The BernaysSchnfinkel Fragment of FirstOrder Autoepistemic Logic

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The Bernays-Schönfinkel Fragment ofFirst-Order
Autoepistemic Logic

• Peter Baumgartner
• MPI Informatik, Saarbrücken

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Motivation Logical Infrastructure for Question
Answering
FrameNet 550 Frames 7000 Lex Units
Text BMW bought Rover from BA
Abstract View Buyer BMW Seller BA
Goods Rover Money unknown
Com GT
Buy
Sell
Linguistic Analysis
BMW Rover
BA Rover
Deduction System
Logic
Which Logic? Which deduction system?
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A Logic Programming Approach
Stable Model Com_GT(e) Buyer(e,BMW) Seller(e,BA) G
oods(e,Rover) Money(e,unknown)
Com GT
Buy
Sell
smodels KRHyper

• Normal Logic Programs
• (Partial) inheritance
• Default values
• Domain restricted

BMW Rover
BA Rover

• Something to work with quickly
• "almost stratified" programs
• Modular, polynomial translation
• "semantical gap" between frame formalism
  and logic program

-gt Not in this talk (Joint paper with
A. Burhardt available)
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Description Logic Autoepistemic Logic

• DL with L-Operator
• Inheritance
• Roles
• Integrity constraints

Com GT
Epistemic Model
Buy
Sell
BMW Rover
BA Rover

• BS-AEL Calculus
• uses decision procedure
• for function-free clause sets e.g. instance based
• method

• Cannot deal with all positive 9 roles in TBox
• High Level Language
• "DLRules" interesting
• for "Semantic Web"

BS-AEL
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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 and complete
for finite ? - 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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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
Scientific Interest

• Goal "efficient" operational treatment of
  BS-AEL, by exploiting known first-order
  techniques
• BS-AEL not operationalized so far. Why?
• Combination DL AEL rule language
• Application areas inferences on FrameNet,
  Semantic Web, Null Values in Databases

Future

• Pressing question decidability with infinite
  domain ? - factor model of finitely many
  equivalence classes- implicit additional
  assumptions, e.g. any ground atom containing an
  element from ? must be unknown- decidability
  without 9 is known (Niemelä 1988)
• Translation into logic programming framework
• Bears some resemblance with MACE-style finite
  model computation for first-order logic. Exploit
  somehow?