The Bernays-Schönfinkel Fragment of First-Order Autoepistemic Logic Peter Baumgartner MPI Informatik, Saarbrücken

The BS Fragment of FO AEL2 Motivation Starting point: Some reasoning tasks on ontologies can naturally be expressed as specific model computation tasks BMW Rover BA Rover BuySell Com GT „BMW buys Rover from BA“ XML Schema

The BS Fragment of FO AEL3 Motivation DL with L -Operator - Inheritance - Roles - Integrity constraints BS-AEL BS-AEL Calculus Decide satisfiability of certain function-free clause sets S 1 … S n Epistemic Model Rules with L -Operator - Transfer of role fillers - Default values - Integrity Constraints BMW Rover BA Rover BuySell Com GT „BMW buys Rover from BA“

The BS Fragment of FO AEL4 Contents Semantics of Propositional Autoepistemic Logic Semantics of First-Order Autoepistemic Logic Transformation of Bernays-Schönfinkel Fragment of Autoepistemic Logic to clausal-like form Calculus to compute epistemic models for clausal-like forms

The BS Fragment of FO AEL5 Propositional Autoepistemic Logic

The BS Fragment of FO AEL6 Propositional Autoepistemic Logic – Examples (1)  = L A ( A "integrity constraint"), does not have an epistemic model: M I1I1 I2I2 AA :B:BB M is sound but not complete: take I :A:A :B:B

The BS Fragment of FO AEL7 Propositional Autoepistemic Logic – Examples (2)  = L A ! A ("select A or not") has two epistemic models M1M1 I1I1 A M 1 is complete: ({ : A }, M 1 ) ² L A ! A M2M2 I1I1 I2I2 A :A:A

The BS Fragment of FO AEL8 Propositional Autoepistemic Logic – Examples (3)  = A ! L A (" A is false by default") has one epistemic model M 1 M1M1 I1I1 :A:A ({ A }, M 1 ) ² A ! L A M3M3 I1I1 I2I2 A :A:A is not sound M2M2 I1I1 A ({ : A }, M 2 ) ² A ! L A is not complete:

The BS Fragment of FO AEL9 First-Order Autoepistemic Logic - Domains Assumptions - Constant domain assumption (CDA): every I 2 M 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 : if s  t then I(s)  I(t) RTA+UNA justifies assumption that  contains all ground  -terms and that every ground  -terms evaluates to itself:  = HU(  ) [  *

The BS Fragment of FO AEL10  = HU(  ) [  * res(h)res(p) 9x acc(x)9y rej(y) hp r1r1 r2r2...  = { h, p }  * countably infinite and  * Å HU(  ) = ; HU(  ) ** - h and p are interpreted the same in every interpretation (rigid designators) - existentially quantified variables may be assigned different values in different interpretations ( I 1 vs. I 2 ) ( ! 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 AEL11 First-Order Autoepistemic Logic - Semantics

The BS Fragment of FO AEL12 First-Order Autoepistemic Logic – Examples (1)  = 9x P(x) Æ :L P(x) ("'Small' domains may not work") I 1 [x ! 0] M1M1 P(0) I 1 [x ! 0] M2M2 P(0) : P(1) is not sound I 2 [x ! 1] : P(0) P(1) I 3 [x ! 1] P(0) P(1) is epistemic model

The BS Fragment of FO AEL13 First-Order Autoepistemic Logic – Examples (2)  = 9x P(x) Æ L P(x) ("Elements from  * can be known"). Models: I 1 [x ! 0] M1M1 P(0) : P(1) P(0) P(1) I 2 [x ! 0] I 1 [x ! 1] M2M2 : P(0) P(1) P(0) P(1) I 2 [x ! 1]

The BS Fragment of FO AEL14 First-Order Autoepistemic Logic – Examples (3)  = P(a) Æ 8x L P(x) ("Herband Theorem does not hold") I 1 [x ! a] M1M1 P(a) I 1 [x ! a] M2M2 P(a) P(0) is a model (  * = ; ) I 1 [x ! 0] P(a) P(0) is not complete because of I = fP(a), :P(0)g

The BS Fragment of FO AEL15 Calculus Given: BS-AEL formula  = 9x 8y  (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 AEL16 Skolemization causes Problems [Baader, Hollunder 95]  (1) implies (2)  But from (1) and (3), (4) does not follow  So, consequences depend from syntax! C D a R Possible Solution (not here) Apply rules to known objects only, those explicitly mentioned:

The BS Fragment of FO AEL17 Transformation to Clausal-like Form (1) Input: BS-AEL formula  = 9x 8y  (x,y) Problem 1: Skolemization (with rigid Skolem constants) is not correct: 9 x P(x) Æ 8y :L P(y) has an epistemic model P(c) Æ 8y :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 (B 1 Æ... Æ B m Æ L B m+1 Æ... Æ L B n ! H 1 Ç... Ç H k Ç L H k+1 Ç... Ç L H l ) where the B's and H's are atoms

The BS Fragment of FO AEL18 Transformation to Clausal-like Form (2) Input: BS-AEL formula  = 9x 8y  (x,y) Output: equivalent formula 9x (8y 1 C 1 (x,y 1 ) Æ... Æ 8y j C j (x,y j )) where each C i is of the form B 1 Æ... Æ B m Æ L B m+1 Æ... Æ L B n ! H 1 Ç... Ç H k Ç L H k+1 Ç... Ç L H l Sketch: use standard algorithm for conversion to CNF augmented with rules: Nested occurences of L : L in front of disjunction: L in front of conjunction: L in front of negation:

The BS Fragment of FO AEL19 L 9y  '(z,y) is Permissible Let  = 9x 8y  (x,y) Suppose  (x,y) contains subformula L 9y  '(z,y) Eliminate it with this rule: Finally move 8y outwards to extend 9 x 8y on the right Example instance:

The BS Fragment of FO AEL20 Model Existence Problem Given: -  and  * (if  * is finite then test below is effective) -  -formula  = 9x (8y 1 C 1 (x,y 1 ) Æ... Æ 8y j C j (x,y j )) in clausal-like form  = 9x f C 1 (x,y 1 ),...,C j (x,y j ) g =: 9x P(x) Algorithm: Guess known/unknown ground atoms and verify: Let  * =  [  * be extended signature, giving names to  * elements Guess knowns K µ HB(  * ) and let unknowns U = HB(  * )nK Let P K/U = f L A j A 2 K g [ f:L A j A 2 U g corresponding (unit) clauses If (1) for all A 2 K and for all d 2  * it holds P K/U [ P(d) ² A (2) for all A 2 U there is a d 2  * such that P K/U [ P(d) ² A then (1) M = f I j there is a d 2  * such that I ² P K/U [ P(d)g is an epistemic model of , and (2) K = f A 2 HB(  * ) j for all I 2 M: I(A) = true g The converse also holds Classical BS problems

The BS Fragment of FO AEL21 Illustration  = 9x f P(x), P(y) ! L P(y) g  * =  * = f 0, 1 g I1I1 M P(0) : P(1) Computing the epistemic model M Guess knowns K = f P(0) g and let unknowns U = f :P(1) g Let P K/U = f L P(0), :L P(1) g corresponding (unit) clauses Test (1): for all A 2 K and for all d 2  * it holds P K/U [ P(d) ² A ? d = 0 : f L P(0), :L P(1), P(0), P(y) ! L P(y)g ² P(0) yes d = 1 : f L P(0), :L P(1), P(1), P(y) ! L P(y)g ² P(0) yes Test (2): for all A 2 U there is a d 2  * such that P K/U [ P(d) ² A ? d = 0 : f L P(0), :L P(1), P(0), P(y) ! L P(y)g ² P(1) yes d = 1 : f L P(0), :L P(1), P(1), P(y) ! L P(y)g ² P(1) no

The BS Fragment of FO AEL22 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 Further Issues 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