rules dependencies in backward chaining of conceptual graphs rules jean-françøis båget lirmm /...
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Rules Dependencies in Backward Chaining of Conceptual Graphs Rules
Jean-Françøis Båget LIRMM / INRIA Rhô[email protected]
Eric Sålvå[email protected]
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
…