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The Bernays-Sch ö nfinkel Fragment of First-Order Autoepistemic Logic

The Bernays-Sch ö nfinkel Fragment of First-Order Autoepistemic Logic. Peter Baumgartner MPI Informatik, Saarbr ü cken. Com GT. Buy. Sell. BMW Rover. BA Rover. Motivation. „BMW buys Rover from BA“. Starting point: Some reasoning tasks on ontologies can naturally

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The Bernays-Sch ö nfinkel Fragment of First-Order Autoepistemic Logic

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  1. The Bernays-Schönfinkel Fragment ofFirst-Order Autoepistemic Logic Peter Baumgartner MPI Informatik, Saarbrücken

  2. Com GT Buy Sell BMW Rover BA Rover Motivation „BMW buys Rover from BA“ Starting point: Some reasoning tasks on ontologies can naturally be expressed as specific model computation tasks XML Schema The BS Fragment of FO AEL

  3. Com GT Buy Sell BMW Rover BA Rover Motivation • DL with L-Operator • Inheritance • Roles • Integrity constraints „BMW buys Rover from BA“ • Rules with L-Operator • Transfer of role fillers • Default values • Integrity Constraints • BS-AEL Calculus • Decide satisfiability of certain function-free clause sets S1 … Sn BS-AEL Epistemic Model The BS Fragment of FO AEL

  4. Contents • Semantics of Propositional Autoepistemic Logic • Semantics of First-Order Autoepistemic Logic • Transformation of Bernays-Schönfinkel Fragment of Autoepistemic Logicto clausal-like form • Calculus to compute epistemic models for clausal-like forms The BS Fragment of FO AEL

  5. Propositional Autoepistemic Logic The BS Fragment of FO AEL

  6. Propositional Autoepistemic Logic – Examples (1)  = LA (A "integrity constraint"), does not have an epistemic model: I I1 I2 Mis sound but not complete: take M :A A A :B :B B The BS Fragment of FO AEL

  7. M1 is complete: ({:A},M1) ²L A ! A Propositional Autoepistemic Logic – Examples (2)  = L A ! A("select A or not") has two epistemic models I1 I2 I1 M2 M1 A A :A The BS Fragment of FO AEL

  8. I1 I1 I2 M2 M3 A A :A is not sound is not complete: ({A},M1) ² A !L A ({:A},M2) ² A !L A Propositional Autoepistemic Logic – Examples (3)  = A !L A("A is false by default") has one epistemic model M1 I1 M1 :A The BS Fragment of FO AEL

  9. First-Order Autoepistemic Logic - Domains Assumptions • Constant domain assumption (CDA): every I2M has the same countable infinite domain |I|= • Rigid term assumption (RTA): every ground -term t evaluates to same value in every interpretation: for all I, J:I(t) = J(t) • Unique name assumption (UNA): different ground -term s, t evaluate to different values: for all I:ifstthenI(s) I(t) RTA+UNA justifies assumption that  contains all ground -termsand that every ground -terms evaluates to itself:  = HU() [* The BS Fragment of FO AEL

  10.  = HU() [* • = {h, p} *countably infinite and *Å HU() = ; * HU() h p r1 r2 ... res(h) res(p) 9x acc(x) 9y rej(y) - h and pare interpreted the same in every interpretation (rigid designators) • existentially quantified variables may be assigned different values in different interpretations (I1 vs. I2 ) • ( ! Skolemization requires flexible designators) - Other options: * = {}or* = {c} - Chosen option seems to be favourable also allows to model "named nullvalues" The BS Fragment of FO AEL

  11. First-Order Autoepistemic Logic - Semantics The BS Fragment of FO AEL

  12. First-Order Autoepistemic Logic – Examples (1)  = 9x P(x) Æ:L P(x) ("'Small' domains may not work") I1[x ! 0] I1[x ! 0] I3[x ! 1] I2[x ! 1] M1 M2 P(0) P(0) :P(1) P(0) P(1) :P(0) P(1) is not sound is epistemic model The BS Fragment of FO AEL

  13. First-Order Autoepistemic Logic – Examples (2)  = 9x P(x) ÆL P(x) ("Elements from * can be known"). Models: I1[x ! 1] I2[x ! 1] I1[x ! 0] I2[x ! 0] M2 M1 :P(0) P(1) P(0) :P(1) P(0) P(1) P(0) P(1) The BS Fragment of FO AEL

  14. First-Order Autoepistemic Logic – Examples (3)  = P(a) Æ8x L P(x) ("Herband Theorem does not hold") I1[x ! a] I1[x ! a] I1[x ! 0] M1 M2 P(a) P(a) P(0) P(a) P(0) is a model (* = ;) is not complete because of I = fP(a), :P(0)g The BS Fragment of FO AEL

  15. Calculus Given: BS-AEL formula  = 9x8y(x,y) Questions: (1) Does  have an epistemic model? If yes, compute some/all (2) Given ' Does ' hold in some/all epistemic models of  ? (undecidable even if ' is a non-modal Bernays-Schönfinkel Formula) Calculus for (1) - sound, complete and terminating for finite * (infinite case can be reduced to finite case with sufficiently large *) - uses calls to decision procedure for function-free clause sets (e.g. any instance-based method) - first step: transformation of  to clausal-like form The BS Fragment of FO AEL

  16. Skolemization causes Problems [Baader, Hollunder 95] D R a C • (1) implies (2) • But from (1) and (3), (4) does not follow • So, consequences depend from syntax! Possible Solution (not here) Apply rules to known objects only, those explicitly mentioned: The BS Fragment of FO AEL

  17. Transformation to Clausal-like Form (1) Input: BS-AEL formula  = 9x8y(x,y) Problem 1: Skolemization (with rigid Skolem constants) is not correct:9x P(x) Æ8y :L P(y)has an epistemic model P(c) Æ8y :L P(y)does not have an epistemic model Therefore convert only 8y(x,y)to clausal form Problem 2: Want to have L only in front of atoms Rationale: view L P(t)as atom L_P(t) But L does not distribute over Ç , nested L's Algorithm: See next slide Result: A conjunction of AEL-clauses equivalent to 8y(x,y), where an AEL-clause is an implication of the form 8y (B1Æ... Æ BmÆL Bm+1Æ ... ÆL Bn! H1Ç ... Ç HkÇL Hk+1Ç ... ÇL Hl ) where the B's and H's are atoms The BS Fragment of FO AEL

  18. Transformation to Clausal-like Form (2) Input: BS-AEL formula  = 9x8y(x,y) Output: equivalent formula 9x (8y1 C1(x,y1) Æ ... Æ8yj Cj(x,yj)) where each Ci is of the form B1Æ... Æ BmÆL Bm+1Æ ... ÆL Bn! H1Ç ... Ç HkÇL Hk+1Ç ... ÇL Hl Sketch: use standard algorithm for conversion to CNF augmented with rules: L in front of conjunction: L in front of disjunction: Nested occurences of L: L in front of negation: The BS Fragment of FO AEL

  19. L9y '(z,y) is Permissible Let  = 9x8y(x,y) Suppose (x,y) contains subformula L9y '(z,y) Eliminate it with this rule: Example instance: Finally move 8youtwards to extend 9x8y on the right The BS Fragment of FO AEL

  20. Model Existence Problem Given: -  and * (if * is finite then test below is effective) - -formula  = 9x (8y1 C1(x,y1) Æ ... Æ8yj Cj(x,yj)) in clausal-like form = 9xf C1(x,y1),...,Cj(x,yj) g =: 9xP(x) Algorithm: Guess known/unknown ground atoms and verify: Let * = [* be extended signature, giving names to * elements Guess knowns Kµ HB(*)and let unknowns U = HB(*)nK Let PK/U = f L A j A 2K g[ f:L A j A 2U g corresponding (unit) clauses If (1) for all A 2Kand for all d2* it holds PK/U[P(d)² A (2) for all A 2Uthere is ad2* such that PK/U[P(d)² A then (1) M = fIjthere is a d2* such that I²PK/U[P(d)g is an epistemic model of , and (2) K = f A 2 HB(*) jfor allI 2 M: I(A) = true g The converse also holds Classical BS problems The BS Fragment of FO AEL

  21. Computing the epistemic modelM Guess knowns K = f P(0) gand let unknowns U = f :P(1) g Let PK/U = f L P(0), :L P(1) g corresponding (unit) clauses Test (1): for all A 2Kand for all d2* it holds PK/U[P(d)² A ? d = 0: f L P(0), :L P(1), P(0), P(y) !L P(y)g² P(0) yes d = 1: f L P(0), :L P(1), P(1), P(y) !L P(y)g² P(0) yes Test (2): for all A 2Uthere is ad2* such that PK/U[P(d)² A ? d = 0: f L P(0), :L P(1), P(0), P(y) !L P(y)g² P(1) yes d = 1: f L P(0), :L P(1), P(1), P(y) !L P(y)g² P(1) no Illustration I1  = 9x f P(x), P(y) !L P(y) g* = * = f 0, 1 g M P(0) :P(1) The BS Fragment of FO AEL

  22. Conclusions • Decidability in presence of infinite domain *- decidability of fragment 8y(y) is known (Tableau Calculus, Niemelä 1988)- factor model of finitely many equivalence classes • Translation (of fragment) into logic programming framework • Goal: "efficient" operational treatment of BS-AEL, by exploiting known first-order techniques and provers (Darwin, DCTP) • BS-AEL not operationalized so far. Why? • Combination DL + AEL + rule language • Application areas: inferences on FrameNet, Semantic Web, Null Values in Databases Further Issues The BS Fragment of FO AEL

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