answer set semantics vs. information term semanticscooml.di.unimi.it/talks/aspslides.pdf ·...
TRANSCRIPT
![Page 1: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/1.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Answer Set Semantics vs. Information TermSemantics
Camillo Fiorentini, Mario Ornaghi
Dipartimento di Scienze dell’Informazione, Universita degli Studi di Milano, Italy
September 13, 2007
![Page 2: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/2.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The context of this talk
Work in progress: a DLV* implementation of CooML (Constructiveobject oriented Modeling Language) snapshot generation.
In this talk: ideas/results from our work in progress comparing CooMLsnapshot semantics and answer sets semanitcs.
![Page 3: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/3.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The Context: Representing OO Modeling Languages inDLV
CooML (Constructive object oriented Modeling Language):
The novelty: semantics based on Fcl(an intermediate constructive logic, Miglioli [1989], similarMedvedev’s Logic of finite problems [1962]).
Logic based tools:
. . .snapshot generation for model validation
![Page 4: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/4.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The Context: Snapshot Generation for Model Validation
To validate an OO model M w.r.t. its informal requirements,show human viable snapshots (representing system states)
Remark: must be empirical
UML+OCL Example: USE (Gogolla 2004), imperative generationlanguage
In CooML: snapshots
specified by class propertiesthrough pieces of informationdeclarative generationUML+OCL can be represented in CooML
![Page 5: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/5.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Outline
1 Representing the CooML snapshot semantics in DLV
2 Fcl semantics vs stable model semantics
3 Conclusion: and now?
![Page 6: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/6.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The syntax of class properties
Class Property (F a first order formula, G a class generator, T theclassical truth operator):
Simple S ::= literal | T (F )
Existential E ::= S | E ∧ E | E ∨ E | ∃x E
Class property P ::= E | ∀ x .G → E
Example
A receipt is empty (no items), or it is not empty and computes a grandtotal. Each item has a price. The grand total is the sum of the prices ofall the items.
class
: ∀x .receipt(x)→T (¬∃i .item(i , x))
∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
class : ∀i , x .item(i , x)→ receipt(x) ∧ ∃p.p = price(i)
constr : T (∀x .receipt(x)→ total(x) = sum(price(i) : item(i , x)))
![Page 7: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/7.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The syntax of class properties
Class Property (F a first order formula, G a class generator, T theclassical truth operator):
Simple S ::= literal | T (F )
Existential E ::= S | E ∧ E | E ∨ E | ∃x E
Class property P ::= E | ∀ x .G → E
Example
A receipt is empty (no items), or it is not empty and computes a grandtotal. Each item has a price. The grand total is the sum of the prices ofall the items.
class : ∀x .receipt(x)→T (¬∃i .item(i , x))
∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
class : ∀i , x .item(i , x)→ receipt(x) ∧ ∃p.p = price(i)
constr : T (∀x .receipt(x)→ total(x) = sum(price(i) : item(i , x)))
![Page 8: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/8.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The syntax of class properties
Class Property (F a first order formula, G a class generator, T theclassical truth operator):
Simple S ::= literal | T (F )
Existential E ::= S | E ∧ E | E ∨ E | ∃x E
Class property P ::= E | ∀ x .G → E
Example
A receipt is empty (no items), or it is not empty and computes a grandtotal. Each item has a price. The grand total is the sum of the prices ofall the items.
class : ∀x .receipt(x)→T (¬∃i .item(i , x))∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
class : ∀i , x .item(i , x)→ receipt(x) ∧ ∃p.p = price(i)
constr : T (∀x .receipt(x)→ total(x) = sum(price(i) : item(i , x)))
![Page 9: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/9.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The syntax of class properties
Class Property (F a first order formula, G a class generator, T theclassical truth operator):
Simple S ::= literal | T (F )
Existential E ::= S | E ∧ E | E ∨ E | ∃x E
Class property P ::= E | ∀ x .G → E
Example
A receipt is empty (no items), or it is not empty and computes a grandtotal. Each item has a price. The grand total is the sum of the prices ofall the items.
class : ∀x .receipt(x)→T (¬∃i .item(i , x))∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
class : ∀i , x .item(i , x)→ receipt(x) ∧ ∃p.p = price(i)
constr : T (∀x .receipt(x)→ total(x) = sum(price(i) : item(i , x)))
![Page 10: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/10.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The syntax of class properties
Class Property (F a first order formula, G a class generator, T theclassical truth operator):
Simple S ::= literal | T (F )
Existential E ::= S | E ∧ E | E ∨ E | ∃x E
Class property P ::= E | ∀ x .G → E
Example
A receipt is empty (no items), or it is not empty and computes a grandtotal. Each item has a price. The grand total is the sum of the prices ofall the items.
class : ∀x .receipt(x)→T (¬∃i .item(i , x))∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
class : ∀i , x .item(i , x)→ receipt(x) ∧ ∃p.p = price(i)
constr : T (∀x .receipt(x)→ total(x) = sum(price(i) : item(i , x)))
![Page 11: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/11.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 12: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/12.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 13: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/13.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 14: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/14.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 15: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/15.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 16: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/16.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 17: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/17.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: an example of piece of information
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
Receipt: r1
Item Priceit1: 5it2: 7
Total: 12
Receipt: r2
![Page 18: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/18.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: pieces of information and their“answer sets”
Piece of information τ : P its I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
with τ1 : P1 and τ2 : P2
[k, τ ] : P1 ∨ P2 ans(τ : Pk)with k ∈ 1..2, τ : Pk
[v , τ ] : ∃x P ans(τ : P(v))with v values for x , τ : P
[ [v1, τ1], . . . , [vn, τn] ] : ∀ x .G → P⋃
j ans(τj : P(v j)) ∪n ≥ 0, v j val. x , τj ∈ it(P) { ∀(G (x) ↔ x ∈ [v1, . . . , vn]) }
![Page 19: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/19.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: pieces of information and their“answer sets”
Piece of information τ : P its I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
with τ1 : P1 and τ2 : P2
[k, τ ] : P1 ∨ P2 ans(τ : Pk)with k ∈ 1..2, τ : Pk
[v , τ ] : ∃x P ans(τ : P(v))with v values for x , τ : P
[ [v1, τ1], . . . , [vn, τn] ] : ∀ x .G → P⋃
j ans(τj : P(v j)) ∪n ≥ 0, v j val. x , τj ∈ it(P) { ∀(G (x) ↔ x ∈ [v1, . . . , vn]) }
![Page 20: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/20.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: pieces of information and their“answer sets”
Piece of information τ : P its I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
with τ1 : P1 and τ2 : P2
[k, τ ] : P1 ∨ P2 ans(τ : Pk)with k ∈ 1..2, τ : Pk
[v , τ ] : ∃x P ans(τ : P(v))with v values for x , τ : P
[ [v1, τ1], . . . , [vn, τn] ] : ∀ x .G → P⋃
j ans(τj : P(v j)) ∪n ≥ 0, v j val. x , τj ∈ it(P) { ∀(G (x) ↔ x ∈ [v1, . . . , vn]) }
![Page 21: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/21.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: pieces of information and their“answer sets”
Piece of information τ : P its I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
with τ1 : P1 and τ2 : P2
[k, τ ] : P1 ∨ P2 ans(τ : Pk)with k ∈ 1..2, τ : Pk
[v , τ ] : ∃x P ans(τ : P(v))with v values for x , τ : P
[ [v1, τ1], . . . , [vn, τn] ] : ∀ x .G → P⋃
j ans(τj : P(v j)) ∪n ≥ 0, v j val. x , τj ∈ it(P) { ∀(G (x) ↔ x ∈ [v1, . . . , vn]) }
![Page 22: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/22.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Snapshot Semantics: pieces of information and their“answer sets”
Piece of information τ : P its I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
with τ1 : P1 and τ2 : P2
[k, τ ] : P1 ∨ P2 ans(τ : Pk)with k ∈ 1..2, τ : Pk
[v , τ ] : ∃x P ans(τ : P(v))with v values for x , τ : P
[ [v1, τ1], . . . , [vn, τn] ] : ∀ x .G → P⋃
j ans(τj : P(v j)) ∪n ≥ 0, v j val. x , τj ∈ it(P) { ∀(G (x) ↔ x ∈ [v1, . . . , vn]) }
![Page 23: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/23.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))
item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 24: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/24.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)
12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))
item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 25: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/25.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)
12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))
item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 26: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/26.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),
T (∃i .item(i , r1)),T (¬∃i .item(i , r2))
item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 27: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/27.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),
T (¬∃i .item(i , r2))
item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 28: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/28.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))
item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 29: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/29.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)
T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 30: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/30.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Example: pieces of information and their I-answer sets
Pieces of information:
[ [ r1, [ [2, [ [12,t],t] ] ], [ r2, [1,t] ] ] :∀x .receipt(x)→T (¬∃i .item(i , x))
∨((∃t.t = total(x)) ∧ T (∃i .item(i , x)))
[ [ (it1, r1), [5,t] ], [ (it2, r1), [7,t] ] ] :∀i , x .item(i , x)→ ∃p.p = price(i)
t : T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
I-answer sets:
receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ]5 = price(it1), 7 = price(it2)T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )
![Page 31: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/31.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
CooML Consistency
CooML consistency: intended models(atoms + generator axioms) |=T-Formulas)⇒ consistency
Example
5 = price(it1), 7 = price(it2), 12 = total(r1)∀(receipt(x) ↔ x ∈ [ r1, r2 ]),∀(item(i , x) ↔ (i , x) ∈ [ (it1, r1), (it2, r1) ])
|=T (∀x .receipt(x) → total(x) = sum(price(i) : item(i , x)) )T (∃i .item(i , r1)), T (¬∃i .item(i , r2))
![Page 32: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/32.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Representing CooML models as DLV programs
Starting from
Example
class : T (¬∃i .item(i ,X )) ∨ ((∃t.total(X , t)) ∧ T (∃i .item(i ,X )))← receipt(X ).
class : receipt(X ) ∧ ∃p.p = price(I )← item(I ,X ).
. . .
replace T -formulas by atoms + constraints
![Page 33: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/33.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Representing CooML models as DLV programs
replacing T (¬∃I .item(I ,X )) by empty(X ):
Example
class : empty(X ) ∨ ((∃t.total(X , t)) ∧ −empty(X ))← receipt(X ).
class : receipt(X ) ∧ ∃p.p = price(I )← item(I ,X ).
constr : false ← ∃I .empty(X ) ∧ item(I ,X ). . .
. . . further translation . . . : DLV programs such that CooML snapshotsare represented by stable models.
![Page 34: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/34.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Representing CooML models as nested programs with ∃
But what happens if we start directly from:
Example
class : ¬ (∃i .item(i ,X )) ∨ ((∃t.total(X , t)) ∧ ¬¬(∃i .item(i ,X )))← receipt(X ).
class : receipt(X ) ∧ ∃p.p = price(I )← item(I ,X ).
. . .
Stable Model Semantics vs Pieces of Information (Fcl) Semantics.
We started from the propositional case
![Page 35: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/35.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Outline
1 Representing the CooML snapshot semantics in DLV
2 Fcl semantics vs stable model semantics
3 Conclusion: and now?
![Page 36: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/36.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The Fcl propositional logic
Propositional syntax, where simple formulas are atomic or negated (canbe extended with Nelson’s negation)
Piece of information τ : P I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
[k, τ ] : P1 ∨ P2 ans(τ : Pk )
f : P1 → P2
⋃τ∈it(P1)
(ans(τ : P1)→ ans(f (τ) : P2))
with f : it(P1) → it(P2)
![Page 37: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/37.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
The Fcl propositional logic
Propositional syntax, where simple formulas are atomic or negated (canbe extended with Nelson’s negation)
Piece of information τ : P I-answer set ans(τ : P)
t : S {S}[τ1, τ2] : P1 ∧ P2 ans(τ1 : P1) ∪ ans(τ2 : P2)
[k, τ ] : P1 ∨ P2 ans(τ : Pk )
f : P1 → P2
⋃τ∈it(P1)
(ans(τ : P1)→ ans(f (τ) : P2))
with f : it(P1) → it(P2)
![Page 38: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/38.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
On the Semantics of Fcl
Theorem (Relationship with classical logic)
A classical interpretation I |= P iff there is τ : P such thatI |= ans(τ : P).
Definition (Constructive validity)
P is Fcl-valid iff there is τ : P such that for every interpretation I ,I |= ans(τ : P).
![Page 39: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/39.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
On the Semantics of Fcl
Theorem (Relationship with classical logic)
A classical interpretation I |= P iff there is τ : P such thatI |= ans(τ : P).
Definition (Constructive validity)
P is Fcl-valid iff there is τ : P such that for every interpretation I ,I |= ans(τ : P).
Example
(¬¬p ∨ ¬p) ∧ (¬¬p → a) ∧ (¬p → b) → a ∨ b
map τ :[ [1,t], λx .t, λx .t ] 7→τ [1,t],[ [2,t], λx .t, λx .t ] 7→τ [2,t]
(¬¬p → a) ∧ (¬p → b) → a ∨ bno valid τ
![Page 40: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/40.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
On the Semantics of Fcl
Theorem (Relationship with classical logic)
A classical interpretation I |= P iff there is τ : P such thatI |= ans(τ : P).
Definition (Constructive validity)
P is Fcl-valid iff there is τ : P such that for every interpretation I ,I |= ans(τ : P).
Example
On the other hand
(¬¬p → a) ∧ (¬p → b) → a ∨ b
holds in ”Here and There”.
![Page 41: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/41.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Are there pieces of information for stable models?
Fcl FHT
Interpretation I 7→ τI : F Stable model M 7→ τM : F
Truth preserving maps maps ...
![Page 42: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/42.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Answer sets for nested expressions using pieces ofinformation
Basic concepts:
Positive answers: ans+(τ : F ) = {p | p ∈ ans(τ : F )},Negative answers: ans−(τ : F ) = {¬H | ¬H ∈ ans(τ : F )}
Default interpretation di(τ : F ): di(τ : F ) |= p iff p ∈ ans+(τ : F )
Default consistency: di(τ : F ) |= ans−(τ : F )... negated formulas constrain the default interpretations.
I-stability: di(τ ′ : F ) ⊂ di(τ : F ) ⇒ di(τ : F ) 6|= ans−(τ ′ : F )
![Page 43: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/43.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Answer sets for nested expressions using pieces ofinformation
Basic concepts:
Positive answers: ans+(τ : F ) = {p | p ∈ ans(τ : F )},Negative answers: ans−(τ : F ) = {¬H | ¬H ∈ ans(τ : F )}Default interpretation di(τ : F ): di(τ : F ) |= p iff p ∈ ans+(τ : F )
Default consistency: di(τ : F ) |= ans−(τ : F )... negated formulas constrain the default interpretations.
I-stability: di(τ ′ : F ) ⊂ di(τ : F ) ⇒ di(τ : F ) 6|= ans−(τ ′ : F )
![Page 44: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/44.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Answer sets for nested expressions using pieces ofinformation
Basic concepts:
Positive answers: ans+(τ : F ) = {p | p ∈ ans(τ : F )},Negative answers: ans−(τ : F ) = {¬H | ¬H ∈ ans(τ : F )}Default interpretation di(τ : F ): di(τ : F ) |= p iff p ∈ ans+(τ : F )
Default consistency: di(τ : F ) |= ans−(τ : F )... negated formulas constrain the default interpretations.
I-stability: di(τ ′ : F ) ⊂ di(τ : F ) ⇒ di(τ : F ) 6|= ans−(τ ′ : F )
ans(t : ¬¬p) = {¬¬p }di(t : ¬¬p) = ∅, ans−(t : ¬¬p) = {¬¬p},
Default inconsistent: ∅ 6|= ¬¬p
![Page 45: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/45.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Answer sets for nested expressions using pieces ofinformation
Basic concepts:Positive answers: ans+(τ : F ) = {p | p ∈ ans(τ : F )},Negative answers: ans−(τ : F ) = {¬H | ¬H ∈ ans(τ : F )}Default interpretation di(τ : F ): di(τ : F ) |= p iff p ∈ ans+(τ : F )Default consistency: di(τ : F ) |= ans−(τ : F )... negated formulas constrain the default interpretations.I-stability: di(τ ′ : F ) ⊂ di(τ : F ) ⇒ di(τ : F ) 6|= ans−(τ ′ : F )
ans([1,t] : p ∨ ¬¬p) = { p }, ans([2,t] : p ∨ ¬¬p) = {¬¬p },di([1,t] : p ∨ ¬¬p) = {p} ans−([1,t] : p ∨ ¬¬p) = ∅,di([2,t] : p ∨ ¬¬p) = ∅ ans−([2,t] : p ∨ ¬¬p) = {¬¬p},
Instability: {p} |= ¬¬p.
![Page 46: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/46.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Answer sets for nested expressions using pieces ofinformation
Basic concepts:
Positive answers: ans+(τ : F ) = {p | p ∈ ans(τ : F )},Negative answers: ans−(τ : F ) = {¬H | ¬H ∈ ans(τ : F )}Default interpretation di(τ : F ): di(τ : F ) |= p iff p ∈ ans+(τ : F )
Default consistency: di(τ : F ) |= ans−(τ : F )... negated formulas constrain the default interpretations.
I-stability: di(τ ′ : F ) ⊂ di(τ : F ) ⇒ di(τ : F ) 6|= ans−(τ ′ : F )
Definition (I-answer sets of nested expressions)
X is an I-answer set of F iff there is a default consistent and I-stableτ : F such that X = di(τ : F ).
![Page 47: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/47.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Equivalence with the standard definition of answer set
Theorem (Equivalence Theorem)
X is an I-answer set of F iff it is an answer set of F .
Example
τi : (p ∨ q) ∧ ¬(p ∧ ¬ q) di(τ : F ) ans−(τ : F )
τ1 = [ [1,t], t ] {p} {¬(p ∧ ¬ q) }τ2 = [ [2,t], t ] {q} {¬(p ∧ ¬ q) }
τ1 not default consistent (di(τ1 : F ) 6|= ans−(τ1 : F ))τ2 default consistent (di(τ2 : F ) |= ans−(τ2 : F )) and I-stable:
{q} is the unique I-answer set.
![Page 48: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/48.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Equivalence with the standard definition of answer set
Theorem (Equivalence Theorem)
X is an I-answer set of F iff it is an answer set of F .
Example
τi : p ∨ ¬¬ p di(τ : F ) ans−(τ : F )
τ1 = [1,t] {p} ∅τ2 = [2,t] ∅ {¬¬p}
τ1 not I-stable: di(τ2 : F ) ⊂ di(τ1 : F ) and di(τ1 : F ) |= ans−(τ2 : F ).τ2 not default consistent (∅ 6|= ¬¬p)
No I-answer set .
![Page 49: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/49.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Proving the Equivalence Theorem by maps preservingans(τ : F )
Ans-preserving map: f : F → G such thatans(τ : F ) = ans(f (τ) : G )
Lemma 1: If there is f : it(F )↔ it(G ) s.t. f and f −1 areans-preserving, then for every H, F ∧ H and G ∧ H have the sameI-answer sets
Lemma 2: There is such a bijective f : F ↔ FN, where FN is adisjunction of conjunctions of simple formulas.
Equivalence Theorem: it sufficies to prove that I-answer sets andanswer sets coincide for FN.
![Page 50: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/50.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Outline
1 Representing the CooML snapshot semantics in DLV
2 Fcl semantics vs stable model semantics
3 Conclusion: and now?
![Page 51: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/51.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
Conclusion
Work in progress: theoretical analysis related to a DLVimplementation of CooML snapshot generation
Next (theoretical) steps: extend our work to nested programs(possibly, with ∃)
idea: extend ans+, di, ans− to the larger language and relateI-stability to the minimality condition in SM[F ](SM defined in: “Stable Models and Circumscription”, Ferraris, Lee,Lifschitz - 2007)
![Page 52: Answer Set Semantics vs. Information Term Semanticscooml.di.unimi.it/talks/aspslides.pdf · Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics](https://reader035.vdocuments.mx/reader035/viewer/2022062607/60473eb9d77ae10a385dfc56/html5/thumbnails/52.jpg)
Representing the CooML snapshot semantics in DLV Fcl semantics vs stable model semantics Conclusion: and now?
And now?
A different characterisation is interesting
Is there a realizability semantics based on stable pieces ofinformation (i.e., maps preserving stable τ : F )?
Discussion: there is no such map f : it(p) → it(¬¬p), butp → ¬¬p is valid in HT (Here and There).HT characterizes strong equivalence, but not “strong implication”.
Is there a reasonable (at least valid) calculus for such a semantics?
Discussion: p,¬¬p ` ¬¬p should not be provable