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Nonmonotonic Extensions of Low Complexity DLs: Complexity Results - - PowerPoint PPT Presentation

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Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods Laura Giordano 1 , Valentina Gliozzi 2 , Nicola Olivetti 3 , and Gian Luca Pozzato 2 1 Dipartimento di Informatica - Universit` a del Piemonte Orientale A.

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Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods

Laura Giordano1, Valentina Gliozzi2, Nicola Olivetti3, and Gian Luca Pozzato2

1 Dipartimento di Informatica - Universit`

a del Piemonte Orientale “A. Avogadro”

2 Dipartimento di Informatica - Universit`

a edgli Studi di Torino

3 LSIS-UMR CNRS 6168 Universit`

e “Paul C´ ezanne” - Aix-Marseille 3

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 1

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Introduction

A non-monotonic extension of the low complexity (lightweight) Description Logics EL⊥ and DL-litec for reasoning about prototypical properties and inheritance with exceptions

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 2

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Introduction

A non-monotonic extension of the low complexity (lightweight) Description Logics EL⊥ and DL-litec for reasoning about prototypical properties and inheritance with exceptions Basic idea: to extend DLs with a typicality operator T

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 2

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Introduction

A non-monotonic extension of the low complexity (lightweight) Description Logics EL⊥ and DL-litec for reasoning about prototypical properties and inheritance with exceptions Basic idea: to extend DLs with a typicality operator T T(C) singles out the “most normal” instances of the concept C

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 2

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Introduction and recall to DLs

knowledge base KB ⇛ two components:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Introduction and recall to DLs

knowledge base KB ⇛ two components: TBox=inclusions relations among concepts

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Introduction and recall to DLs

knowledge base KB ⇛ two components: TBox=inclusions relations among concepts ABox= instances of concepts and roles ⇛ properties and relations of individuals

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Introduction and recall to DLs

knowledge base KB ⇛ two components: TBox=inclusions relations among concepts ABox= instances of concepts and roles ⇛ properties and relations of individuals TBox ⇛ taxonomy of concepts

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Introduction and recall to DLs

knowledge base KB ⇛ two components: TBox=inclusions relations among concepts ABox= instances of concepts and roles ⇛ properties and relations of individuals TBox ⇛ taxonomy of concepts need of representing prototypical properties and of reasoning about defeasible inheritance

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Introduction and recall to DLs

knowledge base KB ⇛ two components: TBox=inclusions relations among concepts ABox= instances of concepts and roles ⇛ properties and relations of individuals TBox ⇛ taxonomy of concepts need of representing prototypical properties and of reasoning about defeasible inheritance to handle defeasible inheritance needs the integration of some kind of nonmonotonic reasoning mechanism [BH95, BLW06, DLN+98, DNR02, ELST, Str93]

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Introduction and recall to DLs

knowledge base KB ⇛ two components: TBox=inclusions relations among concepts ABox= instances of concepts and roles ⇛ properties and relations of individuals TBox ⇛ taxonomy of concepts need of representing prototypical properties and of reasoning about defeasible inheritance to handle defeasible inheritance needs the integration of some kind of nonmonotonic reasoning mechanism [BH95, BLW06, DLN+98, DNR02, ELST, Str93] However, all these methods present some difficulties

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 3

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Logic of typicality

We propose a logic for defeasible reasoning in DLs

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 4

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Logic of typicality

We propose a logic for defeasible reasoning in DLs DL + a typicality operator T

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 4

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Logic of typicality

We propose a logic for defeasible reasoning in DLs DL + a typicality operator T meaning of T: (for any concept C) T(C) singles out the “typical” instances of C

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 4

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Logic of typicality

We propose a logic for defeasible reasoning in DLs DL + a typicality operator T meaning of T: (for any concept C) T(C) singles out the “typical” instances of C semantics of T defined by a set of postulates that are a restatement of Kraus-Lehmann-Magidor axioms of preferential logic P (Representation Theorem [GGOP09])

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 4

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Logic of typicality

A KB comprises, in addition to the standard TBox and ABox, a set of assertions of the type T(C) ⊑ D where D is a concept not mentioning T

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 5

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Logic of typicality

A KB comprises, in addition to the standard TBox and ABox, a set of assertions of the type T(C) ⊑ D where D is a concept not mentioning T “normally students do not pay taxes” ⇛ T(Student) ⊑ ¬TaxPayer

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 5

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Logic of typicality

A KB comprises, in addition to the standard TBox and ABox, a set of assertions of the type T(C) ⊑ D where D is a concept not mentioning T “normally students do not pay taxes” ⇛ T(Student) ⊑ ¬TaxPayer Example: normally a student does not pay taxes, normally a working student pays taxes, but normally a working student having children does not pay taxes (because he is discharged by the government)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 5

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 6

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer T is nonmonotonic = C ⊑ D does not imply T(C) ⊑ T(D)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 6

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer T is nonmonotonic = C ⊑ D does not imply T(C) ⊑ T(D) Which inferences?

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 6

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)
  • 2. Student(john), Worker(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)
  • 2. Student(john), Worker(john)
  • 3. Student(john), Worker(john), ∃HasChild.⊤(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)
  • 2. Student(john), Worker(john)
  • 3. Student(john), Worker(john), ∃HasChild.⊤(john)

expected conclusions:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)
  • 2. Student(john), Worker(john)
  • 3. Student(john), Worker(john), ∃HasChild.⊤(john)

expected conclusions:

  • 1. ¬TaxPayer(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)
  • 2. Student(john), Worker(john)
  • 3. Student(john), Worker(john), ∃HasChild.⊤(john)

expected conclusions:

  • 1. ¬TaxPayer(john)
  • 2. TaxPayer(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

T(Student) ⊑ ¬TaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ ∃HasChild.⊤) ⊑ ¬TaxPayer ABox:

  • 1. Student(john)
  • 2. Student(john), Worker(john)
  • 3. Student(john), Worker(john), ∃HasChild.⊤(john)

expected conclusions:

  • 1. ¬TaxPayer(john)
  • 2. TaxPayer(john)
  • 3. ¬TaxPayer(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 7

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Logic of typicality

We have defined a nonmonotonic inference based on a minimal model semantics For DL + T = ALC + T nonmonotonic inference has a high complexity, namely CO-NEXPNP , comparable however with that one of other NMR DL (circumscription) We are interested in applying our approach to low-complexity DLs EL⊥ and DL-Litecore.

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 8

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The logic EL+⊥T

Extension of our approach to Low Complexity DL EL⊥

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

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The logic EL+⊥T

Extension of our approach to Low Complexity DL EL⊥ Logic EL⊥ of the EL family

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

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The logic EL+⊥T

Extension of our approach to Low Complexity DL EL⊥ Logic EL⊥ of the EL family allows for conjunction (⊓) and existential restriction (∃R.C)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

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The logic EL+⊥T

Extension of our approach to Low Complexity DL EL⊥ Logic EL⊥ of the EL family allows for conjunction (⊓) and existential restriction (∃R.C) allows for ⊥

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

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The logic EL+⊥T

Extension of our approach to Low Complexity DL EL⊥ Logic EL⊥ of the EL family allows for conjunction (⊓) and existential restriction (∃R.C) allows for ⊥ relevant for several applications, in particular in the bio-medical domain (GALEN Medical Knowledge Base, Systemized Nomenclature of Medicine, Gene Ontology) formalized in small extensions of EL

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

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The logic EL+⊥T

Extension of our approach to Low Complexity DL EL⊥ Logic EL⊥ of the EL family allows for conjunction (⊓) and existential restriction (∃R.C) allows for ⊥ relevant for several applications, in particular in the bio-medical domain (GALEN Medical Knowledge Base, Systemized Nomenclature of Medicine, Gene Ontology) formalized in small extensions of EL reasoning in EL is polynomial-time decidable

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 9

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Language of EL⊥Tmin

Alphabet of concept names C role names R individuals O Given A ∈ C and r ∈ R, we define: C := A | ⊤ | ⊥ | C ⊓ C CR := C | CR ⊓ CR | ∃r.C CL := CR | T(C) TBox contains a finite set of concept inclusions CL ⊑ CR

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 10

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Example

The reformulation of the previous example in EL+⊥T gives the following KB: TaxPayer ⊓ NotTaxPayer ⊑ ⊥ Parent ⊑ ∃HasChild.⊤ ∃HasChild.⊤ ⊑ Parent T(Student) ⊑ NotTaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ Parent) ⊑ NotTaxPayer

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 11

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Language of DL-LitecTmin

Alphabet of concept names C role names R individuals O Given A ∈ C and r ∈ R, we define: CL := A | ∃R.⊤ | T(A) R := r | r− CR := A | ¬A | ∃R.⊤ | ¬∃R.⊤ TBox contains a finite set of concept inclusions CL ⊑ CR

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 12

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Monotonic Semantics

A model M is a structure ∆, <, I, where ∆ is the domain and for each extended concept C, CI ⊆ ∆, and for each role R RI ⊆ ∆ × ∆ < is an irreflexive and transitive relation over ∆ satisfying the Smoothness Condition (well-foundness) < is multilinear (or weakly connected): if u < z and v < z, then either u = v or u < v or v < u Semantics of the T operator: (T(C))I = Min<(CI) . For the other

  • perators CI is defined in the usual way ( in particular,

(r−)I = {(a, b) | (b, a) ∈ rI}) A model satisfying a Knowledge Base (TBox,ABox) is defined as usual

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 13

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Modal interpretation

We introduce a new modality

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

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Modal interpretation

We introduce a new modality we interpret the relation < as an accessibility relation

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

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Modal interpretation

We introduce a new modality we interpret the relation < as an accessibility relation by the Smoothness Condition (well-foundness), it turns out that has the properties of Gödel-Löb modal logic G

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

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Modal interpretation

We introduce a new modality we interpret the relation < as an accessibility relation by the Smoothness Condition (well-foundness), it turns out that has the properties of Gödel-Löb modal logic G (C)I = {x ∈ ∆ | for every y ∈ ∆, if y < x then y ∈ CI}

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

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Modal interpretation

We introduce a new modality we interpret the relation < as an accessibility relation by the Smoothness Condition (well-foundness), it turns out that has the properties of Gödel-Löb modal logic G (C)I = {x ∈ ∆ | for every y ∈ ∆, if y < x then y ∈ CI} Thus T(C)I = (C ⊓ ¬C)I

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 14

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Weakness of monotonic semantics

EL+⊥T allows one to reason about typicality

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 15

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Weakness of monotonic semantics

EL+⊥T allows one to reason about typicality e.g. we can consistently express that student, working student and working student with children have a different status as taxpayers

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 15

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Weakness of monotonic semantics

EL+⊥T allows one to reason about typicality e.g. we can consistently express that student, working student and working student with children have a different status as taxpayers but we cannot derive anything about the prototypical properties

  • f a given individual, unless the KB contains explicit tipicality

assumptions concerning this individual

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 15

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Weakness of monotonic semantics

TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T(Student) ⊑ NotTaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ Parent) ⊑ NotTaxPayer

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

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Weakness of monotonic semantics

TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T(Student) ⊑ NotTaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ Parent) ⊑ NotTaxPayer What can we conclude about john?

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

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Weakness of monotonic semantics

TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T(Student) ⊑ NotTaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ Parent) ⊑ NotTaxPayer What can we conclude about john? If T(Student ⊓ Worker ⊓ Parent)(john) ∈ ABox, then in EL+⊥T we can conclude NotTaxPayer(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

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Weakness of monotonic semantics

TaxPayer ⊓ NotTaxPayer ⊑ ⊥ T(Student) ⊑ NotTaxPayer T(Student ⊓ Worker) ⊑ TaxPayer T(Student ⊓ Worker ⊓ Parent) ⊑ NotTaxPayer What can we conclude about john? If T(Student ⊓ Worker ⊓ Parent)(john) ∈ ABox, then in EL+⊥T we can conclude NotTaxPayer(john) If (Student ⊓ Worker ⊓ Parent)(john) ∈ ABox, we cannot derive NotTaxPayer(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 16

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NonMonotonic semantics

We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

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NonMonotonic semantics

We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

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NonMonotonic semantics

We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

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NonMonotonic semantics

We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models we introduce a semantic entailment determined by minimal models

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

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NonMonotonic semantics

We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models we introduce a semantic entailment determined by minimal models Informally, we prefer a model M to a model N if M contains more typical instances of concepts than N

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

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NonMonotonic semantics

We would like to infer that individuals are typical instances of the concepts they belong to, if consistent with the KB In order to maximize the typicality of instances: we define a preference relation on models we introduce a semantic entailment determined by minimal models Informally, we prefer a model M to a model N if M contains more typical instances of concepts than N Given a KB, we consider a finite set LT of concepts occurring in the KB, the typicality of whose instances we want to maximize LT contains at least all concepts C such that T(C) occurs in the KB or in the query

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 17

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NonMonotonic semantics

M−

LT = {(a, ¬¬C) | a ∈ (¬¬C)I, with a ∈ ∆, C ∈ LT }

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

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NonMonotonic semantics

M−

LT = {(a, ¬¬C) | a ∈ (¬¬C)I, with a ∈ ∆, C ∈ LT }

Given two models M = ∆M, <M, IM and N = ∆N , <N , IN of KB, we say that M is preferred to N w.r.t. LT (M <LT N ), if:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

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NonMonotonic semantics

M−

LT = {(a, ¬¬C) | a ∈ (¬¬C)I, with a ∈ ∆, C ∈ LT }

Given two models M = ∆M, <M, IM and N = ∆N , <N , IN

  • f KB, we say that M is preferred to N w.r.t. LT (M <LT N ), if:

∆M = ∆N

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

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NonMonotonic semantics

M−

LT = {(a, ¬¬C) | a ∈ (¬¬C)I, with a ∈ ∆, C ∈ LT }

Given two models M = ∆M, <M, IM and N = ∆N , <N , IN

  • f KB, we say that M is preferred to N w.r.t. LT (M <LT N ), if:

∆M = ∆N M−

LT ⊂ N − LT

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

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NonMonotonic semantics

M−

LT = {(a, ¬¬C) | a ∈ (¬¬C)I, with a ∈ ∆, C ∈ LT }

Given two models M = ∆M, <M, IM and N = ∆N , <N , IN

  • f KB, we say that M is preferred to N w.r.t. LT (M <LT N ), if:

∆M = ∆N M−

LT ⊂ N − LT

aI = aI′ for all a ∈ O

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

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NonMonotonic semantics

M−

LT = {(a, ¬¬C) | a ∈ (¬¬C)I, with a ∈ ∆, C ∈ LT }

Given two models M = ∆M, <M, IM and N = ∆N , <N , IN

  • f KB, we say that M is preferred to N w.r.t. LT (M <LT N ), if:

∆M = ∆N M−

LT ⊂ N − LT

aI = aI′ for all a ∈ O A model M is a minimal model for KB (with respect to LT ) if it is a model of KB and there is no a model M′ of KB such that M′ <LT M

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 18

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Nonmonotonic Semantics

Query F : either a formula C(a) or a subsumption C ⊑ D

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 19

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Nonmonotonic Semantics

Query F : either a formula C(a) or a subsumption C ⊑ D Minimal Entailment in EL⊥Tmin

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 19

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Nonmonotonic Semantics

Query F : either a formula C(a) or a subsumption C ⊑ D Minimal Entailment in EL⊥Tmin A query F is minimally entailed from KB w.r.t. LT : KB | =EL⊥Tmin F if F holds in all models of KB minimal w.r.t. LT

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 19

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Example

Let LT = {Student, Student ⊓ Worker, Student ⊓ Worker ⊓ Parent}

KB ∪ {Student(john)} | =EL⊥Tmin NotTaxPayer(john) KB ∪ {Student(john), Worker(john)} | =EL⊥Tmin TaxPayer(john) KB∪ {Student(john), Worker(john), Parent(john)} | =EL⊥Tmin NotTaxPayer(john)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 20

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Complexity results for EL⊥Tmin

Entailment for EL+⊥T is CoNP , but

Theorem 3.1 in [GGOP]. Entailment in EL⊥Tmin is

EXPTIME-hard . We need further restrctions One possibility: Left Local EL⊥Tmin (considered for circumscriptive extension [BLW06])

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 21

slide-71
SLIDE 71

Language of Left Local EL⊥Tmin

Alphabet of concept names C role names R individuals O Given A ∈ C and r ∈ R, we define: C := A | ⊤ | ⊥ | C ⊓ C CR := C | CR ⊓ CR | ∃r.C CLL

L

:= C | CLL

L

⊓ CLL

L

| ∃r.⊤ | T(C) TBox contains a finite set of concept inclusions CLL

L

⊑ CR

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 22

slide-72
SLIDE 72

Complexity results for Left Local EL⊥Tmin

Small model theorem (Theorem 3.11 in [GGOP]). KB

| =EL⊥Tmin F if and only if F holds in all models of KB

whose size is polynomial in the size of KB.

Theorem 3.12 in [GGOP]. If KB is Left Local, the problem of

deciding whether KB |

=EL⊥Tmin F is in Πp

2.

A small model theorem and a similar complexity result can be proved for DL-LitecTmin [GGOP]

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 23

slide-73
SLIDE 73

Complexity results for Left Local EL⊥Tmin

Small model theorem (Theorem 3.11 in [GGOP]). KB

| =EL⊥Tmin F if and only if F holds in all models of KB

whose size is polynomial in the size of KB.

Theorem 3.12 in [GGOP]. If KB is Left Local, the problem of

deciding whether KB |

=EL⊥Tmin F is in Πp

2.

A small model theorem and a similar complexity result can be proved for DL-LitecTmin [GGOP]

∃R.C ∃R.C C C C R R R ∃R.C ∃R.C C R R R

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 23

slide-74
SLIDE 74

The Tableau calculus TABEL⊥T

min

Tableau calculus TABEL⊥T

min

for deciding whether a query F is minimally entailed from a KB (TBox,ABox)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

slide-75
SLIDE 75

The Tableau calculus TABEL⊥T

min

Tableau calculus TABEL⊥T

min

for deciding whether a query F is minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

slide-76
SLIDE 76

The Tableau calculus TABEL⊥T

min

Tableau calculus TABEL⊥T

min

for deciding whether a query F is minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TABEL⊥T

min

tries to build an open branch representing a minimal model satisfying KB ∪ {¬F}

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

slide-77
SLIDE 77

The Tableau calculus TABEL⊥T

min

Tableau calculus TABEL⊥T

min

for deciding whether a query F is minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TABEL⊥T

min

tries to build an open branch representing a minimal model satisfying KB ∪ {¬F} two-phase computation:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

slide-78
SLIDE 78

The Tableau calculus TABEL⊥T

min

Tableau calculus TABEL⊥T

min

for deciding whether a query F is minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TABEL⊥T

min

tries to build an open branch representing a minimal model satisfying KB ∪ {¬F} two-phase computation:

  • 1. Phase 1: TABEL⊥T

P H1

verifies whether KB ∪{¬F} is satisfiable in an EL+⊥T model, building candidate models

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

slide-79
SLIDE 79

The Tableau calculus TABEL⊥T

min

Tableau calculus TABEL⊥T

min

for deciding whether a query F is minimally entailed from a KB (TBox,ABox) extension of the “standard” tableau calculus for ALC TABEL⊥T

min

tries to build an open branch representing a minimal model satisfying KB ∪ {¬F} two-phase computation:

  • 1. Phase 1: TABEL⊥T

P H1

verifies whether KB ∪{¬F} is satisfiable in an EL+⊥T model, building candidate models

  • 2. Phase 2: TABEL⊥T

P H2

checks whether the candidate models found in Phase 1 are minimal models of KB

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 24

slide-80
SLIDE 80

The Tableau calculus TABEL⊥T

min

Given a knowledge base (TBox,ABox), tableaux nodes of TABEL⊥T

min

are called constraint systems and have the form S | U | W, where :

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

slide-81
SLIDE 81

The Tableau calculus TABEL⊥T

min

Given a knowledge base (TBox,ABox), tableaux nodes of TABEL⊥T

min

are called constraint systems and have the form S | U | W, where : S = {a : C | C(a) ∈ ABox} ∪ {a

R

− → b | aRb ∈ ABox}

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

slide-82
SLIDE 82

The Tableau calculus TABEL⊥T

min

Given a knowledge base (TBox,ABox), tableaux nodes of TABEL⊥T

min

are called constraint systems and have the form S | U | W, where : S = {a : C | C(a) ∈ ABox} ∪ {a

R

− → b | aRb ∈ ABox} U = {C ⊑ D∅ | C ⊑ D ∈ TBox}

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

slide-83
SLIDE 83

The Tableau calculus TABEL⊥T

min

Given a knowledge base (TBox,ABox), tableaux nodes of TABEL⊥T

min

are called constraint systems and have the form S | U | W, where : S = {a : C | C(a) ∈ ABox} ∪ {a

R

− → b | aRb ∈ ABox} U = {C ⊑ D∅ | C ⊑ D ∈ TBox} W is a set of labels xC used by existential rules

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 25

slide-84
SLIDE 84

Special Existential Rules

The rule (∃+) is split in the following two rules:

S, u : ∃R.C | U | W S, u

R

− → xC, xC : C | U | W ∪ {xC} . . . (∃+)1 S, u

R

− → y1, y1 : C | U | W S, u

R

− → ym, ym : C | U | W (∃+)2 S, u : ∃R.C | U | W S, u

R

− → xC | U | W . . . S, u

R

− → y1, y1 : C | U | W S, u

R

− → ym, ym : C | U | W if xC ∈ W and y1, . . . , ym are all the labels occurring in S if xC ∈ W and y1, . . . , ym are all the labels occurring in S

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 26

slide-85
SLIDE 85

Special Rule for (−)

S = S, u : ¬¬C1, . . . , u : ¬¬Cn. SM

u→y = {y : ¬D, y : ¬D | u : ¬D ∈ S} and, for k = 1, 2, . . . , n,

S

−k u→y = {y : ¬¬Cj ⊔ Cj | u : ¬¬Cj ∈ S ∧ j = k}.

S, x : Ck, x : ¬Ck, SM

u→x, S −k u→x | U | W

. . . (−) S, y1 : Ck, y1 : ¬Ck, SM

u→y1, S −k u→y1 | U | W

S, u : ¬¬C1, ¬¬C2, . . . , u : ¬¬Cn | U | W S, ym : Ck, ym : ¬Ck, SM

u→ym, S −k u→ym | U | W

for all k = 1, 2, . . . , n, where y1, . . . , ym are all the labels occurring in S and x is new. Rule (−) contains:

n branches, one for each u : ¬¬Ck in S;

  • ther n × m branches, where m is the number of labels occurring in

S, one for each label yi and for each u : ¬¬Ck in S

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 27

slide-86
SLIDE 86

Phase 1: TABEL⊥T

PH1

S, u : ∃R.C | U | W S, u

R

− → xC, xC : C | U | W ∪ {xC} . . . (∃+)1 if y : ¬C ∈ S S | U, C ⊑ DL | W if x occurs in S and x ∈ L (Unfold) S, x : T(C) | U | W S, x : ¬T(C) | U | W S, x : C, x : ¬C | U | W S, x : ¬C | U | W S, x : ¬¬C | U | W (T+) (T−) (⊓+) (⊓−) (cut) x occurs in S if x : ¬¬C ∈ S and x : ¬C ∈ S C ∈ LT S, x : ¬D | U | W S, x : ¬C | U | W S, x : C, x : D | U | W S, x : C ⊓ D | U | W S, x : ¬(C ⊓ D) | U | W S, x : C, x : ¬C | U | W (Clash)⊥ (Clash)¬⊤ S, x : ¬C | U | W S, x : ¬∃R.C, x

R

− → y, y : ¬C | U | W S, x : ¬∃R.C, x

R

− → y | U | W (∃−) (Clash) S, x : ¬¬C | U | W S | U | W S, x : ⊥ | U | W S, x : ¬⊤ | U | W S, x : ¬C ⊔ D | U, C ⊑ DL,x | W (∃+)2 S, u : ∃R.C | U | W S, u

R

− → xC | U | W S, x : C | U | W S, x : D | U | W S, x : C ⊔ D | U | W (⊔+) S, u

R

− → y1, y1 : C | U | W . . . S, u

R

− → y1, y1 : C | U | W S, u

R

− → ym, ym : C | U | W S, u

R

− → ym, ym : C | U | W S, x : Ck, x : ¬Ck, SM

u→x, S −k u→x | U | W

. . . (−) S, y1 : Ck, y1 : ¬Ck, SM

u→y1, S −k u→y1 | U | W

S, u : ¬¬C1, ¬¬C2, . . . , u : ¬¬Cn | U | W S, ym : Ck, ym : ¬Ck, SM

u→ym, S −k u→ym | U | W

k = 1, 2, . . . , n x new if xC ∈ W and y1, . . . , ym are all the labels occurring in S if xC ∈ W and y1, . . . , ym are all the labels occurring in S if y1, . . . , ym are all the labels occurring in S, y1 = u, . . . , ym = u

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 28

slide-87
SLIDE 87

Phase 2: TABEL⊥T

PH2

for each open branch B built by TABEL⊥T

PH1 , verifies if it is a

minimal model of the KB

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

slide-88
SLIDE 88

Phase 2: TABEL⊥T

PH2

for each open branch B built by TABEL⊥T

PH1 , verifies if it is a

minimal model of the KB Given an open branch B of a tableau built from TABEL⊥T

PH1 , we

define:

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

slide-89
SLIDE 89

Phase 2: TABEL⊥T

PH2

for each open branch B built by TABEL⊥T

PH1 , verifies if it is a

minimal model of the KB Given an open branch B of a tableau built from TABEL⊥T

PH1 , we

define: D(B) as the set of labels occurring on B

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

slide-90
SLIDE 90

Phase 2: TABEL⊥T

PH2

for each open branch B built by TABEL⊥T

PH1 , verifies if it is a

minimal model of the KB Given an open branch B of a tableau built from TABEL⊥T

PH1 , we

define: D(B) as the set of labels occurring on B

B− = {x : ¬¬C | x : ¬¬C occurs in B}

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 29

slide-91
SLIDE 91

Phase 2: TABEL⊥T

PH2

A tableau of TABEL⊥T

PH2

is a tree whose nodes are triples of the form S | U | K

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

slide-92
SLIDE 92

Phase 2: TABEL⊥T

PH2

A tableau of TABEL⊥T

PH2

is a tree whose nodes are triples of the form S | U | K S | U is a constraint system (as in TABEL⊥T

PH1 )

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

slide-93
SLIDE 93

Phase 2: TABEL⊥T

PH2

A tableau of TABEL⊥T

PH2

is a tree whose nodes are triples of the form S | U | K S | U is a constraint system (as in TABEL⊥T

PH1 )

K contains formulas of the form x : ¬¬C, with C ∈ LT

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

slide-94
SLIDE 94

Phase 2: TABEL⊥T

PH2

A tableau of TABEL⊥T

PH2

is a tree whose nodes are triples of the form S | U | K S | U is a constraint system (as in TABEL⊥T

PH1 )

K contains formulas of the form x : ¬¬C, with C ∈ LT Basic idea: given an open B built by TABEL⊥T

PH1 , K is initialized

with B− in order to build smaller models

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 30

slide-95
SLIDE 95

Phase 2: TABEL⊥T

PH2

(∃+) (Unfold) (Clash) S, x : C, x : ¬C | U | K (Clash)∅ (Clash)− S | U | ∅ S, x : ¬¬C | U | K if x : ¬¬C ∈ K S | U, C ⊑ DL | K x ∈ D(B) and x ∈ L S, x : C ⊓ D | U | K S, x : C, x : D | U | K S, x : ¬C | U | K (T+) (T−) (⊓+) (⊓−) (cut)

if x : ¬¬C ∈ S and x : ¬C ∈ S C ∈ LT S, x : ¬D | U | K S, x : ¬(C ⊓ D) | U | K S, x : ¬C | U | K S, x : ¬¬C | U | K S | U | K S, x : ¬T(C) | U | K S, x : ¬C | U | K S, x : ¬¬C | U | K S, x : T(C) | U | K S, x : C, x : ¬C | U | K S, u : ¬¬C1, . . . , u : ¬¬Cn | U | K, u : ¬¬C1, . . . , u : ¬¬Cn (Clash)⊥ S, x : ¬⊤ | U | K (Clash)¬⊤ S, x : ⊥ | U | K (−) S, x : ¬C ⊔ D | U, C ⊑ DL,x | K x ∈ D(B) S, u

R

− → y1, y1 : C | U | K S, u : ∃R.C | U | K S, u

R

− → ym, ym : C | U | K S, ym : Ck, ym : ¬Ck, SM

u→ym, S −k u→ym | U | K

. . . S, y1 : Ck, y1 : ¬Ck, SM

u→y1, S −k u→y1 | U | K . . .

if D(B) = {y1, . . . , ym} if D(B) = {y1, . . . , ym} and y1 = u, . . . , ym = u

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 31

slide-96
SLIDE 96

The Tableau calculus TABEL⊥T

min

S | U | ∅ is the corresponding constraint system of KB F= query S′= set of constraints obtained by adding to S the constraint corresponding to ¬F The calculus TABEL⊥T

min

checks whether a query F is minimally entailed from a KB by means of the following procedure: (phase 1) the calculus TABEL⊥T

PH1

is applied to S′ | U | ∅; if, for each branch B built by TABEL⊥T

PH1 , either

(i) B is closed or (ii) (phase 2) the tableau built by the calculus TABEL⊥T

PH2

for S | U | B− is open, then KB | =EL⊥Tmin F, otherwise KB | =EL⊥Tmin F.

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 32

slide-97
SLIDE 97

An example

{∃hc.S(j), T(S) ⊑ NTP} | =EL⊥Tmin ∃hc.NTP(j)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 33

slide-98
SLIDE 98

An example

{∃hc.S(j), T(S) ⊑ NTP} | =EL⊥Tmin ∃hc.NTP(j)

j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP ∅ | ∅ j : ¬T(S) ⊔ NTP, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬T(S), j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : NTP, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬S, j : NTP, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬¬S, j : NTP, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ j : ¬S, j : ¬¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ (Clash) j : ¬¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ (cut) (cut) (cut) (Unfold) (T−) (⊔+) j

hc

− → xS, xS : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | xS j

hc

− → j, j : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | ∅ (Clash) xS : ¬NTP, j

hc

− → xS, xS : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | xS xS : ¬¬S, xS : ¬NTP, j

hc

− → xS, xS : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | xS j : S, . . . , xS : ¬NTP, j

hc

− → xS, xS : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | xS (Clash) y : S, y : S, xS : ¬NTP, j

hc

− → xS, xS : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j} | xS . . . . . . y : NTP, y : S, y : S, xS : ¬NTP, j

hc

− → xS, xS : S, j : ¬S, j : ¬S, j : ∃hc.S, j : ¬∃hc.NTP | T(S) ⊑ NTP {j,xS,y} | xS

!""#$%&'&#()!*+%&'#+",'&

(−)

'$!$-+#)."&' '$!$-+#)."&'

,/&*#()!*+%

(∃+

1 )

(∃−)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 33

slide-99
SLIDE 99

An example

j : ∃hc.S | T(S) ⊑ NTP ∅ | xS : ¬¬S D(B) = {j, y, xS} j : ¬T(S) ⊔ NTP, j : ∃hc.S | T(S) ⊑ NTP {j} | xS : ¬¬S j : ¬T(S), j : ∃hc.S | T(S) ⊑ NTP {j} | xS : ¬¬S j : NTP, j : ∃hc.S | T(S) ⊑ NTP {j} | xS : ¬¬S j : ¬S, j : ∃hc.S | T(S) ⊑ NTP {j} | xS : ¬¬S j : ¬¬S, j : ∃hc.S | T(S) ⊑ NTP {j} | xS : ¬¬S (Clash)− . . . (Unfold) (⊔+) (T−) j

hc

− → j, j : S, j : ¬S | T(S) ⊑ NTP {j} | xS : ¬¬S j

hc

− → y, y : S, j : ¬S | T(S) ⊑ NTP {j} | xS : ¬¬S j

hc

− → xS, xS : S, j : ¬S | T(S) ⊑ NTP {j} | xS : ¬¬S . . . xS : ¬T(S) ⊔ NTP, j

hc

− → xS, xS : S, j : ¬S | T(S) ⊑ NTP {j,xS} | xS : ¬¬S (Unfold) xS : ¬T(S), j

hc

− → xS, xS : S, j : ¬S | T(S) ⊑ NTP {j,xS} | xS : ¬¬S (⊔+) xS : NTP, j

hc

− → xS, xS : S, j : ¬S | T(S) ⊑ NTP {j,xS} | xS : ¬¬S . . . y : ¬S, j : ¬S, xS : ¬S, xS : NTP, j

hc

− → xS, xS : S, j : ¬S | T(S) ⊑ NTP {j,xS,y} | xS : ¬¬S (∃+) (Clash) (cut) and static rules . . . . . .

!"#$%&'&("$&)(*+,)!-

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 34

slide-100
SLIDE 100

The Tableau calculus TABEL⊥T

min

Theorem: TABEL⊥T

min

is a sound and complete decision procedure for verifying if KB | =EL⊥Tmin F. Proposition: Given a KB and a query F, the problem of checking whether KB ∪{¬F} is satisfiable is in NP. Theorem: The problem of deciding whether KB | =EL⊥Tmin F by means of TABEL⊥T

min

is in Πp

  • 2. (matching known complexity)

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 35

slide-101
SLIDE 101

Conclusions

We have provided a two-phase tableau calculus TABEL⊥T

min

for minimal entailment in the Left Local fragment of the logic EL⊥Tmin of the family of low complexity DLs EL⊥. The proposed calculus matches the known complexity results: Πp

2

A similar tableau procedure can be defined for DL-litecT fragment for which a Πp

2 upper bound for minimal entailment

has been shown [GGOP]. Study optimizations. Find polynomial fragments for minimal entailment, in analogy with circumscription [PFS10].

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 36

slide-102
SLIDE 102

References

[BH95] F . Baader and B. Hollunder. Embedding defaults into terminological knowledge representation formalisms. J. Autom. Reasoning, 14(1):149–180, 1995. [BLW06] P . A. Bonatti, C. Lutz, and F . Wolter. Description logics with circumscription. In KR, pages 400–410, 2006. [DLN+98] F . M. Donini, M. Lenzerini, D. Nardi, W. Nutt, and A. Schaerf. An epistemic

  • perator for description logics. Artif. Intell., 100(1-2):225–274, 1998.

[DNR02] F . M. Donini, D. Nardi, and R. Rosati. Description logics of minimal knowledge and negation as failure. ACM Trans. Comput. Log., 3(2):177–225, 2002. [ELST]

  • T. Eiter, T. Lukasiewicz, R. Schindlauer, and H. Tompits. Combining answer set

programming with description logics for the semantic web. In KR 2004, 141-151. [GGOP]

  • L. Giordano, V. Gliozzi, N. Olivetti, and G. L. Pozzato. Reasoning about typicality

in low complexity DLs: the logics EL⊥Tmin and DL-litecTmin. In IJCAI 2011, 894-899. [GGOP09] L. Giordano, V. Gliozzi, N. Olivetti, and G.L. Pozzato. ALC + Tmin: a preferential extension of description logics. Fundamenta Informaticae, 96:1–32, 2009. [PFS10] P .A.Bonatti, M. Faella, and L. Sauro. EL with default attributes and overriding. In

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 37

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SLIDE 103

Thank you!!!

Nonmonotonic Extensions of Low Complexity DLs: Complexity Results and Proof Methods – p. 38