Rules Dependencies in Backward Chaining of Conceptual Graphs Rules

Jean-Françøis Båget LIRMM / INRIA Rhône-Alpesjean-francois.baget@inrialpes.fr

Eric SålvåtIMERIRsalvat@imerir.com

Context: optimization of deduction with CG Rules

Deduction in SG [Sowa:76]

Deduction in SR [Sowa:84;Salvat,Mugnier:96]

Optimizing deduction in SR : piece unification for Backward Chaining [Salvat, Mugnier:96; Salvat:98]

Optimizing deduction in SR : rules dependencies for Forward Chaining [Baget: 04]

30 !

Caveat

J.-François/Eric paper :=

(Intro . (Definition | Theorem | Proof)* . Concl)

This time it’s even worse:No example !No graph drawing !No poetry …

Caveat

Person: JFB PresentAgnt

member

Team: RCR

Obj

Paper

interest subject

CG

Document subject CG

Document CG

contains

Drawing

subject

IF

THEN

contains

Drawingsubject

G hyp conc

Overview of deduction in SG

Simple GraphG

Simple GraphH

VocabularyV

,

G V H

Projection: a deduction calculus in SG

V

G

H

Theorem [Sowa:84; Chein, Mugnier:96]: G V H iff there is a projection from H into (the normal form of) G.

PROJECTION? is a NP-complete problem

Overview of deduction in SR

Simple GraphG

Simple GraphH

VocabularyV

hyp con

hyp con

Set of Rules R

,

G, R V H

, ,

Forward Chaining: a deduction calculus in SR

V

G

H

hyp con

hyp con R

(normal form)

[R]1(G)

?

(normal form)

[R]2(G)

Theorem [Salvat, Mugnier:96; Salvat:98]: G, R V H iff there exists k s.t. H projects into [R]k(G).

(Un)decidability of deduction in SR

G

H

[R]1(G) [R]2(G) [R]k(G)

?

Theorem [Coulondre, Salvat:98]: Deduction in SR is undecidable (semi-decidable).

V [R]k+1(G)

Definition [Baget, Mugnier:02]: R is a finite expansion set iff G, k / [R]k(G) V [R]k+1(G)

Examples of finite expansion sets Disconnected rules Range restricted rules

The union of 2 f.e.s. is not necessarily a f.e.s.

Dependencies between rules

Definition [Baget:04]: A rule R1 depends upon a rule R2

iff there exists a graph G such that applying R2 on G createsa new application of R1.

R1 R2

G

Suppose now that R1 does not depend upon R2, and use Forward Chaining…

G

[R]1(G)

Precompilation of dependencies reduces the number of applicability tests in Forward Chaining…DEPENDS? is a

NP-complete problem

[R]4(G)

Graph of rules dependencies (GRD)

R

1

2

3

4

5

6

G

[R]1(G)[R]2(G)

[R]3(G)

Theorem [Baget:04]: Deduction in SR is decidable when the GRD contains no circuit.

N2 calls to a NP-hard probem …

Graph of rules dependencies (GRD)

R

1

2

3

4

5

6

G

Disconnected rules

Range-restricted rule

[R]1(G)

[R]k(G)

[R]k’(G)

[R]k’+1(G)

Theorem [Baget:04]: Deduction in SR is decidable when all strongly connected components of the GRD are f.e.s.

Graph of rules dependencies (GRD)

R

1

2

3

4

5

6

G

H

7

8

Using proofs of dependencies

R1 R2

G

G’

is a linear time operator

Theorem [Baget:04]: If ’ is a new projection from hyp(R2) into G’, then ’ extends

Backward Chaining: a deduction calculus in SR

H

R

H’ hyp(R)

Piece unification [Salvat:98]

Backward Chaining: a deduction calculus in SR

hyp con

hyp con RG

H

H

HH

G

Theorem [Salvat, Mugnier:96; Salvat:98]: G, R V H iff there exists a sequence of piece unifications that transforms H into the empty SG.

So, what’s new in this paper ?

Different representations Hypergraphs, colored graphs [Baget:04] Multigraphs, lambda abtractions [Salvat:98]

Different restrictions Lattice as concept types hierarchy [Salvat:98] Poor treatment of individuals in conclusion [Baget:04]

Improving both results Unification of syntaxes Removal of all restrictions Extension to conjunctive concept types (collateral benefit)

=

Using the GRD in Backward Chaining

R

1

2

3

4

5

6

G

H

7

8

H’

Theorem [Baget, Salvat:06]: H’ can only be unified with predecessors of H or predecessors of the rule used to obtain H’.

Using the GRD in Backward Chaining

Reduces the # of rules used in BC as in FC, remove rules that are not on a path from

G to H

Reduces # of unification checks in BC as in FC, only checks for neighbours in the GRD

Reduces the cost of unification checks ? in FC, linearly computes partial projections

to extend. In BC, we should obtain a partial unification

to extend ….

Thanks for your attention

Thankyou

Applause

Questions

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