Strategic Argumentation is NP-complete

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

Strategic Argumentation is NP-Complete Copyright NICTA 2014 1/17

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 thescene 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 cardstolen since facility A does not allow a person to enterwithout an ID card.


Part I

Strategic Argumentation


Strategic Argumentation

• Adversarial two player (Proponent and Opponent) dialogue game toprove/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


Strategic Argumentation Problem

Avoid playing arguments that can be used by the other party to defeatyou

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 ⇒ bOp: f , c ⇒ e, e, f ⇒ ¬bOp wins

Pr: a, a ⇒ g, g ⇒ bPr wins


Part II

Strategic Argumentationin Defeasible Logic


Defeasible Logic (DL)

• Derive (plausible) conclusions with the minimum amount ofinformation.

• 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


Proving Conclusions in DL

1 Give an argument for the conclusion you want to prove

2 Consider all possible counterarguments to it3 Rebut all counterarguments

• Defeat the argument by a stronger one• Undercut the argument by showing that some of the premises do not

hold


Derivations in DL

+∂p

1) there is an applicable rule r pro p2) for all rules t con p either:

2.1) t is not applicable2.2) t is defeated by a rule s pro p

–∂p

1) for all rules r pro p either2.1) r is not applicable, or2.2) there is an applicable rule s con p s.t

there is no rule t pro p that defeats s


Complexity of DL

Theorem (Maher, TPLP, 2001)The extension of a defeasible theory D can be computed in time linear tothe size of the theory.


Strategic Argumentation in DL

At each turn we have a theory

Di = (F , RiCom, Ri

Pr, RiOp, >)

such that

• RiCom = Ri–1

Com ∪ Ri

• Ri ⊆ Ri–1x , Ri

x = Ri–1x \ Ri such that x =

{Pr i is odd

Op i is even

• (F , Ri–1Com, >) ⊢ +∂∼c (if i is odd) (F , Ri–1

Com, >) ⊢ +∂c (if i is even)

• (F , RiCom, >) ⊢ +∂c (if i is odd) (F , Ri

Com, >) ⊢ +∂∼c (if i is even)


Strategic Argumentation Problem

INSTANCE FOR TURN i : Let c be the critical literal and Di–1 ⊢ +∂∼cQUESTION: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 thedispute.


Strategic argumentation is NP-complete

TheoremThe Strategic Argumentation Problem is NP-complete

Proof.Reduction of the Restoring Sociality Problem (Governatori and Rotolo,JAAMAS 2008)


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 ∈ RX [q], X = BEL and p = q, or p = Xq ∈ ModLit} (3)

∪ {rCvx :∪

a∈A(r )

x_pflat(a) ↪→ x_pflat(p)| r ∈ RBELsd [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 ∈ RBEL [p], x ∈ {obl , int}} (6)

∪ {rCfOI :∪

a∈A(r )

flat(a) ; int_pflat(p)|r ∈ ROBL [p]} (7)

∪ {r–xp : x_pflat(p) ⇒ xp|r ∈ RY .¬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)


Part III

Future Work


Strategic Argumentation and ArgumentationSemantics

• Deciding whether a set of rules wins a turn can be computed inlinear 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))


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)

CorollaryDeciding whether a set of arguments justifies a conclusion p underdefeasible semantics can be computed in polynomial time.

CorollaryThe Strategic Argumentation Problem under defeasible semantics isNP-complete


Strategic Argumentation and Grounded Semantics

Theorem (Maher, TPTL 2013)Ambiguity Blocking Defeasible Logic and Ambiguity PropagatingDefeasible Logic can simulate each other in polynomial time.

CorollaryDeciding whether a set of arguments justifies a conclusion p undergrounded semantics can be computed in polynomial time.

TheoremThe Strategic Argumentation Problem under grounded semantics isNP-complete


Conclusions

Play No Games!Price not negotiable


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.


Education

Strategic Argumentation is NP-complete