Formalisme noyau : Graphes Conceptuels de Base

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

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

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

Lets compare BGs … (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 Ex: compute the partial order on the following labels: (Cube, A) (Cube,*) (RegularObject,*) (Ball,*)(RegularObject,B)(Ball,B) First : how to compare labels ? Poset of concept labels

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

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

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

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

Project:P ResearcherResearcher:KResearcher:J Office:#124 Office member in near Query QFact G member Person member Project Q: Are there people working together, who are each member of a project? worksWith member

Project:P ResearcherResearcher:K Researcher:J Office:#124 Office member in worksWith near Query QFact G member worksWith Person member Project

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 H Question : y-a-t-il un homomorphisme de G dans H? (problème NP-complet)

Classical graph homomorphism is a particular case of BG homomorphism A graph homomorphism h 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 a b c d G H

From graph homomorphism to BG homomorphism T T r 1 2 T C = {T} T R ={r} M = {*} Support 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) f From BGs to graphs ? There is a polynomial transformation too…

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

Quelle sémantique pour les BGs ? Intuitivement : un BG représente lexistence dentités 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 Sémantique ensembliste (ou théorie des modèles) Sémantique logique Besoin dune sémantique formelle Ca nest pas assez précis : Combien y-a-t-il dobjets ? 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 » ?

First-order logical semantics ( ) Translation of the Support types (concept/relation)predicates individual markers constants subtype partial orderformulas concept types t < t x t(x) t(x) x Bloc(x) Object(x) relation types r < r x1... xk r(x1,..., xk) r(x1,..., xk) x1 x2 adjoin(x1,x2) near(x1,x2) (S) is the set of the formulas translating the type posets

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

Cube(x) Cube(A) Ball(z) Ball(y) Color(u) onTop(x,A) carac(y,u) carac(z,u) Ball:* Cube:* Ball:* Color:* Cube:A between carac onTop x A y z u between(A,y,z) 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 For each node, an atom

Quelle sémantique pour lhomomorphisme? Sil 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 : (Q) se déduit de (G) et de (S) Et sil nexiste pas dhomomorphisme de Q dans G, que peut-on en conclure?

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

BGs are logically founded Support S t < t r < r Graphs (BGs) predicates, constants x t(x) t(x) x1... xk r(x1,..., xk) r(x1,..., xk) (, ) formulas Soundness: if Q G then (Q) deducible from (G), (S) Completeness: if (Q) deducible from (G), (S) then Q G the BG model is equivalent to the (, ) FOL fragment (without function) (one can get rid off universally quantified formulas associated with the support) BG homomorphism is equivalent to deduction

Une limitation à la complétude Le BG homomorphisme est complet par rapport à la déduction si le graphe cible est sous forme normale T:a r T:b s T:a r T:b s T:a (G) et (H) équivalentes mais G et H incomparables par homomorphisme 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 G H

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