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Rules Dependencies in Backward Chaining of Conceptual Graphs Rules

Rules Dependencies in Backward Chaining of Conceptual Graphs Rules. Jean-Franç ø is B å get LIRMM / INRIA Rhône-Alpes jean-francois.baget@inrialpes.fr Eric S å lv å t IMERIR salvat@imerir.com. Context: optimization of deduction with CG Rules. Deduction in SG [Sowa:76]

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Rules Dependencies in Backward Chaining of Conceptual Graphs Rules

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  1. Rules Dependencies in Backward Chaining of Conceptual Graphs Rules Jean-Françøis Båget LIRMM / INRIA Rhône-Alpes jean-francois.baget@inrialpes.fr Eric Sålvåt IMERIR salvat@imerir.com

  2. 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 !

  3. Caveat • J.-François/Eric paper := (Intro . (Definition | Theorem | Proof)* . Concl) • This time it’s even worse: • No example ! • No graph drawing ! • No poetry …

  4. hyp conc Caveat IF Agnt subject Person: JFB Present Document CG Obj THEN member G Document CG Team: RCR Paper contains subject interest subject contains Drawing CG Drawing subject

  5. Simple Graph G Simple Graph H  G V H   , Overview of deduction in SG Vocabulary V

  6. Projection: a deduction calculus in SG V PROJECTION? is a NP-complete problem H G  Theorem [Sowa:84; Chein, Mugnier:96]: G V H iff there is a projection  from H into (the normal form of) G.

  7. G, RV H Set of Rules R con hyp con hyp   , , , Overview of deduction in SR Vocabulary V Simple Graph H Simple Graph G   

  8. Forward Chaining: a deduction calculus in SR V con hyp con hyp R  Theorem [Salvat, Mugnier:96; Salvat:98]: G, RV H iff there exists k s.t. H projects into [R]k(G).  ? G H (normal form) [R]1(G) (normal form) [R]2(G)

  9. (Un)decidability of deduction in SR [R]2(G) [R]1(G) [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. Definition [Baget, Mugnier:02]: R is a finite expansion set iff  G,  k / [R]k(G) V [R]k+1(G)  ? G Theorem [Coulondre, Salvat:98]: Deduction in SR is undecidable (semi-decidable). H

  10. Dependencies between rules Definition [Baget:04]: A rule R1depends upon a rule R2 iff there exists a graph G such that applying R2 on G creates a new application of R1. R1 R2 Precompilation of dependencies reduces the number of applicability tests in Forward Chaining… DEPENDS? is a NP-complete problem G G [R]1(G) Suppose now that R1 does not depend upon R2, and use Forward Chaining…

  11. Graph of rules dependencies (GRD) [R]3(G) 1 6 4 G 2 [R]1(G) 5 [R]2(G) 3 [R]4(G) R Theorem [Baget:04]: Deduction in SR is decidable when the GRD contains no circuit. N2 calls to a NP-hard probem …

  12. Graph of rules dependencies (GRD) Disconnected rules 1 6 4 G 2 [R]1(G) 5 3 [R]k(G) R Theorem [Baget:04]: Deduction in SR is decidable when all strongly connected components of the GRD are f.e.s. [R]k’(G) Range-restricted rule [R]k’+1(G)

  13. Graph of rules dependencies (GRD) 1 6 4 G  2 5 7 3 R 8 H 

  14. Using proofs of dependencies R1 R2 • is a linear time operator    G Theorem [Baget:04]: If ’ is a new projection from hyp(R2) into G’, then ’ extends   G’

  15. Backward Chaining: a deduction calculus in SR H hyp(R) H’ Piece unification [Salvat:98] R

  16. Backward Chaining: a deduction calculus in SR  con hyp G G con hyp R 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. H H H H

  17. 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) =

  18. Using the GRD in Backward Chaining 1 6 4 G  2 Theorem [Baget, Salvat:06]: H’ can only be unified with predecessors of H or predecessors of the rule used to obtain H’. 5 7 3 R 8 H H’

  19. 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 ….

  20. Thanks for your attention Applause Thank you … Questions

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