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 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


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.


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


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


Outline

1 Representing the CooML snapshot semantics in DLV

2 Fcl semantics vs stable model semantics

3 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)))







Example


class : ∀x .receipt(x)→T (¬∃i .item(i , x))

∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))









Example


class : ∀x .receipt(x)→T (¬∃i .item(i , x))∨ ((∃t.t = total(x)) ∧ T (∃i .item(i , x)))




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


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]) }


Example: pieces of information and their I-answer sets

Pieces of information:



[ [ (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)) )








I-answer sets:

receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)

12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))









I-answer sets:

receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),

T (∃i .item(i , r1)),T (¬∃i .item(i , r2))









I-answer sets:

receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),

T (¬∃i .item(i , r2))









I-answer sets:

receipt(x) ↔ x ∈ [ r1, r2 ] (class generator axiom)12 = total(r1),T (∃i .item(i , r1)),T (¬∃i .item(i , r2))









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)) )








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)) )


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))


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


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 ).


constr : false ← ∃I .empty(X ) ∧ item(I ,X ). . .

. . . further translation . . . : DLV programs such that CooML snapshotsare represented by stable models.


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 ).


. . .

Stable Model Semantics vs Pieces of Information (Fcl) Semantics.

We started from the propositional case


Outline





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)


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







Example

On the other hand

(¬¬p → a) ∧ (¬p → b) → a ∨ b

holds in ”Here and There”.


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 ...


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 )



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 )





Basic concepts:




ans(t : ¬¬p) = {¬¬p }di(t : ¬¬p) = ∅, ans−(t : ¬¬p) = {¬¬p},

Default inconsistent: ∅ 6|= ¬¬p



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.



Basic concepts:




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 ).


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.


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 .


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.


Outline





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)


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

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