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Formalisme noyau : Graphes Conceptuels de Base

Formalisme noyau : Graphes Conceptuels de Base. onTop. between. carac. carac. Basic conceptual graph (BG). Two kinds of nodes : “ concept nodes ” represent entities “relation nodes” represent relationships between these entities. Cube:*. 1. 2. Cube:A. 1. 2. 3. Ball:*.

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Formalisme noyau : Graphes Conceptuels de Base

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  1. Formalisme noyau :Graphes Conceptuels de Base

  2. onTop between carac carac Basic conceptual graph(BG) • Two kinds of nodes : • “concept nodes” represent entities • “relation nodes” represent relationships between these entities Cube:* 1 2 Cube:A 1 2 3 Ball:* Ball:* 1 1 2 2 Color:* Labels are taken in the vocabulary (or support)

  3. The vocabulary (or support) T near(...,...) between(...,...,...) Animate Inanimate adjoin (...,...) Object Property onTopOf (...,...) Bloc Colour RegularObject 1. TC : Poset of concept types Cube Ball 2. TR: Poset of relation types partitioned into types of same arity V or S = (TC, TR, I) 3. I : Set of individual markers * : the generic marker [and : typing of individuals, relation signatures, …]

  4. A onTop between carac carac Basic conceptual graph(BG)G = (C, R, E, l) Cube:* 1 Labels are takenin the support 2 Cube:A 1 2 3 Ball:* Ball:* 1 1 2 2 Color:* “There is a cube, which is on top of cube A, and there are balls, with same color, A being between these balls”

  5. Let’s compare BGs … First : how to compare labels ? (t,m)  (t’,m’)if and only if t  t’ and m  m’ where the order over I{*} is as follows: for all i in I, * > i for all i and j in I, i and j are non comparable Poset of concept labels Ex: compute the partial order on the following labels: (Cube, A) (Cube,*) (RegularObject,*) (Ball,*) (RegularObject,B) (Ball,B)

  6. « Projection » (BG Homomorphism) ? G Q « is the knowledge encoded in graph Q present in graph G? »« does G provides an answer to Q? » • Mapping P from the nodes of Q to the nodes of G, which: • preserves bipartition • preserves edges and their numbering if c-i-r then P(c)-i-P(r) • may specialise labels type subtype generic marker individual marker

  7. carac onTop onTop carac carac Ball d2 Size:big d1 Object c2 Object c1 1 2 1 1 onTop r5 carac r4 r2 r1 2 1 2 2 carac r6 Cube d3 Cube c3 Color:gray c4 1 2 1 1 Color:gray d4 r7 r3 2 2 2 1 Cube d5 r8 Size:big c5 fact G query Q Q: “Are there an object on top of a big cube and a gray object?”

  8. onTop onTop carac carac carac Ball d2 Size:big d1 Object c2 Object c1 1 2 1 1 onTop r5 carac r4 r2 r1 2 1 2 2 carac r6 Cube d3 Cube c3 Color:gray c4 1 1 2 1 Color:gray d4 r7 r3 2 2 2 1 Cube d5 r8 Size:big c5 fact G query Q Image graph 1: there is a ball on top of a big gray cube

  9. onTop onTop carac carac Ball d2 Size:big d1 Object c2 Object c1 1 2 1 1 onTop r5 carac r4 r2 r1 2 1 2 2 carac r6 Cube d3 Cube c3 Color:gray c4 1 1 2 1 Color:gray d4 r7 r3 2 2 2 1 carac Cube d5 r8 Size:big c5 fact G query Q Image graph 2: there is a ball on top of a big cube and there is a gray cube

  10. member in in in Project:P member member Person Person worksWith Researcher Researcher:K Researcher:J member member worksWith Office Project Project Office:#124 near Query Q Fact G Q: “Are there people working together, who are each member of a project?”

  11. member in in in near Project:P member member Person Person worksWith Researcher Researcher:K Researcher:J member member worksWith Office Project Project Office:#124 Query Q Fact G

  12. Specialisation/Generalisation Projection defines a generalisation relation among SGs QG (GQ) if there is a homomorphism from Q to G Q is more general than G G is more specific than Q Problème fondamental : BG-Homomorphisme Données : deux BGs G et HQuestion : y-a-t-il un homomorphisme de G dans H? (problème NP-complet)

  13. Classical graph homomorphism is a particular case of BG homomorphism • A graph homomorphismh from G=(VG, EG) to H=(VH,EH) is a mapping from VG to VH that preserves edges: • if (x,y) is in EG, then (h(x),h(y)) is in EH d 1 2 c G 3 b H a

  14. T 1 r 2 T From graph homomorphism to BG homomorphism Support f TC = {T}TR ={r}M = {*} There is a homomorphism from a graph G to a graph H if and only if there is a BG-homomorphism from f(G) to f(H) From BGs to graphs ? There is a polynomial transformation too…

  15. T p p p p p p p T T T T T T T T Ex : Relationships between these BGs? Specialization is reflexive, transitive but not antisymmetric: it is a preorder

  16. Quelle sémantique pour les BGs ? Intuitivement : un BG représente l’existence d’entités et de relations entre ces entités “There is a cube, which is on top of cube A, and there are balls, with same color, A being between these balls” Ca n’est pas assez précis : Combien y-a-t-il d’objets ? Sont-ils tous différents ?Est-il sous-entendu que ces objets ont tous une couleur?Peut-il y avoir un autre objet sur le cube A? Si on a « onTop » ou « between » a-t-on aussi « near » ? Besoin d’une sémantique formelle Sémantique ensembliste (ou théorie des modèles) Sémantique logique

  17. First-order logical semantics (F) Translation of the Support types (concept/relation) predicatesindividual markers constants ‘subtype’ partial order formulas concept types t < t’x t(x)  t’(x) x Bloc(x)  Object(x) relation typesr < r’x1... xk r(x1,..., xk)  r’(x1,..., xk) x1x2 adjoin(x1,x2)  near(x1,x2) F(S) is the set of the formulas translating the type posets

  18. onTop between carac carac Translation of a BG x Cube:* 1 2 A Cube:A 1 2 3 y Ball:* Ball:* z 1 1 2 2 Color:* u • For each generic concept node, a new variable • For each individual concept node with marker i,the constant assigned to i

  19. onTop between carac carac x • For each node,anatom Cube:* Cube(x) 1 onTop(x,A) 2 A Cube:A Cube(A) between(A,y,z) 1 z y 2 3 Ball:* Ball:* Ball(y) Ball(z) 1 1 2 2 Color:* Color(u) u carac(y,u) carac(z,u) Cube(x)  Cube(A)  Ball(y)  Ball(z)  Color(u) onTop(x,A)  between(A,y,z)  carac(y,u)  carac(z,u) xyzu

  20. Quelle sémantique pour l’homomorphisme? • S’il existe un homomorphisme de Q dans G, cela veut dire quoi? Intuitivement : « la connaissance représentée par Q est aussi présente dans G », « G est plus précis que Q », « on peut déduire Q de G », «G implique Q » Formellement :F(Q) se déduit de F(G) et de F(S) • Et s’il n’existe pas d’homomorphisme de Q dans G, que peut-on en conclure?

  21. f Ensemble de formulesdans une logique K Sémantique logique « fondés par rapport à une logique » A reformuler en termes de BGs • Les raisonnements doivent être fondés par rapport à la déduction dans cette logique • adéquats, corrects(sound) : si i peut être inféré de Kalors f(i) est déductible de f(K) • complets (complete) : si f(i) est déductible de f(K) alors i peut être inféré de K

  22. Support S t < t’r < r’ Graphs (BGs) BGs are logically founded F predicates, constantsx t(x)  t’(x)x1... xk r(x1,..., xk)  r’(x1,..., xk) F ( , )formulas • BG homomorphism is equivalent to deduction Soundness: if Q  G then F(Q) deducible from F(G), F(S) Completeness: if F(Q) deducible from F(G), F(S) then Q  G • the BG model is equivalent to the ( , )FOLfragment(without function) • (one can get rid off universally quantified formulas associated with the support)

  23. Un graphe est sous forme normale s'il n'a pas deux sommets concepts avec le même marqueur individuel T:a r T: b s Une limitation à la complétude T:a r T:b T:a r T:b s T:a s T:b H G F(G) et F(H)équivalentesmais G et H incomparables par homomorphisme Le BG homomorphisme est complet par rapport à la déduction si le graphe cible est sousforme normale

  24. Support (vocabulary) Basic (conceptual) graphsdefined on support The BG model Logical language Formulas Operations BG Homomorphism (« Projection ») Deduction FOL

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