Introduction Related Work Related Work Related Work Our contribution Future Work Epistemic Answer Set Programming Ezgi Iraz Su CILC 2019 @ Trieste, ITALY, June 2019 1 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Outline Motivation 1 Epistemic Specifications (ES) and its K-WVs 2 Epistemic Specifications (ES) and its SE-WVs 3 Epistemic Equilibrium Logic ( EEL) and its AEEMs 4 Epistemic ASP 5 6 Conclusion 2 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work ASP lacks expressivity [Gelfond 1991] Example (Gelfond’s eligibility program Π G , ASP-version ) % university rules to decide eligibility for scholarship ( X : arbitrary applicant) eligible ( X ) ← highGPA ( X ) eligible ( X ) ← fairGPA ( X ) , minority ( X ) ∼ eligible ( X ) ← ∼ highGPA ( X ) , ∼ fairGPA ( X ) % disjunctive info: an applicant data for a specific student called Mike highGPA ( mike ) or fairGPA ( mike ) % if eligibility not determined then interview required (ASP attempt) interview ( X ) ← not eligible ( X ) , not ∼ eligible ( X ) 3 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Quantification problem in ASP Example ( Mike’s eligibility situation, ASP-version ) Π G : eligible ← highGPA 1 eligible ← fairGPA , minority 2 ∼ eligible ← ∼ fairGPA , ∼ highGPA 3 highGPA or fairGPA ← 4 interview ← not eligible , not ∼ eligible 5 has the following answer sets � AS (Π G ) = { highGPA , eligible } , � { fairGPA , interview } . ⇒ eligible ? and ∼ eligible ? undetermined ⇒ interview ? undetermined too... 4 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Quantification problem in ASP Example ( Mike’s eligibility situation, ASP-version ) Π G : eligible ← highGPA 1 eligible ← fairGPA , minority 2 ∼ eligible ← ∼ fairGPA , ∼ highGPA 3 highGPA or fairGPA ← 4 interview ← not eligible , not ∼ eligible 5 has the following answer sets � AS (Π G ) = { highGPA , eligible } , � { fairGPA , interview } . ⇒ eligible ? and ∼ eligible ? undetermined ⇒ interview ? undetermined too... 4 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work So epistemic modalities are required in ASP... Example ( Mike’s eligibility situation, ASP-version ) Π G : eligible ← highGPA 1 eligible ← fairGPA , minority 2 ∼ eligible ← ∼ fairGPA , ∼ highGPA 3 highGPA or fairGPA ← 4 interview ← not eligible , not ∼ eligible 5 Therefore: Π G | � eligible Π G | � ∼ eligible Π G | � interview (counter-intuitive! ) ⇒ wanted: quantification over possible answer sets. . . 5 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Gelfond’s solution [Gelfond 1991] Example (Mike’s scholarship eligibility revisited, ES-version) Π K G : eligible ← highGPA 1 eligible ← minority , fairGPA 2 ∼ eligible ← ∼ fairGPA , ∼ highGPA 3 highGPA or fairGPA ← 4 interview ← not K eligible , not K ∼ eligible 5 will have slightly different answer sets � AS (Π K G ) = { highGPA , eligible , interview } , � { fairGPA , interview } ⇒ eligible ? and ∼ eligible ? unknown ⇒ interview ? YES (intuitive!) 6 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Gelfond’s solution [Gelfond 1991] Example (Mike’s scholarship eligibility revisited, ES-version) Π K G : eligible ← highGPA 1 eligible ← minority , fairGPA 2 ∼ eligible ← ∼ fairGPA , ∼ highGPA 3 highGPA or fairGPA ← 4 interview ← not K eligible , not K ∼ eligible 5 will have slightly different answer sets � AS (Π K G ) = { highGPA , eligible , interview } , � { fairGPA , interview } ⇒ eligible ? and ∼ eligible ? unknown ⇒ interview ? YES (intuitive!) 6 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work ASP lacks expressivity ctd. [Gelfond 2011] Example (Closed World Assumption (CWA) , ASP-version ) % p is assumed to be false if there is no evidence to the contrary (ASP attempt) ∼ p ← not p Consider: Π = { p or q , ∼ p ← not p } has the following answer sets � � AS (Π) = { p } , {∼ p , q } ⇒ p ? unknown ⇒ but also ∼ p ? unknown (counter-intuitive) upshot: again quantification through answer sets is required.... 7 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work ASP lacks expressivity ctd. [Gelfond 2011] Example (Closed World Assumption (CWA) , ASP-version ) % p is assumed to be false if there is no evidence to the contrary (ASP attempt) ∼ p ← not p Consider: Π = { p or q , ∼ p ← not p } has the following answer sets � � AS (Π) = { p } , {∼ p , q } ⇒ p ? unknown ⇒ but also ∼ p ? unknown (counter-intuitive) upshot: again quantification through answer sets is required.... 7 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Two different solutions [Gelfond 2011, Shen et al. 2016] Example (CWA revisited , ES-version ) % p is assumed to be false if there is no evidence to the contrary (ES attempt) ∼ p ← not M p Gelfond’s approach [LPNMR, 2011] ∼ p ← not K p Shen and Eiter’s approach [AIJ, 2016] Consider: K Π = { p or q , ∼ p ← not K p } has the unique answer set � � AS ( K Π) = {∼ p , q } (now, intuitive!) ⇒ Problem ultimately solved? NO , still an open problem... 8 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Two different solutions [Gelfond 2011, Shen et al. 2016] Example (CWA revisited , ES-version ) % p is assumed to be false if there is no evidence to the contrary (ES attempt) ∼ p ← not M p Gelfond’s approach [LPNMR, 2011] ∼ p ← not K p Shen and Eiter’s approach [AIJ, 2016] Consider: K Π = { p or q , ∼ p ← not K p } has the unique answer set � � AS ( K Π) = {∼ p , q } (now, intuitive!) ⇒ Problem ultimately solved? NO , still an open problem... 8 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Outline Motivation 1 Epistemic Specifications (ES) and its K-WVs 2 Epistemic Specifications (ES) and its SE-WVs 3 Epistemic Equilibrium Logic ( EEL) and its AEEMs 4 Epistemic ASP 5 6 Conclusion 9 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Language of ES ( L ES ) [Kahl et al., ICLP 2018] extended the language of ASP by epistemic modalities K and M idea: quantify over all candidate answer sets and correctly represent incomplete information ( non-provability ) K p − − − p is known to be true M p − − − p may be believed to be true atoms: (extended) objective and subjective literals l L g G p | ∼ p l | not l K l | M l g | not g where p ranges over P . strong negation ∼ and default negation (aka, negation as failure) not def def M l = not K notl and M notl = not K l (K and M are dual!) notation: O - Lit — the set of all objective literals S - Lit — the set of all subjective literals 10 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Syntax of ES rule: a logical statement of the form head ← body an ES rule r is of the form l 1 or . . . or l m ← e 1 , . . . , e n head ( r ) : disjunction of objective literals body ( r ) : conjunction of arbitrary literals When m = 0, head ( r ) = ⊥ and r : constraint (headless rule) if body ( r ) of a constraint consists solely of extended sub. literals, i.e., G 1 , . . . , G n , then r : subjective constraint . When n = 0, body ( r ) = ⊤ and r : fact . program: finite collection of rules finite set of ES rules = epistemic specifications 11 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Truth conditions of ES For nonempty A ⊆ 2 O - Lit , l ∈ O - Lit and g ∈ S - Lit , truth conditions: A , A | = l l ∈ A ; if A , A | = not l if l � A ; A , A | = K l l ∈ A for every A ∈ A ; if A , A | = M l if l ∈ A for some A ∈ A ; A , A | = not g A , A �| = g . if equivalences: A | = M l iff A | = not K not l A | = not M l A | = K not l iff ⇒ K and M are (1) dual and (2) interchangeable. 12 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Kahl’s reduct definition [Kahl, PhD thesis 2014] Given A ⊆ 2 O - Lit and an epistemic logic program (ELP) Π : K-reduct r A of an ES rule r w.r.t. A idea: eliminate K and M ( in ASP, we eliminate not ! ) subjective literal ( g ) if true in A if false in A K l replace by l delete rule not K l remove literal replace by not l M l remove literal replace by not not l not M l replace by not l delete rule Π A = { r A : r ∈ Π } remark: K-reduct is rather complex and lacks an intuitive explanation. 13 / 39
Introduction Related Work Related Work Related Work Our contribution Future Work Kahl et al.’s world views (K-WV) [Kahl et al., ICLP 2018] first define: Ep (Π) = { not K l : K l appears in Π } ∪ { M l : M l appears in Π } then take its subset w.r.t. A ⊆ 2 O - Lit Φ A = { G ∈ Ep (Π) : A | = G } finally A is a K-world view (K-WV) of a “constraint-free” Π if: fixed point property A = AS (Π A ) = { A : A is an answer set of Π A } 1 knowledge-minimising property there is no A ′ such that A ′ = AS (Π A ′ ) and Φ A ′ ⊃ Φ A . 2 14 / 39
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