Directions and Applications for Constructive Semantics in ... · Directions and Applications for Constructive Semantics in Description Logics Loris Bozzato DKM - Data and Knowledge

Directions and Applications for ConstructiveSemantics in Description Logics

Loris Bozzato

DKM - Data and Knowledge Management Research Unit,FBK-Irst, Fondazione Bruno Kessler - Trento, Italy

DKM internal seminarMay 2011

L. Bozzato (DKM - FBK-Irst) 05/2011 1 / 62

Acknowledgements

This presentation is a summary of my work with:Mauro FerrariDICOM, Università degli Studi dell’Insubria - Varese, Italy

Camillo FiorentiniDSI, Università degli Studi di Milano - Milano, Italy

Guido FiorinoDIMEQUANT, Università degli Studi di Milano-Bicocca - Milano, Italy

Paola VillaPhD at DICOM, Università degli Studi dell’Insubria - Varese, Italy


Outline

Outline

1 Introduction

2 BCDL: information terms semantics for ALC

3 KALC∞: Kripke-style semantics for ALC

4 KALC: decidable Kripke-style interpretation for ALC

5 Application directions


Introduction

Outline

1 Introduction






Introduction Constructive logics

Constructive logics

Constructive logicsFormalizations of ideas from constructivism

Brouwer-Heyting-Kolmogorov (BHK) or proof interpretation

Semiformal presentation of a constructive semanticsE.g. propositional part:

A proof of A∧ B is composed from proofs of A and BA proof of A∨ B is composed of a proof for A or BA proof of A→B is construction transforming proofs of A in proofs for B⊥ is an unprovable formula (thus A→⊥ = ¬A)

Possible formalizations:IntuitionismRecursive realizabilityInformation terms semantics



Constructive logics

Constructive logicsFormalizations of ideas from constructivism

Brouwer-Heyting-Kolmogorov (BHK) or proof interpretation

Semiformal presentation of a constructive semanticsE.g. propositional part:

A proof of A∧ B is composed from proofs of A and BA proof of A∨ B is composed of a proof for A or BA proof of A→B is construction transforming proofs of A in proofs for B⊥ is an unprovable formula (thus A→⊥ = ¬A)

Possible formalizations:IntuitionismRecursive realizabilityInformation terms semantics



Constructive logics

Characteristic properties:Disjunction property (DP):“Whenever it proves a disjunction formula,it proves one of the disjoints”Explicit definability property (ED):“Whenever it proves an existential formula,it presents a witness of the existence”

Constructivism and Computer Science:Formulas-as-types (Curry-Howard isomorphism)Proofs-as-programs


Introduction Constructive description logics

Constructive description logics

Constructive description logicsConstructive interpretations of description logics

MotivationsComputational interpretation of proofs and formulasUseful in domains with dynamic and incomplete knowledge

Known proposals

[de Paiva, 2005]: translations of DLs in constructive systems[Kaneiwa, 2005]: definitions for different constructive negations in DLs[Odintsov and Wansing, 2003]: inconsistency tolerant version of DLs[Mendler and Scheele, 2010]: Kripke semantics with “fallible” elements



Proposal [de Paiva, 2005]

MotivationExtend proof-theoretical results (Curry-Howard) on DLsDefine a context-sensitive DL

Proposals3 different interpretations of ALC in constructive systems:

IALC: from ALC to IFOL (via ALC → FOL translation)

iALC: from ALC to IK (via ALC → Km translation)

cALC: from ALC to CK (via ALC → Km translation)



Proposal [Kaneiwa, 2005]

MotivationRepresentation of different notions of negative information in DLs(contraries, contradictories and subcontraries)E.g. difference between Happy, Unhappy,¬Happy,¬Unhappy

Proposals2 different extensions to ALC semantics(different interactions between constructive and classical negation)

tableaux algorithm for satisfiability

Similar works: [Kamide, 2010a, Kamide, 2010b]Paraconsistent and temporal versions of ALC, based on a similarsemantics



Proposal [Odintsov and Wansing, 2003]

IdeasParaconsistent versions of ALCConstructive semantics to represent partial information

Proposal [Odintsov and Wansing, 2003]

3 constructive paraconsistent semantics for ALC(Different translations to four valued logic N4)

complete tableaux calculus for each logic

Further work [Odintsov and Wansing, 2008]

Reviews of calculi, tableaux procedure for one of the presented logics



Proposal [Mendler and Scheele, 2010]

Idea

Representation of partial knowledge and consistency under abstraction

Evolving OWA: stages of information with changing properties andabstract individuals

ProposalcALC: Kripke semantics for ALC with fallible entities

fallible entities ⊥I : contradictory domain elements(maximal poset elements or undefined role fillers)

complete and decidable Hilbert and tableaux calculi

Application [Mendler and Scheele, 2009]

Reasoning on data streams in auditing domain



Our proposals

BCDL [Bozzato et al., 2007, Bozzato et al., 2009b, Ferrari et al., 2010]

Information terms semantics + natural deduction calculusÔ computational interpretation of proofs (Proofs as programs)

KALC∞ [Bozzato et al., 2009a, Villa, 2010]

Kripke-style semantics + tableaux calculusÔ possibly infinite models, efficient treatment of implications

KALC [Bozzato et al., 2010, Bozzato, 2011]

Kripke-style semantics + tableaux algorithmÔ finite models, decidability from terminating tableau procedure


BCDL: information terms semantics for ALC

Outline

1 Introduction






BCDL: information terms semantics for ALC Introduction


BCDL: Basic Constructive Description Logic [Ferrari et al., 2010]

Information terms semantics for ALCNatural deduction calculus NDc

Information terms [Miglioli et al., 1989]

Syntactic objects justifying validity of formulas in classical modelsrealization of BHK interpretationrelated to realizability interpretations

Features:Classical reading of DL formulasSimple proof theoretical characterization by NDc

Computational interpretation of proofsNatural notion of state




BCDL: Basic Constructive Description Logic [Ferrari et al., 2010]

Information terms semantics for ALCNatural deduction calculus NDc

Information terms [Miglioli et al., 1989]

Syntactic objects justifying validity of formulas in classical modelsrealization of BHK interpretationrelated to realizability interpretations

Features:Classical reading of DL formulasSimple proof theoretical characterization by NDc

Computational interpretation of proofsNatural notion of state



ALCG syntax and classical semantics

Syntax is the same as ALC, adding a set of generators NG

Concepts

C ::= A | G | ¬C | Cu C | Ct C | ∃R.C | ∀R.C

FormulasK ::= ⊥ | (t, s) : R | t : C | ∀GC

where A ∈ NC, R ∈ NR, t, s ∈ NI and G ∈ NG

Generators NGConcepts over fixed set of individual names dom(G) = {c1, . . . , cn}Ô limited form of subsumption: ∀GC ≡ G v C



ALCG syntax and classical semantics

Validity of formulas: given a model M = (∆M, ·M)

M 6|= ⊥

M |= (t, s) : R iff (tM, sM) ∈ RM

M |= t : H iff tM ∈ HM

M |= ∀GH iff GM = {cM1 , . . . , cMn } ⊆ HM


BCDL: information terms semantics for ALC IT semantics

BCDL information terms semantics

Information terms ITN (K)

Structured objects that constructively justify the truth of a formula K

ITN (K) = {tt}, if K is atomicITN (c : C1 t C2) = {(k, α) | k ∈ {1, 2} and α∈ IT(c : Ck)}

Realizability M� 〈α〉K

Truth of K in a model M justified w.r.t. α

M� 〈tt〉K iff M |= KM� 〈(k, α)〉 c : C1 t C2 iff M� 〈α〉 c : Ck

Theorem (classical and IT semantics)

M |= K iff there exists α ∈ IT(K) such that M� 〈α〉K


























ITN (K) = {tt} for K atomic or negated M� 〈tt〉K iff M |= K

ITN (c : C1 u C2)= IT(c : C1)× IT(c : C2) M� 〈(α, β)〉 c : C1 u C2 iff M� 〈α〉 c : C1and M� 〈β〉 c : C2

ITN (c : C1 t C2)= IT(c : C1) ] IT(c : C2) M� 〈(k, α)〉 c : C1 t C2 iff M� 〈α〉 c : Ck

ITN (c : ∃R.C)= N ×⋃d∈N IT(d : C) M� 〈(d, α)〉 c : ∃R.C iff M |= (c, d) : R

and M� 〈α〉 d : C

ITN (c : ∀R.C)= (⋃

d∈N IT(d : C))N M� 〈φ〉 c : ∀R.C iff M |= c : ∀R.C and,for every d ∈ N , if M |= (c, d) : R thenM� 〈φ(d)〉 d : C

ITN (∀GC)= (⋃

d∈dom(G) IT(d : C))dom(G) M� 〈φ〉 ∀GC iff, for every d ∈ dom(G),M� 〈φ(d)〉 d : C


BCDL: information terms semantics for ALC Natural deduction calculus

Natural deduction calculus ND

Calculus ND : natural deduction calculus for ALCG

Rules

Γ··· π′

t : AktIk

t : A1 tA2

Γ1··· π1

t : ∃R.A

Γ2, [(t, p) : R, p : A]··· π2

K∃E

K

∀GA

Γ′··· π′

t : G∀GE

t : A

Γ, [t : ¬H]··· π′

⊥¬E

t : H

Theorem

ND is sound and complete w.r.t. ALCG

leaving aside generators rules, ND is sound and complete w.r.t. ALC



Natural deduction calculus NDc

Calculus NDc: natural deduction calculus for BCDL

Rules

Every rule fromND

(minus ¬E)

Γ··· π′

t : ¬¬CAt

t : C

Γ··· π′

t : ∀R.¬¬HKUR

t : ¬¬∀R.H

NoteKUR corresponds to the Kuroda axiom schema Kur

Kur ≡ ∀x.¬¬A(x) → ¬¬∀x.A(x)



Soundness of NDc

Operator ΦπN : Given a proof π : Γ ` K over N :

ΦπN : ITN (Γ) → ITN (K)

Note: computable function, inductively defined on depth of π

Example: tIk

If last rule in π is tIk with k ∈ {1, 2}, then:

ΦπN : ITN (Γ) → ITN (t : C1 t C2)

defined as:

ΦπN (γ) = ( k, Φπ′

N (γ) )

tIk ruleΓ··· π′

t : AktIk

t : A1 tA2



Soundness of NDc: computational interpretation

Theorem (Soundness)If π : Γ ` K, then:

Γ |= K.If M� 〈γ〉 Γ then M� 〈Φπ

N (γ)〉K. (constructive consequence)

Computational interpretationGiven a proof π of a formula K. . .. . . its Φπ

N provides a “program” to compute an IT for K



Properties of NDc

Theorem (Completeness)

Γ | BCDL K iff K is a constructive consequence of Γ.

Constructive propertiesFor Γ of Harrop formulas (no t and ∃):

Disjunction property (DP):If Γ | BCDL c : At B, then Γ | BCDL c : A or Γ | BCDL c : B.

Explicit definability property (EDP):If Γ | BCDL c : ∃R.A, then there exists d ∈ NI such thatΓ | BCDL (c, d) : R and Γ | BCDL d : A.


KALC∞ : Kripke-style semantics for ALC

Outline

1 Introduction






KALC∞ : Kripke-style semantics for ALC Introduction

KALC∞: Kripke-style semantics for ALC

KALC∞ [Bozzato et al., 2009a, Villa, 2010]

Kripke-style semantics for ALCTableaux calculus T

Features:Final state conditions on models, related to KurTableaux calculus with efficient treatment of duplications



KALC∞ syntax and classical semantics

Concepts

C ::= A | ¬C | Cu C | Ct C | C→C | ∃R.C | ∀R.C

FormulasK ::= ⊥ | (t, s) : R | t : C | C

where A ∈ NC, R ∈ NR, t, s ∈ NI.

Ô compared to BCDL: implication (unrestricted v) and no generators


M 6|= ⊥ M |= (t, s) : R iff (tM, sM) ∈ RM M |= t : C iff tM ∈ CM

M |= C iff CM = ∆M



KALC∞ syntax and classical semantics

Concepts

C ::= A | ¬C | Cu C | Ct C | C→C | ∃R.C | ∀R.C

FormulasK ::= ⊥ | (t, s) : R | t : C | C

where A ∈ NC, R ∈ NR, t, s ∈ NI.

Ô compared to BCDL: implication (unrestricted v) and no generators


M 6|= ⊥ M |= (t, s) : R iff (tM, sM) ∈ RM M |= t : C iff tM ∈ CM

M |= C iff CM = ∆M


KALC∞ : Kripke-style semantics for ALC Kripke semantics

KALC∞: Kripke semantics

Rooted poset:

K-posets

Given a (rooted) poset (P,≤), final: φ ∈ P such that, for every α ∈ P,φ ≤ α implies φ = α.

K-poset: every poset (P,≤) such that, for every α ∈ P, Fin(α) 6= ∅.



KALC∞-models

KALC∞-model K = 〈P,≤, ρ, ι〉(P,≤) is a K-poset with root ρ;

ι(α) = (Dα, ·α) such that, for every α, β ∈ P with α ≤ β:1 Dα ⊆ Dβ;2 for every c ∈ NI, cα = cβ;3 for every C ∈ NC, Cα ⊆ Cβ;4 for every R ∈ NR, Rα ⊆ Rβ.

Forcing relation (examples)

α‖−−t : A, where A ∈ NC iff tα ∈ Aα

α‖−−t : C→D iff ∀ β ∈ P s.t. α ≤ β, β‖−/−t : C or β‖−−t : D

α‖−−t : ¬C iff ∀ β ∈ P s.t. α ≤ β, β‖−/−t : C

α‖−−t : ∀R.C iff ∀ β ∈ P s.t. α ≤ β and ∀ d ∈ Dβ, β‖−−(t, d) : R ⇒ β‖−−d : C



KALC∞-models

KALC∞-model K = 〈P,≤, ρ, ι〉(P,≤) is a K-poset with root ρ;ι(α) = (Dα, ·α) such that, for every α, β ∈ P with α ≤ β:

1 Dα ⊆ Dβ;2 for every c ∈ NI, cα = cβ;3 for every C ∈ NC, Cα ⊆ Cβ;4 for every R ∈ NR, Rα ⊆ Rβ.








KALC∞-models










KALC∞-models










Kur and final states

KurThe condition on final states can be characterized by Kur:

Kur : ∀R.¬¬A→¬¬∀R.A

Kur axiom schemaLet K be any instance of the axiom schema Kur in L and let

K = 〈P,≤, ρ, ι〉 be a KALC∞-model for L, then ρ‖−−K.

Final worlds φ

Maximal elements w.r.t. ≤ relationφ represents states of complete knowledgeφ behave like classical models: φ‖−−c : At ¬Athis directly comes from forcing of ¬A!


KALC∞ : Kripke-style semantics for ALC Tableau calculus

Tableau calculus T for KALC∞

Tableau calculus TRefutation calculus: the negation of a formula cannot be satisfiedDecompose sets of signed formulas (T, F, Fc) building a proof tree

Signed formulas

T(A): A is true;F(A): A is false;Fc(A): A will never be true;

S is contradictory iff {T(A), F(A)} ⊆ S, {T(A), Fc(A)} ⊆ S

FeaturesSound and complete w.r.t. KALC∞ semanticsKALC∞ Disjunction Property



Relevant rules of T

S, T(t : At B)

S, T(t : A) | S, T(t : B)Tt

S, F(t : At B)

S, F(t : A), F(t : B)Ft

S, Fc(t : At B)

S, Fc(t : A), Fc(t : B)Fct

S, T(t : (At B)→C)

S, T(t : A→C), T(t : B→C)T→t . . .

S, Fc(t : A→B)

Sp, T(t : A), Fc(t : B)Fc→

S, T(t : ∀R.A), T((t, s) : R)

S, T((t, s) : R), T(s : A), T(t : ∀R.A)T∀

S, F(t : ∀R.A)

Sp, T((t, p) : R), F(p : A)F∀∗

S, Fc(t : ∀R.A)

Sp, T((t, p) : R), Fc(p : A)Fc∀∗



Relevant rules of T

S, T(t : At B)

S, T(t : A) | S, T(t : B)Tt

S, F(t : At B)

S, F(t : A), F(t : B)Ft

S, Fc(t : At B)


S, T(t : (At B)→C)

S, T(t : A→C), T(t : B→C)T→t . . .

S, Fc(t : A→B)


S, T(t : ∀R.A), T((t, s) : R)

S, T((t, s) : R), T(s : A), T(t : ∀R.A)T∀

S, F(t : ∀R.A)

Sp, T((t, p) : R), F(p : A)F∀∗

S, Fc(t : ∀R.A)

Sp, T((t, p) : R), Fc(p : A)Fc∀∗



Relevant rules of T

S, T(t : At B)

S, T(t : A) | S, T(t : B)Tt

S, F(t : At B)

S, F(t : A), F(t : B)Ft

S, Fc(t : At B)


S, T(t : (At B)→C)

S, T(t : A→C), T(t : B→C)T→t . . .

S, Fc(t : A→B)


S, T(t : ∀R.A), T((t, s) : R)

S, T((t, s) : R), T(s : A), T(t : ∀R.A)T∀

S, F(t : ∀R.A)

Sp, T((t, p) : R), F(p : A)F∀∗

S, Fc(t : ∀R.A)

Sp, T((t, p) : R), Fc(p : A)Fc∀∗



Properties of T

PropertiesIt can be used to check classic problems on DLs:Concept validity, Subsumption, Instance checking

Disjunction propertyIf At B ∈ KALC∞, then either A ∈ KALC∞ or B ∈ KALC∞

Kur proofWe can exhibit a validity proof for generic instances of KurÔ Essential rule Fc∀: directly corresponds to condition on final states



Properties of T

PropertiesIt can be used to check classic problems on DLs:Concept validity, Subsumption, Instance checking

Disjunction propertyIf At B ∈ KALC∞, then either A ∈ KALC∞ or B ∈ KALC∞

Kur proofWe can exhibit a validity proof for generic instances of KurÔ Essential rule Fc∀: directly corresponds to condition on final states



KALC∞ decidability?

About Finite Model Property

if we omit condition on states, we obtain Int. Kripke semantics for ALC[de Paiva, 2005, Odintsov and Wansing, 2003]

this version has an infinite countermodel for Kur Ô no FMP!since Kur is still valid in every finite model

Still, we have no proof of FMP in KALC∞...

Ô Search for a different Kripke semantics allowing a decidabletableaux procedure...



KALC∞ decidability?

About Finite Model Property

if we omit condition on states, we obtain Int. Kripke semantics for ALC[de Paiva, 2005, Odintsov and Wansing, 2003]

this version has an infinite countermodel for Kur Ô no FMP!since Kur is still valid in every finite model

Still, we have no proof of FMP in KALC∞...

Ô Search for a different Kripke semantics allowing a decidabletableaux procedure...


KALC: decidable Kripke-style interpretation for ALC

Outline

1 Introduction






KALC: decidable Kripke-style interpretation for ALC Introduction

KALC: decidable Kripke-style interpretation for ALC

KALC [Bozzato et al., 2010, Bozzato, 2011]

Kripke-style semantics for ALCTableaux calculus TK and proof search algorithm

Features:Kripke semantics for dynamic and incomplete knowledgeConstructive reading of reasoning problemsReasoning problems solvable by proof search algorithm


KALC: decidable Kripke-style interpretation for ALC Kripke semantics

KALC Kripke semantics

KALC-model K = 〈P,≤, ρ, ι〉(P,≤) is a finite poset with root ρ;ι(α) = (Dα, ·α) such that, for every α, β ∈ P with α ≤ β:

1 Dα ⊆ Dβ;2 for every c ∈ NI, cα = cβ;3 for every A ∈ NC, Aα ⊆ Aβ;4 for every R ∈ NR, Rα ⊆ Rβ.

NotesForcing relation definition does not changeKur is valid also in every KALC-modelSyntax: limitations on formula definitionK ::= (s, t) : R | t : C | C v D(Also, we only consider acyclic TBoxes)


KALC: decidable Kripke-style interpretation for ALC Kripke semantics

KALC Kripke semantics

KALC-model K = 〈P,≤, ρ, ι〉(P,≤) is a finite poset with root ρ;ι(α) = (Dα, ·α) such that, for every α, β ∈ P with α ≤ β:

1 Dα ⊆ Dβ;2 for every c ∈ NI, cα = cβ;3 for every A ∈ NC, Aα ⊆ Aβ;4 for every R ∈ NR, Rα ⊆ Rβ.

NotesForcing relation definition does not changeKur is valid also in every KALC-modelSyntax: limitations on formula definitionK ::= (s, t) : R | t : C | C v D(Also, we only consider acyclic TBoxes)


KALC: decidable Kripke-style interpretation for ALC Auditing example

Auditing example

IdeaKripke semantics can represent partial and evolving knowledgeExample application domain: auditing

Auditing example [Mendler and Scheele, 2010]

Problem domain:There are some companies, some of which may be insolvent

An inference problem:Is a given company credit worthy? (Instance checking)



Auditing example: classical reasoning

Let us consider the following ABox A:

a:Company b:Company c:Company d:Company

(a,b):hasCustomer (a,c):hasCustomer

(b,c):hasCustomer (c,d):hasCustomer

b:Insolvent d:¬Insolvent

A generic model M of A has the form:

I : Insolvent

hasCustomer:

I ¬I

db c

........

a

........

........




A company x is CW (Credit Worthy) if:

x has an insolvent customer ysuch that y can rely on at least one non-insolvent (i.e., solvent)customer z.

I

x z

¬ICW

y

This clause corresponds to the formula

x : D → CW

D ≡ ∃hasCustomer.(Insolventu ∃hasCustomer.¬Insolvent)




A company x is CW (Credit Worthy) if:x has an insolvent customer y

such that y can rely on at least one non-insolvent (i.e., solvent)customer z.

I

x y z

¬ICW

I

x z

¬ICW

y


x : D → CW





A company x is CW (Credit Worthy) if:x has an insolvent customer ysuch that y can rely on at least one non-insolvent (i.e., solvent)customer z.

I

x z

¬ICW

y


x : D → CW





A company x is CW (Credit Worthy) if:x has an insolvent customer ysuch that y can rely on at least one non-insolvent (i.e., solvent)customer z.

I

x z

¬ICW

y


x : D → CW





Let us add to the ABox A the formula

a : D → CW

asserting the above property for a

Instance checking problem

Is the company a Credit Worthy?I ¬I

a

db c

CW ?

A |= a : CW ?




Let us add to the ABox A the formula

a : D → CW

asserting the above property for a

Instance checking problem

Is the company a Credit Worthy?I ¬I

a

db c

CW ?

A |= a : CW ?




We show that in any model M of A we can find y and z such that:

I

y

¬I

a z

CASE 1: c is insolvent CASE 2: c is non-insolvent

¬I

a

db c

II I ¬I

a

db c

¬I

A |= a : ∃hasCustomer.(Insolventu ∃hasCustomer.¬Insolvent)

A |= a : D

A |= a : D→CW (since it belongs to A)

We conclude that: A |= a : CW





I

y

¬I

a z

CASE 1: c is insolvent

CASE 2: c is non-insolvent

¬I

a

db c

II

I ¬I

a

db c

¬I


A |= a : D







I

y

¬I

a z


¬I

a

db c

II I ¬I

a

db c

¬I


A |= a : D







I

y

¬I

a z


¬I

a

db c

II I ¬I

a

db c

¬I


A |= a : D







I

y

¬I

a z


¬I

a

db c

II I ¬I

a

db c

¬I


A |= a : D






NoteWe know that a is CW, but we do not know whyÔ ALC does not support constructive reasoning

ALC models can not represent partial knowledge as:c is neither Insolvent nor ¬Insolvent

Ô We can represent partial and evolving knowledge,by Kripke semantics of KALC




NoteWe know that a is CW, but we do not know whyÔ ALC does not support constructive reasoning

ALC models can not represent partial knowledge as:c is neither Insolvent nor ¬Insolvent

Ô We can represent partial and evolving knowledge,by Kripke semantics of KALC



Auditing example: constructive reasoning

a:Company b:Company c:Company d:Company

(a,b):hasCustomer (a,c):hasCustomer

(b,c):hasCustomer (c,d):hasCustomer

b:Insolvent d:¬Insolvent a:D→CW

Let us consider the KALC-model K of A:

I

a

db c

I

a

db c

I

a

db c

ρ

φ1 φ2

I

<

<




I

a

db c

I

a

db c

I

a

db c

ρ

φ1 φ2

I

<

<

The final worlds φ1 and φ2 are two states of complete knowledge:

φ1‖−−c:¬Insolvent φ2‖−−c:Insolvent

Instead, in ρ we have partial knowledge about c:

ρ‖−/−c:Insolvent ρ‖−/−c:¬Insolvent




We have that:

ρ‖−/−c:Insolventt ¬Insolvent

and thus:

A6|=k c:Insolventt ¬Insolvent

In KALC we are not allowed to prove that a is CW

I

y

¬I

a z

From A, we can not find insolvent y and solvent z in the definition of CW.


KALC: decidable Kripke-style interpretation for ALC Tableau calculus

Tableau calculus TK

To decide KALC-consequence relation |=k we introducethe tableau calculus TKTableau calculus TK

Refutation calculus on signed formulasSigned formulas T, F, Ts with:

Ts(H): H will be true from next statesRealizability relation K, α � W: evaluation of signed formulas in α

K, α � Ts(H) iff α < β implies β‖−−H

TheoremLet F be a set of formulas and q ∈ NI not occurring in F .

F|=k c : C iff T(F ) ∪ { F(c : C) } is not realizable.F |= H iff T(F ) ∪ {Ts(c : ⊥), F(H) } is not realizable.



Tableau calculus TK: relevant rules

∆, T(c : CtD)

∆, T(c : C) | ∆, T(c : D)Tt

∆, F(c : CtD)

∆, F(c : C), F(c : D)Ft

∆, F(c : C → D)

∆, T(c : C), F(c : D) | ∆s, T(c : C), F(c : D)F→

∆, T(c : C → D)

∆, T(c : D) | ∆, F(c : C), Ts(c : D) | ∆s, F(c : C), Ts(c : D)T→

∆, T(c : A), T(A v C)

∆, T(c : A), T(A v C), T(c : C)Tv

∆, T(c : ∃R.C)

∆, T((c, q) : R), T(q : C)T∃

∆, T((c, d) : R), T(c : ∀R.C)

∆, T((c, d) : R), T(c : ∀R.C), T(d : C)T∀

∆s = {T(H) | T(H) ∈ ∆ } ∪ {T(H) | Ts(H) ∈ ∆ }



Tableau calculus TK: main theorem

Theorem (decidability)Let ∆ be a set of signed formulas:

Completeness: ∆ is realizable iff is not provable in TK.Termination: provability of ∆ in TK can be decided in finite time. 2

AlgorithmInput: ∆Output:

Closed proof table for ∆Or KALC-model K such that K, ρ � ∆

Idea:Build a countermodel for ∆ by graph expansion(build a graph for each world of the model)If all attempts fail, we get a closed tableau for ∆



Tableau calculus TK: main theorem

Theorem (decidability)Let ∆ be a set of signed formulas:

Completeness: ∆ is realizable iff is not provable in TK.Termination: provability of ∆ in TK can be decided in finite time. 2

AlgorithmInput: ∆Output:

Closed proof table for ∆Or KALC-model K such that K, ρ � ∆

Idea:Build a countermodel for ∆ by graph expansion(build a graph for each world of the model)If all attempts fail, we get a closed tableau for ∆



Tableau calculus TK: algorithm

REAL(∆){G∆ = 〈N∆, E∆, PF∆, SF∆, TB, DF∆〉;if( CONS(G∆) == “inconsistent” ) then return “not realizable”;else return “realizable”;

}

CONS(G){while ( exp-rules can be applied )and ( PF clash-free ) do begin

pick W ∈ (G);G = exp-rules(W,G);

end

if ( PF contains a clash ) then return “inconsistent”;while ( succ-rules can be applied ) do begin

pick W ∈ (G);G ′ = succ-rules(W,G);if ( CONS(G ′) == “inconsistent” ) then return “inconsistent”;

end

return “consistent”;}



Tableau calculus TK: complexity

PSPACE-hardnessStraightforward translation between KALC and Int[Statman, 1979]: Int decision problem is PSPACE-complete

Ô KALC realizability is PSPACE-hard

PSPACE-completeness?

Conjecture: for TBox = ∅, KALC is in PSPACE

Idea: trace techniqueCountermodel can be exponential w.r.t. starting formulas......but paths are independent and polynomial in size of input

Ô Explore countermodel one path at a timerequires space polynomial in size of input

Still no formal proof for completeness of this strategy!
































KALC: decidable Kripke-style interpretation for ALC Relations with KALC∞

KALC relations with KALC∞

KALC∞ models are KALC models with a possibly infinite posetRelations:

KALC∞ ⊆ KALC, since every KALC model is a KALC∞ modelKALC ⊆ KALC∞?every KALC∞ model has a corresponding KALC model?


Application directions

Outline

1 Introduction






Application directions Introduction


Starting points:BCDL: information terms semantics

Computational interpretation of proofsClassical reading of descriptionsNatural notion of state

KALC: tableau procedure and Kripke semantics

Comparison to tableau procedures for classical DLsBasis for implementationRepresentation of partial and dynamic knowledge

Some current directionsState description and action formalismsSemantic web service compositions and synthesis






















KALC: tableau procedure and Kripke semanticsComparison to tableau procedures for classical DLsBasis for implementationRepresentation of partial and dynamic knowledge







KALC: tableau procedure and Kripke semanticsComparison to tableau procedures for classical DLsBasis for implementationRepresentation of partial and dynamic knowledge



Application directions States and actions

App. directions: IT and states

IT and statesInformation terms encode a natural notion of stateUsed in [Ferrari et al., 2008] to represent system snapshots

Ô Action formalism for ALC [Bozzato et al., 2009b]

An action formalism based on IT semantics of BCDL

System description and statesTheory T: description of a system

TBox: system constraints (general properties)ABox: current state of the system

State: α ∈ IT(T)State consistency: if there is a model M s.t. M� 〈α〉T



App. directions: IT and states

IT and statesInformation terms encode a natural notion of stateUsed in [Ferrari et al., 2008] to represent system snapshots

Ô Action formalism for ALC [Bozzato et al., 2009b]

An action formalism based on IT semantics of BCDL

System description and statesTheory T: description of a system

TBox: system constraints (general properties)ABox: current state of the system

State: α ∈ IT(T)State consistency: if there is a model M s.t. M� 〈α〉T



App. directions: action language

Action: P ⇒ QInformal reading

If the preconditions P hold in a state α, the action can be applied

In the resulting state the postconditions Q must hold

Information content IC(〈α〉T):minimal set of atomic formulas encoding info. from 〈α〉T

Applicability: an action is active if P ⊆ IC(〈α〉T)

Action output Out(α): update IC(〈α〉T) with Q

GENITAlgorithm to build up a state (IT) for a system, given an action outputIt can be used to trace reasons for inconsistency


Application directions Web services composition

App. directions: web services composition

Service composition in BCDL [Bozzato and Ferrari, 2010]

Calculus for definition of Semantic Web Services compositionsRelated to program synthesis in constr. logics [Miglioli et al., 1986]

Services as combined functions "computing" information terms

Composition calculus SC

s(x) :: P ⇒ Q

Π1 : s1(x) :: P1 ⇒ Q1· · ·

Πn : sn(x) :: Pn ⇒ Qn

r

Applicability conditions (AC):constraints for correctness of rule application

Computational interpretation (CI):computational reading of logical rule

ResultIf a composition meets the ACs of its rules, then its computationalinterpretation is sound








s(x) :: P ⇒ Q

Π1 : s1(x) :: P1 ⇒ Q1· · ·


r











s(x) :: P ⇒ Q

Π1 : s1(x) :: P1 ⇒ Q1· · ·


r





Thank you for listening

Directions and Applications for ConstructiveSemantics in Description Logics

Loris [email protected]

https://dkm.fbk.eu/index.php/Loris_Bozzato

DKM - Data and Knowledge Management Research Unit,FBK-Irst, Fondazione Bruno Kessler - Trento, Italy


https://dkm.fbk.eu/index.php/Loris_Bozzato

References I

Bozzato, L. (2011).Kripke semantics and tableau procedures for constructive description logics.PhD thesis, DICOM – Università degli Studi dell’Insubria.

Bozzato, L. and Ferrari, M. (2010).Composition of Semantic Web Services in a Constructive Description Logic.In Proceedings of the 4th International Conference on Web Reasoning and Rule Systems(RR2010), volume 6333 of Lecture Notes in Computer Science, pages 223–226. Springer.

Bozzato, L., Ferrari, M., Fiorentini, C., and Fiorino, G. (2007).A constructive semantics for ALC.volume 250 of CEUR Workshop Proceedings, pages 219–226. CEUR-WS.org.

Bozzato, L., Ferrari, M., Fiorentini, C., and Fiorino, G. (2010).A decidable constructive description logic.In Proceedings of the 12th European Conference on Logics in Artificial Intelligence (JELIA2010), Lecture Notes in Computer Science. Springer.To appear.

Bozzato, L., Ferrari, M., and Villa, P. (2009a).A note on constructive semantics for description logics.Accepted at CILC09 - 24-esimo Convegno Italiano di Logica Computazionale.


References II

Bozzato, L., Ferrari, M., and Villa, P. (2009b).Actions Over a Constructive Semantics for Description Logics.Fundamenta Informaticae, 96(3):253–269.

de Paiva, V. (2005).Constructive description logics: what, why and how.Technical report, Xerox Parc.

Ferrari, M., Fiorentini, C., and Fiorino, G. (2010).BCDL: Basic Constructive Description Logic.Journal of Automated Reasoning, 44(4):371–399.

Ferrari, M., Fiorentini, C., Momigliano, A., and Ornaghi, M. (2008).Snapshot generation in a constructive object-oriented modeling language.In King, A., editor, Logic Based Program Synthesis and Transformation, LOPSTR 2007,Selected Papers, volume 4915 of LNCS, pages 169–184. Springer-Verlag.

Haarslev, V., Toman, D., and Weddell, G., editors (2010).DL2010 - International Workshop on Description Logics, volume 573 of CEUR WorkshopProceedings. CEUR-WS.org.


References III

Kamide, N. (2010a).A compatible approach to temporal description logics.In [Haarslev et al., 2010].

Kamide, N. (2010b).Paraconsistent description logics revisited.In [Haarslev et al., 2010], pages 197–208.

Kaneiwa, K. (2005).Negations in description logic – contraries, contradictories, and subcontraries.In Dau, F., Mugnier, M.-L., and Stumme, G., editors, Common Semantics for SharingKnowledge: Contributions to the 13th International Conference on Conceptual Structures(ICCS ’05), pages 66–79. Kassel University Press.

Mendler, M. and Scheele, S. (2009).Towards a type system for semantic streams.In SR2009 - Stream Reasoning Workshop (ESWC 2009), volume 466 of CEUR WorkshopProceedings. CEUR-WS.org.

Mendler, M. and Scheele, S. (2010).Towards Constructive DL for Abstraction and Refinement.Journal of Automated Reasoning, 44(3):207–243.


References IV

Miglioli, P., Moscato, U., and Ornaghi, M. (1986).PAP: A Logic Programming System Based on a Constructive Logic.In Foundations of Logic and Functional Programming, volume 306 of Lecture Notes inComputer Science, pages 143–156. Springer.

Miglioli, P., Moscato, U., Ornaghi, M., and Usberti, G. (1989).A constructivism based on classical truth.Notre Dame Journal of Formal Logic, 30(1):67–90.

Odintsov, S. and Wansing, H. (2003).Inconsistency-tolerant description logic. Motivation and basic systems.In Hendricks, V. and Malinowski, J., editors, Trends in Logic. 50 Years of Studia Logica,pages 301–335. Kluwer Academic Publishers, Dordrecht.

Odintsov, S. and Wansing, H. (2008).Inconsistency-tolerant description logic. Part II: A tableau algorithm for CALCC.J. Applied Logic, 6(3):343–360.

Statman, R. (1979).Intuitionistic propositional logic is polynomial-space complete.Theoretical Computer Science, 9:67–72.


References V

Villa, P. (2010).Semantics foundations for constructive description logics.PhD thesis, DICOM – Università degli Studi dell’Insubria.


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