This document summarizes a research paper on strategic argumentation and its relationship to defeasible logic. It contains the following key points: 1. Strategic argumentation involves an adversarial dialogue game where players aim to prove or disprove a claim while avoiding playing arguments that could be used against them. 2. Deciding the outcome of an argument at each turn can be computed in polynomial time, but deciding the optimal set of arguments to play (the strategic argumentation problem) is NP-complete. 3. Defeasible logic can be used to model strategic argumentation and compute argument outcomes. The complexity results also apply to defeasible semantics and grounded semantics.

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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