ABA-Based Answer Set Justifications - an overview translate Logic Program ABA framework compute derive answer sets stable extensions 𝑇 𝐹 𝑚 ∈ 𝑇
ABA-Based Answer Set Justifications - an overview translate Logic Program ABA framework compute derive answer sets stable extensions 𝑇 𝐹 construct 𝑚 ∈ 𝑇 … ⊢ 𝑚 ∈ 𝐹
ABA-Based Answer Set Justifications - an overview translate Logic Program ABA framework compute derive answer sets stable extensions 𝑇 𝐹 construct 𝑚 ∈ 𝑇 … ⊢ 𝑚 ∈ 𝐹 construct explains Attack Tree
ABA-Based Answer Set Justifications - an overview translate Logic Program ABA framework compute derive answer sets stable extensions 𝑇 𝐹 construct 𝑚 ∈ 𝑇 … ⊢ 𝑚 ∈ 𝐹 construct explains Attack Tree construct LABAS Justification
ABA-Based Answer Set Justifications - an overview translate Logic Program ABA framework compute derive answer sets stable extensions 𝑇 𝐹 construct 𝑚 ∉ 𝑇 … ⊢ 𝑚 ∉ 𝐹 construct explains Attack Tree construct LABAS Justification
Attack Trees
Attack Trees A − 9 : ( { not ¬ a } , ∅ ) ⊢ a
Attack Trees
Attack Trees A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a
Attack Trees
Attack Trees A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d
Attack Trees
Attack Trees A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + 14 : ( ∅ , { e } ) ⊢ e
Attack Trees
Attack Trees A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a
Attack Trees A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
laser surgery! Answer Set Programming (ASP) intraocular lenses!
Why is laser surgery not part of the solution? Answer Set: { shortSighted , afraidToTouchEyes , student , likesSports , tightOnMoney , correctiveLens , caresAboutPracticality , intraocularLens }
Why is laser surgery not part of the solution? Answer Set: { shortSighted , afraidToTouchEyes , student , likesSports , tightOnMoney , correctiveLens , caresAboutPracticality , intraocularLens } A 1 − : ( { shortSighted } , { not tightOnMoney , not correctiveLens } ) ⊢ laserSurgery A 2+ : ( { student } , { not richParents } ) ⊢ tightOnMoney
Why is intraocular lens part of the solution? Answer Set: { shortSighted , afraidToTouchEyes , student , likesSports , tightOnMoney , correctiveLens , caresAboutPracticality , intraocularLens }
Why is intraocular lens part of the solution? Answer Set: { shortSighted , afraidToTouchEyes , student , likesSports , tightOnMoney , correctiveLens , caresAboutPracticality , intraocularLens } A 3+ : ( { shortSighted } , { not laserSurgery , not glasses , not contactLens } ) ⊢ intraocularLens A 1 − : ( . . . ) ⊢ laserSurgery A − 4 : ( . . . , not afraidToTouchEyes , . . . ) ⊢ contactLens A 2+ : ( . . . ) ⊢ tightOnMoney A + 5 : ( { afraidToTouchEyes } , ∅ ) ⊢ afraidToTouchEyes A − 6 : ( . . . , not caresAboutPracticality , . . . ) ⊢ glasses A + 7 : ( { likesSports } , ∅ ) ⊢ caresAboutPracticality
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11 + + not c + not d + asm asm
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11 + + not c + not d + asm asm − c − A 12
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11 + + not c + not d + asm asm − c − A 12 − not e − asm
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11 + + not c + not d + asm asm − c − A 12 − not e − asm + e + fact
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11 + + not c + not d + asm asm − − c − d − A 12 A 13 − not e − asm + e + fact
Labelled ABA-Based Answer Set (LABAS) Justifications A − 9 : ( { not ¬ a } , ∅ ) ⊢ a A + 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d A + A + 14 : ( ∅ , { e } ) ⊢ e 11 : ( { not c , not d } , ∅ ) ⊢ ¬ a A − A − 12 : ( { not e } , ∅ ) ⊢ c 13 : ( { not ¬ a } , ∅ ) ⊢ d . . . A + 14 : ( ∅ , { e } ) ⊢ e
Labelled ABA-Based Answer Set (LABAS) Justifications a − A 9 − not ¬ a − asm + ¬ a + A 11 + + − not c + not d + asm asm − − c − d − A 12 A 13 − not e − asm + e + fact
laser surgery! Answer Set Programming (ASP) intraocular lenses!
Why is laser surgery not part of the solution? Attack Tree A 1 − : ( { shortSighted } , { not tightOnMoney , not correctiveLens } ) ⊢ laserSurgery A 2+ : ( { student } , { not richParents } ) ⊢ tightOnMoney
Why is laser surgery not part of the solution? Attack Tree A 1 − : ( { shortSighted } , { not tightOnMoney , not correctiveLens } ) ⊢ laserSurgery A 2+ : ( { student } , { not richParents } ) ⊢ tightOnMoney ABA-Based Answer Set (ABAS) Justification laserSurgery − A 1 − not tightOnMoney − asm + tightOnMoney + A 2 + + student + not richParents + asm fact
Why is intraocular lens part of the solution? Attack Tree A 3+ : ( { shortSighted } , { not laserSurgery , not glasses , not contactLens } ) ⊢ intraocularLens A 1 − : ( . . . ) ⊢ laserSurgery A − 4 : ( . . . , not afraidToTouchEyes , . . . ) ⊢ contactLens A 2+ : ( . . . ) ⊢ tightOnMoney A + 5 : ( { afraidToTouchEyes } , ∅ ) ⊢ afraidToTouchEyes A − 6 : ( . . . , not caresAboutPracticality , . . . ) ⊢ glasses A + 7 : ( { likesSports } , ∅ ) ⊢ caresAboutPracticality
Why is intraocular lens part of the solution? ABA-Based Answer Set (ABAS) Justification intraocularLens + A 3 + + + + shortSighted + not contactLens + fact not laserSurgery + not glasses + asm asm asm − − − laserSurgery − contactLens − glasses − A 4 A 1 A 6 − − − not afraidToTouchEyes − not tightOnMoney − not caresAboutPracticality − asm asm asm + + + afraidToTouchEyes + tightOnMoney + caresAboutPracticality + fact A 2 + A 7 + + student + not richParents + fact likesSports + asm fact
Other justificiation approaches LABAS Justification Off-line Justification (Pontelli, Son, Elkhatib) b + A 1 + + not a + e + asm fact − + a − A 2 − not b − asm
Other justificiation approaches LABAS Justification Off-line Justification (Pontelli, Son, Elkhatib) a − A 2 − not b − asm + − b + A 1 + + not a + e + asm fact
Conclusion Answer Set Programming (ASP) + ABA-Based Answer Set Justification =
Conclusion Answer Set Programming (ASP) + ABA-Based Answer Set Justification = laser surgery! intraocular lenses!
Future Work So far: restricted do consistent logic programs
Future Work So far: restricted do consistent logic programs ◮ find source of inconsistency in a logic program ◮ debug the logic program ⇒ more existing literature
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Explaining Answer Sets in Argumentative Terms Questions?!
ASP Semantics Why not simply display the derivation? The answer set of P ( AS ( P )), is the smallest set S ⊆ Lit P s.t.: 1. for any clause l 0 ← l 1 , . . . , l m in P : if l 1 , . . . , l m ∈ S then l 0 ∈ S 2. S = Lit P if S contains complementary literals a and ¬ a . ⇒ For P without NAF literals For P with NAF literals S is an answer set of P if it is the answer set of the reduct P S , i.e. if S = AS ( P S ).
ASP Semantics It all depends on the reduct... For P possibly with NAF literals and for any S ⊆ Lit P The reduct P S is obtained from P by deleting: 1. all clauses with not l in their bodies where l ∈ S 2. all NAF literals in the remaining clauses.
ASP Semantics It all depends on the reduct... For P possibly with NAF literals and for any S ⊆ Lit P The reduct P S is obtained from P by deleting: 1. all clauses with not l in their bodies where l ∈ S 2. all NAF literals in the remaining clauses. Example a ← not ¬ a ← ¬ a , not c , not e a ¬ a ← not c , not d ← c not e d ← not ¬ a ← e
ASP Semantics It all depends on the reduct... For P possibly with NAF literals and for any S ⊆ Lit P The reduct P S is obtained from P by deleting: 1. all clauses with not l in their bodies where l ∈ S 2. all NAF literals in the remaining clauses. Example a ← not ¬ a ← ¬ a , not c , not e a ¬ a ← not c , not d e ∈ S ← c not e d ← not ¬ a ← e
ASP Semantics It all depends on the reduct... For P possibly with NAF literals and for any S ⊆ Lit P The reduct P S is obtained from P by deleting: 1. all clauses with not l in their bodies where l ∈ S 2. all NAF literals in the remaining clauses. Example a ← not ¬ a ← ¬ a , not c , not e a ¬ a ← not c , not d e ∈ S ← c not e d ← not ¬ a ← e
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