Presentation of the paper with the same title given at ECAI 2014.

1. Strategic Argumentation is NP-Complete
Guido Governatori, Francesco Olivieri, Simone Scannapieco,
Antonino Rotolo, Matteo Cristani
ECAI 2014, Prague, 22 August 2014
NICTA Funding and Supporting Members and Partners
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2. A Crime Story
Prosecutor Plaintif
You killed the victim
I did not commit murder! There is no evidence!
There is evidence. We found your ID card near the scene.
It’s not evidence! I had my ID card stolen!’
It is you who killed the victim. Only you were near the
scene at the time of the murder.
I didn’t go there. I was at facility A at that time.
At facility A? Then, it’s impossible to have had your ID card
stolen since facility A does not allow a person to enter
without an ID card.
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3. Part I
Strategic Argumentation
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4. Strategic Argumentation
• Adversarial two player (Proponent and Opponent) dialogue game to
prove/disprove a claim
• At each turn a player plays a set of arguments to:
• prove the claim (Proponent)
• disprove the claim (Opponent)
• Incomplete information
• Set of common arguments (known by both Proponent and Opponent)
• Set of Proponent’s arguments (not known by Opponent)
• Set of Opponent’s arguments (not known by Proponent)
• After an argument has been played it is a common argument
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5. Strategic Argumentation Problem
Avoid playing arguments that can be used by the other party to defeat
you
Example
F = {a, d, f}
RCom = ∅
RPr = {a ⇒ g, g ⇒ b, d ⇒ c, c ⇒ b}
ROp = {c ⇒ e, e, f ⇒ ¬b}.
Pr: d, d ⇒ c, c ⇒ b
Op: f, c ⇒ e, e, f ⇒ ¬b
Op wins
Pr: a, a ⇒ g, g ⇒ b
Pr wins
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6. Part II
Strategic Argumentation
in Defeasible Logic
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7. Defeasible Logic (DL)
• Derive (plausible) conclusions with the minimum amount of
information.
• Definite conclusions
• Defeasible conclusions
• Defeasible Theory
• Facts
• Strict rules (A1, . . . , An → B)
• Defeasible rules (A1, . . . , An ⇒ B)
• Defeaters (A1, . . . , An ; B)
• Superiority relation over rules
• Conclusions
• +∆p: p is definitely provable
• –∆p: p is definitely refuted
• +∂p: p is defeasibly proved
• –∂p: p is defeasibly refuted
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8. Proving Conclusions in DL
1 Give an argument for the conclusion you want to prove
2 Consider all possible counterarguments to it
3 Rebut all counterarguments
• Defeat the argument by a stronger one
• Undercut the argument by showing that some of the premises do not
hold
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9. Derivations in DL
+∂p
1) there is an applicable rule r pro p
2) for all rules t con p either:
2.1) t is not applicable
2.2) t is defeated by a rule s pro p
–∂p
1) for all rules r pro p either
2.1) r is not applicable, or
2.2) there is an applicable rule s con p s.t
there is no rule t pro p that defeats s
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10. Complexity of DL
Theorem (Maher, TPLP, 2001)
The extension of a defeasible theory D can be computed in time linear to
the size of the theory.
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11. Strategic Argumentation in DL
At each turn we have a theory
Di
= (F, Ri
Com, Ri
Pr, Ri
Op, >)
such that
• Ri
Com = Ri–1
Com ∪ Ri
• Ri ⊆ Ri–1
x , Ri
x = Ri–1
x Ri such that x =
{
Pr i is odd
Op i is even
• (F, Ri–1
Com, >) ⊢ +∂∼c (if i is odd) (F, Ri–1
Com, >) ⊢ +∂c (if i is even)
• (F, Ri
Com, >) ⊢ +∂c (if i is odd) (F, Ri
Com, >) ⊢ +∂∼c (if i is even)
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12. Strategic Argumentation Problem
INSTANCE FOR TURN i: Let c be the critical literal and Di–1 ⊢ +∂∼c
QUESTION:
Is there a subset Ri of Ri–1
x (x ∈ {Pr, Op}) such that Di ⊢ +∂c?
Find a subset of your rules such that you change the outcome of the
dispute.
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13. Strategic argumentation is NP-complete
Theorem
The Strategic Argumentation Problem is NP-complete
Proof.
Reduction of the Restoring Sociality Problem (Governatori and Rotolo,
JAAMAS 2008)
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14. Transformation
F = {flat(p)|p ∈ Fsoc, p ∈ Lit or p = OBLq} (1)
R = {rp : → int_pflat(p)|INTp ∈ Fsoc} (2)
∪ {rfl :
∪
a∈A(r)
flat(a) → flat(p)|r ∈ R
X
[q], X = BEL and p = q, or p = Xq ∈ ModLit} (3)
∪ {rCvx :
∪
a∈A(r)
x_pflat(a) → x_pflat(p)| r ∈ R
BEL
sd [p], A(r) ̸= ∅, A(r) ⊆ Lit, x ∈ {obl, int}} (4)
∪ {rCvyCfx :
∪
y_pflat(a)∈A(rCvy )
y_pflat(a) ; x_pflat(p)| (5)
rCvy ∈ R[y_pflat(p)], x, y ∈ {obl, int}, x ̸= y}
∪ {rCfbelx :
∪
a∈A(r)
flat(a) ; x_pflat(p)|r ∈ R
BEL
[p], x ∈ {obl, int}} (6)
∪ {rCfOI :
∪
a∈A(r)
flat(a) ; int_pflat(p)|r ∈ R
OBL
[p]} (7)
∪ {r–xp : x_pflat(p) ⇒ xp|r ∈ R
Y
.¬Xp ∈ A(r)} (8)
∪ {r–negxp : ⇒ ∼xp|r–xp ∈ R} (9)
∪ {rn–xp : ∼xp ⇒ ¬x_pflat(p)|r–negxp ∈ R} (10)
= {(rα, sβ )|(r, s) ∈>soc, α, β ∈ {fl, Cvx, CvxCfy, Cfbelx, CfOI}}
∪ {(rfl , sn–xp)|rfl ∈ R[x_pflat(p)]} ∪ {(r–xp, s–negxp)|r–xp, sdum–negxp ∈ R}. (11)
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15. Part III
Future Work
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16. Strategic Argumentation and Argumentation
Semantics
• Deciding whether a set of rules wins a turn can be computed in
linear time
• Deciding what set of rules to play in a turn is NP-complete
The result covers ambiguity blocking defeasible logic (and Carneades
(Governatori, ICAIL 2011))
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17. DL and Argumentation Semantics
• ambiguity blocking defeasible logic ̸= grounded semantics
• ambiguity blocking defeasible logic = defeasible semantics
(Governatori et al, JLC 2004)
• ambiguity propagating defeasible logic = grounded semantics
(Governatori et al, JLC 2004)
Corollary
Deciding whether a set of arguments justifies a conclusion p under
defeasible semantics can be computed in polynomial time.
Corollary
The Strategic Argumentation Problem under defeasible semantics is
NP-complete
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18. Strategic Argumentation and Grounded Semantics
Theorem (Maher, TPTL 2013)
Ambiguity Blocking Defeasible Logic and Ambiguity Propagating
Defeasible Logic can simulate each other in polynomial time.
Corollary
Deciding whether a set of arguments justifies a conclusion p under
grounded semantics can be computed in polynomial time.
Theorem
The Strategic Argumentation Problem under grounded semantics is
NP-complete
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19. Conclusions
Play No Games!
Price not negotiable
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20. References
Guido Governatori.
On the relationship between Carneades and defeasible logic.
In Proceedings ICAIL 2011, pages 31–40. ACM, 2011.
Guido Governatori and Antonino Rotolo.
BIO logical agents: Norms, beliefs, intentions in defeasible logic.
Journal of Autonomous Agents and Multi Agent Systems, 17(1):36–69, 2008.
Guido Governatori, Michael J. Maher, Grigoris Antoniou, and David Billington.
Argumentation semantics for defeasible logic.
Journal of Logic and Computation, 14(5):675–702, 2004.
Michael J. Maher.
Propositional defeasible logic has linear complexity.
Theory and Practice of Logic Programming, 1(6):691–711, 2001.
Michael J. Maher.
Relative expressiveness of defeasible logics II.
Theory and Practice of Logic Programming, 13:579–592, 2013.
K. Satoh and K. Takahashi.
A semantics of argumentation under incomplete information.
In Proceedings of Jurisn 2011, 2011.
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