Verifiability of Argumentation Semantics - TU Wien: DBAI
Verifiability of Argumentation Semantics Ringo Baumann, Thomas Linsbichler, Stefan Woltran 16th International Workshop on Non-Monotonic Reasoning Cape Town, South Africa
April 22, 2016
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
Thomas Linsbichler, April 22, 2016
e
d
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments stb(F) =
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
stb(F) = {a, d, e},
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
stb(F) = {a, d, e}, {b, c, e}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
stb(F) = {a, d, e}, {b, c, e}
Further semantics: preferred, complete, semi-stable, stage, . . .
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction Conflict-freeness: basic requirement for argumentation semantics.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
F: a
Thomas Linsbichler, April 22, 2016
b
G: a
b
H: a
Verifiability of Argumentation Semantics
b
2
Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
F: a
b
G: a
b
H: a
b
Conflict free sets + their range: (∅, ∅), ({a}, {a, b}), ({b}, {b}) ⇒ enough to compute stage semantics (range-maximal conflict-free sets)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
F: a
b
G: a
b
H: a
b
Conflict free sets + their range: (∅, ∅), ({a}, {a, b}), ({b}, {b}) ⇒ enough to compute stage semantics (range-maximal conflict-free sets)
Which information on top of conflict-free sets has to be added in order to compute a certain semantics? Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction
Systematic comparison of argumentation semantics Principle-based evaluation [Baroni and Giacomin, 2007] ⇒ Hierarchy of verification classes ⇒ Each “rational” semantics is exactly verifiable by one of these classes
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction
Systematic comparison of argumentation semantics Principle-based evaluation [Baroni and Giacomin, 2007] ⇒ Hierarchy of verification classes ⇒ Each “rational” semantics is exactly verifiable by one of these classes
Strong equivalence Central notion in non-monotonic reasoning [Lifschitz et al., 2001, Turner, 2004, Truszczynski, 2006, Baumann and Strass, 2016] Studied for most argumentation semantics [Oikarinen and Woltran, 2011, Baumann, 2016] ⇒ Missing results for naive and strong admissible semantics ⇒ Characterization theorems for intermediate semantics
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Definition An argumentation framework (AF) is a pair (A, R) where
A ⊆ U is a finite set of arguments and R ⊆ A × A is the attack relation representing conflicts.
Definition Given an AF F = (A, R) and S ⊆ A,
S is conflict-free (S ∈ cf(F)) if ∀a, b ∈ S : (a, b) ∈ / R. a ∈ A is defended by S if ∀b ∈ A : (b, a) ∈ R ⇒ ∃c ∈ S : (c, b) ∈ R + SF = S ∪ {a | ∃b ∈ S : (b, a) ∈ R} (the range of S) − SF = S ∪ {a | ∃b ∈ S : (a, b) ∈ R} (the anti-range of S)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Definition Given an AF F = (A, R), a set S ⊆ A is admissible set if S ∈ cf(F) and each a ∈ S is defended by S, complete extension if S ∈ ad(F) and a ∈ S if a is defended by S, naive extension if S ∈ cf(F) and @T ∈ cf(F) : T ⊃ S, + stable extension if S ∈ cf(F) and SF = A, + + stage extension if S ∈ cf(F) and @T ∈ cf(F) : TF ⊃ SF ,
preferred, grounded, semi-stable, ideal, eager, strongly admissible extensions
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}} co(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d, e}, {b, c, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}} co(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d, e}, {b, c, e}} na(F) = {{a, b, e}, {a, d, e}, {b, c, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}} co(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d, e}, {b, c, e}} na(F) = {{a, b, e}, {a, d, e}, {b, c, e}} stb(F) = stg(F) = {{a, d, e}, {b, d, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition n
We call a function rx : 2U × 2U → 2U which is expressible via basic set operations onlya neighborhood function. A neighborhood function rx induces the verification class mapping each AF F to
Fex =
+ − S, rx (SF , SF ) | S ∈ cf(F) .
a x
r (A, B) is in the language X ::= A | B | (X ∪ X) | (X ∩ X) | (X \ X)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition n
We call a function rx : 2U × 2U → 2U which is expressible via basic set operations onlya neighborhood function. A neighborhood function rx induces the verification class mapping each AF F to
Fex =
+ − S, rx (SF , SF ) | S ∈ cf(F) .
a x
r (A, B) is in the language X ::= A | B | (X ∪ X) | (X ∩ X) | (X \ X)
Example F: a
b
c
r+ : rx (A, B) = A Fe+ = {(∅, ∅), ({a}, {a, b}), ({c}, {b, c}), ({a, c}, {a, b, c})} r−± : rx (A, B) = (B, A \ B) Fe−± = {(∅, ∅, ∅), ({a}, {a, b}, ∅), ({c}, {c}, {b}), ({a, c}, {a, b, c}, ∅)} Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Neighborhood functions for n = 1:
r (A, B) = ∅ r+ (A, B) = A r− (A, B) = B r∓ (A, B) = B \ A r± (A, B) = A \ B r∩ (A, B) = A ∩ B r∪ (A, B) = A ∪ B r∆ (A, B) = (A ∪ B) \ (A ∩ B) 27 + 1 syntactically different neighborhood functions rx1 ,...,xn (A, B) ::= (rx1 (A, B), . . . , rxn (A, B)) Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition For neighborhood functions rx and ry , we say thatrx is more minformative n y x y U U than r , short r r , if there is a function δ : 2 → 2 such that for any A, B ⊆ U , it holds that δ (rx (A, B)) = ry (A, B). In case rx ≈ ry (rx ry and ry rx ), we say that rx represents ry .
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition For neighborhood functions rx and ry , we say thatrx is more minformative n y x y U U than r , short r r , if there is a function δ : 2 → 2 such that for any A, B ⊆ U , it holds that δ (rx (A, B)) = ry (A, B). In case rx ≈ ry (rx ry and ry rx ), we say that rx represents ry .
Example δ1 (r+± (A, B)) = δ1 (A, A \ B) =def (A, A \ (A \ B)) = (A, A ∩ B) = r+∩ (A, B); δ2 (r+∩ (A, B)) = δ2 (A, A ∩ B) =def (A \ (A ∩ B), A ∩ B) = (A \ B, A ∩ B) = r±∩ (A, B); δ3 (r±∩ (A, B)) = δ3 (A \ B, A ∩ B) =def ((A \ B) ∪ (A ∩ B), A \ B) = (A, A \ B) = r+± (A, B). ⇒ r+± ≈ r+∩ ≈ r±∩ Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Lemma All neighborhood functions are represented by the ones depicted below and the ≺-relation represented by arcs holds.
+−
+±
+∓
±∓
+
±
∩
∆
∩∪
−±
−∓
∪
∓
−
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability
Definition A semantics σ is verifiable by the verification class induced by the neighborhood function rx (or simply, x-verifiable) iff there is a function n U U U γσ : 2 × 2 → 22 s.t. for every AF F :
γσ Fex , AF = σ(F). Moreover, σ is exactly x-verifiable iff σ is x-verifiable and there is no ry with ry ≺ rx such that σ is y-verifiable.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Proposition Complete semantics is exactly +−-verifiable.
Proof Verifiability:
γco (Fe+− , AF ) = {S | (S, S+ , S− ) ∈ Fe+− , (S− \ S+ ) = ∅, ¯ S¯+ , S¯− ) ∈ Fe+− : S¯ ⊃ S ⇒ (S¯− \ S+ ) 6= ∅)} ∀(S, Exactness:
+± : f1 F
+±
F1 : a
F10 : a
b
f0 = {(∅, ∅, ∅), ({a}, {a}, ∅)} = F 1
b
+±
co(F1 ) = {∅} = 6 {{a}} = co(F10 )
⇒ co is not +±-verifiable Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Proposition Complete semantics is exactly +−-verifiable.
Proof (ctd.) F20 : a
b
b
F30 : a
b
F4 : a
b
F40 : a
b
+∓ :
F5 : a
b
F50 : a
b
∩∪ :
F6 : a
b
F60 : a
b
−∓ :
F2 : a
b
±∓ :
F3 : a
−± :
Thomas Linsbichler, April 22, 2016
c
Verifiability of Argumentation Semantics
c
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Verifiability
+−: co
−±: gr, sad
+∓: ss, eg
+: stb, stg
∓: ad, pr, id
: na
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition We call a semantics σ rational if self-loop-chains are irrelevant. That is, for every AF F it holds that σ(F) = σ(F l ), where F l = (AF , RF \ {(a, b) ∈ RF | (a, a), (b, b) ∈ RF , a 6= b}).
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition We call a semantics σ rational if self-loop-chains are irrelevant. That is, for every AF F it holds that σ(F) = σ(F l ), where F l = (AF , RF \ {(a, b) ∈ RF | (a, a), (b, b) ∈ RF , a 6= b}).
Theorem Every semantics which is rational is exactly verifiable by a verification class induced by one of the neighborhood functions below. +−
+±
+∓
±∓
+
±
∩
∆
∩∪
−±
−∓
∪
∓
−
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence Definition Given semantics σ , two AFs F and G are strongly equivalent w.r.t. σ (F ≡σE G ) iff for all AFs H: σ(F ∪ H) = σ(G ∪ H)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence Definition Given semantics σ , two AFs F and G are strongly equivalent w.r.t. σ (F ≡σE G ) iff for all AFs H: σ(F ∪ H) = σ(G ∪ H)
⇒ syntactical criteria exist
Example (stable semantics) stb-kernel: F k(stb) = (A, R \ {(a, b) | a 6= b, (a, a) ∈ R }). Theorem: F k(stb) = G k(stb) ⇔ F and G are strongly equivalent.
F: a
b
G: a
b
We have F k(stb) = G k(stb) = G . Thus, F and G are strong equivalent.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence
Definition (σ -kernel) Let F = (A, R). We define σ -kernels F k(σ) = A, Rk(σ) whereby
1
Rk(stb) = R \ {(a, b) |a 6= b, (a, a) ∈ R },
2
Rk(ad) = R \ {(a, b) |a 6= b, (a, a) ∈ R, {(b, a), (b, b)} ∩ R 6= ∅ },
3
Rk(gr) = R \ {(a, b) |a 6= b, (b, b) ∈ R, {(a, a), (b, a)} ∩ R 6= ∅ },
4
Rk(co) = R \ {(a, b) |a 6= b, (a, a), (b, b) ∈ R }.
5
Rk(na) = R ∪ {(a, b) | a 6= b, {(a, a), (b, a), (b, b)} ∩ R 6= ∅} . A relation ≡ is characterizable through kernels if there is a kernel k, s.t. F ≡ G ⇔ F k = G k ,
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence
Theorem Strong equivalence is characterizable through kernels (see below).
dstgp dstbp dssp degp dadp dprp didp dgrp dsadp dcop dnap k(stb) k(stb) k(ad) k(ad) k(ad) k(ad) k(ad) k(gr) k(gr) k(co) k(na)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Example “Stagle semantics”: + − + S ∈ sta(F) ⇔ S ∈ cf(F), SF ∪ SF = AF and ∀T ∈ cf (F) : SF 6⊂ TF+
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Example “Stagle semantics”: + − + S ∈ sta(F) ⇔ S ∈ cf(F), SF ∪ SF = AF and ∀T ∈ cf (F) : SF 6⊂ TF+
F: a
b
c
stb(F) = ∅ ⊂ sta(F) = {{b}} ⊂ stg(F) = {{b}, {c}}.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Example “Stagle semantics”: + − + S ∈ sta(F) ⇔ S ∈ cf(F), SF ∪ SF = AF and ∀T ∈ cf (F) : SF 6⊂ TF+
F: a
b
c
stb(F) = ∅ ⊂ sta(F) = {{b}} ⊂ stg(F) = {{b}, {c}}.
F k(stb) : a
b
c
k(stb) , F k(stb) = F k(stb) sta F k(stb) = {{b}, {c}} ⇒ F 6≡sta E F
k(stb)
⇒ Stagle semantics is not compatible with the stable kernel. Thomas Linsbichler, April 22, 2016
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Intermediate Semantics Theorem For each semantics σ which is +-verifiable and stb-stg-intermediate, it holds that
F k(stb) = G k(stb) ⇔ F ≡σE G.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Theorem For each semantics σ which is +-verifiable and stb-stg-intermediate, it holds that
F k(stb) = G k(stb) ⇔ F ≡σE G.
Theorem For each semantics σ which is +∓-verifiable and ρ-ad-intermediate with ρ ∈ {ss, id, eg}, it holds that
F k(ad) = G k(ad) ⇔ F ≡σE G.
Theorem For each semantics σ which is −±-verifiable and gr-sad-intermediate, it holds that
F k(gr) = G k(gr) ⇔ F ≡σE G. Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Conclusion
Summary: Hierarchy of verification classes Each “rational” semantics is exactly verifiable by a certain class Characterization of strong equivalence for intermediate semantics Future work: Semantics not captured by the approach, e.g. cf2 semantics [Baroni et al., 2005] Investigating labelling-based semantics [Caminada and Gabbay, 2009]
Thomas Linsbichler, April 22, 2016
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References I Baroni, P. and Giacomin, M. (2007). On principle-based evaluation of extension-based argumentation semantics. Artif. Intell., 171(10-15):675–700. Baroni, P., Giacomin, M., and Guida, G. (2005). SCC-Recursiveness: A general schema for argumentation semantics. Artif. Intell., 168(1-2):162–210. Baumann, R. (2016). Characterizing equivalence notions for labelling-based semantics. In Principles of Knowledge Representation and Reasoning: Proceedings of the 15th International Conference, pages 22–32. Baumann, R. and Strass, H. (2016). An abstract logical approach to characterizing strong equivalence in logic-based knowledge representation formalisms. In Principles of Knowledge Representation and Reasoning: Proceedings of the 15th International Conference, pages 525–528. Caminada, M. and Gabbay, D. M. (2009). A logical account of formal argumentation. Studia Logica, 93(2):109–145.
Thomas Linsbichler, April 22, 2016
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References II Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell., 77(2):321–357. Lifschitz, V., Pearce, D., and Valverde, A. (2001). Strongly equivalent logic programs. ACM Transactions on Computational Logic, 2(4):526–541. Oikarinen, E. and Woltran, S. (2011). Characterizing strong equivalence for argumentation frameworks. Artif. Intell., 175(14-15):1985–2009. Truszczynski, M. (2006). Strong and uniform equivalence of nonmonotonic theories - an algebraic approach. Annals of Mathematics and Artificial Intelligence, 48(3-4):245–265. Turner, H. (2004). Strong equivalence for causal theories. In 7th International Conference on Logic Programming and Nonmonotonic Reasoning, Proceedings, volume 2923 of Lecture Notes in Computer Science, pages 289–301. Springer.
Thomas Linsbichler, April 22, 2016
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Verifiability
e ) = {S | S ∈ F e, S is ⊆-maximal in F e}; γna (F A e+ ) = {S | (S, S+ ) ∈ F e+ , S+ is ⊆-maximal in {C+ | (C, C+ ) ∈ F e+ }}; γstg (F A e+ ) = {S | (S, S+ ) ∈ F e+ , S+ = A}; γstb (F A e∓ ) = {S | (S, S∓ ) ∈ F e∓ , S∓ = ∅}; γad (F A e∓ ) = {S | S ∈ γad (F e∓ ), S is ⊆-maximal in γad (F e∓ )}; γpr (F A A A e+∓ ) = {S | S ∈ γad (F e∓ ), S+ is ⊆-maximal in {C+ | (C, C+ , C∓ ) ∈ F e+∓ , C ∈ γad (F e∓ )}}; γss (F A A A \ e∓ ) = {S | S is ⊆-maximal in {C | C ∈ γad (F e∓ ), C ⊆ e∓ )}}; γid (F γ ( F pr A A A \ e+∓ )}}; e∓ ), C ⊆ e+∓ ) = {S | S is ⊆-maximal in {C | C ∈ γad (F γ ( F γeg (F ss A A A e−± ) = {S | (S, S− , S± ) ∈ F e−± , ∃(S0 , S− , S± ), . . . , (Sn , S− , S± ) ∈ F e−± : γsad (F n n A 0 0 ± (∅ = S0 ⊂ · · · ⊂ Sn = S ∧ ∀i ∈ {1, . . . , n} : Si− ⊆ Si−1 )};
e−± ) = {S | S ∈ γsad (F e−± ), ∀(¯ e−± : ¯ γgr (F S, ¯ S− , ¯ S± ) ∈ F S⊃S ⇒ (¯ S− \S± )6=∅)}. A A
Thomas Linsbichler, April 22, 2016
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April 22, 2016
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
Thomas Linsbichler, April 22, 2016
e
d
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments stb(F) =
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
stb(F) = {a, d, e},
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
stb(F) = {a, d, e}, {b, c, e}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction
Abstract Argumentation Framework (AF) [Dung, 1995]:
a
c f
b
e
d
Evaluation: argumentation semantics Extension: set of jointly acceptable arguments
stb(F) = {a, d, e}, {b, c, e}
Further semantics: preferred, complete, semi-stable, stage, . . .
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
1
Introduction Conflict-freeness: basic requirement for argumentation semantics.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
2
Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
2
Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
2
Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
F: a
Thomas Linsbichler, April 22, 2016
b
G: a
b
H: a
Verifiability of Argumentation Semantics
b
2
Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
F: a
b
G: a
b
H: a
b
Conflict free sets + their range: (∅, ∅), ({a}, {a, b}), ({b}, {b}) ⇒ enough to compute stage semantics (range-maximal conflict-free sets)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction Conflict-freeness: basic requirement for argumentation semantics.
Example Given conflict-free sets ∅, {a}, {b}. Can we compute semantics based on this? ⇒ only naive semantics (maximal conflict-free sets)
F: a
b
G: a
b
H: a
b
Conflict free sets + their range: (∅, ∅), ({a}, {a, b}), ({b}, {b}) ⇒ enough to compute stage semantics (range-maximal conflict-free sets)
Which information on top of conflict-free sets has to be added in order to compute a certain semantics? Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
2
Introduction
Systematic comparison of argumentation semantics Principle-based evaluation [Baroni and Giacomin, 2007] ⇒ Hierarchy of verification classes ⇒ Each “rational” semantics is exactly verifiable by one of these classes
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Introduction
Systematic comparison of argumentation semantics Principle-based evaluation [Baroni and Giacomin, 2007] ⇒ Hierarchy of verification classes ⇒ Each “rational” semantics is exactly verifiable by one of these classes
Strong equivalence Central notion in non-monotonic reasoning [Lifschitz et al., 2001, Turner, 2004, Truszczynski, 2006, Baumann and Strass, 2016] Studied for most argumentation semantics [Oikarinen and Woltran, 2011, Baumann, 2016] ⇒ Missing results for naive and strong admissible semantics ⇒ Characterization theorems for intermediate semantics
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Definition An argumentation framework (AF) is a pair (A, R) where
A ⊆ U is a finite set of arguments and R ⊆ A × A is the attack relation representing conflicts.
Definition Given an AF F = (A, R) and S ⊆ A,
S is conflict-free (S ∈ cf(F)) if ∀a, b ∈ S : (a, b) ∈ / R. a ∈ A is defended by S if ∀b ∈ A : (b, a) ∈ R ⇒ ∃c ∈ S : (c, b) ∈ R + SF = S ∪ {a | ∃b ∈ S : (b, a) ∈ R} (the range of S) − SF = S ∪ {a | ∃b ∈ S : (a, b) ∈ R} (the anti-range of S)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Definition Given an AF F = (A, R), a set S ⊆ A is admissible set if S ∈ cf(F) and each a ∈ S is defended by S, complete extension if S ∈ ad(F) and a ∈ S if a is defended by S, naive extension if S ∈ cf(F) and @T ∈ cf(F) : T ⊃ S, + stable extension if S ∈ cf(F) and SF = A, + + stage extension if S ∈ cf(F) and @T ∈ cf(F) : TF ⊃ SF ,
preferred, grounded, semi-stable, ideal, eager, strongly admissible extensions
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}} co(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d, e}, {b, c, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}} co(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d, e}, {b, c, e}} na(F) = {{a, b, e}, {a, d, e}, {b, c, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Background
Example a
c f
b
e
d
ad(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {a, d, e}, {b, c, e}} co(F) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, d, e}, {b, c, e}} na(F) = {{a, b, e}, {a, d, e}, {b, c, e}} stb(F) = stg(F) = {{a, d, e}, {b, d, e}}
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition n
We call a function rx : 2U × 2U → 2U which is expressible via basic set operations onlya neighborhood function. A neighborhood function rx induces the verification class mapping each AF F to
Fex =
+ − S, rx (SF , SF ) | S ∈ cf(F) .
a x
r (A, B) is in the language X ::= A | B | (X ∪ X) | (X ∩ X) | (X \ X)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition n
We call a function rx : 2U × 2U → 2U which is expressible via basic set operations onlya neighborhood function. A neighborhood function rx induces the verification class mapping each AF F to
Fex =
+ − S, rx (SF , SF ) | S ∈ cf(F) .
a x
r (A, B) is in the language X ::= A | B | (X ∪ X) | (X ∩ X) | (X \ X)
Example F: a
b
c
r+ : rx (A, B) = A Fe+ = {(∅, ∅), ({a}, {a, b}), ({c}, {b, c}), ({a, c}, {a, b, c})} r−± : rx (A, B) = (B, A \ B) Fe−± = {(∅, ∅, ∅), ({a}, {a, b}, ∅), ({c}, {c}, {b}), ({a, c}, {a, b, c}, ∅)} Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Neighborhood functions for n = 1:
r (A, B) = ∅ r+ (A, B) = A r− (A, B) = B r∓ (A, B) = B \ A r± (A, B) = A \ B r∩ (A, B) = A ∩ B r∪ (A, B) = A ∪ B r∆ (A, B) = (A ∪ B) \ (A ∩ B) 27 + 1 syntactically different neighborhood functions rx1 ,...,xn (A, B) ::= (rx1 (A, B), . . . , rxn (A, B)) Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition For neighborhood functions rx and ry , we say thatrx is more minformative n y x y U U than r , short r r , if there is a function δ : 2 → 2 such that for any A, B ⊆ U , it holds that δ (rx (A, B)) = ry (A, B). In case rx ≈ ry (rx ry and ry rx ), we say that rx represents ry .
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition For neighborhood functions rx and ry , we say thatrx is more minformative n y x y U U than r , short r r , if there is a function δ : 2 → 2 such that for any A, B ⊆ U , it holds that δ (rx (A, B)) = ry (A, B). In case rx ≈ ry (rx ry and ry rx ), we say that rx represents ry .
Example δ1 (r+± (A, B)) = δ1 (A, A \ B) =def (A, A \ (A \ B)) = (A, A ∩ B) = r+∩ (A, B); δ2 (r+∩ (A, B)) = δ2 (A, A ∩ B) =def (A \ (A ∩ B), A ∩ B) = (A \ B, A ∩ B) = r±∩ (A, B); δ3 (r±∩ (A, B)) = δ3 (A \ B, A ∩ B) =def ((A \ B) ∪ (A ∩ B), A \ B) = (A, A \ B) = r+± (A, B). ⇒ r+± ≈ r+∩ ≈ r±∩ Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Lemma All neighborhood functions are represented by the ones depicted below and the ≺-relation represented by arcs holds.
+−
+±
+∓
±∓
+
±
∩
∆
∩∪
−±
−∓
∪
∓
−
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability
Definition A semantics σ is verifiable by the verification class induced by the neighborhood function rx (or simply, x-verifiable) iff there is a function n U U U γσ : 2 × 2 → 22 s.t. for every AF F :
γσ Fex , AF = σ(F). Moreover, σ is exactly x-verifiable iff σ is x-verifiable and there is no ry with ry ≺ rx such that σ is y-verifiable.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Proposition Complete semantics is exactly +−-verifiable.
Proof Verifiability:
γco (Fe+− , AF ) = {S | (S, S+ , S− ) ∈ Fe+− , (S− \ S+ ) = ∅, ¯ S¯+ , S¯− ) ∈ Fe+− : S¯ ⊃ S ⇒ (S¯− \ S+ ) 6= ∅)} ∀(S, Exactness:
+± : f1 F
+±
F1 : a
F10 : a
b
f0 = {(∅, ∅, ∅), ({a}, {a}, ∅)} = F 1
b
+±
co(F1 ) = {∅} = 6 {{a}} = co(F10 )
⇒ co is not +±-verifiable Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Proposition Complete semantics is exactly +−-verifiable.
Proof (ctd.) F20 : a
b
b
F30 : a
b
F4 : a
b
F40 : a
b
+∓ :
F5 : a
b
F50 : a
b
∩∪ :
F6 : a
b
F60 : a
b
−∓ :
F2 : a
b
±∓ :
F3 : a
−± :
Thomas Linsbichler, April 22, 2016
c
Verifiability of Argumentation Semantics
c
13
Verifiability
+−: co
−±: gr, sad
+∓: ss, eg
+: stb, stg
∓: ad, pr, id
: na
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition We call a semantics σ rational if self-loop-chains are irrelevant. That is, for every AF F it holds that σ(F) = σ(F l ), where F l = (AF , RF \ {(a, b) ∈ RF | (a, a), (b, b) ∈ RF , a 6= b}).
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Verifiability Definition We call a semantics σ rational if self-loop-chains are irrelevant. That is, for every AF F it holds that σ(F) = σ(F l ), where F l = (AF , RF \ {(a, b) ∈ RF | (a, a), (b, b) ∈ RF , a 6= b}).
Theorem Every semantics which is rational is exactly verifiable by a verification class induced by one of the neighborhood functions below. +−
+±
+∓
±∓
+
±
∩
∆
∩∪
−±
−∓
∪
∓
−
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence Definition Given semantics σ , two AFs F and G are strongly equivalent w.r.t. σ (F ≡σE G ) iff for all AFs H: σ(F ∪ H) = σ(G ∪ H)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence Definition Given semantics σ , two AFs F and G are strongly equivalent w.r.t. σ (F ≡σE G ) iff for all AFs H: σ(F ∪ H) = σ(G ∪ H)
⇒ syntactical criteria exist
Example (stable semantics) stb-kernel: F k(stb) = (A, R \ {(a, b) | a 6= b, (a, a) ∈ R }). Theorem: F k(stb) = G k(stb) ⇔ F and G are strongly equivalent.
F: a
b
G: a
b
We have F k(stb) = G k(stb) = G . Thus, F and G are strong equivalent.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence
Definition (σ -kernel) Let F = (A, R). We define σ -kernels F k(σ) = A, Rk(σ) whereby
1
Rk(stb) = R \ {(a, b) |a 6= b, (a, a) ∈ R },
2
Rk(ad) = R \ {(a, b) |a 6= b, (a, a) ∈ R, {(b, a), (b, b)} ∩ R 6= ∅ },
3
Rk(gr) = R \ {(a, b) |a 6= b, (b, b) ∈ R, {(a, a), (b, a)} ∩ R 6= ∅ },
4
Rk(co) = R \ {(a, b) |a 6= b, (a, a), (b, b) ∈ R }.
5
Rk(na) = R ∪ {(a, b) | a 6= b, {(a, a), (b, a), (b, b)} ∩ R 6= ∅} . A relation ≡ is characterizable through kernels if there is a kernel k, s.t. F ≡ G ⇔ F k = G k ,
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Strong Equivalence
Theorem Strong equivalence is characterizable through kernels (see below).
dstgp dstbp dssp degp dadp dprp didp dgrp dsadp dcop dnap k(stb) k(stb) k(ad) k(ad) k(ad) k(ad) k(ad) k(gr) k(gr) k(co) k(na)
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Example “Stagle semantics”: + − + S ∈ sta(F) ⇔ S ∈ cf(F), SF ∪ SF = AF and ∀T ∈ cf (F) : SF 6⊂ TF+
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Example “Stagle semantics”: + − + S ∈ sta(F) ⇔ S ∈ cf(F), SF ∪ SF = AF and ∀T ∈ cf (F) : SF 6⊂ TF+
F: a
b
c
stb(F) = ∅ ⊂ sta(F) = {{b}} ⊂ stg(F) = {{b}, {c}}.
Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Intermediate Semantics Note that stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(F) ⊆ σ(F) ⊆ stg(F) for each AF F ?
Example “Stagle semantics”: + − + S ∈ sta(F) ⇔ S ∈ cf(F), SF ∪ SF = AF and ∀T ∈ cf (F) : SF 6⊂ TF+
F: a
b
c
stb(F) = ∅ ⊂ sta(F) = {{b}} ⊂ stg(F) = {{b}, {c}}.
F k(stb) : a
b
c
k(stb) , F k(stb) = F k(stb) sta F k(stb) = {{b}, {c}} ⇒ F 6≡sta E F
k(stb)
⇒ Stagle semantics is not compatible with the stable kernel. Thomas Linsbichler, April 22, 2016
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Intermediate Semantics Theorem For each semantics σ which is +-verifiable and stb-stg-intermediate, it holds that
F k(stb) = G k(stb) ⇔ F ≡σE G.
Thomas Linsbichler, April 22, 2016
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Intermediate Semantics Theorem For each semantics σ which is +-verifiable and stb-stg-intermediate, it holds that
F k(stb) = G k(stb) ⇔ F ≡σE G.
Theorem For each semantics σ which is +∓-verifiable and ρ-ad-intermediate with ρ ∈ {ss, id, eg}, it holds that
F k(ad) = G k(ad) ⇔ F ≡σE G.
Theorem For each semantics σ which is −±-verifiable and gr-sad-intermediate, it holds that
F k(gr) = G k(gr) ⇔ F ≡σE G. Thomas Linsbichler, April 22, 2016
Verifiability of Argumentation Semantics
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Conclusion
Summary: Hierarchy of verification classes Each “rational” semantics is exactly verifiable by a certain class Characterization of strong equivalence for intermediate semantics Future work: Semantics not captured by the approach, e.g. cf2 semantics [Baroni et al., 2005] Investigating labelling-based semantics [Caminada and Gabbay, 2009]
Thomas Linsbichler, April 22, 2016
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References I Baroni, P. and Giacomin, M. (2007). On principle-based evaluation of extension-based argumentation semantics. Artif. Intell., 171(10-15):675–700. Baroni, P., Giacomin, M., and Guida, G. (2005). SCC-Recursiveness: A general schema for argumentation semantics. Artif. Intell., 168(1-2):162–210. Baumann, R. (2016). Characterizing equivalence notions for labelling-based semantics. In Principles of Knowledge Representation and Reasoning: Proceedings of the 15th International Conference, pages 22–32. Baumann, R. and Strass, H. (2016). An abstract logical approach to characterizing strong equivalence in logic-based knowledge representation formalisms. In Principles of Knowledge Representation and Reasoning: Proceedings of the 15th International Conference, pages 525–528. Caminada, M. and Gabbay, D. M. (2009). A logical account of formal argumentation. Studia Logica, 93(2):109–145.
Thomas Linsbichler, April 22, 2016
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References II Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell., 77(2):321–357. Lifschitz, V., Pearce, D., and Valverde, A. (2001). Strongly equivalent logic programs. ACM Transactions on Computational Logic, 2(4):526–541. Oikarinen, E. and Woltran, S. (2011). Characterizing strong equivalence for argumentation frameworks. Artif. Intell., 175(14-15):1985–2009. Truszczynski, M. (2006). Strong and uniform equivalence of nonmonotonic theories - an algebraic approach. Annals of Mathematics and Artificial Intelligence, 48(3-4):245–265. Turner, H. (2004). Strong equivalence for causal theories. In 7th International Conference on Logic Programming and Nonmonotonic Reasoning, Proceedings, volume 2923 of Lecture Notes in Computer Science, pages 289–301. Springer.
Thomas Linsbichler, April 22, 2016
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Verifiability
e ) = {S | S ∈ F e, S is ⊆-maximal in F e}; γna (F A e+ ) = {S | (S, S+ ) ∈ F e+ , S+ is ⊆-maximal in {C+ | (C, C+ ) ∈ F e+ }}; γstg (F A e+ ) = {S | (S, S+ ) ∈ F e+ , S+ = A}; γstb (F A e∓ ) = {S | (S, S∓ ) ∈ F e∓ , S∓ = ∅}; γad (F A e∓ ) = {S | S ∈ γad (F e∓ ), S is ⊆-maximal in γad (F e∓ )}; γpr (F A A A e+∓ ) = {S | S ∈ γad (F e∓ ), S+ is ⊆-maximal in {C+ | (C, C+ , C∓ ) ∈ F e+∓ , C ∈ γad (F e∓ )}}; γss (F A A A \ e∓ ) = {S | S is ⊆-maximal in {C | C ∈ γad (F e∓ ), C ⊆ e∓ )}}; γid (F γ ( F pr A A A \ e+∓ )}}; e∓ ), C ⊆ e+∓ ) = {S | S is ⊆-maximal in {C | C ∈ γad (F γ ( F γeg (F ss A A A e−± ) = {S | (S, S− , S± ) ∈ F e−± , ∃(S0 , S− , S± ), . . . , (Sn , S− , S± ) ∈ F e−± : γsad (F n n A 0 0 ± (∅ = S0 ⊂ · · · ⊂ Sn = S ∧ ∀i ∈ {1, . . . , n} : Si− ⊆ Si−1 )};
e−± ) = {S | S ∈ γsad (F e−± ), ∀(¯ e−± : ¯ γgr (F S, ¯ S− , ¯ S± ) ∈ F S⊃S ⇒ (¯ S− \S± )6=∅)}. A A
Thomas Linsbichler, April 22, 2016
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