the bernays-sch ö nfinkel fragment of first-order autoepistemic logic

The Bernays-Schönfinkel Fragment ofFirst-Order Autoepistemic Logic

Peter Baumgartner

MPI Informatik, Saarbrücken

The BS Fragment of FO AEL 2

Motivation

Starting point:Some reasoning taskson ontologies can naturallybe expressed as specificmodel computation tasks

BMWRover

BARover

Buy Sell

Com GT

„BMW buys Rover from BA“

XML Schema


MotivationDL with L-Operator- Inheritance - Roles- Integrity constraints

BS-AEL BS-AEL Calculus

Decide satisfiability of certain function-free clause sets S1 … Sn

EpistemicModel

Rules with L-Operator- Transfer of role fillers- Default values- Integrity Constraints BMW

RoverBARover

Buy Sell

Com GT

„BMW buys Rover from BA“


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


Propositional Autoepistemic Logic


Propositional Autoepistemic Logic – Examples (1)

= L A (A "integrity constraint"), does not have an epistemic model:

MI1 I2

A A

:B B

M is sound but not complete: take I

:A

:B



= L A ! A ("select A or not") has two epistemic models

M1

I1

A

M1 is complete: ({:A},M1) ² L A ! A

M2

I1 I2

A :A



= A ! L A ("A is false by default") has one epistemic model M1

M1

I1:A

({A},M1) ² A ! L A

M3

I1 I2

A :A

is not sound

M2

I1 A

({:A},M2) ² A ! L A

is not complete:


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 -termsand that every ground -terms evaluates to itself: = HU() [ *


= HU() [ *

res(h) res(p) 9x acc(x) 9y rej(y)

h p r1 r2 ...

= {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 (I1 vs. I2 )

( ! Skolemization requires flexible designators)

- Other options: * = {} or * = {c}- Chosen option seems to be favourable also allows to model "named nullvalues"


First-Order Autoepistemic Logic - Semantics


First-Order Autoepistemic Logic – Examples (1)

= 9x P(x) Æ :L P(x) ("'Small' domains may not work")

I1[x ! 0]M1

P(0)

I1[x ! 0]M2

P(0): P(1)

is not sound

I2[x ! 1]

: P(0) P(1)

I3[x ! 1]

P(0) P(1)

is epistemic model



= 9x P(x) Æ L P(x) ("Elements from * can be known"). Models:

I1[x ! 0]M1 P(0)

: P(1) P(0) P(1)

I2[x ! 0] I1[x ! 1]M2 : P(0)

P(1) P(0) P(1)

I2[x ! 1]



= P(a) Æ 8x L P(x) ("Herband Theorem does not hold")

I1[x ! a]M1

P(a)

I1[x ! a]M2

P(a) P(0)

is a model (* = ;)

I1[x ! 0]

P(a) P(0)

is not complete because of I = fP(a), :P(0)g


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


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

aR

Possible Solution (not here)

Apply rules to known objects only,those explicitly mentioned:


Transformation to Clausal-like Form (1)

Input: BS-AEL formula = 9x 8y (x,y)

Problem 1: Skolemization (with rigid Skolem constants) is not correct:

9x 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 (B1 Æ ... Æ Bm Æ L Bm+1 Æ ... Æ L Bn ! H1 Ç ... Ç Hk Ç L Hk+1 Ç ... Ç L Hl )

where the B's and H's are atoms


Transformation to Clausal-like Form (2)

Input: BS-AEL formula = 9x 8y (x,y)

Output: equivalent formula 9x (8y1 C1(x,y1) Æ ... Æ 8yj Cj(x,yj)) where each Ci is of the form

B1 Æ ... Æ Bm Æ L Bm+1 Æ ... Æ L Bn ! H1 Ç ... Ç Hk Ç L Hk+1 Ç ... Ç L Hl

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:


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 9x 8y on the right

Example instance:


Model Existence ProblemGiven: - and * (if * is finite then test below is effective) - -formula = 9x (8y1 C1(x,y1) Æ ... Æ 8yj Cj(x,yj)) in clausal-like form

= 9x f C1(x,y1),...,Cj(x,yj) 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 PK/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 PK/U [ P(d) ² A

(2) for all A 2 U there is a d 2 * such that PK/U [ P(d) ² A

then

(1) M = f I j there is a d 2 * such that I ² PK/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


Illustration

= 9x f P(x), P(y) ! L P(y) g * = * = f 0, 1 gI1M P(0)

: P(1)

Computing the epistemic model MGuess knowns K = f P(0) g and 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 2 K and for all d 2 * it holds PK/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 PK/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


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

the bernays-sch ö nfinkel fragment of first-order autoepistemic logic

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