The%20Bernays-Sch

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The Bernays-Schönfinkel Fragment ofFirst-Order
Autoepistemic Logic

• Peter Baumgartner
• MPI Informatik, Saarbrücken

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Motivation
BMW buys Rover from BA
Starting point Some reasoning tasks on
ontologies can naturally be expressed as
specific model computation tasks
XML Schema
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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
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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

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Propositional Autoepistemic Logic
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Propositional Autoepistemic Logic Examples (1)
? L A (A "integrity constraint"), does
not have an epistemic model
I
I1
I2
M is sound but not complete take
M
A
A
A
B
B
B
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Propositional Autoepistemic Logic Examples (2)
? L A ! A ("select A or not") has two
epistemic models
I1
I1
I2
M1
M2
A
A
A
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Propositional Autoepistemic Logic Examples (3)
? A ! L A ("A is false by default") has
one epistemic model M1
I1
M1
A
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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)

RTAUNA justifies assumption that ? contains all
ground ?-termsand that every ground ?-terms
evaluates to itself ? HU(?) ?
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? 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 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"
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First-Order Autoepistemic Logic - Semantics
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First-Order Autoepistemic Logic Examples (1)
? 9x P(x) Æ L P(x) ("'Small' domains may not
work")
I1x ! 0
I1x ! 0
I3x ! 1
I2x ! 1
M1
M2
P(0)
P(0) P(1)
P(0) P(1)
P(0) P(1)
is not sound
is epistemic model
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First-Order Autoepistemic Logic Examples (2)
? 9x P(x) Æ L P(x) ("Elements from ? can be
known"). Models
I1x ! 1
I2x ! 1
I1x ! 0
I2x ! 0
M2
M1
P(0) P(1)
P(0) P(1)
P(0) P(1)
P(0) P(1)
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First-Order Autoepistemic Logic Examples (3)
? P(a) Æ 8x L P(x) ("Herband Theorem does not
hold")
I1x ! a
I1x ! a
I1x ! 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
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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
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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
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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 Bm1 Æ ... Æ L Bn ! H1 Ç
... Ç Hk Ç L Hk1 Ç ... Ç L Hl )
where the B's and H's are atoms
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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 Bm1 Æ ... Æ L Bn !
H1 Ç ... Ç Hk Ç L Hk1 Ç ... Ç L Hl Sketch use
standard algorithm for conversion to CNF
augmented with rules
L in front of disjunction
L in front of conjunction
Nested occurences of L
L in front of negation
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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
Example instance
Finally move 8y outwards to extend 9x 8y on the
right
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Model Existence Problem
Given - ? 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 fL 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
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Illustration
I1
? 9x f P(x), P(y) ! L P(y) g ? ? f 0,
1 g
M
P(0) P(1)
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Conclusions

• 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

• 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