Cerutti--TAFA 2011

Computing with InfiniteArgumentation Frameworks:

the Case of AFRAs

Pietro Baroni1, Federico Cerutti1,Paul E. Dunne2, Massimiliano Giacomin1

1Dipartimento di Ingegneria dell’Informazione, Universita di BresciaVia Branze 38, I-25123 Brescia, Italy

2Department of Computer Science, Ashton Building, University of LiverpoolLiverpool, L69 7ZF, United Kingdom

July 17th, 2011

First International Workshop on the Theory and Applications of Formal Argumentation (TAFA-11)

Infinite Argumentation Frameworks

({}, even(0))

({}, q(0))

({¬even(0)}, even(1))

({¬even(n− 1)}, even(n))

({¬even(0)},¬even(0))

({¬even(1)},¬even(1))

({¬even(n)},¬even(n))

({¬even(0)}, q(0))

({¬q(0)},¬q(0)) ({¬q(0)}, p)

({¬even(0)}, q(1))

({¬even(1)}, q(1))

({¬q(1)},¬q(1)) ({¬q(1)}, p)

({¬p}, r)

({¬r},¬r)

({¬p},¬p)({¬even(n− 1)}, q(n))

({¬even(n)}, q(n))

({¬q(n)},¬q(n)) ({¬q(n)}, p) r ← ¬pp← ¬q(x)q(x)← even(x)q(x)← ¬even(x)even(s(x))← ¬even(x)even(0)←

Preliminary BackgroundFormal languages

Representing afra with a regular languageComputing the grounded extension

Conclusions and Future works

The Argumentation Framework [Dung, 1995]

Definition

An argumentation framework (af) is a pair 〈X ,A〉, in which X is aset of arguments and A ⊆ X × X is the attack relationship.

A pair 〈x, y〉 ∈ A is referred to as ‘y is attacked by x’ or ‘x attacks y’;x ∈ X is acceptable with respect to S ⊆ X if for every y ∈ X thatattacks x there is some z ∈ S that attacks y.

The characteristic function, F : 2X → 2X is the mapping which,given S ⊆ X , returns the set of y ∈ X for which y is acceptable to S.For any set S we define F0(S) = ∅ and for k ≥ 1Fk(S) = F(Fk−1(S)).

The grounded extension is the (unique) least fixed point of F . Wedenote by GE(〈X ,A〉) ⊆ X the grounded extension of 〈X ,A〉.

The Argumentation Framework with RecursiveAttacks [Baroni et al., 2011]

Definition

An Argumentation Framework with Recursive Attacks (afra) isdescribed by a pair 〈X ,R〉 where X is a (finite) set of arguments andR consists of pairs of the form 〈x, α〉 where x ∈ X and α ∈ X ∪R.

For α = 〈x, β〉 ∈ R, the source (src) and target (trg) of α are definedby src(α) = x and trg(α) = β.

xk xk−1 xk−2 · · · x2 x1 ∈ R if {x1, . . . , xk} ⊆ X , 〈x2, x1〉 ∈ R and〈xj〈xj−1 〈· · · x1〉〉〉 ∈ R, with 2 < j ≤ k.

Letting C = R∪ X , for α ∈ R and β ∈ C, α is said to defeat β(α→ β) whenever any of the following hold:

1. trg(α) = β

2. trg(α) = src(β) with β ∈ R, α = xy and β = yγ (y ∈ X ).

β direct defeats Cγ direct defeats α

γ = CABsrc(γ) = C

β indirectly defeats γ

Definition

Let Γ = 〈A,R〉 be an AFRA, the corresponding AF ΓAF = 〈A, R〉 isdefined as follows:

A = A ∪R;

R = {(V ,W )|V ,W ∈ A ∪R and V defeats W }.

Proposition

Let Γ = 〈A,R〉 an AFRA and ΓAF = 〈A, R〉 its corresponding AF ,S ⊆ A ∪R [. . . ]:

S is the grounded extension for Γ iff S is the D-groundedextension for ΓAF ;

Infinite Argumentation Frameworks

Why considering AFRA?

Finite number of argument

Infinite number of attacks

We can describe an infinite number of attacksgiven a finite number of arguments

When R is infinite what characterisessuitable specification mechanisms for describing R?

Preliminary Background

Formal languagesRepresenting afra with a regular language

Computing the grounded extensionConclusions and Future works

Terminology

Definition

For X a finite set of arguments, we denote by X ∗ the set of all finitelength sequences (or words) that can be formed using arguments in X(noting this includes ε the so-called empty sequence comprising noarguments). Given w ∈ X ∗ we will denote as w the sequence obtainedby reversing the order of the symbols in w, namely, givenw = x1x2 . . . xn, w = xn . . . x2x1.Given u = u1u2 . . . ur and v = v1v2 . . . vk ∈ Σ∗ we denote by u · v (orsimply uv) the word w of length k + r defined by u1u2 . . . urv1v2 . . . vk.We note that w · ε = ε · w = w. We say that L ⊆ X ∗ is an attacklanguage over X if L satisfies, ∀ w ∈ L w = xu with x ∈ X and either|u| = 1 or u ∈ L.

Regular languages

Definition

A deterministic finite automaton (dfa) is defined via a 5-tuple,M = 〈Σ, Q, q0, F, δ〉 where Σ = {σ1, . . . , σk} is a finite set of inputsymbols, Q = {q0, q1, . . . , qm} a finite set of states; q0 ∈ Q the initialstate; F ⊆ Q the set of accepting states; and δ : Q× Σ → Q thestate transition function. For q ∈ Q and w ∈ Σ∗, the reachable statefrom q on input w is

ρ(q, w) =

q if w = εδ(q, w) if |w| = 1δ(ρ(q, u), x) if w = u · x

Definition

A sequence w = w1w2 . . . wn ∈ Σ∗ is accepted by the dfa〈Σ, Q, q0, F, δ〉 if ρ(q0, w) = ρ(q0, wnwn−1 . . . w1) ∈ F . For a dfa, M ,L(M) is the subset of Σ∗ accepted by M .

Representing afra with aregular languageComputing the grounded extension

The dfa+ representation of an afra

Definition

Given an afra 〈X ,R〉 where R ⊂ X ∗ is a regular languagerepresented as a dfaM, its dfa+ is a representation of 〈X ,R〉 as asingle dfaM+ = 〈X , QM+ , q0, FM+ , δ+〉 such that for any w ∈ X ∗ itholds w ∈ L(M+) if and only if w ∈ X ∪R.

Notations on the dfa+

For p ∈ QM+ we define:

state−out(p) = { q ∈ QM+ : ∃ x ∈ X for which q = δ+(p, x)}sym− in(p) = {x ∈ X : ∃ q ∈ QM+ for which p = δ+(q, x)}state− in(p) = { q ∈ QM+ : ∃ x ∈ X for which p = δ+(q, x)}

Argument states

Definition

Let 〈X ,R〉 be an afra, and be 〈X , QM+Γ, q0, FM+

Γ, δ+〉 a dfa+

representation of it. ∀x ∈ X ∃q = argst(x) ∈ FM+ such thatρ(q0, x) = q and sym− in(q) = {x} and if q = argst(x) we will saythat x = reparg(q). For the whole set of symbols X in a dfa+

representation ArgS(M+) , {argst(x) | x ∈ X}.

Argument states

The set of direct defeaters of an argument x isdirdef(x) , {y ∈ X | δ+(argst(x), y) ∈ FM+}An argument x is unattacked in afra if and only if dirdef(x) = ∅The set of unattacked arguments will be denoted asunatt− args(M+)

Attack states

Definition

The set of attack states in a dfa+ is defined as AttS(M+) ,FM+ \ArgS(M+); every attack state q corresponds to a (possiblyinfinite) subset of R AttL(q) (∀q ∈ AttS(M+) AttL(q) ,{r ∈ R | ρ(q0, r) = q}). Given r ∈ AttL(q) we will say that q is therepresentative state of r, denoted as q = repst(r).

Attack states

dirdef(q) , {x ∈ X | δ+(q, x) ∈ FM+}indirdef(q) ,

⋃r∈AttL(q) indirdef(r) =

⋃x∈sym−in(q) dirdef(x)

totdef(r) , dirdef(r) ∪ indirdef(r)

an attack state q is unattacked if totdef(q) = ∅

Splitting

totdef(r) = ∅ ⇒ indirdef(repst(r)) = ∅ if |sym− in(repst(r))| = 1.Under this condition r ∈ R is unattacked ⇔ repst(r) is unattacked.

Splitting

Definition

An attack state p is splittable if |sym− in(p)| > 1. The set ofsplittable states of a dfa+ M+ will be denoted as split− states(M+).

Definition

Let p be a splittable state with sym− in(p) = {x1, . . . , xn}, (n > 1).The dfa+ resulting by splitting p, split(M+, p) is obtained by:

S1. QsplM+ = QM+ ∪ {p2, . . . , pn} where p2, . . . , pn in F spl

S2. Letting p1 = p the transition function δ+spl has, for i = 1 . . . n:δspl(q′, xi) = pi if q′ ∈ state− in(p) ∧ δ(q′, xi) = p,δspl(pi, y) = δ(p, y), δspl(q, y) = δ(q, y) if q ∈ QM+ \ state− in(p)

Splitting

Definition

Splitting

Definition

Representing afra with a regular language

Computing the groundedextension

Grounded extension in finitary frameworks

For any afra where X is finite, the corresponding af 〈X , R〉 isfinitary:

the attackers of each element x of X ∩ X correspond to the directdefeaters of x in afra, which are at most |X |;the attackers of each element r of X ∩ R correspond to the directand indirect defeaters of r in afra, which are at most 2 ∗ |X |.

Proposition

If an argumentation framework af is finitary then GE(af) =⋃i=1...∞F i(∅) where F is the characteristic function of af.

Computing the grounded extension

1: Input: dfa+ M+ = 〈X , QM+ , q0, FM+ , δ+〉 with α ∈ L(M+)⇔ α ∈ X ∪R.2: Output: dfaMG = 〈X , QG, q0, FG, δG〉 with α ∈ L(MG)⇔ α ∈ GE(〈X , R〉)3: i := 04: Mi := csplit(M+); withMi = 〈X , Qi, q0, Fi, δi〉5: repeat6: i := i+ 1;Mi := Mi−1;7: For each (unmarked) unattacked state q ofMi mark q as in(i).8: for each unattacked state q and every q′ ∈ state− in(q) ∩ Fi do9: Mark q′ as out and remove q′ from Fi.10: end for11: for each x ∈ X s.t. argst(x) is marked out do12: For each state q ∈ Fi with x ∈ sym− in(q) mark q as out and remove q

from Fi.13: end for14: untilMi =Mi−1

15: for any q ∈ Fi which is not marked in() do16: remove q from Fi

17: end for18: return 〈X , Qi, q0, Fi, δi〉

Line 1:

Line 4 (M0):

Line 6 (M1):

Line 7 (M1):

Line 9 (M1):

Line 11 (M1):

Line 7 (M2):

Line 9 (M2):

Line 7 (M3):

Line 9 (M3):

Theorem

Let M+ = 〈X , QM+ , q0, FM+ , δ+〉 with α ∈ L(M+)⇔ α ∈ X ∪R bea dfa+ describing the afra, 〈X ,R〉 with corresponding af 〈X , R〉. Itis possible to construct in polynomial time a dfaMG = 〈X , QG, q0, FG, δG〉 with α ∈ L(MG)⇔ α ∈ GE(〈X , R〉).

Representing afra with a regular languageComputing the grounded extension

Conclusions and Futureworks

Conclusions

Methodology and initial results in the field of computing withinfinite argumentation frameworks

Main idea of drawing correspondences between the specificationof argumentation frameworks and well-known notions and resultsin formal language theory

While there are cases of infinite attacks which can not berepresented with formal grammars, dfas provide a convenientway to represent infinite attack relations

With the dfa representation the problem of computing thegrounded extension (tractable in the finite case) preserves itstractability in the infinite case

Future works

Extension of this kind of analysis to other standardcomputational problems in abstract argumentation:

conflict-freenessadmissibilitystable extensions

Extending the approach by considering general Dung’s af withinfinite arguments [Baroni et al., under submission]

Computing with InfiniteArgumentation Frameworks:

the Case of AFRAs

Thank you

References I

[Baroni et al., under submission] Baroni, P., Cerutti, F., Dunne, P. E., and Giacomin, M. (undersubmission).Automata for infinite argumentation structures.Artificial Intelligence.

[Baroni et al., 2011] Baroni, P., Cerutti, F., Giacomin, M., and Giovanni, G. (2011).AFRA: Argumentation framework with recursive attacks.International Journal of Approximate Reasoning, 52(1):19 – 37.

[Dung, 1995] Dung, P. M. (1995).On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logicprogramming, and n-person games.Artificial Intelligence, 77(2):321–357.

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