Γ, Ab ψ. A cumulative relation is preferential, if it satisfies: Distribution (OR): If Γ, Ab, φ ρ and Γ, Ab, ψ ρ then Γ, Ab, φ ψ ρ. 26 / 34

36 Results for Skeptical Entailments Proposition Let ABF = L, Γ, Ab, be a simple contrapositive ABF. Then Sem is preferential for Sem {Naive, Prf, Stb}. If F Ab, then Grd is also preferential. 27 / 34

37 Results for Credulous Entailments Example Let L = CL, Γ =, and Ab = {r (q p), r (t p)}. Note that: MCS( L, Γ, Ab {q}, ) = {{r (q p), q}, { r (t p), q}} MCS( L, Γ, Ab {t}, ) = {{r (q p), t}, { r (t p), t}} MCS( L, Γ, Ab {q t}, ) = {{r (q p), q t}, { r (t p), q t}} Then Ab, q p and Ab, t p but Ab, q t p for every entailment of the form Sem where Sem {Naive, Prf, Stb}. Proposition Let ABF = L, Γ, Ab, be a simple contrapositive ABF. Then Sem is cumulative for Sem {Naive, Prf, Stb}. 28 / 34

38 Discussion (in view of related work) Much work has been done on classical respectively Tarskian logic instantiations of Dung argumentation [1, 2, 4, 8]. However, for ABA such a study was missing. For grounded semantics, some care has to be taken (similar problems have been discussed in [5]). Other semantics work as expected. A benefit of ABA is that for a finite knowledge base we obtain a finite argumentation graph (which is not the case for many other formalisms). 29 / 34

39 Future work Disjunctive attacks (e.g. { p q} attacks {p, q}). Closure requirements (turn out to be redundant). Modal Logics. 30 / 34

40 Thank you for your attention Questions? 31 / 34

41 Bibliography I Leila Amgoud and Philippe Besnard. Logical limits of abstract argumentation frameworks. J. Applied Non-Classical Logic, 23(3): , Ofer Arieli, Annemarie Borg, and Christian Straßer. Argumentative approaches to reasoning with consistent subsets of premises. In Proc. IEA/AIE 2017, pages , Andrei Bondarenko, Phan Minh Dung, Robert Kowalski, and Francesca Toni. An abstract, argumentation-theoretic approach to default reasoning. Artif. Intell., 93(1):63 101, / 34

42 Bibliography II Claudette Cayrol. On the relation between argumentation and non-monotonic coherence-based entailment. In Proc. IJCAI 95, pages , Kristijonas Čyras, Xiuyi Fan, Claudia Schulz, and Francesca Toni. Assumption-based argumentation: Disputes, explanations, preferences. Handbook of Formal Argumentation, pages , Sarit Kraus, Daniel Lehmann, and Menachem Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1): , / 34

43 Bibliography III Claudia Schulz. On stable labellings and odd-length cycles in abstract argumentation frameworks. Srdjan Vesic. Identifying the class of maxi-consistent operators in argumentation. Journal of Artificial Intelligence Research, 47:71 93, / 34