A Tableau Prover for Hybrid Logic. Daniel Götzmann Graduate Seminar Programming Systems Lab Advisor: Mark Kaminski

Transcription

1 A Tbleu Prover for Hybrid Logic Dniel Götzmnn Grdute Seminr Progrmming Systems Lb Advisor: Mrk Kminski

2 Bsic Modl Logic s ::= p s s s s s propositionl logic

3 Bsic Modl Logic s ::= p s s s s s <r> s [r] s propositionl logic + modl opertors

4 Bsic Modl Logic s ::= p s s s s s <r> s [r] s propositionl logic + modl opertors s : S B s x predictes on sttes s holds on stte x

5 Bsic Modl Logic s ::= p s s s s s <r> s [r] s propositionl logic + modl opertors s : S B s x predictes on sttes s holds on stte x <r> s x := y. r x y s y <r> s r s

6 Bsic Modl Logic s ::= p s s s s s <r> s [r] s propositionl logic + modl opertors s : S B s x predictes on sttes s holds on stte x <r> s x := y. r x y s y <r> s r s [r] s x := y. r x y s y [r] s r r s s

7 Hybrid Logic s ::= p s s s s s <r> s [r] s x s bsic modl logic + y y s x := x = y := s y stte x is nmed y s holds on the stte nmed y

8 Hybrid Logic s ::= p s s s s s <r> s [r] s bsic modl logic + x s A s E s + globl modlities y y s x A s x E s x := x = y := s y := y.s y := y.s y stte x is nmed y s holds on the stte nmed y s holds on ll sttes s holds on some stte

9 Applictions nd Systems Knowledge representtion semntic web description logics (nottionl vrints of hybrid logics)

10 Applictions nd Systems Knowledge representtion semntic web description logics (nottionl vrints of hybrid logics) DL resoners Fct++ pellet Rcer other implementtions HTAB ( ( ( ( HyLoRes ( HyLoTb (

11 Tbleu Rules disjunction dimond universl equlity (s t) x s x t x <r> t x r x y, A t x x = y s x s y y fresh y on brnch conflict s x s x

12 Tbleu Rules disjunction (s t) x s x t x conjunction (s t) x s x, t x dimond <r> t x r x y, y fresh [r] t x r x y box universl A t x y on brnch existentil E t x y fresh equlity x = y s x y t x conflict s x s x nominls y x x = y

13 Tsk Implement tbleu-bsed prover for hybrid logic with globl modlities Investigte implementtion nd optimiztion techniques Prover should be resonbly efficient

14 Comprison Bsic Modl Logic Hybrid Logic with globl modlities tree structure propgtion to successors only lwys terminting

15 Comprison Bsic Modl Logic Hybrid Logic with globl modlities E s s <r>x,... <r>x,... x, t,... x t tree structure propgtion to successors only lwys terminting typiclly not tree cretion of new sources (E) propgtion to remote sttes merging equivlent sttes termintion non-trivil

16 Queue Problem: keep trck of unexpnded dimonds, disjunctions,... Wnted: flexible expnsion order Ide: bcktrckble priority queue

17 Queue Problem: keep trck of unexpnded dimonds, disjunctions,... Wnted: flexible expnsion order Ide: bcktrckble priority queue <r>s 2 y (s 3 s 4 ) x <r>s 0 x (s 1 s 8 ) y <r>s 5 z Es 9 FIFO <r>s 2 y <r>s 0 x <r>s 5 z (s 3 s 4 ) x (s 1 s 8 ) y Es 9 <> (FIFO) before (FIFO) (s 1 s 8 ) y (s 3 s 4 s 7 ) x <r>s 5 z <r>s 2 y <r>s 0 x Es 9 (incr. size) before <> (new sttes first)

18 Termintion A (<r> p) <r> t x r x y, A t x [r] t x r x y y fresh y on brnch

19 Termintion A (<r> p) universl modlities: <r> t x r x y, y fresh A t x y on brnch [r] t x r x y

20 Termintion universl modlities: <r> p <r> t x r x y, A t x [r] t x r x y y fresh y on brnch

21 Termintion <r> p universl modlities: <r> t x r x y, y fresh <r> p A t x y on brnch [r] t x r x y

22 Termintion <r> p universl modlities: <r> t x r x y, y fresh p r <r> p A t x [r] t x r x y y on brnch

23 Termintion <r> p universl modlities: <r> t x r x y, y fresh r p, <r> p <r> p A t x [r] t x r x y y on brnch

24 Termintion <r> p universl modlities: <r> t x r x y, y fresh r p, <r> p r <r> p A t x [r] t x r x y y on brnch p, <r> p nïve pproch: not terminting!

25 Termintion <r> p universl modlities: <r> t x r x y, y fresh r p, <r> p self loop: terminting! r <r> p A t x [r] t x r x y y on brnch

26 Pttern-bsed Blocking <r> p, [r] q 1, [r] q 2 universl modlities: <r> t x r x y, y fresh <r> p A t x y on brnch [r] t x r x y blocked ptterns:

27 Pttern-bsed Blocking <r> p, [r] q 1, [r] q 2 universl modlities: <r> t x r x y, y fresh r <r> p A t x y on brnch p, q 1, q 2 [r] t x r x y blocked ptterns: <r> p, [r] q 1, [r] q 2

28 Pttern-bsed Blocking <r> p, [r] q 1, [r] q 2 universl modlities: <r> t x r x y, y fresh r p, q 1, q 2, <r> p <r> p A t x [r] t x r x y y on brnch blocked ptterns: <r> p, [r] q 1, [r] q 2

29 Pttern-bsed Blocking <r> p, [r] q 1, [r] q 2 universl modlities: <r> t x r x y, y fresh r p, q 1, q 2, <r> p <r> p is blocked, hence not expnded <r> p blocked ptterns: A t x [r] t x r x y y on brnch <r> p, [r] q 1, [r] q 2

30 Pttern-bsed Blocking <r> p, [r] q 1, [r] q 2 universl modlities: <r> t x r x y, y fresh r p, q 1, q 2, <r> p <r> p is blocked, hence not expnded r <r> p blocked ptterns: A t x [r] t x r x y y on brnch <r> p, [r] q 1, [r] q 2

31 Pttern-bsed Blocking Necessry to gurntee termintion Useful optimiztion Efficiency needed for inserting ptterns removing ptterns (when bcktrcking) mtching ptterns (superset mtching) e.g., tree structure (Hoffmnn nd Koehler, 1999) bitvector-bsed structure (Giunchigli, Tcchell, 2000)

32 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1 b s 2 c s 1, s 3 d s 4 e s 4, s 5

33 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1, s 4 b s 2 c s 1, s 3 d s 4 e s 4, s 5 = d

34 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1, s 4 b s 2 c s 1, s 3, s 4, s 5 d s 4 e s 4, s 5 = d c = e

35 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1, s 4, s 3, s 5 b s 2 c s 1, s 3, s 4, s 5 d s 4 e s 4, s 5 = d c = e = e

36 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1, s 4, s 3, s 5 b s 2 c s 1, s 3, s 4, s 5 d s 4 e s 4, s 5 = d c = e = e Sttes re equivlent if they re in the sme tree Representtive of equivlence clss = root in forest Informtion shifted to representtive

37 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1, s 4, s 3, s 5, s 6 b s 2 c s 1, s 3, s 4, s 5 d s 4 e s 4, s 5 = d c = e = e s 6 e Sttes re equivlent if they re in the sme tree Representtive of equivlence clss = root in forest Informtion shifted to representtive Propgte new terms only to representtive (less redundncy)

38 Stte Equivlence Disjoint set forest represents stte equivlence clsses s 1, s 4, s 3, s 5, s 6 b s 2 c s 1, s 3, s 4, s 5 d s 4 e s 4, s 5 = d c = e = e s 6 e Sttes re equivlent if they re in the sme tree Representtive of equivlence clss = root in forest Informtion shifted to representtive Propgte new terms only to representtive (less redundncy)

39 Conclusion nd Outlook Implementing efficient decision methods chllenging Prototype implementtion exists for modl logic with globl modlities Preliminry results encourging Still much room for optimiztion

40 References Hybrid Logic C. Areces nd B. ten Cte. Hybrid Logics. In P. Blckburn, J. vn Benthem, nd F. Wolter, editors, Hndbook of Modl Logic. Elsevier, Pttern-bsed blocking M. Kminski nd G. Smolk. Hybrid Tbleux for the Difference Modlity. In Proc. 5th Workshop on Methods for Modlities (M4M-5), pp , Cchn, Frnce, November To Apper in ENTCS M. Kminski nd G. Smolk. Terminting Tbleu Systems for Modl Logic with Equlity. Drft. April Superset-mtching J. Hoffmnn nd J. Koehler. A New Method to Index nd Query Sets. In Proceedings of the 16th Interntionl Joint Conference on Artificil Intelligence, pp , E. Giunchigli nd O. Tcchell. A subset-mtching size-bounded cche for stisfibility of modl logics. In Proceedings Interntionl Conference Tbleux 2000, pp Disjoint Set Forests T. H. Cormen, C. E. Leiserson, R. L. Rivest, nd C. Stein. Introduction to Algorithms. MIT Press, 2001.