Epistemic Answer Set Programming Ezgi Iraz Su CILC 2019 @ Trieste, - - PowerPoint PPT Presentation

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

1

Motivation

2

Epistemic Specifications (ES) and its K-WVs

3

Epistemic Specifications (ES) and its SE-WVs

4

Epistemic Equilibrium Logic ( EEL) and its AEEMs

5

Epistemic ASP

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 ΠG, 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 ) ΠG :

1

eligible ← highGPA

2

eligible ← fairGPA, minority

3

∼eligible ← ∼fairGPA, ∼highGPA

4

highGPA or fairGPA ←

5

interview ← not eligible, not ∼eligible

has the following answer sets AS(ΠG) =

• {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 ) ΠG :

1

eligible ← highGPA

2

eligible ← fairGPA, minority

3

∼eligible ← ∼fairGPA, ∼highGPA

4

highGPA or fairGPA ←

5

interview ← not eligible, not ∼eligible

has the following answer sets AS(ΠG) =

• {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 ) ΠG :

1

eligible ← highGPA

2

eligible ← fairGPA, minority

3

∼eligible ← ∼fairGPA, ∼highGPA

4

highGPA or fairGPA ←

5

interview ← not eligible, not ∼eligible

Therefore:

ΠG | eligible ΠG | ∼eligible ΠG | 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) ΠK G :

1

eligible ← highGPA

2

eligible ← minority, fairGPA

3

∼eligible ← ∼fairGPA, ∼highGPA

4

highGPA or fairGPA ←

5

interview ← not K eligible, not K ∼eligible

will have slightly different answer sets AS(ΠK 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) ΠK G :

1

eligible ← highGPA

2

eligible ← minority, fairGPA

3

∼eligible ← ∼fairGPA, ∼highGPA

4

highGPA or fairGPA ←

5

interview ← not K eligible, not K ∼eligible

will have slightly different answer sets AS(ΠK 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

1

Motivation

2

Epistemic Specifications (ES) and its K-WVs

3

Epistemic Specifications (ES) and its SE-WVs

4

Epistemic Equilibrium Logic ( EEL) and its AEEMs

5

Epistemic ASP

6

Conclusion

9 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Language of ES (LES) [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 M l

def

= not K notl and M notl

def

= 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

l1 or . . . or lm ← e1 , . . . , en

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., G1 , . . . , Gn, 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 ⊆ 2O-Lit, l ∈ O-Lit and g ∈ S-Lit, truth conditions:

A, A |= l

if

l ∈ A; A, A |= not l

if

l A; A, A |= K l

if

l ∈ A for every A ∈ A; A, A |= M l

if

l ∈ A for some A ∈ A; A, A |= not g

if

A, A |= g.

equivalences:

A |= M l

iff

A |= not K not l A |= not M l

iff

A |= K not l ⇒ 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 ⊆ 2O-Lit and an epistemic logic program (ELP) Π: K-reduct rA 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

ΠA = {rA : 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 ⊆ 2O-Lit

ΦA = {G ∈ Ep(Π) : A |= G}

finally A is a K-world view (K-WV) of a “constraint-free” Π if:

fixed point property

1

A = AS(ΠA) = {A : A is an answer set of ΠA} knowledge-minimising property

2

there is no A′ such that A′= AS(ΠA′) and ΦA′ ⊃ ΦA.

14 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Some new language constructs

[Kahl et al., ICLP 2018]

effect of a constraint r over answer sets in ASP: (at most) rule it out when it violates r in ES: also an additive or subtractive effect (not reqular!) Solution by Kahl and Leclerc: world view constraints (WVCs) in the form of subjective constraints replace ← by

wv

←

wv

←ϕ is read: “it is not a world view if it satisfies ϕ” Ex:

wv

←notK p: “it is not a world view if p is not known”

(any world view satisfying notK p should be eliminated)

15 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

WVCs can solve the constraint problem?

...to some extent! because only works for subjective constraints what about for

← K p , q? Definition (Kahl and Leclerc’s restricted solution)

Let Π be an ELP containing WVCs such that Π = Π0 ∪ Πwvc

Πwvc: set of all WVCs occurring in Π Π0 is a constraint-free part of Π.

Then A is a K-WV of Π if A is a K-WV of Π0 and A does not violate any constraint in Πwvc.

16 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Outline

1

Motivation

2

Epistemic Specifications (ES) and its K-WVs

3

Epistemic Specifications (ES) and its SE-WVs

4

Epistemic Equilibrium Logic ( EEL) and its AEEMs

5

Epistemic ASP

6

Conclusion

17 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Shen and Eiter’s (SE’s) reduct definition

[SE, AIJ 2016]

SE use not K (epistemic negation) to minimise knowledge first remember:

Ep(Π) = {not K l : K l appears in Π}∪{M l : M l appears in Π}

then take its subset Φ ⊆ Ep(Π) (they call it a guess) and take SE-reduct rΦ for each r ∈ Π: given A ⊆ 2O-Lit,

SE-reduct rΦ of an ES rule r w.r.t. Φ

idea: eliminate K and M (aligning with K-reduct)

epistemic-negated sub. literal (G) if G ∈ Φ if G ∈ EpΠ \ Φ not K l replace by ⊤ replace by not l M l replace by ⊤ replace by not not l

next form

ΠA = {rA : r ∈ Π}

finally consider ΦA = {G ∈ Ep(Π) : A |= G}

18 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

SE’s world views (SE-WV)

[SE, AIJ 2016]

A is a SE-world view (SE-WV) of a “constraint-free” Π if: fixed point property

1

A = AS(ΠΦ) = {A : A is an answer set of ΠΦ};

2

ΦA agrees with Φ, i.e., ΦA = Φ; knowledge-minimising property

3

Φ is maximal, i.e., there is no bigger guess Φ′ ⊃ Φ such that A′ = AS(ΠΦ′) and ΦA′ = Φ′ for some nonempty collection A′

• f consistent sets of objective literals.

⇒ but SE-WVs cannot handle ES including constraints either...

19 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Outline

1

Motivation

2

Epistemic Specifications (ES) and its K-WVs

3

Epistemic Specifications (ES) and its SE-WVs

4

Epistemic Equilibrium Logic ( EEL) and its AEEMs

5

Epistemic ASP

6

Conclusion

20 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Epistemic equilibrium logic ( EEL)

[FHS, IJCAI 2015]

Equilibrium logic ( EL): today’s general purpose nonmonotonic reasoning first proposed by Pearce [1996] as a logical foundation of answer set programming (ASP) [Lifschitz,Gelfond,. . . ] based on here-and-there logic (HT):

a well-known nonclassical monotonic logic [Heyting,G¨

• del]

Epistemic equilibrium logic ( EEL) in a nutshell: extend EL with (nondual) epistemic modal operators K and ˆ K

semantics via EEMs (minimise truth just as in EL) EEMs are not strong enough as a new semantics for ES

so, FHS define a selection process over EEMs and propose AEEMs (min. knowledge/max. ignorance)

inspired by Moore’s “Autoepistemic logic” [1980] and Levesque’s “All-that-I-know logic” [1990]

21 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Epistemic equilibrium logic ( EEL)

[FHS, IJCAI 2015]

Equilibrium logic ( EL): today’s general purpose nonmonotonic reasoning first proposed by Pearce [1996] as a logical foundation of answer set programming (ASP) [Lifschitz,Gelfond,. . . ] based on here-and-there logic (HT):

a well-known nonclassical monotonic logic [Heyting,G¨

• del]

Epistemic equilibrium logic ( EEL) in a nutshell: extend EL with (nondual) epistemic modal operators K and ˆ K

semantics via EEMs (minimise truth just as in EL) EEMs are not strong enough as a new semantics for ES

so, FHS define a selection process over EEMs and propose AEEMs (min. knowledge/max. ignorance)

inspired by Moore’s “Autoepistemic logic” [1980] and Levesque’s “All-that-I-know logic” [1990]

21 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Advantages and disadvantages of EEL

Good points: The AEEM approach can handle a more general language EEMs minimise truth in a standard way (aligning with EMs) Problems: AEEMs depend on 2 different orderings

set inclusion ⊆ and a preference ordering ≤ϕ

the latter is a bit complex no order between these orderings fortunately not any clash has been found so far

cannot cope with ES including constraints

22 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Outline

1

Motivation

2

Epistemic Specifications (ES) and its K-WVs

3

Epistemic Specifications (ES) and its SE-WVs

4

Epistemic Equilibrium Logic ( EEL) and its AEEMs

5

Epistemic ASP

6

Conclusion

23 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Novelty offered by Epistemic ASP (EASP)

EASP allows only K , but K may also appear in head

not can only appear in front of literals and not included in a

literal formation

• ur semantics enjoys a 2-fold computation procedure
• ur reduct definition is oriented to eliminate not
• ur approach can handle programs with constraints

24 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Our aim and motivation

aim: solve the problem with ELPs including constraints propose a more natural generalisation of ASP motivating example:

• epis. spec. Π

K-WVs SE-WVs AEEMs Π1 : p ← not q q ← not p

• {p, r}, {q, r}, {p, s}, {q, s}
• same

same r ∨ s ← not K p Π2 : p ← not q q ← not p r ∨ s ← not K p

• {p}
• {p}
• {p}
• ← r

← s

What would we expect? no world views/AEEMs (intuitive!)

25 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Language of EASP (LEASP)

extended the language of ASP with epistemic modal operator K literals (λ) :objective literals (l) and subjective literals (g)

l g p | ∼p K p | K ∼p

where p ranges over P. EASP rules are of the following form:

λ1 or . . . or λk ← λk+1 , . . . , λm , not λm+1 , . . . , not λn

positive rules — without negation as failure (NAF) not

(pos.) EASP program: finite collection of (pos.) EASP rules

⇒ ASP: EASP in which literals are restricted to objective literals

26 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Constraint-free and positive EASP programs

semantics: via epistemic stable models (ESMs)

Definition (weakening of a point in an S5 model A ⊆ 2O-Lit)

Given a (subset) map s : A → 2O-Lit such that

s(A) ⊆ A for every A ∈ A, s id on A and s|A\{A} = id, s[A], s(A): weakening of A at a point A ∈ A.

notation:s[A], s(A) ⊳ A, A. Ex:

• ∅, {q, r}
• ⊳
• {p}, {q, r}
• .

Definition (nonmono. satisfaction reln |=∗ minimising truth)

Given a pointed S5 model A, A and an EASP program Π,

A, A|=∗Π iff

1

A, A |= Π and

2

s[A], s(A) |= Π for every map s viz. s[A], s(A) ⊳ A, A.

Ex:

• {p}, {r}
• |=∗ q or r.

27 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Definition (generalisation of answer set defn in ASP to EASP) A is a minimal model of Π if A, A |=∗ Π for every A ∈ A. Example

Consider the following (pos. & constraint-free) EASP program Σ: p or q ← s ← q r ← K p Claim:

• {p}, {q, s}
• is a minimal model of Σ: indeed,
• {p}, {q, s}
• |= Σ while its only weakening
• ∅, {q, s}
• |= Σ.
• {p}, {q, s}
• |= Σ while all its weakenings, i.e,
• {p}, {q}
• ,
• {p}, {s}
• and
• {p}, ∅
• do not satisfy it.

{{p, r}} and {{q, s}} are the other (unintended) minimal models of Σ. ⇒ minimality of truth does not guarantee intuitive results.

28 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Definition (epistemic stable model)

Let A be a nonempty collection of consistent sets of objective

• literals. Then A is an epistemic stable model (ESM) of a

constraint-free and positive EASP program Π if

truth-minimising condition

1

A is a minimal model of Π; knowledge-minimising condition

2

there is no minimal model A′ of Π such that ΦA ⊂ ΦA′;

(it means that A makes max. possible sub. literals in Π false)

3

there is no minimal model A′ of Π such that A ⊂ A′.

⇒ however, 2nd may not be the right approach (but it works) Why?

In ASP, we do not follow such a procedure with not l... Consider p or not p in ASP! ∅ and {p} are the answer sets, but we do not choose ∅. Consider also K p or not K p!

29 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Constraint-free, positive EASP programs ctd.

Example

Consider the following EASP program Σ once again: p or q ← s ← q r ← K p

Σ has 3 min. models: A1={{p}, {q, s}}, A2={{p, r}} & A3={{q, s}}. ΦA1 = ΦA3 = {K p} ⊃ ΦA2 = ∅. A1 ⊃ A3. ∴ ESM(Σ) = {A1}.

30 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

What if Π contains constraints?

first, we separate Π into 2 disjoint subprograms: T(Π) = {r ∈ Π : r is a constraint}

(“top”)

where we evaluate candidate ESMs.

B(Π) = Π \ T(Π) — constraint-free (main) part of Π (“bottom”)

where we decide candidate ESMs.

⇒ we ensure constraints to behave regularly as in ASP!

then, we compute ESM(B(Π)) finally, we evaluate each A ∈ ESM(B(Π)) wrt. their behaviour

• n T(Π): accept, refute or reorganise!

31 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

How do we evaluate candidate ESMs on T(Π)?

Let ϕ =

r∈T(Π) body(r).

Then, for every A ∈ ESM(B(Π)) and every A ∈ A,

• if A, A |= ϕ, then we accept A and call it Aaccept;
• if A, A |= ϕ, then we eliminate A.
• Finally, we reorganise the rest in such a way that we take the

biggest possible subset Anew ⊆ A viz. Anew is still a minimal model of B(Π) and Anew, A |= ϕ for every A ∈ Anew. In other words, Anew turns into Aaccept. remark: If T(Π) only contains sub. constraints then we either refute or accept the ESMs of B(Π).

32 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Let’s turn back our example!

Example

Consider the following EASP program Σ once again: p or q ← s ← q r ← K p remember: A1 = {{p}, {q, s}} is the unique ESM of Σ. Take Σ0 = {← r}: {{p}, {q, s}} |= r and {{p}, {q, s}} |= r. So, A1 passes our test (accept!). ∴ ESM(Σ ∪ Σ0) = A1. Take Σ1 = {← not K q}: A1 |= not K q So, A1 fails to be an ESM of Σ ∪ Σ1 (refute!). ∴ ESM(Σ ∪ Σ1) = ∅. Take Σ2 = {← p}. Since {{p}, {q, s}} |= p, we delete {p} from

A1 and result in Anew = {{q, s}}. Anew is still a minimal model

• f Σ, so we accept it (reorganise!). ∴ ESM(Σ ∪ Σ2) = {{q, s}}.

33 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

What if Π is not positive?

then we first take the reduct!

• ur reduct defn is oriented to eliminate NAF only as in ASP!

Definition (generalisation of the reduct definition of ASP)

Let Π be a “constraint-free” EASP program. Let A ⊆ 2O-Lit be nonempty and A ∈ A.Then, the reduct ΠA,A of Π w.r.t. A, A is given by replacing every

• ccurrence of notλ with

⊥ if A, A |= λ (for λ = l if A |= l; for λ = K l if A |= K l); ⊤ if A, A |= λ (for λ = l if A |= l; for λ = K l if A |= K l). Thus, A is a minimal model of Π if

A, A |=∗ ΠA,A for every A ∈ A.

34 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Let’s see an example!

Example

Consider the following EASP program Γ: p ← not ∼q

∼q ← not p

r ← not K p Claim: A =

• {p, r}, {∼q, r}
• is a minimal model of Γ: indeed,

Γ{{p,r},{∼q,r}} : p ← ⊤ Γ{{p,r},{∼q,r}} :

p ← ⊥

∼q ← ⊥ ∼q ← ⊤

r ← ⊤ r ← ⊤

• {p, r}, {∼q, r}
• |= Γ{{p,r},{∼q,r}}, but all its weakenings do not.
• {p, r}, {∼q, r}
• |= Γ{{p,r},{∼q,r}}, but all its weakenings do not.

35 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Comparison with K-WVs, SE-WVs, AEEMs and ESMs

where we differ? mainly for EASP programs including constraints...

EASP program Π K-WVs SE-WVs AEEMs ESMs Π1 : p or q

• {p}, {q}
• {p}, {q}
• {p}, {q}
• {p}, {q}
• p or q

none

• {p}
• {p}
• none

← not K p Π2 : p ← not q

• {p}, {q}
• {p}, {q}
• {p}, {q}
• {p}, {q}
• q ← not p

r ← K p p ← not q

• {p, r}
• {p, r}
• {p, r}
• none

q ← not p r ← K p ← not r

36 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Discussion: inclusion of belief operator

let’s call our belief operator B :

1

can consider B as dual of K (same as M in ES), i.e.,

B is equivalent to not K not can treat it neither positive nor negative construct (similar to notnot in ASP) ? shoud we take its reduct? probably YES! complicated because then we have to define how to take the reduct of K not

2

can consider B (similar to ˆ K in EEL) as non-dual of K

reasonable because EASP is a 3-valued formalism treat it as a positive subjective literal like K p and we do not take its reduct ? but then p ← B p has a unique ESM {∅}. Intuitive?

37 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

Outline

1

Motivation

2

Epistemic Specifications (ES) and its K-WVs

3

Epistemic Specifications (ES) and its SE-WVs

4

Epistemic Equilibrium Logic ( EEL) and its AEEMs

5

Epistemic ASP

6

Conclusion

38 / 39

Introduction Related Work Related Work Related Work Our contribution Future Work

To sum it up

many different semantics approaches for ES

most of them are obsolete today:

[Gelfond 1991,1994,2011; Kahl et al. 2014,2016, Wang&Zhang 2005,...]

successful candidates (to some extent):

[Kahl 2018, SE 2016, FHS 2015] cannot cope with programs including constraints

Our approach:

propose a more standard generalisation of ASP

• ffer a solution to the constraint problem

still, it is not fully satisfactory.....

Future Work: improve knowledge-minimising property of ESMs better solution for programs including constraints make it more expressive inserting a belief operator Thank you!

39 / 39