A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision
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The document presents a four-state labelling semantics designed to integrate abstract argumentation frameworks with belief revision. It outlines various types of semantics including conflict-free, admissible, complete, and grounded, and discusses mappings to existing labellings. Future work is suggested to enhance the semantics, particularly concerning weighted argumentation frameworks.
A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision
- 1. ICTCS 2021 A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision Stefano Bistarelli, Carlo Taticchi
- 2. ARGUMENTATION LOGIC POLITICS CYBERSECURITY PERSUASION DIALECTICS PHILOSOPHY A.I. speech debate strong clusters meaning defeasible reasoning beliefs secure argument talk analysis big data information concurrency tool semantics coalitions partial order topic frameworks robustness extensions debate attacks conflict subject admissible invariant operators grounds accepted interest
- 3. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • Consist of a pair • Consider : ‣ is set of arguments attacked by ‣ is set of arguments attacking ‣ is defended by when ⟨ 𝒜 , ℛ⟩ a ∈ 𝒜 a+ a a− a a D ⊆ 𝒜 ∀b ∈ 𝒜 ∣ b ∈ a− . ∃d ∈ D ∣ d ∈ b− Argumentation Frameworks 3
- 4. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • is: ‣ con fl ict-free if ‣ admissible, if it is con fl ict-free and each is defended by ‣ complete, if it is admissible and defended by , ‣ grounded, if it is complete and it is minimal with respect to set inclusion E ⊆ 𝒜 ∄a, b ∈ E ∣ (a, b) ∈ ℛ a ∈ E E ∀a ∈ 𝒜 E a ∈ E Extension-based semantics 4 {}, {a}, {b}, {c}, {d}, {a,c}, {a,d}, {b,d } {}, {a}, {b}, {c}, {d}, {a,c}, {a,d}, {b,d} {}, {a}, {b}, {c}, {d}, {a,c}, {a,d}, {b,d} {}, {a}, {b}, {c}, {d}, {a,c}, {a,d}, {b,d}
- 5. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 Labelling-based semantics 5 IN-OUT-UNDEC labelling • • L(a) = IN ⟹ ∀b ∈ 𝒜 ∣ (b, a) ∈ ℛ . L(b) = OUT L(a) = OUT ⟺ ∃b ∈ 𝒜 ∣ (b, a) ∈ ℛ ∧ L(b) = IN IN OUT IN OUT UNDEC is a labelling of in which the set of IN arguments ( ) is an extension of LF σ F 𝒜 ↓IN σ
- 6. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 Labelling-based semantics 6 +-− labelling • • − ∈ M(a) ⟹ ∃b ∈ 𝒜 ∣ (b, a) ∈ ℛ ∧ + ∈ M(b) + ∈ M(a) ⟹ ∀b ∈ 𝒜 ∣ (b, a) ∈ ℛ . − ∈ M(b) ∧ + ∈ M(a) ⟹ ∀c ∈ 𝒜 ∣ (a, c) ∈ ℛ . − ∈ M(c) ∅ {−} {+} {−} {+,−}
- 7. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 Labelling-based semantics 7 Grounded IN-OUT-UNDEC-OFF labelling • • ∀a ∈ 𝒜 ∖ 𝒮 . Ngde(a) = OFF ∀a ∈ 𝒮 . Ngde(a) = Lgde(a) IN IN OUT UNDEC in the example 𝒮 = {a, c, d, e} OFF
- 9. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 DK OUT IN DC ↑ • A four-state labelling is a partial function such that • Both and DC represent ignored arguments, but are syntactically di ff erent ‣ denotes arguments in ‣ DC arguments are all in LF LF : 𝒰 ⇀ {IN, OUT, DK, DC} ∀a ∈ 𝒰 ∖ 𝒜 . L(a) = ↑ ↑ ↑ 𝒰 ∖ 𝒜 𝒜 Four-state labelling 9
- 10. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 con fl ict-free admissible • is con fl ict-free when: • is admissible when: L L Four-state labelling 10 Labelling-based semantics ‣ and ‣ ‣ and ‣ L(a) = IN ⟹ ∀b ∈ a− . L(b) ≠ IN L(a) = OUT ⟹ ∃b ∈ a− ∣ L(b) = IN L(a) = IN ⟹ ∀b ∈ a− . L(b) = OUT L(a) = OUT ⟺ ∃b ∈ a− ∣ L(b) = IN DK OUT IN DC ↑
- 11. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • Labellings seen before can be translated into the four-state labelling ‣ is a grounded IN-OUT-UNDEC-OFF labelling with respect to if and only if is grounded, and with and L 𝒮 L 𝒜 ↓DC = ∅ 𝒜 = 𝒮 ↑ ≡ OFF DK ≡ UNDEC Four-state labelling 11 Mapping with other labellings ↑ DK DK OUT IN Example: 𝒜 = 𝒮 = {a, b, c, d} OFF UNDEC UNDEC OUT IN Four-state Grounded IN-OUT-UNDEC-OFF
- 12. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • • Single labelling argument expansion computes • satis fi es AGM-style expansion postulates ↑ ≺ DC ≺ IN/OUT ≺ DK ⊕LF σ : AF × 𝒰 → AF F′  = F ⊕LF σ a ∣ LF′  σ (a) ≽ LF σ (a) ⊕LF σ Revision of Labels in AFs 12 Labels as belief states
- 13. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • Four-state labelling semantics for AFs • Correspondence to extension-based semantics ‣ con fl ict-free, admissible, complete, stable, preferred, grounded • Mapping with other labellings ‣ IN-OUT-UNDEC, +-−, Grounded IN-OUT-UNDEC-OFF, IN-OUT-BOTH-NONE • Connection with the AGM framework ‣ Expansion operator Conclusion 13
- 14. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • Use a pessimistic interpretation for DC arguments • Extend the four-state labelling to weighted AFs Future Work 14 optimistic pessimistic
- 15. Thank you for your attention! A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision Stefano Bistarelli, Carlo Taticchi ICTCS 2021
- 16. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 NONE IN OUT IN BOTH 16 IN-OUT-BOTH-NONE labelling • • • • OE(a) = IN ⟸ a ∈ E ∧ a ∉ E+ OE(a) = OUT ⟸ a ∉ E ∧ a ∈ E+ OE(a) = BOTH ⟸ a ∈ E ∧ a ∈ E+ OE(a) = NONE ⟸ a ∉ E ∧ a ∉ E+ in the example E = {b, d, e}
- 17. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 con fl ict-free admissible • is con fl ict-free when: • is admissible when: L L 17 Labelling-based semantics ‣ and ‣ ‣ and ‣ L(a) = IN ⟹ ∀b ∈ a− . L(b) ≠ IN L(a) = OUT ⟹ ∃b ∈ a− ∣ L(b) = IN L(a) = IN ⟹ ∀b ∈ a− . L(b) = OUT L(a) = OUT ⟺ ∃b ∈ a− ∣ L(b) = IN DK OUT IN DC ↑
- 18. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • is complete when: • is grounded when: L L 18 Labelling-based semantics ‣ and ‣ ‣ is a complete labelling and ‣ is the minimal complete labelling L(a) = IN ⟺ ∀b ∈ a− . L(b) ∈ {OUT, DC} L(a) = OUT ⟺ ∃b ∈ a− ∣ L(b) = IN L 𝒜 ↓IN ↑ OUT IN OUT IN complete grounded
- 19. A Unifying Four-State Labelling Semantics for Bridging Abstract Argumentation Frameworks and Belief Revision ICTCS 2021 • We say that when 1. is an AF 2. given , 3. given , 4. if , then 5. if , then 6. given , if , then , and if , then G ⊆a Lσ F LG σ (a) ≼kb LF σ (a) F ⊕LF σ a F′  = F ⊕LF σ a LF′  σ (a) ≻kb ↑ F′  = F ⊕LF σ a LF′  σ (a) ≽kb LF σ (a) LF σ (a) = DK F ⊕LF σ a = F G ⊆a Lσ F G ⊕LG σ a ⊆a Lσ F ⊕LF σ a F′  = F ⊕LF σ a LF σ (a) = ↑ LF′  σ (a) = IN/OUT LF σ (a) = IN/OUT LF′  σ (a) = DK 19 Single labelling argument expansion postulates