Optimization and Relaxation in Constraint Logic Languages

Optimization
Kannan
Dept.
and Relaxation
Govindarajan
of
Computer
SUNY
at
Buffalo,
Bharat
Science
Dept.
Buffalo
NY
govin-ktlcs.
f alo.
edu
Computer
SUNY
at
NY
bharat@cs
and
naturall~
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gramming
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optimal
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concept
Essentially,
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Technology
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problems
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proposed
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1
Optimization
Languages
Surya
Science
Abstract
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Logic
Jayaraman
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in Constraint
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91
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truth
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computes
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timization
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timization
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introduce
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definitions
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with
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ate
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92
a total
a class
as the
with
for
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22]
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includes
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setwo
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Ross
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To
[6].
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operations,
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Sudarshan
of
with
models
By
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[13]
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aggregation
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aggregate
Stuckey
that
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[23]
aggregation.
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aggregation
first-order
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[20] provide
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providing
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To compute
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alent
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of deductive
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prefercontrast,
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direct
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recent
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and
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to
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by
with
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provide
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et al [5] considered
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To
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transformation
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of optimization
(PTSLD)
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predicates
in
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among
field
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at
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predicates
predicates
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in a simple
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discuss
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minimizing
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are interested
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optimal
semantics
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ordering
we consider
relax
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to
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[16]
into
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programs
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PLPisgiven
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also
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and
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lution:
notion
preferences.
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example
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as well
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distinguishing
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approaches
[18, 3, 4], the advantage
gives an explicit
treatment
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that
finding
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provide
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solutions
optimization
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ordering
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with
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offers
and
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tion.
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Marriott
allows
3] only
pressed
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treating
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criteria
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[24].
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worlds.
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by
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ence
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restricting
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optimal
consequence
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translating
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incorporate
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mantics
RELAX
either
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local
ness for
our
to
propose
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erence
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ifies
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ation
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to be the
set
program
P
a correct
predicate
.411 the
{A
and
optimal
call
graph
of constraints,
tion
atom
suffers
the
following
program
such
P
that
wzth
and
the set
respect
solutzons
relaxed
for
correct
to P
p(o
that
valuatz.
ns
empty.
( -)
1.
clause,
, p(~#2}.
, p(q6’3}.
instance-the
P
similar
among
the
m
a cycle
this
to
the
only
get
assigned
generahzed
of the
number
and
C( ii)O,
are satisfiable.
order
move
to
the
are not
96
that
tive
In
Her-
[12]
obtain
worlds
solutions
to the
$3 is also
notice
not
this
that
using
an optimiza-
intended
preference
of p(~:
valuation
WO1M
relaxable
a relaxed
because
wit h the
query
correct
would
t), such
valuation.
hold
that
th
is L93. By
irrespec-
both
p(~d,
❑
a satisfactory
have
of the
solution
any
argument
of substitutions
that
c(U).
.91 is the
correct
than
62 and
satisfy
instances
a relaxed
is better
solution
reasoning,
Furthermore,
O-predicates
62 is not
UQ —which
of p(~);
that
that
in the
following
{p(f)@l
solution
p( ~
that
are worlds
the
{p(i)o,
G,
for
there
contain
~
world
for
solution
further
p is defined
~
a goal
of P.
Since
w,
in
optimal
Suppose
loss of generality
2. w~
The
a valu-
c(u,).
solutions
of them.
that
81 is the
satisfy
nomoptimal
both
model
of the
Suppose
not
without
I P & A}.
prechcates
refers
two
c BP
valuation
consequence
01 does
pruned
n levels
consequence
Sketch:
Assume
model
an
Proof
clearly,
93 are
ordinal
Base
theory
valuation
a relaxation
defining
preference
A is a preferential
logic
model
of clauses.
Proposition
1.
k + 1 con-
clauses
solutions
proposition.
for
k +
ordering
examples
Herbrand
is defined
d is said
bra,,cl
for
model
G if GO is a preferential
same
The
stage-
of hierarchic]
k + 1.
programs
We begin
from
P ~ A) if A is a preferential
consequence
model
at level
n.
The declarative
se-
a preference
topologlcally
a model
clauses
optimal
at level
with
solutions
the
notion
the
Relaxation
definition
pref-
level
conflict
Furthermore,
the
main-
orderings
k + 1.
a preference
whose
we say
can
model
k + 1 ex-
at level
the
1.
the
of P is the
program
(written
of the preference
Dp,
k +
n levels
model
(1-predicates
3\Ve
in
O-pred~cates
at
model
model
level
to
of the
of
O-predicates
that
at
P with
preference
the
worlds
is referred
program
and
at
of the
becomes
we
optimal
of Relaxation
4.3.1
tion
of the
at level
If
the
might
opti-
orderings
too.
LS empty,
worlds
preference
instances
world
the
at level
description
Given
k with
bodies
laxation
clauses
are true
consequences
each
is enforced
O-predicates
mantics,
the
levels.
Theory
logic
Definition
prefer-
that
model
preference
those
worlds
A c Bp,
the
ordering
consequences
preference
defining
the
n.
1 so
defining
O-predicates
set of atoms
in
L’-
at level
possible
the
a
of the
intended
level.
the
the
as in [2],
level
the
a
have
arbiter
all the
world
at, level
O-predicates
at level
The
1 in
preference
The
the
defining
1 is the
k + 1 so that
the
tailed
by
1.
clauses
the
intended
reader
program.
1 contains
of preferential
clauses
tains
level
optimal
model
at level
The
clauses
only
to
disallowing
captures
solutions
to programs
clauses
1.
the
the
O-predicates
set of preferential
at level
worlds
tend
ation
the
strongly
The
model
is enforced
at
model
minimal
the
terms
they
in the
lower
develop
C-predicates
clauses,
the
in
at level
the
world
for
1 in
at level
that
at level
and
a model
O-predmztes
some
of
definite
Each
program
logic
only
model.
becomes
only
Since
minimal
it
defined
O-predicates
mannerj.
unique
that
The
1 are
other
recursive
yi-eclicates
-.‘
level
at the
because
to preference
in n
ory
at
higher
at one
thereby
Model
4.3
goals
O-predicates
clauses
level,
to the
regard
all levels
at differ-
obtain
the
as follows:
1. The
at
for
construction
constructed
us to
without
clauses
arbiter
worlds
enables
a
assigned
Suppose
arbiter
by
contribute
of
level
set of worlds
optimization
stages.
the
This
one
at a lower
wise
program
ordinal
to
in
The
in
in the
in
01,
to
Programs
at
one
enforced
approach,
different.
by
tained
w’ different
w ~ w’.
Models
are
solutions
those
4.2
above
levels
mal
+.
w:
no world
In
such
preference
minimizes
optimalif
that
is a formula
A supported
model
to be strongly
from
A.
instances
C-predicate.
definition,
of the
we need
O-predicate
to
rethat
Definition
z
franle
.4 preference
OJ a preference
<I subset
the
of the
worlds
relntzon
set
among
among
frame
F1
is sad
to
be a sub-
frame
Fz
if the set of worlds
o,f worlds
in
F2
the
the
worlds
worlds
in
and
FI
~n FI
the
is
relation
worlds
logic
program
of
M,.
which
3
kwulde
Gtuen
query
G=
erence
nloclel
tended
jJreference
a preference
RELAXp(~
for
P
WRT c(z),
the
(if
m FI.
and
G
modelM
the
P
the WOVICIS in M such that
the
appear
ZII each (oorla’ correspond
such
only
to
a re-
relaxecl
M,
that
c is a constraint,
in-
contains
instances
valuat~ons
ofp(t)
that
the
that
satisfy
laxed
c(u).
1
Given
laxation
query
ists
is unique.
and
G,
Proof
Sketch:
tendecl
preference
preference
ence
aprejerence
the
any
intended
preference
moclel
model
P
and
preference
logic
is unique
is constructed
Since
the
interpreted
uniformly
the
intended
relaxed
relaxed
program
at any
erence
a re-
model
intended
modeiex-
and
[8].
The
from
the
C-predicatein
across
the
program
preference
relaxed
showing
model
is well
is
relaxed
in
are used
the
❑
defined.
example,
tionland
consider
the
the
relaxation
WRT notinpath(c,
P).
mode]
only
will
such
contain
that
in
the
paths
b, C,P)
path
query?The
is
are
relaxed
preference
between
a andb
in
model
that
true.
will
instances
optimal
correspond
do not
pass
C ,P)
whose
level
cates
at level
we can
of
used
c.
Programs
We
now
grams
extend
where
The
difficulty
in
programs
each
tnstance
body
might,
and
Let
previous
clause
to
consult
in the
than
goal
of each
that
of 02.
to the
relaxation
goal
head.
of level
bodies
level
a model
as the
the
O-predicate)
is de-
Therefore
each
for
and
definitions
O-predicate
n + 1. ’31e
relaxation
the
goals
the
world
core
of the
in
bodies
predzcates,
relaxed
M,
and
of
logic
the
clauses,
a relaxation
preference
of the
intended
P
predicates
level
4
2)
but
only
query
with
G = RELAX
mode]
preference
for
P
and
model
of
WRT
G’ is
3(
where
each
member
c(Z))
of R is
c is a C-predzcate
1 of the
index,
(or
a
of the
set of
P.
A set
program
is an instance
2 is
programs
is the
set S is an extension
be
2, the
at level
of the
(P(~j,
may
(not
necessarily
of p(f)
that
query
are
RELAX
relaxed
p(fi
consequences
WRT
C( ii,),
goals)
A
w =
q is an
O-predicate
by a clause
(w~th
m
in
the
ordinary
program
and
n
re-
if
the
of the form:
world
(S, R)
holds:
‘J
M sazd
For
that
to S,
if
q(Z)O
A
Al
q(~)~
dl(til
), . . . . dl(tit
))
that
to
n
l...
rj (U3 )OuJ
such
a clause
instance
belongs
~=
such
ground
satisfies
mandul,
Zs a valuation
to satisfy
every
S,
such
then
that
is satisfiable,
rJ (VJ)6UJ
belows
(where
of the
there
the
0
head
exzst
II, ,
conjunction
i.e.,
to the
,for
all
j,
set in R
at level
level
P(9
model
base
at level
Suppose
~%pb(Fc)8vL
it be-
symbols,
relaxation
one
that
at level
of the
1, and
to R if p(i)
defined
laxation
to belong
1 =1...
at
with
so that
without
body,
as outlined
goals
preference
where
at level
relaxation
is a valuation
in the
semantics
predicates
program
., ~redi.
in the
models
O-predicates
set of sets,
set of instances
‘Based
a preference
in the
consequences
followtng
O-
is constructed
D-predicates
defining
in a
at the
an
2,
O.predicates
be easily
appears
is at least
any
i.e.,
model
an
can
(at
level
O-predicate
containing
that
them.
as before,
least
(or
have
O-predicates
for
that
2 in
at level
goals
relaxation
S is a subset
as R(P(r),C(ti)),
Definition
O-
program.
Given
frame
the
here
goal
an
the
for
k
illustrate
at level
clauses
k.
1, the
in
appears
is less than
of the
as in CILP ). The
of P and
model
that
defining
presented
level
a level
the
O-predicate
a D-predicate
defining
of the
1 in the
same
1 do not
same
extending
comes
the
of 01
is the
O-predicates
01
k +
model
preference
we
with
up to level
semantics
the
the
ground)
of an O-predicate
(at level 1) and c(ii) is an instance
of a C-predicate
(or a constraint,
as in CLP)6
and R(r( t),ctti))
the
goal.
construct
world
I?(P( ~),.~ ~)) belongs
preference
the
of a clause
level
of a relaxation
of clauses
instances
the
is at the
at level
I is the
body
the
to
O-predicate
semantics
n, it is assigned
predicates
goals
The
if
m terms
that
and
any
the
relaxation
levels
program
case when
However,
fined
of the
ordinal
in
level
1. We now
relaxation
of D- and
O-predicate
preferential
is that
goal
relaxed
p is an
pro-
preference
for
a relaxation
denoted
constraint
bodies
the
to
any
track
R is an indexed
a set,
Since
relaxed
definitions
(S, R.) where
of clauses.
in
a different
that
in
Oz is such
with
bodies
theory
goals
truth
assign
us assume
extendecl
by
the
we
above
in the
a model
relaxation
and
1.
Once
are of level
at
consequences
have
the
worlds
models
queries
the
pref-
define
paragraph.
to keep
in the
structure
Goals
presented
occur
providing
the
a relaxation
~Jredwate
goals
of
D-predicates
stages.
theory
need
before,
Relaxation
model
with
to ascertain
As
in
the
relaxation
logic
model
with
most
the
preference
preference
O-
defined,
O-predicates
the
relaxed
1 cannot
In order
a pair
4.3.2
is at
been
k+
in
an
the
worlcls.
describing
the
determine
in the
worlds
to shortest
through
to know
We
re-
goals
intended
the
at level
defining
the
model
may
with
k has
of relaxation
worlds
by
For
query
level
whose
G
the
relaxation
orders
intencled
models
the
levels
geuerai,
by defining
relaxed
goals
process
detail.
we need
intended
(a ,b,
The
in strongly
sec-
b, C,P)
relaxed
of sh-path
also
true
from
sh-path(a,
the
instances
P)
that
program
RELAX
worlds
those
notinpath(c,
sh-path(a,
shortest
various
relaxed
query
higher
defined.
at a level
at
correspond
set of atonls
in the
In
once
been
arbiter
the
to define
above
has
for
G’.
at
(with
only
present
a relaxation
goals.
be defined
also
should
a relaxation
model
preference
greater
For
the
to relaxation
are
world
P and
a program
can
constructing
those
The
D-predicates
level
the
the
are
of P and
model
how
determine
IVhen
goal
set of worlds
level
that
C(U)
optimal
and
preference
respect
prefer-
relaxation
the
can
in-
intended
intended
the
worlds,
P, its
of
at that
such
world
of p(f)
of relaxation
preference
iutendecl
For
model.
logic
and
terms
of clauses)
preclicate
Theorenl
strongly
O-prediccdes
bodies
instances
of a program
consequences
the
of
consequences
in
M
each
C(Z) is satisfiable).
in some
relaxed
be defined
in
in
instances
that
model
The
all
worlds
appear
the
such
are true
ale
the
corresponding
preference
pref-
of the
M,
that
to valuations
P and
is a sub-frame
for
all
of p(f)
the
that
Definition
contains
instances
among
a restr~ctzon
to the
that
zn F1 w
for
61t
O-
on
the
rather
should
pre-mterpretatlon
than
be
noted
the
that
ful enough
to capture
the
crlterlon
M an O-vredtcate
c(ti],
pred]cate
a sub-
paper
P such
97
We
h;ve,
that
Herhrand
the
Interprets
the
constraint
pre-mterpretat]on,
semantics
presented
here
M power-
meaning
of programs
where
the relaxation
that
IS at a lower
level
than
the relaxable
however,
not
discussed
such
programs
in
this
indexed
such
by (r] (vj )19, Cl(ii]
that
p,(~,
Definition
(at
)$q,
5
level
2)
laxation
Suppose
defined
goals)
q(~)
)$),
q M an
world
lon~s
j
=
~,
is satd
For
that
a
for
belongs
=
k, I),
valuations
curs
in
and
are no
in
world
(S, R) in the intended
corresponding
the
In
of the
assign
the
a clause,
the
The
instance
example
truth
R
the
{
n-sh-path(l,
a, b,5,
10,
u],
n-sh-path(l,a,c,
15, [e(a,b)
Notice
by
that
later
r~ (ti~ )0 (or
if a variable
the
in
the
CJ(iiJ )0)
clause,
same
ordinary
the
set S and
section
is not
thatforthe
instance
to be present,
we need
goal
RELAX
6
The
of all possible
predicates
preference
worlds
at level
as supported
by the
as determined
7
level
2 M the
some
strongly
The
set
that
that
instances
set
satisfy
the
the
The
arbiter.
by the
Definition
at lel~el
(S, R)
2 such
at
P =
[e(a,c)].
to
C >
15 )is{sh_path(a,c,25,
goals
valuation
optimal
world
the
set in
of P
Once
that
the
preference
respect
model
relaxed
level
able
2 consists
clauses
exzsts
definzng
the
worlds
The
the
set S.
without
in
consequences
members
at
of the
model
to relaxation
also
be
and
a relaxation
are
sub-frames
the
set
with
goal
of the
G
predicate
(whose
model
of G such
criterion
that
at level
also
for
of
at level
(S, Ii)
fies
of
in the
the
k, each
world
O-predicates
2 such
preference
clauses
model
defining
at level
predicates
relaxed
k is such
k.
that
n as described
For
edges
example,
{edge
(a, b,5)
set of preferential
{edge
(a, b,5),
, edge(b,
c,lO)
consequences
edge(b,
n-sh.path
c,l
O),
in
path(a,
[e(a,b)]),
c,25,
[e(a,
path(b,
c,lO,
the
logic
set
program
relaxed
intended
case for
preference
[17,
S
tn
some
model
P wzth
relax-
preference
model
logic
programs
8].
the
operational
semantics
relaxation
operational
goals
semantics
bodies
of prefer-
[8],
for
we present
programs
with
of clauses.
Semarrtics
the
tains
it satzs-
review
which
of
a derivation
stands
for
computing
in
[8].
of definite
Optimization
sense
can
and
of
we
[10,
11];
Tree
optimal
assume
optimization
be extended
schernecalled
Pruned
the
Below
constraints
5.1.1
with
c,25)}.
the
valuattonsto
that
clauses
as goals
the
the
con-
constraints
describe
case where
in bodies
queries
program
without
we subsequently
to the
PTSLD-
SLD-derivation,
how
this
program
com
of clauses.
P be
The
all
[e(b,
a definite
clause
SLD-tree
for
of whose
Pu
c,25)}
l? TSLD-Derivations
A partial
{G}.
a substitution
c)]),
c)]),
98
branches
Because
a substitution,
U
found
{path(a,b,5,
to
~n themtendedprejerence
without
in the
briefly
scheme
world
1 is:
edge(a,
belongs
aground
If a preference
program
, edge(a,
at level
P,
consequence
Semantics
of the
efficiently
Let
the
to the
reviewing
p~esented
above.
consider
the
goals
goals
now
for
logic program
has n levels of predicate
symbols,
the relaxed
intendecl
preference
model
is the preference
model
at
level
body,
programs
derivation,
con-
k – 1. Each
at level
zf A
Operational
We
that
of the
(satisfiable)
to the
at most
)}.
logzcprogram
Foranypreference
briefly
logic
5.1
instances
present
has access
of level
is C = 25,
preferential
(S, R)
Operational
sists
Now
the
only
2)
in S.
sequences
In
the
by (sh_path(a,c,C,P),
[e(a,c)]
A)
is similar
relaxation
P
level
instances
corresponding
of G are
re-
2 can
models
is
onlv . those
contains
model,
)
the
IS unkque.
an extension
with
of level
relaxed
O-predicate
preference
world
relaxation
the
~
relaxation
After
determined,
program
O-predicates
Essentially
S in each
relaxable
of the
queries
determined.
been
of the
2
proof
5
set S zn
the preference
2 has
of the
WRT C > 15.
sh-path(a,c,C,P)
world
zn the
and
among
models
P
optimal
goals
ordering
at level
preference
[e(a,c)]
truth
of P.
the
2.
various
,a,c,25,
the
preference
to be arelaxed
(written
ence
the
for
M satd
strongly
relaxation
worlds
in
n.sh_path(2
Gzvenapreference
aton~A
the
(S, R)
2. The
S are:
)}
Thesetinllindexed
Definition
in
truth
among
are
in
ground.
of O-predicates
that
at level
one
),
intended
optimal
relation
of preferential
of atoms
is only
,e(b,c)]),
sh-path(a,c,C,P)
oc-
the
illust~ates
model
model
to determine
relaxed
Theorem
Definition
c)]),
a world
of the
necessarily
[e(b,
[e(a,b)]),
[e(a,c)]
the
value
whether
instance
2, there
are present
[e(b,c)]
n_sh.path(2,a,c,25,
such
by consulting
relevant
that
n-sh-path(l,b,c,
indexed
of the
is determined
by
that
to determine
by consulting
goals
is indexed
goal.
to
have
words,
is determined
relaxation
R that
of
other
satisfies
body
body
valuations
variable.
(S, R)
the
in
the
in
level
preference
of n-sh-path
c,lO,
c)l)}
O-predzcatesat
be-
laxable
goals
sh.path(b,
is a valua-
valuatzon
be consistent
clifferent
U
b) ,e(b,
the
of S.
must
two
e(a,b)])}
[e(a,
there
corresponding
The
c),
Since
A
~1
set
is a
if
q(I)O
1 . . . m
all]
to the
all
a clause
q(i)$j
~,p%(%,)$~,
where,
for
M a member
v,,
c,15,
set of instances
such
con~unction
and,
[e(b,
[e(a,b)]),
sh-path(a,
n re-
...
instance
valuations
r] (VJ)8C3
C,(.ii,)t))j
c,i5,
{sh-path(a,b,5,
program
and
WRT c~(ti~).
ground
is satzsfiablel
p,(~t)d?l,
the
ordtnary
to satisfy
every
exist
.t}tat
the
I . . . IL stdch
such
in
m
WRT CI(Z1),.
Zf there
r,(tij)$aj
tlon
D-predicate
r~(fi~)
(S, R)
holds:
(rj(fij)(i,
path(a,
i, q, is a valuation
of the form:
w =
to S
all
(wzth
*pl(:tl),RELAXrl(til)
followzng
for
of S.
by a clause
p,,,(~~),RELAX
A
and,
is a member
on the
program
Pu
every
edge
path
is the
from
the
each
node
to the
not
derivations
is labeled
in the
of the
root
goal.
SLD-tree,
SLD-tree
composition
node
a positive
or failed
inthe
with
G
is a finite
are successful
we associate
which
and
{G},
SLD-tree
substitutions
of the
tree.
of
by
Definition
Given
Pu{G},
Tl
we define
by choosing
o,f T1 , Am
two
TI
+
partial
a non-empty
bezng
the
SLD-trees
Tz to mean
selected
that
TI
cmd
Tz
Tz is derived
leaf
1 =+
Al,..
goal,
and
creating
jor
A
from
., An,,
. . . . Ah
chddren
A
of 1
for
(A,
,...,
general
A
B,,,,
+-
unifier
be expanded
in
and
deraoation)
G
.,.,
TI
to
get
be a goal.
T2.
a Preference
Logic
derivation
derivation
for
node
every
most
the
or
that
leaf
0
M
description
_
-
such
the
of
to
above
as we
would
children
for
all
appear
clause
for
of the
the
body
by
the
arbiter
proofs;
the
of
to
for
at
the
one
met
the
goal
variable
for
Definition
original
=-
at
differ.
be com-
or pruned.
if
result
the
it
ends
of the
in
a
complete
respect
to P.
at
another.
we simply
If
for
argument
of
...,
a partzal
A,
in T
SLD-tree
an
and
is said
B1 , . . . . Bk
Dl,
...,
such
Dn.
Dn,
is p
t)where
that
and
nl
to be pruned
and
nz
node
and
By
induction
soundness
for
we can
<
answer
answer
preference
logic
PTSLD
and
this
ing
from
nodes
nl
p(~tt2
is an instance
and
n2
02 are
such
that
of p(iil
the
Stratified
out
that
p(~OI
) and
are
not
is
~ from
some
least
appears
JQZ, then
of
zs sub~’ect
instance
with
in-
■
a cor-
exists
-y such
that
for
can
q~.
because
be completed,
PTSLD
search
is infinite
but
to the
.9 =
arbitrary
arise
cannot
the
a computed
tree
there
emanatis a well-
goal.
Logic
Programs
the
PTSLD
that
logic
holds:
to the
an instance
body
of a ~
< ~(p..).
For
tree
is finite,
optimization
preference
following
if
search
the
set of O-predicates
preference
in
that
O-predicates.
that
is
A
if the
n, such
in the
f(l’1)
the
because
There
set
of an
clause
defining
any O-predzcate
predi-
program
P
is
is a map-
{1, . . . . n},
for
O-predicate
PI
an O-predtcate
P, its rank
is
to be f(P).
as the
of p(d),
constraint
when
arise
stratified.
ping
associated
an
even
n =-
restrictive
the following
Since
G, and
complete
Preference
can
if there
. . ..Ln
substitutions
not
goal
answer
O-
interpreted
to P, we say that
if there
the
goal
optimal
to be stratified
of
and
when
to
when
P, a goal
respect
some
SLD-
solutions
semantics.
Incompleteness
an optimization
PTLSD
of
are
to
is sound.
a substitution
for
happen
the
for
answer
uninstantiated
are
program
answer
of the
clauses
program
programs.
correct
All
~
is complete
derivation
can
levels
pruning
derivations
logic
optimal
operational
9 of G with
q and
PTSLD
o,f
to
to the program
completeness
[15].
the
logic
semantics
However,
defined
p(fi)+L1,
,for
of computed
optimal
the
and
by the
that
a preference
optimal
on
to be sufficiently
show
set
respect
a preference
as the
— clauses
have
be the
a correct
clauses
considered
as the
Stratified
t+
definite
are
to
G with
0 u
the
and
said
of ~~e form.
p(ti)
then
Sketch:
It turns
a
are descendants
p is an O-predicate,
to P,
PTSLD-dertvatton
Given
tree
5.1.2
by
P U {G},
an internat
0 is said
we
position
T for
G,
respect
the
by
p,
wtth
$ zs a computed
Proof
defiued
this
replace
considered
of solving
this
is
use
a goal
a complete
search
the
are computed
over
purpose
of in-
made
and
to G
zf 0 c 6.
suppose
wdh
optimal
unbound
pair
requirement
and
of p being
binding
Given
Al,
to an arbiter
G,
G
rect
achieve
Gd
to P,
operational
creating
with
the
This
view
~
respect
Given
exactly
to
P
El is sa~d
wzth
voked,
hand
in
of
order
positions
run-time,
instance
and
where
solution
n~ =—
. . .. D.~,
P
incompleteness
10
addition
to
failed
complete
the
answer
3 (Soundness)
exactly
of each
right
purpose
clause
exz.sts a node
In
is
be
to the query
G
cates
IL, where
to
and
zn G}.
wrzte
predicates
optimal
— clause
in
these
optimization
prefer
the
be invoked
clause
values
is not
on
the
positions
arbiter
the
We
clerivations
heads
unification.”
nl
is said
successful,
a program
P
answers
P cmd a goal
goal
the
bodies
quantified,
treat
p must
argument
the
only
Furthermore,
O-predicate
by
the
the
with
Given
Gzven
Theorem
a
simplifies
and
unifies
existentially
a P
of p in any
to
and
is supported
only
GO.
O-predicates
a node.
because
node
is said
optimal
vc{r~ab/es
P.
TSLD-
selected
O-predzcate
restriction
instances
treat
needed
enforce
T,
12
optimal
p, we assume
it is a candidate
that
are
stances
argument
then
A),
is an
of the
semantics
we can
an
requirement
of an
restriction,
at those
unbound
derivation
either
To, . . . . T,
tree.
be a correct
the
To =
succeed.
of
sounclness,
variables
are
P U {G}
with result T., let @ = {6114 is the composition
the swb.stztutions
along
a successful
path
in T. restricted
clause
that
To,
O-predzcate
operational
variables
— clauses
of the
head
This
appropriate
— clauses
Since
the
(7c,
P U {G}
of an O-predicate
clauses,
if the
P =
is an
clause.
of the
only
11’hen
expanded
if instance
of multiple
of the
a PTSLD
paths
PTSLD-derivation
~f P ~
(TSLD
sequence
P U {G}
the
1 is said
be a definite
tnflnite
for
instance
one
otherwise,
answer
be
ground
at
“back
P
Program
Tc- A TO.
to
by
the
such
to
T,+l.
the
in
its
Definition
SLD-derivation
for
C;, a, TSLD
sides
Ak)$
The
Let
is a finite
SLD-trees
P
A,
A Tree
Given
that
in
and
cluery
in
B,
of Am
of partial
TL +
Bq, Arn+,,n,Ak)$.,
BI,
of P u {G}
{i, TI,
i,
A,,,
clause
program
all
PTSLD-derivation.
every
most
occuring
if
complete
o,f the form:
–
tree
plete
need
programs
do
However,
dynamic
allow
there
are
many
programs,
are
definitions
programs,
such
of shortest,
of O-predicates.
stratified
above
recursive
formulation
definitions
in locally
presented
not
programming
recursive
interested
logic
they
Hence
described
path,
we are
below.
satisfiable:
Definition
{p(i)@,
The
= P(ti)
substitution
Definition
11
p(7)&
Oz as said
Given
TSLD-derivation
irk whzch,
at each
scenda,nt
,
= p(z)}
U
to be better
a preference
U,{
than
log~c program
L#Itl
Z}
locally
$1.
There
P,
A preference
a Prunecl
an
is a mappzng
nocle.
There
ground
99
for
O-predicate
defining
logic
progrclm
if the following
{1, . . . . n}
(F’TSLIl-derivation)
is a ‘TSLD-derivation
step,
the leaf to be expanded
as not a cle-
of a prwned
13
stratified
an
the
P1
f
least
M a well-founded
of all
the
such
n
appears
O-predicate
znstunces
from
two
in
P2,
P
is said
conditions
set
of
that
the
then
f (Pl
ordertng
+k
O-predicates
O-predicates
zf an
body
to
be
hold:
to
instance
of a -
of
clause
) < f (P2 ).
over
of rank
the
set
of
k, defined
as follows:
(i)
predkcates
+~:
and
hale
ground
of rank
k
(ZZ) ground
the property
of rank
k
o,f the
tnstances
all
each
appears
O-predtcate
of
to
znstances
that
that
map
base
the
1
the
of rank
of
k that
O-pred~cate
is +~
the
appears
in
nocle
than
one
CLP
framework
the
head.
that
in
positions
biter
defining
of a ground
clauses
4
PTSLD
preference
logic
Proof
Sketch:
<k
of each
and
rank
one
are
induction
soundness
finite
on
the
used
and
ar-
for
locally
of
strati-
trees.
of SLD-resolution.
first-order
main
difference
node
by
in
the
a set
the
is not
the
the
solutions,
of constraints
clause
by
Pruning
node
when
program.
The
case
a single
is that
the
rule
out
the
associated
with
that
another
one
adding
can
solution
14
purhal
Gtven
SLD-trees
tomean
that
TI
program
and
T2 for
T2 is derived
from
P,
P U {G},
G and
we define
T1 by choosing
set
., A,n,
of T1 , choosing
T1 +
T2
a non-empty
goal
ct D-predicate,),
and
a clause
creating
({ Al,
Am
P
chitdren
A
~
head
C;,
is a C-predtcate
. . . . c;,
Bl,
or
. . . . Bq
Given
the
addition
Note
of
further
not
we can
that
be different
may
define
with
block
without
PTSLD-
constraints
PTSLD
in
derivations
constraints
a preference
a computed
ts is
P
optimal
a vuluatzon
node
in
for
bodies
of
@ that
zn a tree T
logic
program
valuation
satisfies
at the end
P
and
to G’ wzth
the
a
respect
constraints
at
of a PTSLD
derzvatzon
completeness
theorems
a
crnd G.
also formulate
soundness
similar
5
suppose
to the
Given
to P,
respect
to P.
proof
and
definite
clause
a preference
then
optimal
0 is a correct
is similar
to the
case,
logzc program
$ M a computed
respect
Theorem
in
P,
6
a goal
to
(whose
programs
16
G,
and
Ak}, {Cj})Cj})
a goal
above,
to
from
Al,..
The
optzmal
definite
P
valuation
and
answer
clause
u goal
for
G
wzth
to G
wtth
case.
two
kflf
/=({
more
in
set of constraints.
definitions
logic
a com
in
SLD-tree
need
similar
preference
has
set of constraints
a manner
G,
have
to the
a goal
to the
the
in
Theorem
solution.
a CLP
definition
to
the
bodies
the
-y.
programs
The
Definition
the
solution
logic
We can
solution.
a node
nz in
the
preference
in a manne~
but
by
pruned
in
a set of solutions.
solution
two
node
for
for
each
substitution
is accomplished
that
it is possible
prunes
semantics
logic
definite
labeled
set of constraints
multiple
operational
a constraint
of constraints.
to
the
is
from
tree
constraints
Since
describe
theory
each
blocks
solution
successful
Y\Te no~v briefly
be satisfiable
derivations
to
❑
the
framework
may
abstracts
one
Definition
of O-predicates
completeness
CLP
which
Therefore
n I and
Using
in the
it
clauses.
O-predicates,
instances
with
{m-y}
i.e.
another
by the
search
rank
ground
argument
another.
are complete
with
on the
those
not
over
derlwcttions
defined
only
that
solution
programs
By
the ordering
we consider
instance
to prefer
Theorem
fied
<k
a SLD-tree
way.
nodes
nodes,
Note
m
associated
a constraint
instance
the
Each
straint
clauses
of an
body
O-
order?ng
of optzmtzatton
instance
in
facts
of the
P
G
the
Given
and
wzth
that
that
P,
PTSLD
there
PTSLD
then
to
logic
there
that
program
derivation
tf 8 M a correct
then
tree
exasts
preference
the
G is finite,
respect
successful
u such
a stratified
C; such
is
optzmal
answer
a successful
is satisfied
a substitution
P
emanatmg
node
in
by an
valuat~on
that
$ = crq.
q such
of 1 of the form:
. . .. A.l,Bl,l,
Bq, A~+l,~+l,
. . .. Ah}.
The
{CJ}U{C}U
proof
is similar
to the
clefinite
clause
case.
{c:})
z.f {c~}
u {C}
U {cl}
constraints
generated
tn
the
body
of the
to be expanded
De firlition
is soivable7j
where
bg the equation
Am
clause
are
15
Gtven
Al,
. . .. AJ}.
{C,,,})
and
an
{Cn]],
such
that
w
p(t)where
biter
n,
M an
of the form:
p(z)
{p(i)
the
= p(z)}
We
{C,.,
the
that
then
in
the
constraints
‘F
n
where
({Dl,
p(~)
a variant
Z and
is an
the
where
f?l,
of n,
Definition
relaxation
,Dn,
},
relwr(p(i),
Dm
tncluding,
., ., Ln.
to
In
an
ar-
u~,{L,}
tree
q such
of the
{Cj},
etc
chffer
} U {Cn,
the
{C~l
~ is said
)
for
by renammg
new
of node
nl
that
states
that
to
clauses
— p~(~~),
and
a
for
. . . ,p,, (,tn]
p iy
for
p,
of clauses:
2.
-
Z #i
I p,(Tl),...
c,rbzter
clause
relux_p_ca(%)
—
for
LI,
y
ables.
and
flv
100
. . . ,pn(in),
c(ii)
)p(En)n)
of p of the form
c(i))
znclucles
~ relaz-p~.cz(%’z)
. . ,J5,1U11T2,
@l and
that
i81
The
and
C(W1),
OZ are
and
p(fl
) 3
the followzng
fQI , and
-
C(IW2),
variable
@2 do not
substitutions
al
crz is the
renamtng
share
zs the
most
substitutions
any
most
common
general
genera!
variunifier
unljier
of F2
Ft+z.
relax_pF_ca(Fl)
vamables
retax(p(i),
-p~-cn(~$l)
where
positions
L,,,
pI(ZI)i
clauses:
such
to
every
. . . .
Llalcmj.
and
solutions
argument
pair
p(i)
P
the function
the set of relaxed
clause
I
arbiter
of constraints
those
returns
program
c(u),
--+ Z = t
of ?I
a set
c(z))
to con~-
to queries.
logic
p(t>WRT
= RELAX
every
valuations
a preference
G
for
optimal
Gkuen
of Relaxation
of PTSLD-derivations
1. reluz-p-cv(z)
p(i2)
get pruned.
stand
17
relax
}
the
correct
query
Furthermore,
projection
to be pruned,
zf all
relaxed
the followzng
of p(~).
set
Semantics
an extension
addition
zs such that P(~)-Y M an
to the variables
tn {Cn, },
to be pruned
, to
with
that
is a constraint
node
of i obtained
al
}
tnstance
solution
is sazd
U {Cn,
present
the
a
where
is subject
--- LI,
constraint
{-Iy}
The
Operational
. . .. Bk}.
P U {G},
=({
. . . ,Dm,.
and
substitution
p(F)a
update
7\Ve use {C},
of i
n =
var~a~les
in {cn~
and the projection
a solution.
a node
~
T for
arsodenz
are descendants
= p(?@}
by the
) U {~-j},
M not
}),
O-predicate,
u {p(~)
(?1 Of v to the
instance
o,fp(ii)
is such
n,
We now
1 as sazd
constraint
zs satzsj$abie
o,
node
and
5.2
leaf
pute
SLD-tree
{C~,
znternal
The
is the set of
and the C~s
T2.
a lJartial
p
suppose
constraints.
in T1 to get
=({
nodenl
{C}
= A,
E#f2).
s
retax_pF_cQ(fz)
—
Ll,
. . . . L,l,
(t
#
The
relaxeci
For
query
example,
?-
corresponding
given
RELAX
the
query
WRT notinpath(c,p),
(X,
C, P), notinpath(c,
j,
Y)
a,bC,P)_noti
=
(a,
b)
P)) returns
the follow-
npath(C,c)(X,
1 path(X,Y,
y
,C,P),
C,P)
Y)#(a,
b)
-
npath(c,P)
not:
relax_sh-dist(a,b,~,Pl_notinpathLc,c~(X,y
(X,
C,P)
lpath(X,Y,C,P).
relax-sh-dist(.,b,~,p)-notinpa’thf.,~j(a,b
,C2,F’2)
—
notinpath(c,Pl),
notinpath(c,P2).
relax_sh_dist(G,b,C,p)notinpath(C,~~(X
,Y,C1,P1)
~
relaxsh.dist(a,b,c,p)notlnpa’thfc,c)(x
,y,C2,p2)
-
C2 <
Cl,
(X,
Y)
#
(a,
with
respect
set
relaz(p(t),
c(ti))
rela. ~_pt_cti
into
the
tainecl
the
clauses
from
plicable
to p(aint
C(U)
to
the
p(f)
are
applicable
c(z),
clauses
modified.
to p(t~
\,ersion,
one
and
clauses
for
are
instance
not
are
that
1
applicable
key difference
two
clauses
the
applicable
presentation
in
laxation
goals
version
of the
ing
on the
not
allow
and
therefore
not
for
in the
body
p is
relaxed
containing
toiwith
the
a very
with
to
operational
specific
which
occur
relaxed
the
allow
re-
the
relaxed
name
depend-
it is invoked.
in
the
semantics
of the
we now
of clauses,
and
bodies
[9] did
the
PTSLD-derivations
when
there
of an
is captured
of pairs
by
by enabling
a relaxation
Sketch:
By induction
P.
For
the
of
O-predicates
in
the
program
relax _pF_cfi
for
presenting
of program
and
the
_yi_cti (t)a
ence
model
(C’, T)
18
enforce
p satisfy
C.
inductive
case
where
Cisacollection
PTSLD-tree.
(C,, T,),
pair
and
timal
valuation
lfthe
T,+l)
following
and
Proof
Sketch:
O-predicc~tes
for
the
not
selected
toang-
in
a relaxalde
respect
in
goal,
to clauses
the
leaf
C,+l
=
to
Tis
apartial
from
thepazr
the
selected
is a relaxable
relux(p(i),
goal
c(ti)),
relaz_pr_cU(t>,
in
the
leaf
p(~)
T% +
relatable
and T, +
T,+l
to
T,
T,+l
goal
with
a stratified
qoalG’,
pairs
a relatable
(P, G),
clause.
is derived
(CI,
a preference
in
C,+l
=
is replaced
respect
T,
Sketch:
program.
the
G
. . ..{ C., T,),...
in
the
(C,,
PTSLD-tree
and
prefer-
clauses
the
argument
is
for
solutions
works
a preference
to
for
the
Jogic
computed
by
on
induction
program.
It
proop-
the
makes
levels
use
computed
of
for
the
relax-
■
of clauses.
preference
that
of P,
node
and
there
at the
8’
program
P and
PTSLD-derivation
suppose
then
tree
that
iogic
the relaxecl
G is finite,
PTSLD
GO is a relaxed
is a successful
end
of the
is a valuation
that
furthermore,
By induction
P
C,
is
each (C,+l,
a set
T,+l
the
RELAX
p(f)
and
nocle
relaxed
PT-
satzsfies
0 =
the
was
the
such
arbiter
jor
8’1)
relaxed
the
some
the
the
the
the
by
our
are
for
the
complete
are
correct
the
program
no
Therefore
optimal
is finite
operational
semantics
semantics
answers
was complete
to
was
rela.t.p;
answers.
for
.cti( ~
)
T,).
to
the
result.
101
relaxable
The
query
inductive
ale
concerned,
case is similar.
original
program.
we
comof
and
Therefore
the
declarative
semantics
of the transformed
program
with the relaxed
intended
preference
model
as far
transformed
C(Z)),
original
is capable
gram,
the
seman-
(the
of
for
answer
operational
of
it is complete
relaxation
P U relaz(p(f),
c(ii)
sub-optimal
semantics
relaxation
O-predicates.
operational
potential
O-predicates
the
Furthermore,
as there
by
from
of the
Suppose
as any
is computed
that
prunes
operational
p(~
defining
emanating
Therefore,
all
cluery
as general
p(F)
tree
case:
WRT c(ii).
of clauses
at least
search
on the level
base
PTSLD-derivations
and
query
plete).
a
the
bodies
program
and
For
Consider
the
by
2s a sequence
where
and
the
was
P
tics.
C; U
to the clauses
program
PTSLD-derivation
TI),
and T, is a partial
from
logic
instance
q.
Proof
the
Given
P
such
at
computing
19
arbiter
when
of answers
G such
derivation
goals
in C,+. I.
Definition
the
intended
.$ is a relaxed
in bodies
consequence
program
is
with
be erpanded
WRT c(fi),
is
the
query
from
an answer
the
RELAX
then
show
relaxed
GO.
proof
in
appear
successful
query
in C’%.
goal
P ~
Given
in
for
>.
‘~ If
we can
the form
be e.rpandedin
C% and
instance
preference
in the
in the
The
P
that
soundness
that
2
the
SLD
holds:
goal
If
such
The
above
goals
G,
prefer-
the
world
and
is an
intended
Similarly,
same
of levels
there
intended
ordering
the
O-predicates
Clearly,
world
C( u,)).
❑
then
assumptions
1. If
same
G is a goal
constraints
wclerived
P
prefer-
too.
of
the
the
c(z)).
7 (Soundness)
gram
preferential
as a deriva-
lsapairof
ofclauses
Thepair(C,+l,
the
Essentially
a relaxation
SLD-trees.
Aprograrn-tree
of
at
a relax-
number
in
in any
in some
of the
Suppose
is true.
model
occurs
and
relaz(p(~,
1.
relaxed
occurs
con-
af relaz_pF_cti(Qb’
the
C( ii )).
of P U relrsx(p(t],
substitution
Definition
p(fja
P
PU
case
a world
G as c(~)t)
preference
relax
only
of
is
in the
the
is a pair
is a rela~ed
on the levels
in
a world
of P and
program
if and
P U r-eluz(p(~),
to
if an instance
Lemma
did
is encountered.
derivation
partial
(1)8
a
to G
satdsfies
(C, T)
p(t~@
base
and
].
consequence
program
ation
O-predicate
them
goal
G,
the
belongs
@ that
logic
of P and
P
valuation
where
c(z),
WRT
Proof
emanat~ng
program
if there
a successful
of clauses
described
version
p(t~
in
lemma
specialized.
FVeextend
tion
gets
goals
name
Since
bodies
here
is
program
derivation
a preference
preferential
model
logtc
in T
PTSLD
consequence
Theorem
thebody
presentation
following.
arguments
the
our
in the
O-prechcafe
exact
rec[uire
This
is the
to occur
relaxation
to beso
ment
~]
between
Given
relux_pi_cz
to
T
optinlal
node
G = RELAX
intended
ob-
is ap-
as before.
The
a successful
goal
that
by adding
of a clause
to ~with
isnot
which
forp
a preference
is is a valuation
of a relaxed
end
p(~tl
O-pred/cate
is modified
p that
we obtain
that
for
clause
goal
is applicable
other
a new
Every
If some
then
that
the
the
forp.
herelaxation
body;
not
introduces
program,
to P
the
model
The
where
computed
at
ence
to be successful
derivation,
Given
straints
instance
b).
is said
the
a relaxed
is a relaxed
~
4.
G,
ential
,C1,P1)
Cl,
20
goal
ation
—
at
tree.
Lemuna
.1. relax-sh-dist(a,b,~,p~notinpathtc,~)(a,b
C2 <
PTSLD-derivation
(C, T)
Definition
set of clauses:
1. relax_sh_dist(
A relaxed
is a pair
PTSLD
sh_dj.st(a,b,C,p)
re~u~(sh_dist(a,b,
ing
to G is relax.p(f).
have
if
pro-
Since
the
coincides
as answers
our
desired
■
Theorem
ence
8 (Conlpleteness)
logic
ond
program
C; is a goal
anating
from
such
it
a successful
that
the
that
satisfies
P
and
constraints
that
then
end
at the
there
that
node
in
is
the
Sketch:
The
is a
Trondheim
there
O-predtcatesin
the
is by
induction
program.
on
It makes
the
levels
of
Brown,
ematacs
6
Conclusions
The
concept
problems
earlier
work
provides
requiring
[8],
be specified
a unifying
approach
optimization
we showed
how
declaratively
logic
programming.
@ showing
how
the
and
of
7th
Intl.
Systems,
Symp.
LNAI
689,
and
a
T.
Wakayama.
Unified
Approach
Reasoning.
Preferto
Annuls
Intelligence,
Non-
of Afath-
10:233–280,
1994.
On
optimization
in the
[4]
problems
paradigm
of preference
This
paper
extends
our
paradigm
can also capture
notion
of
given
by practical
preference
a query
considerations,
relaxation
RELAX p(f)
c is a C-predicate,
satisfies
c.
c and
to answer
the
the
best
we must
cluery.
We
for such
notion
of
preferential
relaxed
consisted
changing
the
defining
p ensure
LNCS
erential
the
program
to
transformed
The
if
for
to the
with
to
start
with,
they
arbiter
were
are
for computing
also
[7]
We
to
are investigating
efficient.
Given
operational
so that
the
[8]
clauses
for
preferential
better
query
to it.
For
paths,
force
in
for
every
evenlength
paths
that
are
a constraint
composed
such
the
not
c is not
is not
any
into
the
predicate
a node
if it
c(ti)
when
over
the
satisfy
in
inpath
distributes
some
of paths
path
as there
of odd
as evenlength
We
try
‘Jle
programs
programs
laxation
as the
goal
than
investigating
O-predicates
to be the
with
in this
the
level
the
in
same
can
level
in
the
relaxation
paper
of
body
of the
semantics
relaxation
as the
level
goals
be termed
the
re-
[11]
is passed
of a -
[12]
Therefore,
[13]
over
of clauses
when
goals
of the
in
had
to
in
be
head.
we allow
the
level
bodies
of the
-
O-predicate
at the
C.
Op-
PLILP
Zaniolo.
Logic
’94,
Minimum
Programming.
Principles
of
The
In
and
In
Database
and
Proc.
Systems,
model
A.
Proc.
5th
Symposzum
1081–1086,
B.
stable
Robert
editors,
on
seman-
Kowalski
,Jo~nt
Logzc
and
InternaProgranz-
1988.
Jayaraman,
Grammars.
Science,
Govindarajan,
and
Technical
SUNY
B.
Logic
at
S. Mantha.
Report
Buffalo,
Jayaraman,
Programming.
and
In
Programming,
K.
Goviudarajan,
in
pages
port
95-22,
falo,
1995.
Dept.
J. Jaffar
and
ming.
Proc.
In
B.
Pref-
94-27,
Dept.
1994.
S. Mantha.
Proc.
12th
731–745,
Jayaraman,
Constraint
Logic
Pref-
Intl,
Conf.
1995.
and
S. Ma,ntha.
Languages.
Technical
of Computer
J. L.
Lassez.
Idth
Science,
Constraint
A Chf
Languages,
J. Jaffar
and
M.
Symp.
pages
J. Maher.
A Survey.
SIJNY
Logic
Re-
111-119,
Re-
at
Buf-
Program-
on Principles
of Pro-
1987.
Constraint
Journal
P. C. Kannelakis,
of Logic
D.
G. M.
Query
Prznczples
length
relaxation
at the
K.
straint
em
in bodies
clause
O-predicate
Preferences
Relational
Symposium
Lifschitz.
Bowen,
Logic
ming:
a graph
distribute
present
with
Logic
Program-
Programming,
1994.
over
stratified
O-predicate
V.
C+ovindarajan,
gramming
B.
Kemp
and
with
[14]
P.
J. K.
D. B. Shmoys
a Guzded
Tour
We
and
Chichester,
Sons,
of
The
ACM
pages
Stuckey.
Con-
SYmp.
299-313,
Semantics
In Proc.
on
1990.
of
Logic
International
Log2c
1991.
Lenstra,
(eds. ).
strictly
are
J.
P. Z. Revesz.
Proc.
Systems,
Symposium,
E. L. Lawler,
and
In
Aggregates.
Programming
a re-
Kuper,
Languages.
of Database
Programs
paths.
considered
Handling
call,
are even
not
[10]
to
cannot
length.
does
on
Conference
laxation
the
the
P in
of P.
and
programming.
pages
on Logic
do not
in
that
A.
the
of p
recursive
structure
sub-path
sub-path
C(U),
we can
the
Sci-
1991.
and
Kenneth
erence
more
C(Z)
C-predicate
Sola.
Intl.
in
Symp.
logic
K.
T.
6th
S. Greco,
for
for
opera-
definition
recursively,
by pushing
example,
only
do not
the
WRT
the
Computer
con-
be made
p(f)
RELAX
if p is defined
“distributes”
which
can
C(V) into
that
is possible
because
if and
to p(~
efficiency
this
goal
“pushes”
solutions
Intuitively,
under
relaxation
a relaxable
However,
laxable
conditions
preference
semantics
succeed.
gain
for
Theoretical
Programming
Predicates
of Computer
[9]
semantics
Foundations
on
1994.
Gelfond
erence
sequences.
tloual
Predicates
Conf.
pref-
complete
complete
relaxed
by
respect
M.
ming,
clauses
relaxed
the
PTSLD-derivations
program
~
1.9th
and
Proc.
154-163,
tional
the
program
computing
changes
oper-
operational
original
PTSLD-derivations
are sound
and
844,
ACM
tics
for p
we introduced
The
[6]
may
answers
consequence.
The
p that
own
theoretic
and
changes
that
that
model
of p.
consequences.
p ensure
sub-optimal
relaxations
10th
for
In
S. Ganguly,
pages
p on its
of Optimization
Proc.
and
Logic
Maximum
the
O-predtcate
answer
for
the
program
for
best
J. Fowler,
Constraint
Essentially,
p is an
of i,ransforming
clefinition
resulting
the
provided
semantics
paper.
answer
consider
ational
semantics
this
In
Technology
timization.
previous
work
the notion
of
we introduced
where
we want
However,
satisfy
in
WRT c(ii),
Semantics
1993.
F. Fages,
in
[5]
Motivated
the
Languages.
to
relaxation.
relaxation.
lower
Mantha,
.4rtificzal
of Software
of preference
formulating
the
Logical
Hierarchies
1993.
in Deductive
F. Fages.
ence,
not
Proc.
Intelligent
Towards
and
in CLP
and
S.
Wakayama.
Relaxation
In
for
Logics:
T.
❑
[3]
can
and
Constraint
Norway,
Monotonicity
use of thelemrna
above.
In
of
Methodologies
[2] A.
Q = tl’q.
proof
Mantha,
Programming.
on
ence
Proof’
S.
Logic
PT-
8’
and
Brown,
Reconstruction
that
of the
G such
[1] A.
emsuch
of P,
at the
References
preferof clauses
PTSLD-derivation
tree
from
the
such
Ij
in bodzes
if O is a valuation
PTSLD
emanating
ts a valuation
ZS cz stratzj$ed
consequence
zn the
derivation
P
goals
relaxed
then
preferential
node
valuation
If
selaxatton
w finite,
G@ is a relaxed
SLD
with
A. H. G. R.innooy
Traveling
Combinatorial
United
Salesman
Optimization.
Kingclorn,
I<an,
Wiley
1985.
of the
clauses
[15]
head.
J.
W.
Lloyd.
Springer-verlag,
102
Foundations
1987.
of
Logic
and
Problem:
Programming.
[16]
M. J. Maher
and
in Constraint
Lnsk
P. J. Stuckey.
Logic
and
R.
A.
Overbeek,
~can Conference
19,39
[17]
S. Mantha.
on
[18]
ber
199].
Ii.
Marriott
st, ra,int
Logic
PhD
Preference
P.
J.
Programs
Programming
Proc.
Progrcammingj
thesis,
and
Logic
Query
Languages,
editors,
First-Order
Appltcutions.
Expanding
Programming
Amer-
pages
20–36,
and
of Utah,
Stuckey.
with
and
Systems,
and
H. D.
Warren.
their
Novem-
Semantics
of
Optimization.
Languages
E. L.
North
Theories
University
Power
In
(30n-
Letters
on
2(1-4):181–196,
1993.
[19]
F. (2. N.
Pereira
C+rammars
malism
for
and
Artificial
K. A. Ross
ductive
and
In
Systems,
S. Sndarshan
Relevance
and
in
with
Y. Sagiv.
R.
the International
Aggregation
Databases.
Conference
on
in De-
on Principles
1992,
Ramakrishnan.
Deductive
1980.
Aggregation
Symp.
114–126,
For-
Transition
13:231 -27’8,
ACM
pages
Clause
of the
Augmented
Monotonic
Proc.
Definite
- A Survey
Intell~gencej
Databases.
[If Database
[21]
Analysis
a Comparison
Networks.
[ZU]
D.
Language
In
Verg
and
Proceedings
Large
of
Databases,
1991.
[’?]
S.
Sudarshan,
(0. Beeri.
mantics
for
D.
Srivast
Extending
the
A.
van
Sets
Programs.
[24]
M.
Wilson
Logic
K.
JACM,
and
Pro..
pages
Ross,
Well-Fonnded
Programming.
1G:277-318.
In
Symposium,
Gelder,
and
R.
and
J.S.
Semantics
Borning.
Journal
and
and
Valid
International
590-608,
38(3):620-650,
A.
Ramakrishnan,
Well-Founded
Aggregation.
Programming
[23]
ava,
Schlipf.
for
SeLogic
1993.
Unfounded
General
Logic
1991.
Hierarchical
of
Logic
Constraint
Programming,
1993.
103