Characterizations

and

Implementation

of

Static

Semantics

of

Disjunctive

Programs

Stefan

Brass

Univ.

Hannover

Inst

f.

Informatik

Lange

Laube

Hannover

BRD

J?urgen

Dix

Univ.

Koblenz

Inst

f.

Informatik

Rheinau

Koblenz

BRD

Teodor

C.

Przymusinskiy

Dept.

of

CS

University

of

California

Riverside

CA

USA

Abstract

In

recent

years

much

work

was

devoted

to

the

study

of

theoretical

foundations

of

Disjunctive

Logic

Programs

and

Disjunctive

Deductive

Databases

While

the

semantics

of

non-disjunctive

programs

is

fairly

well

understood

and

there

exist

several

implementations

of

the

best

known

semantics

-LRB-

well-founded

semantics

and

stable

semantics

-RRB-

the

declarative

and

computational

foundations

of

disjunctive

programming

proved

to

be

much

more

elusive

and

difficult

Consequently

as

of

now

there

is

no

general

agreement

concerning

the

proper

semantics

of

disjunctive

programs

Quite

recently

however

two

new

and

very

promising

semantics

have

been

proposed

for

the

class

of

disjunctive

logic

programs

Both

of

them

extend

the

well-founded

semantics

of

normal

programs

The

first

one

is

the

static

semantics

proposed

by

Przymusinski

and

the

other

is

the

D

WFS

semantics

proposed

by

Brass

and

Dix

Although

the

two

semantics

are

based

on

very

different

ideas

they

turn

out

to

be

quite

closely

related

and

in

fact

they

are

completely

equivalent

when

restricted

to

the

suitable

common

language

This

paper

is

devoted

to

an

in-depth

study

of

the

static

semantics

in

the

class

of

so

called

superprograms

which

properly

contains

the

class

of

disjunctive

logic

programs

We

first

establish

three

powerful

syntactic

and

semantic

characterizations

of

the

static

semantics

significantly

simplifying

the

original

definition

Subsequently

we

demonstrate

the

applicability

and

importance

of

the

above

characterizations

by

using

one

of

them

as

an

inference

engine

in

a

Prolog

implementation

of

an

query-answering

interpreter

for

static

semantics

in

the

class

of

super-programs

Introduction

In

natural

discourse

as

well

as

in

various

programming

applications

we

often

use

disjunctive

statements

One

particular

example

of

such

a

situation

is

reasoning

by

cases

That

is

why

considerable

interest

and

research

effort

has

been

recently

devoted

to

disjunctive

logic

programming

disjunctive

deductive

databases

and

to

various

extensions

of

non-monotonic

formalisms

ensuring

a

proper

treatment

of

disjunctive

information

There

are

good

reasons

justifying

this

research

effort

Formalisms

promoting

disjunctive

reasoning

are

more

expressive

and

natural

to

use

since

they

permit

direct

translation

of

disjunctive

statements

from

natural

language

and

from

informal

specifications

The

additional

expressive

power

of

disjunctive

logic

programs

significantly

simplifies

the

problem

of

translation

of

non-monotonic

formalisms

into

Responsible

Author

E-mail

dix@informatik.uni-koblenz.de,

phone

261 9119 420

fax

+49 261 9119 496

yPartially

supported

by

the

National

Science

Foundation

grant

#IRI

logic

programs

and

consequently

facilitates

using

logic

programming

as

an

inference

engine

for

nonmonotonic

reasoning

Finally

extensive

recent

work

devoted

to

theoretic

and

algorithmic

foundations

of

disjunctive

programming

suggests

that

there

are

good

prospects

for

extending

the

logic

programming

paradigm

to

disjunctive

programs

However

the

issue

of

finding

a

suitable

semantics

for

disjunctive

programs

and

databases

proved

to

be

far

more

complex

than

it

was

in

the

case

of

normal

non-disjunctive

Quite

recently

however

two

new

and

very

promising

semantics

have

been

proposed

for

the

class

of

disjunctive

logic

programs

Both

of

them

extend

the

well-founded

semantics

of

normal

programs

The

first

one

is

the

static

semantics

proposed

by

Przymusinski

in

-LSB-

-RSB-

and

the

other

is

the

D-WFS

semantics

proposed

by

Brass

and

Dix

in

-LSB-

-RSB-

The

two

semantics

are

based

on

very

different

intuitions

and

are

constructed

by

completely

different

means

While

D-WFS

is

defined

as

the

least

semantics

that

is

invariant

under

certain

natural

program

transformations

the

static

semantics

is

obtained

by

first

translating

a

logic

program

into

a

belief

theory

in

the

Autoepistemic

Logic

of

Beliefs

-LRB-

AEB

-RRB-

and

then

constructing

its

least

static

expansion

-LRB-

defined

as

the

least

fixed

point

of

a

natural

monotonic

operator

-RRB-

In

spite

of

the

fact

that

the

two

semantics

attempt

to

provide

meaning

to

disjunctive

programs

based

on

very

different

ideas

they

turn

out

to

be

quite

closely

related

and

in

fact

they

are

completely

equivalent

when

restricted

to

the

appropriate

common

language

-LSB-

-RSB-

This

paper

is

devoted

to

an

in-depth

study

of

the

static

semantics

in

the

class

of

so

called

superprograms

which

properly

contains

the

class

of

disjunctive

logic

programs

We

first

establish

three

powerful

syntactic

and

semantic

characterizations

of

static

semantics

significantly

simplifying

the

original

definition

Subsequently

we

use

the

obtained

characterizations

to

implement

an

query-answering

interpreter

for

static

semantic

in

the

class

of

super-programs

The

relationship

between

static

semantics

and

D-WFS

is

studied

elsewhere

-LSB-

-RSB-

Although

the

definition

of

the

least

static

expansion

in

AEB

is

quite

natural

and

elegant

due

to

the

fact

that

it

applies

to

a

very

general

class

of

theories

namely

to

all

belief

theories

in

AEB

its

construction

is

quite

complex

and

involves

a

number

of

iterations

Our

first

syntactic

characterization

shows

that

the

least

static

expansion

of

a

super-program

P

can

be

obtained

in

just

one

step

by

augmenting

P

with

the

set

of

formulae

fBF

F

compl

-LRB-

Pres

-RRB-

g

and

closing

it

under

derivation

in

the

logic

AEB

Here

Pres

denotes

the

residuum

of

the

program

P

-LSB-

-RSB-

and

compl

-LRB-

Pres

-RRB-

denotes

a

disjunctive

version

of

Clark

's

completion

of

Pres.

This

result

not

only

eliminates

the

need

for

multiple

iterations

in

the

computation

of

static

expansions

of

super-programs

but

it

also

underscores

a

very

interesting

and

somewhat

intriguing

relationship

between

static

semantics

and

Clark

's

completion

Due

to

the

fact

that

the

logic

AEB

allows

arbitrarily

deep

nesting

of

beliefs

even

if

the

original

theory

is

finite

the

construction

of

the

least

static

expansion

in

general

involves

an

infinite

and

vast

language

of

nested

beliefs

Our

two

subsequent

characterizations

one

of

which

is

syntactic

and

the

other

semantic

demonstrate

that

in

the

class

of

super-programs

the

iterative

construction

of

the

least

static

expansion

can

be

significantly

simplified

by

eliminating

nested

beliefs

and

by

restricting

the

computation

to

only

those

formulae

that

appear

in

queries

to

such

programs

We

demonstrate

the

applicability

and

the

importance

of

the

above

characterizations

by

using

one

of

them

as

an

inference

engine

in

a

Prolog

implementation

of

an

query-answering

interpreter

for

static

semantics

in

the

class

of

The

paper

is

organized

as

follows

In

the

next

Section

we

recall

the

definition

and

basic

properties

of

static

expansions

in

the

Autoepistemic

Logic

of

Beliefs

In

Section

we

introduce

the

class

of

super

logic

programs

which

properly

contains

the

class

of

disjunctive

logic

programs

and

we

show

how

to

translate

super-programs

into

belief

theories

in

AEB

thus

defining

static

semantics

of

such

programs

In

Sections

and

we

establish

the

three

characterizations

of

static

semantics

of

super-programs

discussed

above

In

Section

we

briefly

discuss

the

implementation

of

the

query-answering

interpreter

book

by

Minker

et

al.

-LSB-

-RSB-

provides

a

detailed

and

well-organized

account

of

the

extensive

research

effort

in

this

area

See

also

-LSB-

-RSB-

interpreter

is

available

from

ftp

via

FTP

and

WWW

for

static

semantic

in

the

class

of

super-programs

We

conclude

with

some

final

remarks

Static

Expansions

in

the

Autoepistemic

Logic

of

Beliefs

First

we

briefly

recall

the

definition

and

basic

properties

of

the

Autoepistemic

Logic

of

Beliefs

AEB

introduced

by

Przymusinski

in

-LSB-

-RSB-

Consider

a

fixed

propositional

language

L

with

standard

connectives

-LRB-

:-RRB-

and

the

propositional

letter

-LRB-

denoting

false

-RRB-

We

denote

the

set

of

its

propositions

by

AtL

and

assume

that

it

is

finite

The

language

of

AEB

is

a

propositional

modal

language

LAEB

obtained

by

augmenting

the

language

L

with

a

modal

operator

B

called

the

belief

operator

The

atomic

formulae

of

the

form

BF

where

F

is

an

arbitrary

formula

of

LAEB

are

called

belief

atoms

The

formulae

of

the

original

language

L

are

called

objective

Any

theory

T

in

the

language

LAEB

is

called

a

belief

theory

Observe

that

arbitrarily

deep

level

of

nested

beliefs

is

allowed

in

belief

theories

We

assume

the

following

two

simple

axiom

schemata

and

one

inference

rule

describing

the

arguably

obvious

properties

of

belief

atoms

-LRB-

D

-RRB-

Consistency

Axiom

B

-LRB-

K

-RRB-

Normality

Axiom

For

any

formulae

F

and

G

B

-LRB-

F

G

-RRB-

-LRB-

BF

BG

-RRB-

-LRB-

N

-RRB-

Necessitation

Rule

For

any

formula

F

FBF

The

first

axiom

states

that

tautologically

false

formulae

are

not

believed

The

second

axiom

states

that

if

we

believe

that

a

formula

F

implies

a

formula

G

and

if

we

believe

that

F

is

true

then

we

believe

that

G

is

true

as

well

The

necessitation

inference

rule

states

that

if

a

formula

F

has

been

proven

to

be

true

then

F

is

believed

to

be

true

Definition

-LRB-

Formulae

Derivable

from

a

Belief

Theory

-RRB-

For

any

belief

theory

T

we

denote

by

CnAEB

-LRB-

T

-RRB-

the

smallest

set

of

formulae

of

the

language

LAEB

which

contains

the

theory

T

all

the

-LRB-

substitution

instances

of

-RRB-

the

axioms

-LRB-

K

-RRB-

and

-LRB-

D

-RRB-

and

is

closed

under

both

standard

propositional

consequence

and

the

necessitation

rule

-LRB-

N

-RRB-

We

say

that

a

formula

F

is

derivable

from

theory

T

in

the

logic

AEB

if

F

belongs

to

CnAEB

-LRB-

T

-RRB-

A

belief

theory

T

is

consistent

if

the

theory

CnAEB

-LRB-

T

-RRB-

is

consistent

In

the

presence

of

axiom

-LRB-

K

-RRB-

the

axiom

-LRB-

D

-RRB-

is

equivalent

to

the

axiom

BF

B

F

stating

that

if

we

believe

in

a

formula

F

then

we

do

not

believe

in

F

Moreover

it

is

easy

to

see

that

the

axioms

imply

that

the

belief

operator

B

obeys

the

distributive

law

for

conjunctions

B

-LRB-

F

G

-RRB-

BF

BG

-LRB-

see

-LSB-

-RSB-

-RRB-

For

readers

familiar

with

modal

logics

it

should

be

clear

by

now

that

we

are

in

effect

considering

here

a

normal

modal

logic

with

one

modality

B

which

satisfies

the

consistency

axiom

-LRB-

D

-RRB-

-LSB-

-RSB-

Definition

-LRB-

Minimal

Models

-LSB-

-RSB-

-RRB-

By

a

minimal

model

of

a

belief

theory

T

we

mean

a

model

M

of

T

with

the

property

that

there

is

no

smaller

model

N

of

T

which

coincides

with

M

on

belief

atoms

BF

If

a

formula

F

is

true

in

all

minimal

models

of

T

then

we

write

T

j

min

F

and

say

that

F

is

minimally

by

T

readers

familiar

with

circumscription

this

means

that

we

are

considering

predicate

circumscription

CIRC

-LRB-

T

K

-RRB-

of

the

theory

T

in

which

atoms

from

the

objective

language

L

are

minimized

while

the

belief

atoms

BF

are

fixed

i.e.

T

j

min

F

CIRC

-LRB-

T

L

-RRB-

j

F

In

other

words

minimal

models

are

obtained

by

first

assigning

arbitrary

truth

values

to

the

belief

atoms

and

then

minimizing

objective

atoms

Static

Autoepistemic

Expansions

The

intended

meaning

of

belief

atoms

BF

is

based

on

the

principle

of

predicate

minimization

BF

F

is

minimally

entailed

F

is

true

in

all

minimal

models

Accordingly

beliefs

in

AEB

can

be

called

minimal

beliefs

-LRB-

see

-LSB-

-RSB-

and

-LSB-

-RSB-

-RRB-

As

in

Moore

's

Autoepistemic

Logic

also

in

the

Autoepistemic

Logic

of

Beliefs

the

intended

meaning

of

belief

atoms

is

implemented

by

defining

plausible

sets

of

beliefs

that

an

ideally

rational

and

introspective

agent

may

hold

given

a

set

of

premises

T

Definition

-LRB-

Static

Autoepistemic

Expansion

-LSB-

-RSB-

-RRB-

A

belief

theory

T

is

called

a

static

autoepistemic

expansion

of

a

belief

theory

T

if

it

satisfies

the

following

fixed-point

equation

T

CnAEB

-LRB-

T

-LSB-

fBF

T

j

min

Fg

-RRB-

where

F

ranges

over

all

formulae

of

LAEB

The

definition

of

static

autoepistemic

expansions

is

based

on

the

idea

of

building

an

expansion

T

of

a

belief

theory

T

by

closing

it

with

respect

to

-LRB-

i

-RRB-

the

derivability

in

the

logic

AEB

and

-LRB-

ii

-RRB-

the

addition

of

belief

atoms

BF

satisfying

the

condition

that

the

formula

F

is

minimally

entailed

by

T

Consequently

the

definition

of

static

expansions

enforces

the

intended

meaning

of

belief

atoms

described

above

Note

that

negations

BF

of

the

remaining

belief

atoms

are

not

explicitly

added

to

the

expansion

although

some

of

them

will

be

forced

in

by

the

Normality

and

Consistency

Axioms

It

turns

out

that

every

belief

theory

T

in

AEB

has

the

least

-LRB-

in

the

sense

of

set-theoretic

inclusion

-RRB-

static

expansion

T

which

has

an

iterative

definition

as

the

least

fixed

point

of

the

belief

closure

operator

T

-LRB-

S

-RRB-

CnAEB

-LRB-

T

-LSB-

fBF

S

j

min

Fg

-RRB-

where

S

is

an

arbitrary

belief

theory

and

the

F

's

range

over

all

formulae

of

LAEB

Theorem

-LRB-

Least

Static

Expansion

-LSB-

-RSB-

-RRB-

Every

belief

theory

T

in

AEB

has

the

least

static

expansion

namely

the

least

fixed

point

T

of

the

monotonic

belief

closure

operator

T

More

precisely

the

least

static

expansion

T

of

a

belief

theory

T

can

be

constructed

as

follows

Let

T

CnAEB

-LRB-

T

-RRB-

and

suppose

that

T

has

already

been

defined

for

any

ordinal

number

If

is

a

successor

ordinal

then

define

T

T

-LRB-

T

-RRB-

CnAEB

-LRB-

T

-LSB-

fBF

T

j

min

Fg

-RRB-

where

F

ranges

over

all

formulae

in

LAEB

Else

if

is

a

limit

ordinal

then

define

T

S

ff

fi

T

The

sequence

fT

g

is

monotonically

increasing

and

has

a

unique

fixed

point

T

T

T

-LRB-

T

>=-RRB-

for

some

ordinal

For

finite

theories

T

the

fixed

point

T

is

reached

after

finitely

many

steps

Observe

that

the

least

static

autoepistemic

expansion

T

of

T

contains

those

and

only

those

formulae

which

are

true

in

all

static

autoepistemic

expansions

of

T

It

defines

the

so

called

static

semantics

of

a

belief

theory

T

It

is

easy

to

verify

that

a

belief

theory

T

either

has

a

consistent

least

static

expansion

T

or

it

does

not

have

any

consistent

static

expansions

at

all

speaking

the

operator

is

only

monotone

on

the

lattice

of

all

theories

of

the

form

T

-LSB-

Mbel

where

T

is

fixed

and

Mbel

ranges

over

all

belief

formulae

Super

Logic

Programs

We

now

introduce

the

class

of

super

logic

programs

briefly

called

super

programs

This

powerful

class

of

programs

contains

the

class

of

disjunctive

logic

programs

as

a

proper

subclass

As

usual

in

the

definition

of

logic

programs

we

use

the

operator

not

to

denote

the

negation

by

default

often

referred

to

as

negation-by-failure

The

precise

meaning

of

negation

by

default

depends

on

the

semantics

assigned

to

it

In

our

approach

its

meaning

will

be

defined

by

translating

not

F

into

the

formula

B

F

in

AEB

thus

providing

not

F

with

the

intended

meaning

of

F

is

minimally

entailed

or

F

is

false

in

all

minimal

models

Definition

-LRB-

Super-Programs

-RRB-

A

Super

Logic

Program

is

a

-LRB-

finite

or

infinite

-RRB-

set

of

super-clauses

of

the

form

F

G

not

H

where

F

G

and

H

are

arbitrary

positive

from

L

and

F

is

not

-LRB-

tautologically

-RRB-

Observe

that

since

a

positive

formula

F

is

a

finite

conjunction

of

disjunctions

of

atoms

we

can

always

replace

a

super-clause

by

a

finite

set

of

clauses

whose

heads

are

disjunctions

Ak

of

atoms

Similarly

since

the

positive

formula

G

can

be

represented

as

a

disjunction

of

conjunctions

we

can

always

replace

a

super-clause

by

a

finite

set

of

clauses

whose

positive

premises

are

conjunctions

Bk

of

atoms

Finally

assuming

that

the

operator

not

obeys

the

distributive

law

for

disjunction

not

-LRB-

K

L

-RRB-

not

K

not

L

as

it

does

under

our

translation

the

formula

not

H

can

be

replaced

by

the

formula

not

not

Cn

where

each

of

the

Ci

's

is

a

conjunction

of

atoms

The

above

remarks

allow

us

to

establish

the

following

useful

fact

Lemma

A

Super

Logic

Program

can

be

equivalently

defined

as

a

-LRB-

finite

or

infinite

-RRB-

set

of

clauses

of

the

form

Ak

Bm

not

not

Cn

where

Ai

's

and

Bi

's

are

atoms

Ci

's

are

conjunctions

of

atoms

and

k

Clearly

the

class

of

super-programs

is

syntactically

significantly

broader

than

the

class

of

normal

logic

programs

In

fact

it

is

broader

than

the

class

of

programs

usually

referred

to

as

disjunctive

logic

programs

because

it

allows

premises

of

the

form

not

C

where

C

is

not

just

an

atom

but

a

conjunction

of

atoms

Since

the

class

of

super-programs

is

significantly

broader

than

the

class

of

normal

programs

the

class

of

allowed

queries

about

such

programs

-LRB-

we

will

call

them

super-queries

-RRB-

must

also

be

broader

than

the

class

of

standard

queries

to

normal

programs

In

fact

the

class

of

allowed

queries

must

closely

correspond

to

the

syntax

of

such

extended

logic

programs

In

the

following

definition

by

a

positive

literal

we

mean

an

arbitrary

objective

propositional

symbol

-LRB-

atom

-RRB-

p

and

by

a

negative

literal

we

mean

not

C

where

C

is

a

conjunction

of

atoms

Definition

-LRB-

Super-Queries

-RRB-

A

super-query

is

an

arbitrary

conjunction

-LRB-

respectively

disjunction

-RRB-

of

-LRB-

positive

or

negative

-RRB-

literals

Example

Consider

the

super-program

V

isitEurope

V

isitAustralia

Happy

V

isitEurope

V

isitAustralia

Bankrupt

V

isitEurope

V

isitAustralia

P

rudent

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

Disappointed

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

e.

they

are

conjunctions

of

disjunctions

of

atoms

from

L.

e.

all

disjunctions

in

F

are

non-empty

Given

the

usual

exclusive

interpretation

of

disjunctions

we

would

most

likely

conclude

that

we

will

visit

either

Australia

or

Europe

but

not

both

and

thus

we

are

likely

to

expect

the

answer

to

the

disjunctive

query

V

isitEurope

V

isitAustralia

to

be

positive

resulting

in

a

positive

answer

to

the

query

Happy

Similarly

we

expect

the

answer

to

the

negative

query

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

to

be

positive

resulting

in

a

positive

answer

to

the

query

P

rudent

On

the

other

hand

we

expect

the

answer

to

the

conjunctive

query

V

isitEurope

V

isitAustralia

to

be

negative

thus

resulting

in

a

negative

answer

to

the

query

Bankrupt

Similarly

we

expect

the

answer

to

the

negative

query

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

or

equivalently

not

V

isitEurope

not

V

isitAustralia

to

be

negative

thus

resulting

in

a

negative

answer

to

the

query

Disappointed

Observe

that

the

query

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

intuitively

means

can

it

be

assumed

by

default

that

we

do

n't

visit

both

Europe

and

Australia

and

thus

is

different

from

the

query

not

V

isitEurope

not

V

isitAustralia

which

says

can

it

be

assumed

by

default

that

we

do

n't

visit

Europe

and

can

it

be

assumed

by

default

that

we

do

n't

visit

Australia

The

meaning

of

negation

in

super-programs

is

based

on

the

notion

of

minimal

belief

defined

in

the

Autoepistemic

Logic

of

Beliefs

and

is

given

by

the

formula

not

F

B

F

thus

resulting

in

its

intended

meaning

not

F

F

is

false

in

all

minimal

models

F

is

minimally

entailed

The

above

definition

allows

us

to

translate

any

super-program

P

into

a

belief

theory

TP

in

AEB

and

thus

assign

to

it

the

meaning

provided

by

the

static

semantics

of

TP

Observe

that

since

the

belief

operator

B

is

distributive

w.r.t.

conjunctions

the

operator

not

obeys

the

distributive

law

for

disjunctions

not

-LRB-

K

L

-RRB-

not

K

not

L.

However

in

general

it

does

not

satisfy

the

distributive

law

for

conjunctions

Example

Consider

the

super-program

discussed

in

example

It

is

easy

to

check

that

both

V

isitEurope

V

isitAustralia

and

V

isitEurope

V

isitAustralia

hold

in

all

minimal

models

of

the

program

and

therefore

by

the

definition

of

minimal

beliefs

both

B

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

and

B

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

must

be

true

Since

by

definition

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

B

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

we

conclude

that

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

is

true

in

our

semantics

On

the

other

hand

from

B

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

and

from

the

Consistency

axiom

it

follows

that

B

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

holds

Since

by

definition

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

B

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

this

shows

that

not

-LRB-

V

isitEurope

V

isitAustralia

-RRB-

must

be

false

As

a

result

the

so

defined

semantics

results

in

the

intuitively

correct

meaning

of

the

program

described

in

the

previous

section

The

following

theorem

proved

by

Przymusinski

in

-LSB-

-RSB-

summarizes

the

basic

properties

of

superprograms

and

their

semantics

Theorem

-LRB-

Static

Semantics

of

Super-Programs

-LSB-

-RSB-

-RRB-

ffl

Every

super-program

has

a

uniquely

defined

and

consistent

static

semantics

ffl

When

restricted

to

normal

logic

programs

the

semantics

coincides

with

the

well-founded

semantics

ffl

When

restricted

to

positive

disjunctive

logic

programs

the

semantics

coincides

with

the

minimal

model

semantics

A

Syntactic

Characterization

via

Clark

's

Completion

In

this

section

we

present

our

first

syntactic

characterization

of

static

semantic

which

shows

that

the

least

static

expansion

of

a

super-program

P

can

be

obtained

in

just

one

step

by

closing

the

theory

Pcomp

P

-LSB-

fBF

F

compl

-LRB-

Pres

-RRB-

g

under

derivation

in

the

logic

AEB

Here

Pres

denotes

the

residuum

of

the

program

P

-LSB-

-RSB-

and

compl

-LRB-

Pres

-RRB-

denotes

-LRB-

the

disjunctive

version

of

-RRB-

Clark

's

completion

of

Pres.

This

result

not

only

eliminates

the

need

for

multiple

iterations

in

the

computation

of

static

expansions

of

super-programs

but

it

also

underscores

a

very

interesting

and

somewhat

intriguing

relationship

between

static

semantics

and

Clark

's

completion

Since

the

theory

Pcomp

is

easily

computable

this

result

reduces

reasoning

under

the

static

semantics

to

theorem-proving

in

the

modal

logic

LAEB

-LRB-

either

by

hand

or

with

an

automated

theorem

prover

-RRB-

In

particular

the

fixpoint

construction

of

the

least

static

expansion

can

be

altogether

avoided

The

idea

is

to

add

to

P

the

set

of

formulae

which

ensure

that

models

seen

through

the

modality

B

are

in

fact

minimal

An

appropriate

generalization

of

Clark

's

completion

accomplishes

this

but

only

for

programs

without

positive

body

literals

Such

residual

programs

were

introduced

and

investigated

in

-LSB-

-RSB-

Fortunately

it

turns

out

that

every

program

P

can

be

transformed

into

a

residual

program

Pres

without

changing

the

set

of

minimal

models

In

fact

the

two

elementary

program

transformations

unfolding

-LRB-

or

partial

evaluation

GPPE

-LSB-

-RSB-

-RRB-

and

elimination

of

tautologies

are

sufficient

to

obtain

a

residual

program

The

efficient

computation

of

the

residual

program

and

the

relation

to

elementary

program

transformations

is

explained

in

-LSB-

-RSB-

Here

we

only

give

an

example

Consider

the

following

program

P

p

q

B

-LRB-

r

-RRB-

q

B

-LRB-

q

-RRB-

r

q

The

first

two

rules

already

have

the

required

form

In

the

last

rule

we

unfold

the

body

literal

q

and

obtain

the

residual

program

Pres

p

q

B

-LRB-

r

-RRB-

q

B

-LRB-

q

-RRB-

p

r

B

-LRB-

r

-RRB-

r

B

-LRB-

q

-RRB-

This

is

in

fact

nothing

else

but

a

form

of

hyperresolution

We

now

define

an

appropriate

version

of

Clark

's

completion

for

residual

programs

As

usual

the

goal

is

to

ensure

that

an

objective

atom

A

is

true

only

if

it

really

has

to

be

true

i.e.

if

there

is

a

rule

with

A

in

the

head

in

which

the

body

is

true

and

all

the

other

head

literals

are

false

Definition

-LRB-

Clark

's

Completion

of

Residual

Programs

-RRB-

Let

Pres

be

a

super

program

without

positive

body

literals

Let

A

AtL

be

an

objective

atom

and

A

B

-LRB-

-RRB-

B

-LRB-

-RRB-

A

Bk

mk

B

-LRB-

Ck

-RRB-

B

-LRB-

nk

-RRB-

be

all

rules

in

Pres

with

A

in

the

head

Then

the

completion

axiom

for

A

is

the

formula

A

k

i

Bi

Bi

mi

B

-LRB-

Ci

-RRB-

B

-LRB-

Ci

ni

-RRB-

As

usual

Clark

's

completion

comp

-LRB-

Pres

-RRB-

of

the

whole

residual

program

Pres

is

the

set

of

all

completion

axioms

for

every

A

AtL

We

also

use

the

shorthand

B?comp

-LRB-

Pres

-RRB-

B

-LRB-

F

-RRB-

F

comp

-LRB-

Pres

-RRB-

Lemma

An

interpretation

I

is

a

minimal

model

of

a

super

program

P

iff

it

is

a

model

of

comp

-LRB-

Pres

-RRB-

It

is

interesting

to

note

that

when

we

replace

B

-LRB-

Ci

j

-RRB-

by

Ci

j

we

obtain

the

stable

models

of

P

We

used

this

fact

in

-LSB-

-RSB-

to

compute

the

stable

semantics

of

disjunctive

programs

Theorem

-LRB-

Characterization

Through

Clark

's

Completion

-RRB-

For

every

super

program

P

the

least

static

expansion

of

P

is

P

CnAEB

-LRB-

P

-LSB-

B?comp

-LRB-

Pres

-RRB-

-RRB-

A

surprising

corollary

of

the

above

Theorem

is

that

for

super

programs

a

single

application

of

the

belief

closure

operator

T

is

sufficient

Indeed

due

to

Lemma

the

completion

axioms

hold

in

all

minimal

models

so

they

are

already

believed

in

the

first

iteration

of

T

A

Model-Theoretic

Characterization

In

this

section

we

give

a

model-theoretic

characterization

of

the

least

static

expansion

Our

main

idea

is

not

to

describe

arbitrary

interpretations

but

only

reduced

interpretations

While

much

simpler

than

full

interpretations

reduced

interpretations

are

sufficiently

powerful

to

determine

answers

to

arbitrary

queries

about

super-programs

-LRB-

super-queries

-RRB-

and

thus

are

well-suited

as

a

basis

for

a

query-answering

mechanism

-LRB-

see

Section

-RRB-

Note

that

although

our

underlying

set

of

atoms

AtL

is

finite

the

language

LAEB

contains

infinitely

many

belief

atoms

-LRB-

because

beliefs

may

be

nested

to

arbitrary

depth

-RRB-

Since

interpretations

are

infinite

they

can

not

be

easily

handled

For

general

belief

theories

this

seems

unavoidable

but

super

programs

contain

only

very

special

belief

atoms

namely

belief

atoms

of

the

form

B

-LRB-

pn

-RRB-

Since

the

axioms

of

LAEB

ensure

that

beliefs

do

not

distinguish

between

equivalent

formulas

there

are

only

finitely

many

-LRB-

equivalence

classes

of

-RRB-

belief

atoms

of

this

form

We

will

show

that

for

super

programs

it

suffices

-LRB-

in

some

precise

sense

-RRB-

to

consider

only

valuations

of

the

set

AtL

-LSB-

B

-LRB-

pn

-RRB-

pi

AtL

Definition

-LRB-

Reduced

Interpretation

-RRB-

An

objective

interpretation

Iobj

is

a

propositional

valuation

of

AtL

Let

Obj

be

the

set

of

all

objective

interpretations

A

belief

interpretation

Ibel

is

a

propositional

valuation

of

B

-LRB-

pn

-RRB-

pi

AtL

We

let

Bel

denote

the

set

of

all

belief

interpretations

A

reduced

interpretation

I

is

the

combination

of

an

objective

interpretation

Iobj

and

a

belief

interpretation

Ibel

written

as

I

Iobj

-LSB-

Ibel

I.e.

a

reduced

interpretation

is

a

propositional

valuation

of

AtL

-LSB-

B

-LRB-

pn

-RRB-

pi

AtL

In

order

to

emphasize

the

difference

we

sometimes

call

an

interpretation

of

the

complete

language

LAEB

a

full

interpretation

Since

a

reduced

interpretation

assigns

a

truth

value

to

all

-LRB-

objective

and

belief

-RRB-

atoms

occurring

in

a

super

program

it

makes

sense

to

say

that

a

reduced

interpretation

is

a

model

of

a

super

program

It

is

also

clear

what

a

minimal

reduced

model

is

namely

there

is

no

smaller

reduced

model

with

the

same

interpretation

of

the

belief

atoms

This

notion

is

completely

compatible

with

the

corresponding

notion

for

full

interpretations

assume

that

disjunctions

pn

that

differ

only

in

ordering

or

multiplicity

are

viewed

as

the

same

Finally

given

a

full

interpretation

I

which

is

a

valuation

for

AtL

-LSB-

B

-LRB-

F

-RRB-

F

L

we

can

of

course

restrict

the

domain

to

the

subset

AtL

-LSB-

B

-LRB-

pn

-RRB-

pi

AtL

We

call

the

resulting

reduced

the

reduct

of

I

Now

our

goal

in

this

section

is

to

characterize

the

reducts

of

the

full

models

of

the

least

static

expansion

P

of

a

super

program

P

In

fact

the

objective

parts

of

these

reducts

are

easy

once

we

have

the

possible

belief

interpretations

The

idea

how

to

compute

the

possible

belief

interpretations

is

closely

related

to

Kripke

structures

-LRB-

which

is

not

surprising

for

a

modal

logic

-RRB-

The

worlds

in

such

a

Kripke

structure

are

marked

with

objective

interpretations

and

the

formula

B

-LRB-

pn

-RRB-

is

true

in

a

world

w

iff

pn

is

true

in

all

worlds

which

can

be

seen

from

w.

Due

to

the

consistency

axiom

every

world

w

must

see

at

least

one

world

Now

all

the

static

semantics

does

is

to

ensure

that

every

world

w

sees

only

worlds

marked

with

minimal

objective

models

So

given

a

set

Mobj

of

minimal

objective

models

a

belief

interpretation

Ibel

is

possible

iff

there

is

some

subset

ae

Mobj

-LRB-

namely

the

objective

interpretation

in

the

worlds

seen

from

w

-RRB-

such

that

Ibel

j

B

-LRB-

pn

-RRB-

iff

Iobj

j

pn

for

all

Iobj

But

the

minimal

objective

models

Mobj

also

depend

conversely

on

the

possible

belief

interpretations

-LRB-

since

a

minimal

model

contains

an

objective

part

and

a

belief

part

-RRB-

So

when

we

restrict

the

possible

belief

interpretations

we

also

get

less

minimal

objective

models

This

obviously

results

in

a

fixpoint

computation

formalized

by

the

following

operators

Definition

-LRB-

Possible

Belief

Interpretations

-RRB-

Let

P

be

a

super

program

The

operator

P

yields

the

minimal

objective

models

given

a

set

of

possible

belief

interpretations

For

every

Mbel

Bel

let

P

-LRB-

Mbel

-RRB-

Iobj

Obj

there

is

a

Ibel

Mbel

such

that

I

Iobj

-LSB-

Ibel

is

a

minimal

model

of

P

The

operator

P

yields

the

possible

belief

interpretations

given

a

set

of

minimal

objective

models

For

every

Mobj

Obj

let

P

-LRB-

Mobj

-RRB-

Ibel

Bel

there

is

a

non-empty

Mobj

such

that

for

every

pn

AtL

Ibel

j

B

-LRB-

pn

-RRB-

-LRB-

-RRB-

for

every

Iobj

Iobj

j

pn

The

operator

P

is

the

composition

P

P

ffi

P

of

P

and

P

Theorem

-LRB-

Model-Theoretic

Characterization

-RRB-

The

operator

P

is

monotone

and

thus

its

iteration

beginning

from

the

set

Bel

of

all

belief

interpretations

has

a

fixed

point

Bel

Bel

fIbel

Bel

Ibel

is

the

belief

part

of

a

full

model

I

of

P

g

A

reduced

interpretation

Iobj

-LSB-

Ibel

is

reduct

of

a

full

model

I

of

P

iff

Ibel

Bel

and

Iobj

-LSB-

Ibel

j

P

the

equivalence

requirement

for

disjunctive

belief

atoms

might

be

violated

Our

results

hold

equally

well

without

this

requirement

but

then

we

would

have

an

infinite

set

of

reduced

interpretations

However

a

full

model

of

the

least

static

expansion

trivially

satisfies

the

equivalence

requirement

so

it

suffices

to

consider

such

I.

Example

Let

us

consider

the

following

logic

program

P

p

q

B

-LRB-

r

-RRB-

q

B

-LRB-

q

-RRB-

r

q

Since

only

belief

atoms

with

a

single

proposition

occur

in

P

it

suffices

to

consider

only

such

belief

atoms

We

start

with

the

set

Bel

containing

all

eight

possible

valuations

of

the

three

belief

atoms

B

-LRB-

:p

-RRB-

B

-LRB-

q

-RRB-

and

B

-LRB-

r

-RRB-

These

can

be

extended

to

the

minimal

reduced

models

listed

in

the

following

table

Note

that

models

number

to

are

equal

to

models

number

to

except

that

B

-LRB-

:p

-RRB-

is

true

in

them

No

B

-LRB-

:p

-RRB-

B

-LRB-

q

-RRB-

B

-LRB-

r

-RRB-

p

q

r

So

we

see

that

there

are

only

three

possible

valuations

for

the

objective

parts

i.e.

P

-LRB-

Bel

-RRB-

consists

of

the

valuation

which

makes

all

p

q

r

false

the

valuation

which

only

makes

p

true

and

the

valuation

which

makes

only

p

false

Now

these

three

minimal

objective

interpretations

are

the

basis

for

the

next

round

of

belief

interpretations

Of

course

from

every

minimal

objective

interpretation

we

immediately

get

a

possible

belief

belief

interpretation

if

we

translate

the

truth

of

p

to

the

falsity

of

B

-LRB-

:p

-RRB-

But

we

can

also

combine

minimal

objective

parts

conjunctively

and

let

B

-LRB-

:p

-RRB-

be

true

only

if

:p

is

true

in

all

elements

of

some

set

of

minimal

models

Thus

we

get

the

following

four

valuations

for

the

belief

atoms

in

P

P

-LRB-

Bel

-RRB-

B

-LRB-

:p

-RRB-

B

-LRB-

q

-RRB-

B

-LRB-

r

-RRB-

This

means

that

only

the

models

number

remain

possible

given

the

current

knowledge

about

the

beliefs

Now

p

is

false

in

all

of

these

models

so

B

-LRB-

:p

-RRB-

must

be

assumed

and

only

the

following

two

valuations

of

the

belief

atoms

are

possible

in

the

fixed

point

Bel

B

-LRB-

:p

-RRB-

B

-LRB-

q

-RRB-

B

-LRB-

r

-RRB-

The

reduced

models

of

the

least

static

expansion

P

are

all

reduced

interpretations

which

satisfy

the

rules

of

P

and

have

one

of

these

two

belief

parts

Taking

the

reduct

of

a

full

interpretation

was

easy

However

with

the

following

Kripke

structure

K

hW

R

V

i

we

can

extend

the

reduced

interpretations

back

to

full

interpretations

which

are

models

of

the

least

static

expansion

ffl

The

set

of

worlds

W

are

the

reduced

interpretations

I

Iobj

-LSB-

Ibel

which

are

models

of

P

and

satisfying

Ibel

Bel

ffl

R

-LRB-

I

I

-RRB-

i.e.

I

sees

I

iff

I

is

a

minimal

model

of

P

and

for

all

pn

AtL

I

j

B

-LRB-

pn

-RRB-

=-RRB-

I

j

pn

ffl

The

valuation

V

-LRB-

I

-RRB-

of

a

reduced

interpretation

I

Iobj

-LSB-

Ibel

is

the

objective

part

Iobj

Now

in

order

to

get

from

a

reduced

interpretation

I

W

to

a

full

interpretation

we

simply

assign

the

formulas

B

-LRB-

F

-RRB-

the

truth

values

they

have

in

the

world

I

However

let

us

finally

note

that

not

all

full

models

of

the

least

static

expansion

are

reconstructed

in

this

way

only

one

representative

from

every

equivalence

class

with

the

same

reduct

Already

the

program

P

fp

B

-LRB-

:p

-RRB-

g

has

infinitely

many

different

full

models

A

Syntactic

Characterization

via

a

Fixpoint

Operator

If

we

are

only

interested

in

the

reduced

models

of

the

least

static

expansion

of

super

programs

and

this

is

all

we

need

in

order

to

answer

super-queries

we

can

significantly

simplify

the

belief

closure

operator

T

Instead

of

generating

arbitrary

formulae

which

hold

in

all

minimal

models

it

suffices

to

consider

only

formulas

of

the

form

-LRB-

-RRB-

-LRB-

pm

pm

km

-RRB-

-LRB-

qn

-RRB-

In

particular

this

means

that

we

do

not

have

to

consider

nested

beliefs

Furthermore

instead

of

having

to

compute

logical

consequences

CnAEB

in

the

modal

logic

AEB

it

suffices

to

compute

only

standard

propositional

consequences

These

two

simplifications

greatly

enhance

the

implementability

of

the

belief

closure

operator

Definition

-LRB-

Restricted

Belief

Closure

Operator

-RRB-

Given

a

super

program

P

the

restricted

belief

closure

operator

P

is

defined

by

P

-LRB-

S

-RRB-

Cn

P

-LSB-

B

-LRB-

-RRB-

B

-LRB-

Cn

-RRB-

B

-LRB-

-RRB-

S

j

min

Cn

where

Cn

denotes

the

consequences

of

propositional

logic

and

the

Ci

are

conjunctions

of

objective

atoms

We

allow

the

special

case

that

is

the

empty

conjunction

-LRB-

i.e.

true

-RRB-

and

identify

B

-LRB-

-RRB-

with

Since

we

do

not

consider

nested

beliefs

we

of

course

can

not

expect

the

least

fixed

point

of

P

to

be

the

least

static

expansion

P

However

when

we

consider

only

reduced

models

i.e.

if

we

consider

only

the

propositional

language

generated

by

the

atoms

AtL

-LSB-

B

-LRB-

C

-RRB-

C

is

a

conjunction

of

elements

from

AtL

then

the

least

fixed

point

of

P

is

equivalent

to

P

Theorem

-LRB-

Characterization

via

a

Simplified

Fixpoint

Operator

-RRB-

The

operator

P

is

and

has

a

fixed

point

P

which

is

equivalent

to

the

least

static

expansion

P

of

P

namely

a

reduced

interpretation

I

is

reduct

of

a

full

model

I

of

P

iff

I

j

P

we

apply

it

only

to

theories

of

the

form

P

-LSB-

Mbel

where

P

is

fixed

and

Mbel

ranges

over

belief

formulae

An

Implementation

of

the

Static

Semantics

We

demonstrated

the

practical

applicability

and

importance

of

the

above

characterizations

by

implementing

a

query-answering

interpreter

for

static

semantics

in

the

class

of

super-programs

based

on

one

of

these

In

fact

we

believe

that

all

three

of

our

characterizations

can

be

directly

implemented

although

at

the

moment

we

have

implemented

only

the

model-theoretic

characterization

from

Section

We

intend

to

produce

alternative

implementations

based

on

the

results

presented

in

this

paper

The

only

other

project

to

implement

the

static

semantics

of

which

we

know

is

described

in

-LSB-

-RSB-

The

DisLoP

project

in

Koblenz

which

is

tightly

related

to

this

research

also

aims

at

the

development

of

implementations

of

disjunctive

logic

programs

under

various

Let

us

explain

in

a

little

more

detail

how

our

prototype

implementation

works

First

because

of

the

large

number

of

possible

belief

atoms

it

seems

impossible

to

construct

explicitly

the

set

Bel

of

all

valuations

it

would

have

elements

where

n

is

the

number

of

objective

atoms

However

it

suffices

to

consider

only

the

belief

atoms

which

occur

explicitly

in

the

program

other

belief

atoms

have

no

influence

on

the

minimal

models

and

we

can

always

extend

a

valuation

of

these

critical

belief

atoms

consistently

to

a

valuation

of

all

belief

atoms

Furthermore

it

is

possible

to

reduce

the

number

of

belief

atoms

occurring

in

the

program

by

evaluating

the

simple

cases

directly

For

example

when

it

is

possible

to

derive

the

disjunctive

fact

VisitEurope

VisitAustralia

-LRB-

in

the

example

it

is

directly

given

-RRB-

we

can

delete

a

rule

like

Disappointed

not

-LRB-

VisitEurope

-RRB-

not

-LRB-

VisitAustralia

-RRB-

because

we

know

that

the

body

can

never

be

true

-LRB-

this

is

a

special

case

of

the

negative

reduction

transformation

used

in

-LSB-

-RSB-

-RRB-

In

this

example

deleting

the

rule

also

eliminates

two

belief

atoms

So

the

first

step

of

our

algorithm

is

to

compute

the

residual

program

and

applying

the

reduction

operators

defined

in

-LSB-

-RSB-

This

transformation

preserves

the

minimal

models

and

often

significantly

reduces

the

number

of

belief

atoms

we

have

to

consider

-LRB-

for

instance

no

belief

atoms

remain

if

the

input

program

is

stratified

and

non-disjunctive

-RRB-

Then

we

apply

the

definition

of

the

operators

P

and

P

from

Section

quite

literally

However

two

points

are

worth

mentioning

In

order

to

compute

the

minimal

models

for

the

P

operator

we

use

again

Clark

's

completion

Given

an

interpretation

of

the

belief

atoms

occurring

in

the

residual

program

we

evaluate

the

rule

bodies

so

we

get

a

set

of

positive

disjunctions

pn

Then

we

use

the

model

generation

algorithm

described

in

-LSB-

-RSB-

which

never

runs

into

dead

ends

Of

course

any

other

model

generation

algorithm

could

also

be

used

Next

it

would

be

very

inefficient

to

consider

all

subsets

Mobj

as

required

in

the

definition

of

the

P

operator

However

in

order

to

check

whether

a

given

belief

interpretation

Ibel

is

selected

by

P

only

the

maximal

is

interesting

We

construct

it

as

Iobj

Mobj

for

every

belief

atom

B

-LRB-

pn

-RRB-

if

Ibel

j

B

-LRB-

pn

-RRB-

then

Iobj

j

pn

Obviously

the

inclusion

of

other

Iobj

into

would

immediately

destroy

the

required

property

Next

we

check

wether

the

so

constructed

is

non-empty

Finally

we

check

that

for

every

belief

atom

B

-LRB-

pn

-RRB-

which

is

false

in

Ibel

there

is

an

Iobj

with

Iobj

pn

The

program

prints

the

fixed

point

of

P

-LRB-

only

the

evaluations

of

the

critical

belief

atoms

-RRB-

and

then

the

minimal

reduced

models

the

belief

part

of

which

is

contained

in

P

It

then

waits

for

queries

We

allow

arbitrary

super-queries

which

are

answered

entirely

on

the

basis

of

these

minimal

models

interpreter

is

available

from

ftp

via

FTP

and

WWW

-LSB-

-RSB-

and

which

also

contains

links

to

related

projects

Conclusion

While

the

semantics

of

non-disjunctive

programs

is

fairly

well

understood

and

there

exist

several

implementations

of

the

best

known

semantics

-LRB-

well-founded

semantics

and

stable

semantics

-RRB-

the

declarative

and

computational

foundations

of

disjunctive

programming

proved

to

be

much

more

elusive

and

difficult

However

the

recent

introduction

of

the

two

new

promising

semantics

for

the

class

of

disjunctive

logic

programs

namely

the

static

semantics

proposed

by

Przymusinski

-LSB-

-RSB-

and

the

D-WFS

semantics

proposed

by

Brass

and

Dix

-LSB-

-RSB-

seems

to

significantly

enhance

our

understanding

of

the

theoretical

and

algorithmic

aspects

of

disjunctive

logic

programs

Both

semantics

extend

the

well-founded

semantics

of

normal

programs

and

have

very

natural

and

regular

features

analogous

to

those

shared

by

the

well-founded

semantics

Although

the

two

semantics

are

based

on

very

different

ideas

they

turn

out

to

be

quite

closely

related

and

in

fact

they

are

completely

equivalent

when

restricted

to

the

suitable

common

language

Our

forthcoming

paper

-LSB-

-RSB-

is

devoted

for

an

in-depth

study

of

this

close

relationship

In

this

paper

we

conducted

a

detailed

study

of

the

static

semantics

in

the

class

of

so

called

superprograms

The

newly

introduced

class

of

super-programs

properly

extends

the

class

of

disjunctive

logic

programs

and

is

interesting

in

itself

We

proved

three

powerful

syntactic

and

semantic

characterizations

of

static

semantics

significantly

simplifying

the

original

definition

Subsequently

we

demonstrated

the

applicability

and

importance

of

the

above

characterizations

by

using

one

of

them

as

an

inference

engine

in

a

Prolog

implementation

of

an

query-answering

interpreter

for

static

semantics

in

the

class

of

superprograms

References

-LSB-

-RSB-

Stefan

Brass

and

J?urgen

Dix

A

General

Approach

to

Bottom-Up

Computation

of

Disjunctive

Semantics

In

J.

Dix

L.

Pereira

and

T.

Przymusinski

editors

Nonmonotonic

Extensions

of

Logic

Programming

LNAI

pages

-LCB-

Springer

Berlin

-LSB-

-RSB-

Stefan

Brass

and

J?urgen

Dix

Disjunctive

Semantics

based

upon

Partial

and

Bottom-Up

Evaluation

In

Leon

Sterling

editor

Proceedings

of

the

Int

Conf

on

Logic

Programming

Tokyo

pages

-LCB-

MIT

Press

June

-LSB-

-RSB-

Stefan

Brass

J?urgen

Dix

and

Teodor

C.

Przymusinski

A

comparison

of

static

semantics

and

D-WFS

Technical

Report

TR

University

of

Koblenz

Department

of

Computer

Science

Rheinau

February

-LSB-

-RSB-

F.

Bry

Logic

programming

as

constructivism

A

formalization

and

its

application

to

databases

In

Proc

of

the

Eighth

ACM

SIGACT-SIGMOD-SIGART

Symposium

on

Principles

of

Database

Systems

-LRB-

PODS

-RRB-

-LCB-

-LSB-

-RSB-

F.

Bry

Negation

in

logic

programming

A

formalization

in

constructive

logic

In

D.

Karagiannis

-LRB-

ed

-RRB-

Information

Systems

and

Artificial

Intelligence

Integration

Aspects

-LCB-

Springer

-LSB-

-RSB-

S.

Costantini

G.

A.

Lanzarone

Static

semantics

as

program

transformation

and

well-founded

computation

In

J.

Dix

L.

M.

Pereira

T.

Przymusinski

-LRB-

eds

-RRB-

Non-monotonic

Extensions

of

Logic

Programming

Theory

Implementation

and

Applications

LNAI

pages

Springer

-LSB-

-RSB-

J?urgen

Dix

Semantics

of

Logic

Programs

Their

Intuitions

and

Formal

Properties

An

Overview

In

Andre

Fuhrmann

and

Hans

Rott

editors

Logic

Action

and

Information

-LCB-

Essays

on

Logic

in

Philosophy

and

Artificial

Intelligence

pages

-LCB-

DeGruyter

-LSB-

-RSB-

J.

Dix

and

U.

Furbach

The

DFG-Project

DisLoP

on

Disjunctive

Logic

Programming

Computational

Logic

-LCB-

-LSB-

-RSB-

P.

M.

Dung

K.

Kanchansut

A

fixpoint

approach

to

declarative

semantics

of

logic

programs

In

Proc

NACLP

-LCB-

-LSB-

-RSB-

P.

M.

Dung

K.

Kanchansut

A

natural

semantics

of

logic

programs

with

negation

In

Proc

of

the

Ninth

Conf

on

Foundataions

of

Software

Technology

and

Theoretical

Computer

Science

-LCB-

-LSB-

-RSB-

Michael

Gelfond

Halina

Przymusinski

and

Teodor

C.

Przymusinski

On

the

relationship

between

circumscription

and

negation

as

failure

Journal

of

Artificial

Intelligence

-LRB-

-RRB-

-LCB-

February

-LSB-

-RSB-

J.

Lobo

J.

Minker

and

A.

Rajasekar

Foundations

of

Disjunctive

Logic

Programming

MIT

Press

Cambridge

Massachusetts

-LSB-

-RSB-

J.

McCarthy

Circumscription

-LCB-

a

form

of

non-monotonic

reasoning

Journal

of

Artificial

Intelligence

-LCB-

-LSB-

-RSB-

J.

Minker

On

indefinite

data

bases

and

the

closed

world

assumption

In

Proc

Conference

on

Automated

Deduction

pages

-LCB-

New

York

Springer

Verlag

-LSB-

-RSB-

Jack

Minker

An

Overview

of

Nonmonotonic

Reasoning

and

Logic

Programming

Journal

of

Logic

Programming

Special

Issue

-LSB-

-RSB-

W.

Marek

and

M.

Truszczynski

Non-Monotonic

Logic

Springer

Verlag

-LSB-

-RSB-

T.

C.

Przymusinski

Autoepistemic

logic

of

knowledge

and

beliefs

-LRB-

In

preparation

-RRB-

University

of

California

at

Riverside

-LRB-

Extended

abstract

appeared

in

A

knowledge

representation

framework

based

on

autoepistemic

logic

of

minimal

beliefs

In

Proceedings

of

the

Twelfth

National

Conference

on

Artificial

Intelligence

Seattle

Washington

August

pages

Los

Altos

CA

American

Association

for

Artificial

Intelligence

Morgan

Kaufmann

-RRB-

-LSB-

-RSB-

Chiaki

Sakama

and

Hirohisa

Seki

Partial

Deduction

of

Disjunctive

Logic

Programs

A

Declarative

Approach

In

Logic

Program

Synthesis

and

Transformation

-LCB-

Meta

Programming

in

Logic

LNCS

pages

-LCB-

Berlin

Springer

A

Proof

of

Characterization

via

Completion

-LRB-

Theorem

-RRB-

First

we

need

that

for

residual

programs

Clark

's

completion

characterizes

exactly

the

minimal

models

Lemma

A.

Let

Pres

be

a

program

without

positive

body

literals

A

full

interpretation

I

-LRB-

i.e.

a

LAEB

interpretation

-RRB-

is

a

minimal

model

of

Pres

iff

I

j

comp

-LRB-

Pres

-RRB-

Proof

ffl

Suppose

I

is

not

a

model

of

comp

-LRB-

Pres

-RRB-

so

there

is

a

completion

axiom

A

k

i

Bi

Bi

mi

B

-LRB-

Ci

-RRB-

B

-LRB-

Ci

ni

-RRB-

which

is

not

satisfied

in

I

If

the

direction

is

not

satisfied

I

is

obviously

not

even

a

model

of

Pres

So

assume

that

this

direction

is

satisfied

for

all

completion

axioms

i.e.

I

is

a

model

of

Pres

Now

if

the

direction

is

not

satisfied

this

means

that

I

j

A

but

for

every

rule

A

Bi

Bi

mi

B

-LRB-

Ci

-RRB-

B

-LRB-

Ci

ni

-RRB-

with

A

in

the

head

one

of

the

Bi

j

is

true

or

one

of

the

B

-LRB-

Ci

j

-RRB-

is

false

But

then

the

interpretation

I

which

differs

from

I

only

in

I

A

is

also

a

model

of

Pres

so

I

is

not

minimal

ffl

Suppose

conversely

that

I

is

not

a

minimal

model

of

Pres

If

I

is

not

even

a

model

i.e.

does

not

satisfy

some

rule

Am

B

-LRB-

-RRB-

B

-LRB-

Cn

-RRB-

then

obviously

the

direction

of

the

completion

axioms

for

every

Ai

are

violated

So

let

us

assume

that

I

is

a

model

but

not

minimal

so

there

is

a

smaller

model

I

We

can

choose

I

in

such

a

way

that

it

is

minimal

Let

A

be

an

objective

atom

with

I

j

A

and

I

A.

Let

us

consider

the

completion

axiom

for

A

A

k

i

Bi

Bi

mi

B

-LRB-

Ci

-RRB-

B

-LRB-

Ci

ni

-RRB-

By

the

contraposition

of

the

already

proven

other

direction

the

minimal

model

I

is

a

model

of

comp

-LRB-

Pres

-RRB-

so

it

satisfies

the

completion

axiom

Since

I

A

for

every

i

there

is

a

j

such

that

I

j

Bi

j

or

I

B

-LRB-

Ci

j

-RRB-

But

since

I

is

smaller

than

I

I

j

Bi

j

=-RRB-

I

j

Bi

j

and

I

B

-LRB-

Ci

j

-RRB-

-LRB-

-RRB-

I

B

-LRB-

Ci

j

-RRB-

Thus

the

right

hand

side

is

also

false

in

I

but

A

is

true

in

I

and

we

get

I

comp

-LRB-

Pres

-RRB-

Now

we

want

of

course

that

comp

-LRB-

Pres

-RRB-

characterizes

not

only

the

minimal

models

of

Pres

but

the

minimal

models

of

the

original

program

P

This

follows

from

the

fact

that

every

program

P

can

be

transformed

into

a

residual

program

Pres

by

applying

only

the

two

transformations

unfolding

-LRB-

GPPE

-RRB-

and

elimination

of

tautologies

-LSB-

-RSB-

and

the

following

lemma

Lemma

A.

Let

and

be

two

super

programs

such

that

results

from

by

an

application

of

GPPE

or

deletion

of

tautologies

Then

and

have

the

same

minimal

-LRB-

full

-RRB-

models

Proof

The

deletion

of

tautologies

is

an

equivalence

transformation

so

it

does

not

change

the

set

of

models

and

thus

also

not

the

set

of

minimal

models

Now

suppose

that

we

apply

GPPE

to

a

rule

A

B

R

and

B

Ai

Ri

i

n

are

all

rules

about

B.

ffl

Since

the

new

rules

A

Ai

R

Ri

are

logical

consequences

of

the

rules

in

every

model

of

is

also

a

model

of

ffl

Let

I

be

a

minimal

model

of

We

first

show

that

it

is

a

model

of

-LRB-

not

necessarily

minimal

-RRB-

Suppose

this

were

not

the

case

so

the

rule

A

B

R

were

violated

in

I

-LRB-

this

is

the

only

rule

in

n

-RRB-

This

means

that

I

j

B

I

j

R

and

I

A

Since

I

is

a

minimal

model

of

the

interpretation

I

with

I

B

-LRB-

and

otherwise

equal

to

I

-RRB-

can

not

be

a

model

of

Since

objective

atoms

do

not

occur

negated

in

rule

bodies

this

means

that

there

is

a

rule

with

B

in

the

head

i.e.

one

of

the

B

Ai

such

that

I

j

Ri

and

I

Ai

Since

I

j

Ri

the

atom

B

can

not

be

contained

in

Ri

and

since

we

exclude

multiple

occurrences

in

the

head

it

is

also

not

contained

in

Ai

Thus

I

j

Ri

and

I

Ai

But

now

we

have

that

the

new

rule

R

Ri

contained

in

is

violated

by

I

which

is

a

contradiction

We

now

have

to

show

that

the

sets

of

minimal

models

are

equal

ffl

Suppose

that

I

is

a

minimal

model

of

By

the

first

half

above

we

know

that

it

is

a

model

of

If

it

were

not

minimal

there

must

be

a

smaller

minimal

model

I

of

But

by

the

second

half

above

this

would

also

be

a

model

of

contradicting

the

assumed

minimality

of

I

as

a

model

of

ffl

Suppose

that

I

is

a

minimal

model

of

Then

it

is

a

model

of

If

there

were

a

smaller

model

I

of

this

would

be

also

a

model

of

again

a

contradiction

Thus

we

can

already

conclude

that

each

rule

F

in

comp

-LRB-

Pres

-RRB-

is

minimally

implied

by

P

and

thus

B

-LRB-

F

-RRB-

is

contained

in

the

least

static

expansion

So

we

immediately

get

the

easier

half

of

Theorem

P

CnAEB

P

-LSB-

B

-LRB-

comp

-LRB-

Pres

-RRB-

-RRB-

In

order

to

show

the

inclusion

we

must

show

that

CnAEB

P

-LSB-

B

-LRB-

comp

-LRB-

Pres

-RRB-

-RRB-

is

already

a

static

expansion

of

P

i.e.

another

application

of

the

operator

P

does

not

yield

anything

new

The

following

lemma

allows

us

to

characterize

the

minimal

models

of

P

-LSB-

B?comp

-LRB-

Pres

-RRB-

Lemma

A.

Let

T

be

any

belief

theory

and

TB

be

a

set

of

formulas

which

contain

only

belief

atoms

Then

the

minimal

models

of

T

-LSB-

TB

are

the

minimal

models

of

T

which

satisfy

TB

Proof

ffl

Let

I

be

a

minimal

model

of

T

-LSB-

TB

Of

course

it

is

a

model

of

T

and

of

TB

Now

suppose

that

there

were

a

smaller

model

I

of

T

Since

I

and

I

do

not

differ

in

the

interpretation

of

the

belief

atoms

I

is

also

a

model

of

TB

and

thus

a

model

of

T

-LSB-

TB

But

this

contradicts

the

assumed

minimality

of

I.

ffl

Let

I

be

a

minimal

model

of

T

which

also

satisfies

TB

Then

I

is

of

course

a

model

of

T

-LSB-

TB

and

the

existence

of

a

smaller

model

I

would

immediately

contradict

the

assumed

minimality

of

I

since

I

is

of

course

also

a

model

of

T

Lemma

A.

Let

P

CnAEB

P

-LSB-

B

-LRB-

comp

-LRB-

Pres

-RRB-

-RRB-

Suppose

that

F

holds

in

all

minimal

models

of

P

Then

B

-LRB-

F

-RRB-

is

already

contained

in

P

Proof

Let

TB

be

the

subset

of

P

which

consists

entirely

of

belief

atoms

i.e.

B?comp

-LRB-

Pres

-RRB-

together

with

all

instances

of

the

axioms

of

CnAEB

including

the

formulas

B

-LRB-

F

-RRB-

following

by

necessitation

Then

the

-LRB-

full

-RRB-

models

of

P

are

exactly

the

-LRB-

full

-RRB-

models

of

P

-LSB-

TB

-LRB-

the

required

closure

under

logical

consequences

does

not

change

the

set

of

models

-RRB-

Now

the

minimal

models

of

P

-LSB-

TB

are

the

minimal

models

of

P

which

satisfy

TB

-LRB-

by

Lemma

A.

-RRB-

But

the

minimal

models

of

P

are

the

models

of

comp

-LRB-

P

-RRB-

Thus

F

holds

in

all

models

of

comp

-LRB-

P

-RRB-

-LSB-

TB

is

not

a

new

rule

A

Ai

R

Ri

because

then

A

must

contain

B

and

the

unfolded

rule

A

B

R

would

be

a

tautology

and

could

never

be

violated

i.e.

it

is

a

propositional

logic

consequence

By

the

compactness

of

propositional

logic

there

is

a

finite

subset

Fng

comp

-LRB-

P

-RRB-

-LSB-

TB

such

that

Fng

j

F

But

then

Fn

F

is

a

tautology

thus

B

-LRB-

-RRB-

B

-LRB-

Fn

-RRB-

B

-LRB-

F

-RRB-

is

contained

in

P

If

Fi

is

from

comp

-LRB-

P

-RRB-

then

B

-LRB-

Fi

-RRB-

is

obviously

contained

in

P

and

otherwise

we

get

B

-LRB-

Fi

-RRB-

by

necessitation

Thus

by

the

closure

under

classical

consequences

we

get

B

-LRB-

F

-RRB-

P

This

shows

that

P

CnAEB

P

-LSB-

B

-LRB-

comp

-LRB-

Pres

-RRB-

-RRB-

is

itself

a

fixpoint

of

the

belief

closure

operator

Since

the

least

static

expansion

P

is

the

least

static

expansion

we

can

conclude

that

P

CnAEB

P

-LSB-

B

-LRB-

comp

-LRB-

Pres

-RRB-

-RRB-

B

Kripke

Models

of

Static

Expansions

Before

we

prove

the

modeltheoretic

characterization

-LRB-

Theorem

-RRB-

we

prove

here

a

Theorem

which

easily

allows

us

to

construct

models

of

the

least

static

expansion

from

Kripke

structures

It

is

used

as

a

lemma

in

the

proof

to

Theorem

but

it

is

of

its

own

interest

Although

we

later

need

only

super

programs

and

reduced

models

we

allow

in

this

section

arbitrary

belief

theories

and

consider

full

models

In

order

to

be

precise

and

self-contained

let

us

briefly

repeat

the

definition

of

Kripke

structures

-LSB-

-RSB-

Since

the

static

semantics

contains

the

normality

axiom

it

suffices

to

consider

normal

Kripke

structures

i.e.

triples

K

hW

R

V

i

consisting

of

ffl

a

nonempty

set

W

the

elements

w

W

are

called

worlds

ffl

a

relation

R

W

W

if

R

-LRB-

w

-RRB-

we

say

that

world

w

sees

world

ffl

a

mapping

V

W

Obj

which

assigns

to

every

world

w

a

valuation

Iobj

V

-LRB-

w

-RRB-

of

the

objective

atoms

AtL

The

validity

of

a

formula

F

in

a

world

w

given

Kripke

structure

K

is

defined

by

ffl

If

F

is

a

proposition

p

AtL

then

-LRB-

K

w

-RRB-

j

F

-LRB-

-RRB-

V

-LRB-

w

-RRB-

j

p.

ffl

If

F

is

a

negation

F

then

-LRB-

K

w

-RRB-

j

F

-LRB-

-RRB-

-LRB-

K

w

-RRB-

F

ffl

-LRB-

and

further

propositional

connectives

as

usual

-RRB-

ffl

If

F

is

a

belief

atom

B

-LRB-

F

-RRB-

then

-LRB-

K

w

-RRB-

j

F

-LRB-

-RRB-

for

all

W

with

R

-LRB-

w

-RRB-

-LRB-

K

-RRB-

j

F

Now

given

such

a

Kripke

structure

K

we

get

from

every

world

w

a

propositional

interpretation

I

K

-LRB-

w

-RRB-

of

LAEB

i.e.

a

valuation

of

AtL

-LSB-

B

-LRB-

F

-RRB-

F

LAEB

We

simply

make

an

objective

or

belief

atom

A

true

in

I

iff

-LRB-

K

w

-RRB-

j

A

Since

the

propositional

connectives

are

defined

in

a

Kripke

structure

like

in

standard

propositional

logic

we

obviously

have

I

j

F

-LRB-

-RRB-

-LRB-

K

w

-RRB-

j

F

for

all

F

LAEB

Theorem

B.

-LRB-

Kripke

Structures

Yield

Static

Expansions

-RRB-

Let

T

be

an

arbitrary

belief

theory

and

K

hW

R

V

i

be

a

normal

Kripke

structure

satisfying

ffl

For

every

w

W

there

is

a

W

with

R

-LRB-

w

-RRB-

-LRB-

consistency

-RRB-

ffl

For

every

w

W

-LRB-

K

w

-RRB-

j

T

i.e.

the

interpretation

I

K

-LRB-

w

-RRB-

is

a

model

of

T

ffl

For

every

w

W

with

R

-LRB-

w

-RRB-

the

interpretation

I

K

-LRB-

-RRB-

is

a

minimal

model

of

T

-LRB-

only

minimal

models

are

seen

-RRB-

Then

T

fF

LAEB

for

every

w

W

w

j

Fg

is

a

static

expansion

of

T

Proof

ffl

Consistency

Axiom

Let

any

w

W

be

given

We

have

to

show

that

-LRB-

K

w

-RRB-

j

B

-LRB-

-RRB-

By

the

first

requirement

of

the

theorem

there

is

W

with

R

-LRB-

w

-RRB-

Now

-LRB-

K

-RRB-

and

therefore

-LRB-

K

w

-RRB-

j

B

-LRB-

-RRB-

ffl

Normality

Axiom

We

have

to

show

that

for

all

formulas

F

G

LAEB

and

all

w

W

the

normality

axiom

holds

in

w

-LRB-

K

w

-RRB-

j

B

-LRB-

F

G

-RRB-

B

-LRB-

F

-RRB-

B

-LRB-

G

-RRB-

Suppose

that

-LRB-

K

w

-RRB-

j

B

-LRB-

F

G

-RRB-

-LRB-

otherwise

the

implication

is

trivially

satisfied

-RRB-

This

means

that

for

all

W

with

R

-LRB-

w

-RRB-

if

-LRB-

K

-RRB-

j

F

then

-LRB-

K

-RRB-

j

G.

Furthermore

suppose

that

-LRB-

K

w

-RRB-

j

B

-LRB-

F

-RRB-

-LRB-

again

the

formula

is

otherwise

trivially

satisfied

-RRB-

This

means

that

for

all

of

the

above

it

is

true

that

-LRB-

K

-RRB-

j

F

Then

obviously

it

follows

that

-LRB-

K

-RRB-

j

G

But

this

means

-LRB-

K

w

-RRB-

j

B

-LRB-

G

-RRB-

ffl

Necessitation

Rule

Let

F

T

i.e.

for

every

w

W

we

have

-LRB-

K

w

-RRB-

j

F

But

then

also

-LRB-

K

w

-RRB-

j

B

-LRB-

F

-RRB-

holds

for

every

w

W

since

-LRB-

K

-RRB-

j

F

holds

for

every

W

with

R

-LRB-

w

-RRB-

ffl

Closure

under

propositional

consequences

Let

F

LAEB

a

formula

which

is

a

propositional

consequence

of

Fn

T

Now

Fi

T

means

that

-LRB-

K

w

-RRB-

j

Fi

for

every

w

W

But

the

formulas

valid

in

one

world

are

of

course

closed

under

propositional

consequences

So

we

get

-LRB-

K

w

-RRB-

j

F

and

thus

F

T

ffl

Now

suppose

that

T

j

min

F

i.e.

F

is

true

in

all

minimal

models

of

T

We

have

to

show

that

B

-LRB-

F

-RRB-

T

i.e.

-LRB-

K

w

-RRB-

j

B

-LRB-

F

-RRB-

for

every

w

W

This

means

that

-LRB-

K

-RRB-

j

F

for

every

W

with

R

-LRB-

w

-RRB-

Let

such

a

be

given

and

let

I

K

-LRB-

-RRB-

By

the

last

condition

of

the

theorem

we

know

that

I

is

a

minimal

model

of

T

By

construction

it

is

also

a

model

of

T

and

since

T

T

there

can

be

no

smaller

model

Thus

I

is

a

minimal

model

of

T

and

therefore

satisfies

F

This

theorem

gives

us

a

simple

way

to

construct

models

of

the

least

static

expansion

T

Obviously

for

every

w

W

the

interpretation

I

K

-LRB-

w

-RRB-

is

a

model

of

T

But

the

least

static

expansion

is

a

subset

of

every

other

static

expansion

i.e.

T

T

and

therefore

we

have

I

j

T

Example

B.

Let

us

consider

the

theory

P

fp

B

-LRB-

:p

-RRB-

g

corresponding

to

a

well-known

logic

program

We

claimed

in

Section

that

the

least

static

expansion

P

of

this

program

has

infinitely

many

different

models

But

let

us

first

look

a

Kripke

structure

which

generates

only

two

models

:p

p

oe

I.e.

the

set

of

worlds

W

is

the

visibility

relation

R

is

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

each

world

can

see

the

other

one

but

not

itself

-RRB-

and

p

is

false

in

world

and

true

in

world

In

world

B

-LRB-

:p

-RRB-

is

false

and

in

world

it

is

true

But

there

are

also

quite

different

Kripke

models

The

construction

in

Section

will

yield

:p

R

p

oe

ffifl

fflfip

R

Note

that

in

world

the

belief

atom

B

-LRB-

:p

-RRB-

is

false

so

it

is

non-minimal

-LRB-

there

is

no

need

to

make

p

true

-RRB-

Thus

there

can

be

no

incoming

edges

Let

us

now

finally

present

the

Kripke

structure

which

yields

infinitely

many

models

:p

p

oe

:p

oe

p

oe

:p

oe

oe

Now

we

of

course

have

to

explain

that

different

worlds

really

yield

different

interpretations

The

trick

is

that

world

is

the

only

world

in

which

neither

B

-LRB-

:p

-RRB-

nor

B

-LRB-

p

-RRB-

are

true

Now

other

worlds

can

be

identified

with

the

number

of

nested

beliefs

necessary

to

get

to

world

So

the

formula

Bn

B

-LRB-

p

-RRB-

B

-LRB-

:p

-RRB-

Bn

B

-LRB-

p

-RRB-

B

-LRB-

:p

-RRB-

is

true

in

world

n

but

no

other

world

C

Proof

of

Modeltheoretic

Characterization

-LRB-

Theorem

-RRB-

The

monotonicity

of

P

is

obvious

The

more

belief

interpretations

we

have

the

more

minimal

objective

models

we

get

But

then

there

are

more

possibilities

to

choose

the

set

in

the

definition

of

P

We

will

first

prove

that

a

reduced

model

I

Iobj

-LSB-

Ibel

with

I

j

P

and

Ibel

Bel

can

be

extended

to

a

full

model

I

of

the

least

static

expansion

P

Of

course

this

proof

is

based

on

the

Kripke

structure

K

hW

R

V

i

already

mentioned

in

Section

ffl

The

set

of

worlds

W

are

the

reduced

interpretations

I

Iobj

-LSB-

Ibel

we

are

interested

in

i.e.

statisfying

I

j

P

and

Ibel

Bel

ffl

R

-LRB-

I

I

-RRB-

i.e.

I

sees

I

iff

I

is

a

minimal

model

of

P

and

for

all

pn

AtL

I

j

B

-LRB-

pn

-RRB-

=-RRB-

I

j

pn

ffl

The

valuation

V

-LRB-

I

-RRB-

of

a

reduced

interpretation

I

Iobj

-LSB-

Ibel

is

the

objective

part

Iobj

As

before

we

denote

by

K

-LRB-

I

-RRB-

the

full

interpretation

satisfying

the

belief

formulas

true

in

world

I

i.e.

K

-LRB-

I

-RRB-

j

B

-LRB-

F

-RRB-

-LRB-

-RRB-

-LRB-

K

I

-RRB-

j

B

-LRB-

F

-RRB-

-LRB-

-RRB-

for

all

I

W

with

R

-LRB-

I

I

-RRB-

-LRB-

K

I

-RRB-

j

F

Now

the

reduced

interpretation

I

Iobj

-LSB-

Ibel

assigns

a

truth

value

to

the

atoms

B

-LRB-

pn

-RRB-

and

the

Kripke

structure

also

assigns

a

truth

value

to

them

But

the

construction

guarantees

that

the

truth

values

always

agree

Lemma

C.

For

I

-LRB-

Iobj

-LSB-

Ibel

-RRB-

W

and

all

pn

AtL

K

-LRB-

I

-RRB-

j

B

-LRB-

pn

-RRB-

-LRB-

-RRB-

Ibel

j

B

-LRB-

pn

-RRB-

Proof

ffl

Let

Ibel

j

B

-LRB-

pn

-RRB-

By

the

construction

I

j

pn

holds

for

all

worlds

I

seen

by

I

i.e.

satisfying

R

-LRB-

I

I

-RRB-

Thus

K

-LRB-

I

-RRB-

j

B

-LRB-

pn

-RRB-

ffl

Let

Ibel

B

-LRB-

pn

-RRB-

Since

Bel

is

a

fixpoint

of

P

we

have

Bel

P

P

-LRB-

Bel

-RRB-

By

the

definition

of

P

there

is

a

non-empty

P

-LRB-

Bel

-RRB-

such

that

the

belief

atoms

true

in

Ibel

are

the

intersection

of

the

corresponding

true

disjunctions

in

Thus

there

is

I

obj

with

I

obj

pn

Furthermore

for

all

qm

with

Ibel

j

B

-LRB-

qm

-RRB-

we

have

I

obj

j

qm

Now

by

the

definition

of

P

there

is

a

I

bel

Bel

such

that

I

I

obj

-LSB-

I

bel

is

a

minimal

model

of

P

It

follows

that

R

-LRB-

I

I

-RRB-

holds

i.e.

I

sees

a

world

with

valuation

I

obj

in

which

pn

is

false

Thus

K

-LRB-

I

-RRB-

B

-LRB-

pn

-RRB-

Lemma

C.

For

all

reduced

interpretations

I

W

the

full

interpretation

I

K

-LRB-

I

-RRB-

is

a

model

of

the

least

static

expansion

P

Proof

We

show

that

the

conditions

of

Theorem

B.

are

satisfied

ffl

We

first

have

to

show

that

every

world

I

Iobj

-LSB-

Ibel

sees

at

least

one

world

I

This

follows

from

Ibel

Bel

i.e.

Ibel

P

P

-LRB-

Bel

-RRB-

By

the

definition

of

P

there

is

a

non-empty

subset

Mobj

P

-LRB-

Bel

-RRB-

such

that

every

I

obj

Mobj

satisfies

for

all

pn

AtL

Ibel

j

B

-LRB-

pn

-RRB-

=-RRB-

I

obj

j

pn

Now

by

the

definition

of

P

there

is

a

I

bel

Bel

such

that

I

I

obj

-LSB-

I

bel

is

a

minimal

model

of

P

But

then

R

-LRB-

I

I

-RRB-

holds

ffl

The

reduced

interpretations

I

W

satisfy

the

program

P

and

by

Lemma

C.

I

and

I

K

-LRB-

I

-RRB-

agree

on

the

atoms

occurring

in

P

Thus

also

I

is

a

model

of

P

ffl

For

every

-LRB-

I

I

-RRB-

R

the

reduced

interpretation

I

is

a

minimal

model

of

P

If

the

full

interpretation

I

K

-LRB-

I

-RRB-

were

not

a

minimal

model

of

P

i.e.

there

were

a

smaller

model

I

of

P

then

its

reduct

I

would

be

a

smaller

model

than

I

-LRB-

using

again

Lemma

C.

in

order

to

conclude

that

I

and

I

and

thus

I

and

I

agree

on

the

belief

atoms

-RRB-

Now

Theorem

B.

allows

us

to

conclude

that

the

formulas

true

in

all

worlds

of

W

are

a

static

expansion

of

P

Of

course

every

I

K

-LRB-

I

-RRB-

is

a

model

of

this

static

expansion

But

the

least

static

expansion

P

is

a

subset

so

I

is

also

a

model

of

P

Lemma

C.

Let

I

be

a

full

model

of

the

least

static

expansion

P

and

let

I

Iobj

-LSB-

Ibel

be

its

reduct

Then

Ibel

Bel

Proof

We

show

by

induction

on

k

that

the

belief

part

Ibel

of

a

model

I

of

the

least

static

expansion

P

is

contained

in

kP

-LRB-

Bel

-RRB-

For

k

this

is

trivial

since

Bel

is

the

complete

set

of

belief

interpretations

Let

us

assume

that

Ibel

kP

-LRB-

Bel

-RRB-

We

have

to

show

that

Ibel

is

not

filtered

out

by

one

further

application

of

P

Let

Mbel

kP

-LRB-

Bel

-RRB-

Mobj

P

-LRB-

Mbel

-RRB-

and

Iobj

Mobj

for

every

belief

atom

B

-LRB-

pn

-RRB-

if

Ibel

j

B

-LRB-

pn

-RRB-

then

Iobj

j

pn

We

will

now

show

that

has

the

properties

required

in

the

definition

of

P

namely

Ibel

j

B

-LRB-

pn

-RRB-

-LRB-

-RRB-

for

all

I

obj

I

obj

j

pn

That

is

non-empty

follows

trivially

if

we

allow

also

n

i.e.

the

empty

disjunction

which

is

logically

false

Because

of

the

consistency

axiom

Ibel

B

-LRB-

-RRB-

But

then

there

must

be

at

least

one

I

obj

which

of

course

satisfies

I

obj

Also

the

direction

=-RRB-

follows

trivially

from

the

construction

Now

we

have

to

show

that

for

every

belief

atom

B

-LRB-

pn

-RRB-

which

is

false

in

Ibel

there

is

I

obj

with

I

obj

pn

Let

B

-LRB-

qi

qi

ni

-RRB-

i

m

be

all

belief

atoms

true

in

Ibel

Since

I

j

P

the

formula

m

i

B

-LRB-

qi

qi

ni

-RRB-

B

-LRB-

pn

-RRB-

can

not

be

contained

in

P

-LRB-

it

is

violated

by

Ibel

and

thus

by

I

-RRB-

However

using

the

tautology

-LRB-

Fm

-RRB-

G

-LRB-

-LRB-

Fm

G

-RRB-

:-RRB-

and

the

normality

axiom

we

see

that

the

above

formula

would

follow

from

B

m

i

-LRB-

qi

qi

ni

-RRB-

-LRB-

pn

-RRB-

Thus

this

formula

is

also

not

contained

in

P

But

the

static

semantics

requires

that

B

-LRB-

F

-RRB-

P

if

P

j

min

F

So

there

must

be

a

minimal

model

I

of

P

violating

m

i

-LRB-

qi

qi

ni

-RRB-

-LRB-

pn

-RRB-

i.e.

with

I

j

qi

qi

ni

for

i

n

and

I

pn

But

by

our

inductive

hypothesis

the

belief

part

I

bel

of

I

is

contained

in

Mbel

and

thus

the

objective

part

I

obj

is

in

Mobj

Since

I

obj

j

qi

qi

ni

it

is

contained

in

Thus

I

obj

is

the

required

element

Now

Theorem

follows

directly

from

Lemma

C.

and

Lemma

C.

Part

of

Theorem

can

also

be

written

as

For

every

Ibel

Bel

There

is

a

full

model

I

of

P

with

belief

part

Ibel

-LRB-

-RRB-

Ibel

Bel

The

direction

=-RRB-

is

Lemma

C.

and

the

direction

-LRB-

follows

from

Lemma

C.

and

the

fact

that

any

belief

interpretation

Ibel

can

be

extended

to

a

reduced

model

I

Iobj

-LSB-

Ibel

of

P

-LRB-

by

making

all

objective

atoms

AtL

true

in

Iobj

-RRB-

But

then

I

W

Part

of

Theorem

was

For

every

reduced

interpretation

I

Iobj

-LSB-

Ibel

There

is

a

full

model

I

of

P

with

reduct

I

-LRB-

-RRB-

Ibel

Bel

and

Iobj

-LSB-

Ibel

j

P

Here

the

direction

-LRB-

is

Lemma

C.

and

the

direction

=-RRB-

follows

trivially

from

Lemma

C.

and

P

P

D

Proof

of

Fixpoint

Characterization

-LRB-

Theorem

-RRB-

Unfortunately

I

do

n't

have

time

at

the

moment

to

do

this

carefully

Lemma

D.

A

belief

interpretation

Ibel

is

a

model

of

the

belief

formulas

iP

-LRB-

P

-RRB-

nP

iff

Ibel

iP

-LRB-

Bel

-RRB-

Proof

The

proof

is

by

induction

on

i

The

case

i

is

trivial

-LRB-

no

belief

formulas

all

belief

interpretations

-RRB-

Let

us

use

the

shorthand

Bel

i

iP

-LRB-

Bel

-RRB-

ffl

Suppose

that

Ibel

-LRB-

Bel

i

-RRB-

and

S

iP

-LRB-

P

-RRB-

minimally

implies

Cn

We

have

to

show

that

Ibel

j

B

-LRB-

-RRB-

B

-LRB-

Cn

-RRB-

B

-LRB-

-RRB-

Suppose

this

were

not

the

case

Since

Ibel

P

P

-LRB-

Bel

i

-RRB-

we

would

need

a

model

I

obj

P

-LRB-

Bel

i

-RRB-

which

satisfies

Cj

for

j

n

but

violates

Let

I

bel

Bel

i

be

the

belief

interpretation

on

which

the

minimal

model

is

based

i.e.

such

that

I

I

obj

-LSB-

I

bel

is

a

minimal

model

of

P

But

the

indeuctive

hypothesis

gives

us

Ibel

j

iP

-LRB-

P

-RRB-

n

P

and

since

I

is

a

minimal

model

of

P

Lemma

A.

allows

us

to

conclude

that

I

is

a

minimal

model

of

S

iP

-LRB-

P

-RRB-

But

this

is

a

contradiction

since

Cn

is

minimally

implied

ffl

Let

Ibel

be

a

model

of

i

P

-LRB-

P

-RRB-

n

P

By

Lemma

A.

this

is

a

superset

of

iP

-LRB-

P

-RRB-

n

P

so

Ibel

is

also

a

model

of

the

subset

and

thus

the

inductive

hypothesis

gives

Ibel

Bel

i.

Let

B

-LRB-

Cj

-RRB-

j

n

be

all

belief

atoms

true

in

Ibel

and

B

-LRB-

-RRB-

be

any

belief

atom

false

in

Ibel

Then

i

P

-LRB-

P

-RRB-

can

not

contain

B

-LRB-

-RRB-

B

-LRB-

Cn

-RRB-

B

-LRB-

-RRB-

But

this

means

that

there

is

a

minimal

model

I

I

obj

-LSB-

I

bel

of

S

iP

-LRB-

P

-RRB-

with

I

obj

j

Cj

j

n

and

I

obj

By

the

inductive

hypothesis

I

bel

Bel

i

and

thus

I

obj

P

-LRB-

Bel

i

-RRB-

-LRB-

using

again

Lemma

A.

-RRB-

Now

the

set

containing

-LRB-

at

least

-RRB-

one

such

I

obj

for

every

not

satisfied

in

Ibel

shows

that

Ibel

P

P

-LRB-

Bel

i

-RRB-

fLast

document

revision

February

at

MET

g

disjunctive logic programming
implementation

non-monotonic reasoning
semantics and foundations