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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)
c© 2011 Federico Cerutti <[email protected]>
Infinite Argumentation Frameworks
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 2
({}, 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
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 3
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〉.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 4
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 ).
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 5
The Argumentation Framework with RecursiveAttacks [Baroni et al., 2011]
β direct defeats Cγ direct defeats α
γ = CABsrc(γ) = C
β indirectly defeats γ
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 5
The Argumentation Framework with RecursiveAttacks [Baroni et al., 2011]
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 ;
. . .
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 5
Infinite Argumentation Frameworks
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 6
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?
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 7
Preliminary Background
Formal languagesRepresenting afra with a regular language
Computing the grounded extensionConclusions and Future works
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 8
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.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 9
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 .
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 10
Preliminary BackgroundFormal languages
Representing afra with aregular languageComputing the grounded extension
Conclusions and Future works
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 11
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.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 12
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)}
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 13
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}.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 14
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+)
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 14
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).
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 15
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) = ∅
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 15
Splitting
totdef(r) = ∅ ⇒ indirdef(repst(r)) = ∅ if |sym− in(repst(r))| = 1.Under this condition r ∈ R is unattacked ⇔ repst(r) is unattacked.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 16
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
M+.
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)
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 16
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
M+.
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)
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 16
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
M+.
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)
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 16
Preliminary BackgroundFormal languages
Representing afra with a regular language
Computing the groundedextension
Conclusions and Future works
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 17
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.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 18
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 1:
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 4 (M0):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 6 (M1):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 7 (M1):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 9 (M1):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 11 (M1):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 7 (M2):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 9 (M2):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 7 (M3):
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〉
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 19
Line 9 (M3):
Computing the grounded extension
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Computing the grounded extension
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〉).
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Preliminary BackgroundFormal languages
Representing afra with a regular languageComputing the grounded extension
Conclusions and Futureworks
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 20
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
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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]
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Computing with InfiniteArgumentation Frameworks:
the Case of AFRAs
Thank you
c© 2011 Federico Cerutti <[email protected]>
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.
c© 2011 Federico Cerutti <[email protected]> July 17th, 2011 24