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Constructive Semantics for Description LogicsASP Based Generation of Information Terms for
Constructive EL
Loris Bozzato
DKM - Data and Knowledge Management Research Unit,FBK - Fondazione Bruno Kessler, Trento, Italy
UniVR Logic Seminar
February 2, 2017 – Verona, Italy
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 1 / 54
Description logics
Description logics (DL)A family of logic based Knowledge Representation formalisms
Main features:Expressive but decidable fragments of FOLFormally defined semantics⇒ ReasoningEfficient implementations for key problemsRelevant applications (Semantic Web)
Elements:Concepts: classes of objects HumanRoles: binary relations between objects hasChild
Complex descriptions:Concept constructors (u, t, ¬) Squaret ¬RoundRole restrictions (∃, ∀, >) > 2hasChild
∀hasBrother.( ∃isChildOf.Father )
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 2 / 54
Description logics
Description logics (DL)A family of logic based Knowledge Representation formalisms
Main features:Expressive but decidable fragments of FOLFormally defined semantics⇒ ReasoningEfficient implementations for key problemsRelevant applications (Semantic Web)
Elements:Concepts: classes of objects HumanRoles: binary relations between objects hasChild
Complex descriptions:Concept constructors (u, t, ¬) Squaret ¬RoundRole restrictions (∃, ∀, >) > 2hasChild
∀hasBrother.( ∃isChildOf.Father )
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 2 / 54
Description logics
Description logics (DL)A family of logic based Knowledge Representation formalisms
Main features:Expressive but decidable fragments of FOLFormally defined semantics⇒ ReasoningEfficient implementations for key problemsRelevant applications (Semantic Web)
Elements:Concepts: classes of objects HumanRoles: binary relations between objects hasChild
Complex descriptions:Concept constructors (u, t, ¬) Squaret ¬RoundRole restrictions (∃, ∀, >) > 2hasChild
∀hasBrother.( ∃isChildOf.Father )
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 2 / 54
Constructive logics
Constructive logicsFormalizations of ideas from constructivism
Brouwer-Heyting-Kolmogorov (BHK) or proof interpretationSemiformal 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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 3 / 54
Constructive logics
Constructive logicsFormalizations of ideas from constructivism
Brouwer-Heyting-Kolmogorov (BHK) or proof interpretationSemiformal 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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 3 / 54
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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 4 / 54
Constructive description logics
Constructive description logicsConstructive interpretations of description logics
MotivationsComputational interpretation of proofs and formulasUseful in domains with dynamic and incomplete knowledge
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
BCDL [Ferrari et al., 2010]: Information terms semantics + natural deduction
KALC [Bozzato, 2011]: Kripke-style semantics + tableaux algorithm
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 5 / 54
Constructive DLs and applications
Constructive DLs mostly studied from formal point of view...Limited proposals for application in KR and Semantic Web languages
Applications (examples)
[Mendler and Scheele, 2009]: reasoning over incomplete data streams[Haeusler et al., 2011]: conflict management on legal ontologies[Hilia et al., 2012]: semantic services compositions (on BCDL)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 6 / 54
Idea: ...let’s try to bridge the gap!
ProposalTheory: study relations between IT and ASPPractice: prototype over “off the shelf” tools (OWL API, dlv)
ContributionsELc: IT semantics for description logic ELASP and IT semantics: formal relation and datalog rewritingAsp-it prototype: ASP based IT generator for OWL-EL ontologies
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 7 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 7 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 7 / 54
Proposals
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
BCDL [Ferrari et al., 2010]: Information terms semantics + ND calculusKALC [Bozzato, 2011]: Kripke-style semantics + tableaux algorithm
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 8 / 54
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)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 9 / 54
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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 10 / 54
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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 11 / 54
Proposal [Mendler and Scheele, 2010]
IdeaRepresentation 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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 12 / 54
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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 13 / 54
BCDL: information terms semantics for ALC
BCDL: Basic Constructive Description Logic [Ferrari et al., 2010]
Information terms semantics for ALCNatural deduction calculus NDc
Information terms (IT) [Miglioli et al., 1989]
Syntactic objects justifying validity of formulas in classical modelsrealization of BHK interpretationrelated to realizability interpretations
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 14 / 54
BCDL: information terms semantics for ALC
Information terms (IT) justificationE.g.: Truth of ∃R.C(a) inM justified by IT (b, α) s.t.
M |= R(a, b) andα justifies truth of C(b)
Features:Classical reading of DL formulasSimple proof theoretical characterization by NDc
Computational interpretation of proofsNatural notion of state
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 15 / 54
ALCG syntax and classical semantics
Syntax is the same as ALC, adding a set of generators NG
ConceptsC ::= A | G | ¬C | Cu C | Ct C | ∃R.C | ∀R.C
FormulasK ::= ⊥ | R(t, s) | C(t) | ∀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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 16 / 54
ALCG syntax and classical semantics
Validity of formulas: given a modelM = (∆M, ·M)
M 6|= ⊥
M |= R(t, s) iff (tM, sM) ∈ RM
M |= H(t) iff tM ∈ HM
M |= ∀GH iff GM = {cM1 , . . . , cMn } ⊆ HM
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 17 / 54
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 (C1 t C2(c)) = {(k, α) | k ∈ {1, 2} and α∈ IT(Ck(c))}
RealizabilityMB 〈α〉KTruth of K in a modelM justified w.r.t. α
MB 〈tt〉K iffM |= KMB 〈(k, α)〉C1 t C2(c) iffMB 〈α〉 Ck(c)
Theorem (classical and IT semantics)M |= K iff there exists α ∈ IT(K) such thatMB 〈α〉K
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 18 / 54
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 (C1 t C2(c)) = {(k, α) | k ∈ {1, 2} and α∈ IT(Ck(c))}
RealizabilityMB 〈α〉KTruth of K in a modelM justified w.r.t. α
MB 〈tt〉K iffM |= KMB 〈(k, α)〉C1 t C2(c) iffMB 〈α〉 Ck(c)
Theorem (classical and IT semantics)M |= K iff there exists α ∈ IT(K) such thatMB 〈α〉K
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 18 / 54
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 (C1 t C2(c)) = {(k, α) | k ∈ {1, 2} and α∈ IT(Ck(c))}
RealizabilityMB 〈α〉KTruth of K in a modelM justified w.r.t. α
MB 〈tt〉K iffM |= KMB 〈(k, α)〉C1 t C2(c) iffMB 〈α〉 Ck(c)
Theorem (classical and IT semantics)M |= K iff there exists α ∈ IT(K) such thatMB 〈α〉K
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 18 / 54
BCDL information terms semantics
ITN (K) = {tt} for K atomic or negated MB 〈tt〉K iffM |= K
ITN (C1 u C2(c))= IT(C1(c))× IT(C2(c)) MB 〈(α, β)〉C1 u C2(c) iffMB 〈α〉C1(c)andMB 〈β〉C2(c)
ITN (C1 t C2(c))= IT(C1(c)) ] IT(C2(c)) MB 〈(k, α)〉C1 t C2(c) iffMB 〈α〉Ck(c)
ITN (∃R.C(c))= N ×⋃d∈N IT(C(d)) MB 〈(d, α)〉 ∃R.C(c) iffM |= R(c, d)
andMB 〈α〉C(d)
ITN (∀R.C(c))= (⋃
d∈N IT(C(d)))N MB 〈φ〉 ∀R.C(c) iffM |= ∀R.C(c) and,for every d ∈ N , if M |= R(c, d) thenMB 〈φ(d)〉C(d)
ITN (∀GC)= (⋃
d∈dom(G) IT(C(d)))dom(G) MB 〈φ〉 ∀GC iff, for every d ∈ dom(G),MB 〈φ(d)〉C(d)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 19 / 54
Natural deduction calculus ND
Calculus ND : natural deduction calculus for ALCG
Rules
Γ··· π′
Ak(t)tIk
A1 tA2(t)
Γ1··· π1
∃R.A(t)
Γ2, [R(t, p), A(p)]··· π2
K∃E
K
∀GA
Γ′··· π′
G(t)∀GE
A(t)
Γ, [¬H(t)]··· π′
⊥¬E
H(t)
TheoremND is sound and complete w.r.t. ALCG
leaving aside generators rules, ND is sound and complete w.r.t. ALC
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 20 / 54
Natural deduction calculus NDc
Calculus NDc: natural deduction calculus for BCDL
Rules
Every rule fromND
(minus ¬E)
Γ··· π′
¬¬C(t)At
C(t)
Γ··· π′
∀R.¬¬H(t)KUR
¬¬∀R.H(t)
NoteKUR corresponds to the Kuroda axiom schema Kur
Kur ≡ ∀x.¬¬A(x) → ¬¬∀x.A(x)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 21 / 54
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 (C1 t C2(t))
defined as:
ΦπN (γ) = ( k, Φπ′
N (γ) )
tIk ruleΓ··· π′
Ak(t)tIk
A1 tA2(t)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 22 / 54
Soundness of NDc: computational interpretation
Theorem (Soundness)If π : Γ ` K, then:
Γ |= K.IfMB 〈γ〉 Γ thenMB 〈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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 23 / 54
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 At B(c), then Γ | BCDL A(c) or Γ | BCDL B(c).
Explicit definability property (EDP):If Γ | BCDL ∃R.A(c), then there exists d ∈ NI such thatΓ | BCDL R(c, d) and Γ | BCDL A(d).
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 24 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 24 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 24 / 54
ELc
ELc: information terms semantics for EL [Baader, 2003]
Simple restriction of BCDL to EL
Why EL?The most simple DL s.t. semantics withconstructive properties (ED) can be definedWell-known and used DL language, base of OWL EL profile
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 25 / 54
DL language L
Syntax is the same as EL, adding a set of generators NG
ConceptsC ::= A | G | Cu C | ∃R.C
FormulasK ::= R(s, t) | C(t) | ∀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
> operator: special generator >N s.t. DOM(>N ) = N
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 26 / 54
DL language L
Syntax is the same as EL, adding a set of generators NG
ConceptsC ::= A | G | Cu C | ∃R.C
FormulasK ::= R(s, t) | C(t) | ∀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
> operator: special generator >N s.t. DOM(>N ) = N
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 26 / 54
EL: classical semantics
Classical model: M = (∆M, ·M)
Individuals: cM ∈ ∆M
Atomic concepts: AM ⊆ ∆M
Roles: RM ⊆ ∆M × ∆M
Generators: GM = {cM1 , . . . , cMn }
Non-atomic concepts:
(CuD)M = CM ∩DM
(∃R.C)M = { c ∈ ∆M | there is d s.t. (c, d) ∈ RM and d ∈ CM }
Validity of formulas:
M |= R(s, t) iff (sM, tM) ∈ RM
M |= C(t) iff tM ∈ CM
M |= ∀GC iff GM ⊆ CM
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 27 / 54
Example: Food and Wines
Scenario: food and wines knowledge base [Brachman et al., 1991]
Task: pair each food with the correct wine
Conditions:Pairings:
Meat with red winesFish with white wines
For every food, a correct wine colorFor every wine color, at least one wine
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 28 / 54
Example: Food and Wines KB
Knowledge base KW:
TBox T :
(Ax1) : ∀Food∃goesWith.Color ≡ Food v ∃goesWith.Color
(Ax2) : ∀Color∃isColorOf.Wine ≡ Color v ∃isColorOf.Wine
DOM(Food) = {fish,meat} DOM(Color) = {red,white}
ABox A:
Wine(barolo)
Wine(chardonnay)
isColorOf(red,barolo)
isColorOf(white,chardonnay)
goesWith(fish,white)
goesWith(meat,red)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 29 / 54
Example: Food and Wines KB
Knowledge base KW:
TBox T :
(Ax1) : ∀Food∃goesWith.Color ≡ Food v ∃goesWith.Color
(Ax2) : ∀Color∃isColorOf.Wine ≡ Color v ∃isColorOf.Wine
DOM(Food) = {fish,meat} DOM(Color) = {red,white}
ABox A:
Wine(barolo)
Wine(chardonnay)
isColorOf(red,barolo)
isColorOf(white,chardonnay)
goesWith(fish,white)
goesWith(meat,red)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 29 / 54
ELc: information terms semantics
Given N ⊆ NI finite and a closed formula K ∈ LN
Information terms ITN (K)Structured objects that constructively justify the truth of a formula K
RealizabilityMB 〈α〉KTruth of K in a modelM justified w.r.t. α
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 30 / 54
ELc: information terms semantics
Given N ⊆ NI finite and a closed formula K ∈ LN
Information terms ITN (K)Structured objects that constructively justify the truth of a formula K
RealizabilityMB 〈α〉KTruth of K in a modelM justified w.r.t. α
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 30 / 54
ELc: information terms semantics
Given N ⊆ NI finite and a closed formula K ∈ LN
Information terms ITN (K)ITN (K) = {tt}, if K is atomic
ITN (C1 u C2(c)) = {(α, β) | α∈ IT(C1(c)) and β∈ IT(C2(c))}ITN (∃R.C(c)) = {(d, α) | d ∈ N and α∈ IT(C(d))}ITN (∀GC) = { φ : DOM(G)→ ⋃
d∈dom(G) ITN (C(d)) | φ(d) ∈ ITN (C(d)) }
RealizabilityMB 〈α〉KMB 〈tt〉K iffM |= K
MB 〈(α, β)〉C1 u C2(c) iffMB 〈α〉C1(c) andMB 〈β〉C2(c)
MB 〈(d, α)〉 ∃R.C(c) iffM |= R(c, d) andMB 〈α〉C(d)
MB 〈φ〉 ∀GC iff, for every d ∈ DOM(G),MB 〈φ(d)〉C(d)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 31 / 54
ELc: information terms semantics
Given N ⊆ NI finite and a closed formula K ∈ LN
Information terms ITN (K)ITN (K) = {tt}, if K is atomic
ITN (C1 u C2(c)) = {(α, β) | α∈ IT(C1(c)) and β∈ IT(C2(c))}
ITN (∃R.C(c)) = {(d, α) | d ∈ N and α∈ IT(C(d))}ITN (∀GC) = { φ : DOM(G)→ ⋃
d∈dom(G) ITN (C(d)) | φ(d) ∈ ITN (C(d)) }
RealizabilityMB 〈α〉KMB 〈tt〉K iffM |= K
MB 〈(α, β)〉C1 u C2(c) iffMB 〈α〉C1(c) andMB 〈β〉C2(c)
MB 〈(d, α)〉 ∃R.C(c) iffM |= R(c, d) andMB 〈α〉C(d)
MB 〈φ〉 ∀GC iff, for every d ∈ DOM(G),MB 〈φ(d)〉C(d)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 31 / 54
ELc: information terms semantics
Given N ⊆ NI finite and a closed formula K ∈ LN
Information terms ITN (K)ITN (K) = {tt}, if K is atomic
ITN (C1 u C2(c)) = {(α, β) | α∈ IT(C1(c)) and β∈ IT(C2(c))}ITN (∃R.C(c)) = {(d, α) | d ∈ N and α∈ IT(C(d))}
ITN (∀GC) = { φ : DOM(G)→ ⋃d∈dom(G) ITN (C(d)) | φ(d) ∈ ITN (C(d)) }
RealizabilityMB 〈α〉KMB 〈tt〉K iffM |= K
MB 〈(α, β)〉C1 u C2(c) iffMB 〈α〉C1(c) andMB 〈β〉C2(c)
MB 〈(d, α)〉 ∃R.C(c) iffM |= R(c, d) andMB 〈α〉C(d)
MB 〈φ〉 ∀GC iff, for every d ∈ DOM(G),MB 〈φ(d)〉C(d)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 31 / 54
ELc: information terms semantics
Given N ⊆ NI finite and a closed formula K ∈ LN
Information terms ITN (K)ITN (K) = {tt}, if K is atomic
ITN (C1 u C2(c)) = {(α, β) | α∈ IT(C1(c)) and β∈ IT(C2(c))}ITN (∃R.C(c)) = {(d, α) | d ∈ N and α∈ IT(C(d))}ITN (∀GC) = { φ : DOM(G)→ ⋃
d∈dom(G) ITN (C(d)) | φ(d) ∈ ITN (C(d)) }
RealizabilityMB 〈α〉KMB 〈tt〉K iffM |= K
MB 〈(α, β)〉C1 u C2(c) iffMB 〈α〉C1(c) andMB 〈β〉C2(c)
MB 〈(d, α)〉 ∃R.C(c) iffM |= R(c, d) andMB 〈α〉C(d)
MB 〈φ〉 ∀GC iff, for every d ∈ DOM(G),MB 〈φ(d)〉C(d)
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 31 / 54
Example: IT for Food and Wines
(Ax1) :
∀Food∃goesWith.Color ≡ Food v ∃goesWith.Color
φ1 ∈ ITN (Ax1) :
[ fish 7→ (white,tt), meat 7→ (red,tt) ]
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Example: IT for Food and Wines
(Ax2) :
∀Color∃isColorOf.Wine ≡ Color v ∃isColorOf.Wine
φ2 ∈ ITN (Ax2) :
[ red 7→ (barolo,tt), white 7→ (chardonnay,tt) ]
LetM be a model of KW: then,MB 〈(φ1, φ2)〉 T
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Constructive consequence
Theorem (classical and IT semantics)If α ∈ IT(K),MB 〈α〉K impliesM |= K
Constructive consequence Γ|=c K:Γ|=c K iffMB 〈γ〉 Γ impliesMB 〈η〉K
Computational interpretation
Γ|=c K implicitly defines a semantic map ΦN such that:
ifMB 〈γ〉 Γ thenMB 〈ΦN (γ)〉K
In BCDL we can extract ΦN from natural deduction proofs[Bozzato et al., 2007, Ferrari et al., 2010]
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Constructive consequence
Theorem (classical and IT semantics)If α ∈ IT(K),MB 〈α〉K impliesM |= K
Constructive consequence Γ|=c K:Γ|=c K iffMB 〈γ〉 Γ impliesMB 〈η〉K
Computational interpretation
Γ|=c K implicitly defines a semantic map ΦN such that:
ifMB 〈γ〉 Γ thenMB 〈ΦN (γ)〉K
In BCDL we can extract ΦN from natural deduction proofs[Bozzato et al., 2007, Ferrari et al., 2010]
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 34 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
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Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
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Answer sets for formulas and information terms
TaskCompute information terms of input KB Γ in EL
IdeaUse relations across IT and Answer Sets semantics[Fiorentini and Ornaghi, 2007] on propositional nested expressions
Ô We extend these results to ELc formulas
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Answer set
lp-interpretationSet of closed atomic formulas in LN
Given a closed K ∈ LN :
I |= K, iff K ∈ I and K is atomicI |= CuD(c) iff I |= C(c) and I |= D(c)I |= ∃R.C(c) iff R(c, d) ∈ I for d ∈ N and I |= C(d)
I |= ∀GC iff for every e ∈ DOM(G), I |= C(e)I |= Γ iff I |= K for K ∈ Γ
Answer setI answer set for set of closed formulas Γ ⊆ LN iff:
I |= Γfor every I′ ⊆ I, I′ |= Γ implies I′ = I (minimality)
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Answer set
lp-interpretationSet of closed atomic formulas in LN
Given a closed K ∈ LN :
I |= K, iff K ∈ I and K is atomicI |= CuD(c) iff I |= C(c) and I |= D(c)I |= ∃R.C(c) iff R(c, d) ∈ I for d ∈ N and I |= C(d)
I |= ∀GC iff for every e ∈ DOM(G), I |= C(e)I |= Γ iff I |= K for K ∈ Γ
Answer setI answer set for set of closed formulas Γ ⊆ LN iff:
I |= Γfor every I′ ⊆ I, I′ |= Γ implies I′ = I (minimality)
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Answers of pieces of information
Piece of information (POI)〈η〉K with closed K ∈ LN and η ∈ ITN (K)(Idea: “state” of K defined by η)
Answers ans(〈η〉K)Set of closed atomic formulas: (Idea: “information content” of 〈η〉K)
ans(〈tt〉K) = {K}, with K atomic
ans(〈(α, β)〉A1 uA2(c)) = ans(〈α〉A1(c)) ∪ ans(〈β〉A2(c))
ans(〈(d, α)〉∃R.A(c)) = {R(c, d)} ∪ ans(〈α〉A(d))
ans(〈φ〉∀GA) =⋃
d∈DOM(G) ans(〈φ(d)〉A(d))
Minimal POI〈η〉K is minimal iff @µ with ans(〈µ〉K) ⊂ ans(〈η〉K)
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Answers of pieces of information
Piece of information (POI)〈η〉K with closed K ∈ LN and η ∈ ITN (K)(Idea: “state” of K defined by η)
Answers ans(〈η〉K)Set of closed atomic formulas: (Idea: “information content” of 〈η〉K)
ans(〈tt〉K) = {K}, with K atomic
ans(〈(α, β)〉A1 uA2(c)) = ans(〈α〉A1(c)) ∪ ans(〈β〉A2(c))
ans(〈(d, α)〉∃R.A(c)) = {R(c, d)} ∪ ans(〈α〉A(d))
ans(〈φ〉∀GA) =⋃
d∈DOM(G) ans(〈φ(d)〉A(d))
Minimal POI〈η〉K is minimal iff @µ with ans(〈µ〉K) ⊂ ans(〈η〉K)
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Answers of pieces of information
Piece of information (POI)〈η〉K with closed K ∈ LN and η ∈ ITN (K)(Idea: “state” of K defined by η)
Answers ans(〈η〉K)Set of closed atomic formulas: (Idea: “information content” of 〈η〉K)
ans(〈tt〉K) = {K}, with K atomic
ans(〈(α, β)〉A1 uA2(c)) = ans(〈α〉A1(c)) ∪ ans(〈β〉A2(c))
ans(〈(d, α)〉∃R.A(c)) = {R(c, d)} ∪ ans(〈α〉A(d))
ans(〈φ〉∀GA) =⋃
d∈DOM(G) ans(〈φ(d)〉A(d))
Minimal POI〈η〉K is minimal iff @µ with ans(〈µ〉K) ⊂ ans(〈η〉K)
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Example: answers of POIs
(Ax1) : ∀Food∃goesWith.Color ≡ Food v ∃goesWith.Color
φ1 ∈ ITN (Ax1) : [ fish 7→ (white,tt), meat 7→ (red,tt) ]
ans(〈φ1〉Ax1) = {Color(white), goesWith(fish,white),Color(red), goesWith(meat,red)}
(Ax2) : ∀Color∃isColorOf.Wine ≡ Color v ∃isColorOf.Wine
φ2 ∈ ITN (Ax2) : [ red 7→ (barolo,tt), white 7→ (chardonnay,tt) ]
ans(〈φ2〉Ax2) = {Wine(barolo), isColorOf(red,barolo),Wine(chardonnay),isColorOf(white,chardonnay)}
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Example: answers of POIs
(Ax1) : ∀Food∃goesWith.Color ≡ Food v ∃goesWith.Color
φ1 ∈ ITN (Ax1) : [ fish 7→ (white,tt), meat 7→ (red,tt) ]
ans(〈φ1〉Ax1) = {Color(white), goesWith(fish,white),Color(red), goesWith(meat,red)}
(Ax2) : ∀Color∃isColorOf.Wine ≡ Color v ∃isColorOf.Wine
φ2 ∈ ITN (Ax2) : [ red 7→ (barolo,tt), white 7→ (chardonnay,tt) ]
ans(〈φ2〉Ax2) = {Wine(barolo), isColorOf(red,barolo),Wine(chardonnay),isColorOf(white,chardonnay)}
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Result: answer sets and minimal POI
TheoremFor every modelM,MB 〈η〉K iffM |= ans(〈η〉K)
TheoremI answer set for K iff∃ a minimal 〈η〉K such that I = ans(〈η〉K)
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Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 39 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 39 / 54
ASP based generation of information terms
Solution to generate (minimal) IT:1 Compute answer sets of input KB Γ2 For each formula K ∈ Γ, use recursive definition of ans(〈η〉K) to
reconstruct IT η
Idea: computation of answer setsTranslate input KB to datalog program
Ô Translation includes rules recursively constructing ITs(generate-and-test approach)
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Model generating rewriting
Model generating rewriting (P1)Generates interpretations for input EL formulas:
A(b) 7→ { is(b, A)← is(b, lA). }R(a, b) 7→ { rel(a, R, b). }
CuD(a) 7→ { is(a, lC)← is(a, lCuD).is(a, lD)← is(a, lCuD). } ∪ P1(C(a)) ∪ P1(D(a))
∃R.C(a) 7→ { is(x, lC)← rel(a, R, x), is(a, l∃R.C). } ∪ P1(C(x))
∀GC 7→ { is(x, lC)← is(x, G). } ∪ P1(C(x))
NotesFixed set R of roles assertions from input KBLabelling for complex concepts lCAtomic assertions and generator domains added as facts
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 41 / 54
IT generating rewriting
IT generating rewriting (P2)Retrieves IT as complex terms, using definition of ans(〈η〉K):
A(b) 7→ { is_it(tt, b, lA)← is(b, A). }R(a, b) 7→ { rel_it(tt, a, R, b)← rel(a, R, b). }
CuD(a) 7→ { is_it([x, y], a, lCuD)←is_it(x, a, lC), is_it(y, a, lD). } ∪ P2(C(a)) ∪ P2(D(a))
∃R.C(a) 7→ { is_it([x, y], a, l∃R.C)←rel_it(tt, a, R, x), is_it(y, x, lC). } ∪ P2(C(x))
∀GC 7→ { isa_it([x, y], G, lC)← is(x, G), is_it(y, x, lC). } ∪ P2(C(x))
Complete rewriting (P)P(Γ) = P1(Γ) ∪ P2(Γ)
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Correctness
Let IT(K, I) be the set of IT “returned” by P2 for formula K andinterpretation I for P(Γ)
TheoremLet I be the (unique) answer set for P(Γ).If η ∈ IT(K, I), then ∃ lp-interpretation I′ for Γ s.t. ans(〈η〉K) ⊆ I′
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Example
Suppose we add:
Wine(teroldego) isColorOf(red,teroldego)
Applying P2(Ax2) to the model computed by P1(KW):
[red, [barolo,tt]], [red, [teroldego,tt]], [white, [chardonnay,tt]]
(Ax3) : ∀Food∃goesWith.(Coloru ∃isColorOf.Wine)
Applying P2(Ax3):
[fish, [white, [tt, [chardonnay,tt]]]],
[meat, [red, [tt, [barolo,tt]]]],
[meat, [red, [tt, [teroldego,tt]]]]
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Example
Suppose we add:
Wine(teroldego) isColorOf(red,teroldego)
Applying P2(Ax2) to the model computed by P1(KW):
[red, [barolo,tt]], [red, [teroldego,tt]], [white, [chardonnay,tt]]
(Ax3) : ∀Food∃goesWith.(Coloru ∃isColorOf.Wine)
Applying P2(Ax3):
[fish, [white, [tt, [chardonnay,tt]]]],
[meat, [red, [tt, [barolo,tt]]]],
[meat, [red, [tt, [teroldego,tt]]]]
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Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 44 / 54
Overview
1 Constructive semantics for DLs
2 ELc: constructive semantics for EL
3 Answer sets and IT semantics
4 ASP based generation of IT
5 Asp-it prototype
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 44 / 54
Asp-it prototype
Asp-itJava-based command line application:
Input: OWL-EL ontology ΓOutput: ontology Γ′ annotated with IT (elc:hasIT)
ToolsOWL API: ontology I/O, axioms annotationDLV: models computation (via DLVWrapper)
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IT generation process
DLV system
Output
ontology
Γ'
Input
ontology
Γ
Rewrite
P(Γ)
Annotate
hasIT
Datalog
rewriting
P(Γ)
Retrieve
Info.Terms
Prototype and examples available at:https://github.com/dkmfbk/asp-it
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 46 / 54
Conclusions
Summary:ELc: IT semantics for description logic ELASP and IT semantics: formal relation and datalog rewritingAsp-it prototype: ASP based IT generator for OWL-EL ontologies
Future works:Integrate procedures for transformation of IT(Calculus and proofs-as-programs)
Applications: synthesis of SemanticServices [Bozzato and Ferrari, 2010a]
Extend to larger DLs: SROEL (full OWL EL), ALC (BCDL)
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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 modelM s.t. MB 〈α〉T
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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 modelM s.t. MB 〈α〉T
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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
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App. directions: web services composition
Service composition in BCDL [Bozzato and Ferrari, 2010b]
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
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App. directions: web services composition
Service composition in BCDL [Bozzato and Ferrari, 2010b]
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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 50 / 54
App. directions: web services composition
Service composition in BCDL [Bozzato and Ferrari, 2010b]
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
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 50 / 54
Thank you for listening
Constructive Semantics for Description LogicsASP Based Generation of Information Terms for Constructive EL
Loris [email protected]
https://dkm.fbk.eu/
DKM - Data and Knowledge Management Research Unit,FBK - Fondazione Bruno Kessler, Trento, Italy
L. Bozzato (DKM - FBK) Constructive Semantics for DLs 02/02/2017 51 / 54
References I
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Bozzato, L. (2011).Kripke semantics and tableau procedures for constructive description logics.PhD thesis, DICOM – Università degli Studi dell’Insubria.
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Bozzato, L., Ferrari, M., Fiorentini, C., and Fiorino, G. (2007).A constructive semantics for ALC.In DL2007, volume 250 of CEUR-WP, pages 219–226. CEUR-WS.org.
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References II
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