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Upward Refinement for Conceptual Blending in Description Logic An ASP-based Approach and Case Study in EL ++ Roberto Confalonieri 1 Manfred Eppe 1 , 2 Marco Schorlemmer 1 Oliver Kutz 3 Rafael Pealoza 3 Enric Plaza 1 1 IIIA-CSIC,

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Upward Refinement for Conceptual Blending in Description Logic — An ASP-based Approach and Case Study in EL++—

Roberto Confalonieri1 Manfred Eppe1,2 Marco Schorlemmer1 Oliver Kutz3 Rafael Peñaloza3 Enric Plaza1

1IIIA-CSIC, Barcelona, Spain 2International Computer Science Institute, Berkeley, USA 3Free University of Bozen-Bolzano, Italy

ONTOLP 2015, 25 July, 2015, Buenos Aires, Argentina

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Outline

1

Introduction

2

Computational Conceptual Blending Overview

3

Upward Refinement Operator for EL++

4

Generalisation as Search Problem in Answer Set Programming

5

Conclusion and Future Works

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 2 / 23

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Introduction

COINVENT Project goals:

◮ Computationally feasible model of conceptual blending ⋆ Cognitive theory described by Fauconnier and Turner [1998, 2002].

The universal creative engine of human thinking.

⋆ Model concept invention ◮ Symbolic approach, based on formal logic (CASL, OWL) ◮ Applications areas: Mathematics, Music, and Computer Icon Design

This paper goals:

◮ Define an upward refinement operator for EL++ ◮ Use ASP to generate and search for EL++ concept generalisations Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 3 / 23

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Introduction

COINVENT Project goals:

◮ Computationally feasible model of conceptual blending ⋆ Cognitive theory described by Fauconnier and Turner [1998, 2002].

The universal creative engine of human thinking.

⋆ Model concept invention ◮ Symbolic approach, based on formal logic (CASL, OWL) ◮ Applications areas: Mathematics, Music, and Computer Icon Design

This paper goals:

◮ Define an upward refinement operator for EL++ ◮ Use ASP to generate and search for EL++ concept generalisations Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 3 / 23

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Creating Icons by Conceptual Blending

ICON LIBRARY OWL Conceptual Blending Engine

Designer

“Give me an icon with meaning Preview- Document” Amalgamation OWL2ASP ASP2OWL Evaluation

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 4 / 23

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Computational Model of Conceptual Blending - Amalgamation

Amalgamation originates from the notion of amalgam Ontañón and Plaza [2010] in case-based reasoning It applies to any language L such that L, ⊑ is a poset I1 I2 ¯ I2 ¯ I1 B G An amalgam of two input concepts is a new concept that combines parts from the

  • riginal descriptions

◮ Find Generic Space (G) of input concepts (commonalities) and try to combine

non-common elements in I1 and I2

◮ Often, input concepts I1 and I2 cannot be combined directly (inconsistency or

insatisfaction of some properties)

◮ Input concepts have to be first generalised into I ′ 1 and I ′ 2 ◮ I ′ 1 and I ′ 2 can be finally blended to obtain a ‘good’ B Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 5 / 23

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Creating Icons by Conceptual Blending

Blend Input 1 Input 2 Generic Space

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 6 / 23

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Modeling Computer Icons in EL++

NC = {Icon, Sign, Document, MagnifyingGlass, Pen, HardDisk} Nr = {hasSign, isAbove, isLeft, isRight, isBelow, isSpatialRelation}

Background Knowledge

Icon ⊑ Thing domain(isInSpatialRelation) ⊑ Sign Sign ⊑ Thing range(isInSpatialRelation) ⊑ Sign Document ⊑ Sign . . . HardDisk ⊑ Sign . . . MagnifyingGlass ⊑ Sign isAbove ⊑ isInSpatialRelation Pen ⊑ Sign isBehind ⊑ isInSpatialRelation domain(hasSign) ⊑ Icon isLeft ⊑ isInSpatialRelation range(hasSign) ⊑ Sign isRight ⊑ isInSpatialRelation

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 7 / 23

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Modeling Computer Icons in EL++

NC = {Icon, Sign, Document, MagnifyingGlass, Pen, HardDisk} Nr = {hasSign, isAbove, isLeft, isRight, isBelow, isSpatialRelation}

Background Knowledge

Icon ⊑ Thing domain(isInSpatialRelation) ⊑ Sign Sign ⊑ Thing range(isInSpatialRelation) ⊑ Sign Document ⊑ Sign . . . HardDisk ⊑ Sign . . . MagnifyingGlass ⊑ Sign isAbove ⊑ isInSpatialRelation Pen ⊑ Sign isBehind ⊑ isInSpatialRelation domain(hasSign) ⊑ Icon isLeft ⊑ isInSpatialRelation range(hasSign) ⊑ Sign isRight ⊑ isInSpatialRelation

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 7 / 23

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Modeling Computer Icons in EL++

NC = {Icon, Sign, Document, MagnifyingGlass, Pen, HardDisk} Nr = {hasSign, isAbove, isLeft, isRight, isBelow, isSpatialRelation}

Background Knowledge

Icon ⊑ Thing domain(isInSpatialRelation) ⊑ Sign Sign ⊑ Thing range(isInSpatialRelation) ⊑ Sign Document ⊑ Sign . . . HardDisk ⊑ Sign . . . MagnifyingGlass ⊑ Sign isAbove ⊑ isInSpatialRelation Pen ⊑ Sign isBehind ⊑ isInSpatialRelation domain(hasSign) ⊑ Icon isLeft ⊑ isInSpatialRelation range(hasSign) ⊑ Sign isRight ⊑ isInSpatialRelation

Domain Knowledge

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk)

Input 1

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

Input 2

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 7 / 23

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Blending Computer Icons

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk)

Blend Input 1

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

Input 2

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk) ⊓ Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

?

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 8 / 23

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Generalising Icon Concepts

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk)

Input 1

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

Input 2

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 9 / 23

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Generalising Icon Concepts

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk)

Input 1

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

Input 2 Generic Space

Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign. (Sign ⊓ ∃isAbove.Sign)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 9 / 23

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Generalising Icon Concepts

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk)

Generalisation Input 1

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

Input 2 Generalisation Generic Space

Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign. (Sign ⊓ ∃isAbove.Sign) Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.Sign) Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Sign ⊓ ∃isAbove.Document)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 9 / 23

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Generalisation and Refinement operators

The generalisation in the amalgamation algorithm is based on a search in the poset L(T ), ⊑T The generalisation of an EL++ concept can be done through an upward refinement operator γ Refinement operator properties Local finiteness Properness Completeness

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 10 / 23

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Generalisation of EL++ concepts

The upward refinement operator generalises an EL++ concept by:

◮ generalising a concept ◮ generalising the concept filling the range of a role ◮ generalising a role ◮ ‘removing’ a role/concept

Properties Trade-off between completeness and finiteness

◮ The operator is finite but not proper and complete ◮ It is possible that the generic space is not least general ◮ Not a big issue for conceptual blending, the important thing is to find

the commonalities between the concepts

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 11 / 23

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Generalisation in ASP

The search for generalisations is modeled as an ASP search problem where the ‘goal’ is to find a generic space for two input EL++ concepts:

1

EL++ concept in background and domain knowledge are translated to ASP facts (base part)

2

Generalisation operators are implemented as a step-wise process to generalise EL++ concepts in the domain knowledge until they are not equal (cumulative part)

3

Each ASP stable model returns a generalisation path from the input specifications to a generic space

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 12 / 23

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Modeling EL++ concepts in ASP

Background knowledge in ASP

Sign ⊑ Thing concept(concept_Sign). subConcept(concept_Document,concept_Thing). Document ⊑ Sign concept(concept_Document). subConcept(concept_Document,concept_Sign). . . . . . . domain(hasSign) ⊑ Icon role(role_hasSign). range(hasSign) ⊑ Sign domain(role_hasSign,concept_Icon). range(role_hasSign,concept_Sign). . . . . . . isAbove ⊑ isInSpatialRelation subRole(role_isAbove,role_isInSpatialRelation). . . . . . .

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 13 / 23

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Modeling EL++ concepts in ASP II

Domain knowledge in ASP - EditDocument

Icon ⊓ ∃hasSign.Document ⊓ ∃ hasSign.(Pen ⊓ ∃isAbove.Document) spec(spec_EditDocument). hasConjunct(spec_EditDocument,root,concept_Icon,0). hasConjunct(spec_EditDocument,root,roleEx1,0). roleExHasRoleAndConcept(spec_EditDocument,roleEx1,role_hasSign,concept_Document,0). hasConjunct(spec_EditDocument,root,roleEx2,0). roleExHasRoleAndConcept(spec_EditDocument,roleEx2,role_hasSign,conjunction1,0). hasConjunct(spec_EditDocument,conjunction1,concept_Pen,0). hasConjunct(spec_EditDocument,conjunction1,roleEx3,0). roleExHasRoleAndConcept(spec_EditDocument,roleEx3,role_isAbove,concept_Document,0).

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 14 / 23

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Generalisation in ASP

A generalisation operator is defined via:

◮ precondition rule: states when it is possible to execute a generalisation

  • peration (poss/3)

◮ inertia rule: states when an element of a specification stays as it is

after the execution (noninertial/3)

◮ effect rule: models how a generalisation operator changes an input

specification ( noninertial/3 - only for renaming operators)

The execution of a generalisation operator is modeled via:

◮ exec(γ, s, t): a generalisation operator γ is applied to s at a step t Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 15 / 23

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Generalisation of Atomic concepts

◮ exec(genConcept(Ex, C_sub, C_super), s, t) denotes the generalisation of a concept C_sub to a concept C_super in Ex of an icon specification s at step t using γ(A). The precondition rule for generalising A is:

poss(genConcept(Ex, C_sub, C_super), s1, t) ← hasConjunct(s1, Ex, C_sub, t), subConcept(C_sub, C_super), not roleExHasRoleAndConcept(s1, C_sub, _, _, t), not hasConjunct(s1, C_sub,, t), conjunctNotEq(s1, s2, C_sub, t), not exec(genConcept(Ex, C_sub, _), s2, t), spec(s2).

poss(genConcept(conjunction1,concept_Pen,concept_Sign),EditDoc,0)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 16 / 23

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Generalisation path

A generalisation path P = {exec(γ1, s, t1), · · · , exec(γn, s, tn)} of s is a sequence of generalisation operator steps applied in s Generalisation paths are generated with the following choice rule, that allows one or zero generalisation operations per specification at t. 0{exec(a, s, t) : poss(a, s, t)}1 ← not genericReached(t), spec(s)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 17 / 23

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Generalisation path

PEditDoc = { exec(genConceptInRole(roleEx3, role_isAbove, concept_Document, concept_Sign), EditDoc, 0), exec(genConcept(conjunction1, concept_Pen, concept_Sign), EditDoc, 1), exec(genConceptInRole(roleEx1, role_hasSign, concept_Document, concept_Sign), EditDoc, 2) }

EditDoc

  • 0. Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign.(Pen ⊓ ∃isAbove.Sign)
  • 1. Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign.(Sign ⊓ ∃isAbove.Sign)
  • 2. Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign.(Sign ⊓ ∃isAbove.Sign)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 18 / 23

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Blends in EL++

Icon ⊓ ∃hasSign.HardDisk ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.HardDisk)

Generalisation MGS Blend Input 1

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Pen ⊓ ∃isAbove.Document)

Input 2 Generalisation Generic Space

Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.Document) Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign. (Sign ⊓ ∃isAbove.Sign) Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign. (MagnifyingGlass ⊓ ∃isAbove.Sign) Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign. (Sign ⊓ ∃isAbove.Document)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 19 / 23

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Blends in EL++

Blends are computed as MGS of pairs of generalised concepts In EL++, the MGS is defined by ⊓

Generalisations C ′

1 = Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign.(MagnifyingGlass ⊓ ∃isAbove.Sign)

C ′

2 = Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign.(Sign ⊓ ∃isAbove.Document)

MGS C ′

1 ⊓ C ′ 2 = Icon ⊓ ∃hasSign.Sign ⊓ ∃hasSign.(MagnifyingGlass ⊓ ∃isAbove.Sign)

⊓ Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign.(Sign ⊓ ∃isAbove.Document)

Needed: Normalisation rules for rewriting the blend C ′

1 ⊓ C ′ 2 into a

‘normal’ form (no trivial)

Blend B = Icon ⊓ ∃hasSign.Document ⊓ ∃hasSign.(MagnifyingGlass ⊓ ∃isAbove.Document)

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 20 / 23

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Conclusion and Future Works

Achieved Results Upward refinement operator for EL++ ASP implementation Future Works To extend the icon ontology to include icon meanings To identify normalisation rules (for blend rewriting and operator properness) To extend the approach to a richer DL (without ⊔) To incorporate blend evaluation

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 21 / 23

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References I

  • M. Eppe, R. Confalonieri, E. MacLean, M. Kaliakatsos, E. Cambouropoulos, M. Schorlemmer, and K.-U.

Kühnberger. Computational invention of cadences and chord progressions by conceptual chord-blending. In Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI), 2015a.

  • M. Eppe, E. Maclean, R. Confalonieri, O. Kutz, M. Schorlemmer, and E. Plaza. ASP, Amalgamation, and the

Conceptual Blending Workflow. In Proceedings of the 13th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR), 2015b.

  • G. Fauconnier and M. Turner. Conceptual integration networks. Cognitive Science, 22(2):133–187, 4 1998.
  • G. Fauconnier and M. Turner. The Way We Think: Conceptual Blending And The Mind’s Hidden Complexities.

Basic Books, 2002.

  • S. Ontañón and E. Plaza. Amalgams: A formal approach for combining multiple case solutions. In ICCBR, 2010.

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 22 / 23

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Thanks for the attention!

Confalonieri, et al Upward Refinement for EL++ in ASP 25/07/2015 23 / 23