The Bernays-Schönfinkel Fragment of First-Order Autoepistemic Logic Peter Baumgartner MPI...
-
Upload
imogen-robbins -
Category
Documents
-
view
218 -
download
0
Transcript of The Bernays-Schönfinkel Fragment of First-Order Autoepistemic Logic Peter Baumgartner MPI...
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
The BS Fragment of FO AEL 3
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“
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)
= 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
The BS Fragment of FO AEL 7
Propositional Autoepistemic Logic – Examples (2)
= 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
The BS Fragment of FO AEL 8
Propositional Autoepistemic Logic – Examples (3)
= 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:
The BS Fragment of FO AEL 9
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() [ *
The BS Fragment of FO AEL 10
= 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"
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]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
The BS Fragment of FO AEL 13
First-Order Autoepistemic Logic – Examples (2)
= 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]
The BS Fragment of FO AEL 14
First-Order Autoepistemic Logic – Examples (3)
= 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
The BS Fragment of FO AEL 15
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 AEL 16
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:
The BS Fragment of FO AEL 17
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
The BS Fragment of FO AEL 18
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:
The BS Fragment of FO AEL 19
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:
The BS Fragment of FO AEL 20
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
The BS Fragment of FO AEL 21
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
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
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