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TRANSCRIPT
Introductionto
NeighborhoodSemanticsfor
ModalLogic
EricPacuit
January
7,2007
ILLC,University
ofAmsterdam
staff.science.uva.nl/∼epacuit
Introduction
1.Motivation
2.NeighborhoodSem
antics
forModalLogic
3.BriefSurvey
ofResults
4.Bisimulations
Game�Forcing�Operator
LetG
beanextensivegameandsanodeinG.
Wesayanagenticanforceaform
ulaφats(w
ritten
s|=
�iφ)
provided
1.icanmoveats
2.thereisastrategy
forisuch
thatforallstrategieschosenbythe
other
players,φwillbecometrue.
Game�Forcing�Operator
lr
LR
lr
A
BB
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hoo
dM
odel
s
Eri
cPac
uit
Jan
uar
y6,
2007
1In
troducti
on
p
q Inth
ese
not
esw
ew
illst
udy
the
model
theo
ryof
modal
logi
cusi
ng
nei
ghbor
-hood
model
s.N
eigh
bor
hood
model
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vente
dby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
modal
logi
c(s
ee[4
,2]
for
mor
ein
form
atio
n)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revie
wof
modal
logi
can
dit
sse
man
tics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
senttw
oex
ample
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
nta
ins
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hood
fram
es.
Inth
ere
mai
nin
gse
ctio
nsw
ew
illsy
stem
atic
ally
study
the
model
theo
ryof
modal
logi
cw
ith
resp
ect
tonei
ghbor
hood
model
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sente
nce
sis
gener
ated
by
the
follow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬
!¬
!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Game�Forcing�Operator
lr
lr
A
BB
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
anti
csin
term
sof
Kri
pke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lationalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lational
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
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aliz
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nof
the
stan
dar
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ripke
,or
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tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
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ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
anti
csin
term
sof
Kri
pke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
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age,
den
oted
byL
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,w
her
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set
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ence
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dM
onta
gue
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dep
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tly
in[5
]an
d[3
]).
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assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
modal
logi
c(s
ee[4
,2]
for
mor
ein
form
atio
n)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revie
wof
modal
logi
can
dit
sse
man
tics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
senttw
oex
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sw
hic
hw
illm
otiv
ate
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rst
udy.
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tion
2co
nta
ins
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hood
fram
es.
Inth
ere
mai
nin
gse
ctio
nsw
ew
illsy
stem
atic
ally
study
the
model
theo
ryof
modal
logi
cw
ith
resp
ect
tonei
ghbor
hood
model
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
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nce
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gener
ated
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odal
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vent
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ttan
dM
onta
gue
(indep
en-
den
tly
in[?
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d[?
]).
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assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
mod
allo
gic
(see
[?,?]
for
mor
ein
form
atio
n)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
cture
s.T
hes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
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mod
allo
gic
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its
sem
anti
csin
term
sof
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pke
stru
cture
s.A
fter
that
,w
ew
illpre
sent
two
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sw
hic
hw
illm
otiv
ate
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rst
udy.
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tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
c-ti
ons
we
willsy
stem
atic
ally
study
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
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ence
sis
gener
ated
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efo
llow
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gram
mar
1:
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2.
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troducti
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theo
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modal
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s.N
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sar
ea
gener
aliz
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nof
the
stan
dar
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ripke
,or
rela
tion
al,se
man
tics
form
odal
logi
cin
vente
dby
Sco
ttan
dM
onta
gue
(indep
en-
den
tly
in[?
]an
d[?
]).
We
assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
modal
logi
c(s
ee[?
,?]
for
mor
ein
form
atio
n)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
cture
s.T
hes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revie
wof
modal
logi
can
dit
sse
man
tics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
illpre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
nta
ins
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hood
fram
es.
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ere
mai
nin
gse
c-ti
ons
we
willsy
stem
atic
ally
study
the
model
theo
ryof
modal
logi
cw
ith
resp
ect
tonei
ghbor
hood
model
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
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sente
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sis
gener
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the
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dm
odel
s.N
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bor
hoo
dm
odel
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ea
gener
aliz
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nof
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stan
dar
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ripke
,or
rela
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al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
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dep
enden
tly
in[5
]an
d[3
]).
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assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
anti
csin
term
sof
Kri
pke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
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tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
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gram
mar
1:
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s.N
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bor
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gener
aliz
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nof
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stan
dar
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ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
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assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
anti
csin
term
sof
Kri
pke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
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tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
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edon
afr
ame
Fis
apai
r#F
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mula
sfr
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rpre
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take
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ecl
ear
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Sect
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2.
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ghbor
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cPac
uit
Janu
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6,20
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troduct
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p
q Inth
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illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
wit
hbas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
anti
csin
term
sof
Kri
pke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
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1:
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note
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gue
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en-
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tly
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eass
um
efa
milia
rity
wit
hbasi
cm
odel
theo
ryof
modal
logic
(see
[?,?]
for
more
info
rmati
on)
usi
ng
Kripke,
or
rela
tional,
stru
cture
s.T
hes
enote
sare
org
aniz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revie
wof
modal
logic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that,
we
willpre
sent
two
exam
ple
sw
hic
hw
illm
oti
vate
the
rest
ofour
study.
Sec
tion
2co
nta
ins
som
ebasi
cte
rmin
olo
gy
and
resu
lts
about
nei
ghborh
ood
fram
es.
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ere
main
ing
sec-
tions
we
willsy
stem
ati
cally
study
the
model
theo
ryof
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gic
with
resp
ect
tonei
ghborh
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model
s.T
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basi
cm
odal
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den
ote
dbyL
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,w
her
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sis
gen
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dby
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mm
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ypic
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eot
her
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ampl
e"!
isso
met
imes
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edto
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!¬
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h!
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ason
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hich
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rla
ter
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ctio
n2.
1
Nei
ghbor
hood
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s
Eri
cPacu
it
January
6,2007
1In
troduction
p
q p q Inth
ese
note
sw
ew
illst
udy
the
model
theo
ryofm
odallo
gic
usi
ng
nei
ghbor-
hood
model
s.N
eighborh
ood
model
sare
agen
eraliza
tion
ofth
est
andard
Kri
pke,
orre
lati
onal,
sem
anti
csfo
rm
odallo
gic
inven
ted
by
Sco
ttand
Monta
gue
(indep
en-
den
tly
in[?
]and
[?])
.W
eass
um
efa
milia
rity
wit
hbasi
cm
odel
theo
ryof
modal
logic
(see
[?,?]
for
more
info
rmati
on)
usi
ng
Kripke,
or
rela
tional,
stru
cture
s.T
hes
enote
sare
org
aniz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revie
wof
modal
logic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that,
we
willpre
sent
two
exam
ple
sw
hic
hw
illm
oti
vate
the
rest
ofour
study.
Sec
tion
2co
nta
ins
som
ebasi
cte
rmin
olo
gy
and
resu
lts
about
nei
ghborh
ood
fram
es.
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ere
main
ing
sec-
tions
we
willsy
stem
ati
cally
study
the
model
theo
ryof
modallo
gic
with
resp
ect
tonei
ghborh
ood
model
s.T
he
basi
cm
odal
language,
den
ote
dbyL
(At)
,w
her
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set
of
ato
mic
sente
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gen
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dby
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gra
mm
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:
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of!
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"is
take
nas
prim
itiv
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dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
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!¬
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h!
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hich
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elth
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odal
logi
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nei
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s.N
eigh
bor
hoo
dm
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aliz
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nof
the
stan
dar
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ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
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d[3
]).
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assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
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tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
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lati
onalfr
am
eis
apai
r#W
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isa
rela
tion
onW
(i.e
.,R%
W&
W).
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lati
onal
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bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
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luat
ion
funct
ion.
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mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
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,fo
rex
ampl
e"!
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imes
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chw
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ecl
ear
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Sect
ion
2.
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Nei
ghbor
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els
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cPac
uit
Janu
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6,20
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troduct
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p
q Inth
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the
mod
elth
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odal
logi
cusi
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nei
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dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
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nof
the
stan
dar
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ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
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assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
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opte
dto
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both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
07
1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
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lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
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have
opte
dto
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both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hood
Mod
els
Eri
cPac
uit
Janu
ary
6,20
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1In
troduct
ion
p
q Inth
ese
not
esw
ew
illst
udy
the
mod
elth
eory
ofm
odal
logi
cusi
ng
nei
ghbor
-hoo
dm
odel
s.N
eigh
bor
hoo
dm
odel
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vent
edby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
mod
allo
gic
(see
[4,
2]fo
rm
ore
info
rmat
ion)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revi
ewof
mod
allo
gic
and
its
sem
antics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
sent
two
exam
ple
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
ntai
ns
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hoo
dfr
ames
.In
the
rem
ainin
gse
ctio
nsw
ew
ills
yste
mat
ical
lyst
udy
the
mod
elth
eory
ofm
odal
logi
cw
ith
resp
ect
tonei
ghbor
hoo
dm
odel
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sent
ence
sis
gener
ated
byth
efo
llow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬!
¬!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
Nei
ghbor
hoo
dM
odel
s
Eri
cPac
uit
Jan
uar
y6,
2007
1In
troducti
on
p
q Inth
ese
not
esw
ew
illst
udy
the
model
theo
ryof
modal
logi
cusi
ng
nei
ghbor
-hood
model
s.N
eigh
bor
hood
model
sar
ea
gener
aliz
atio
nof
the
stan
dar
dK
ripke
,or
rela
tion
al,
sem
anti
csfo
rm
odal
logi
cin
vente
dby
Sco
ttan
dM
onta
gue
(in-
dep
enden
tly
in[5
]an
d[3
]).
We
assu
me
fam
ilia
rity
with
bas
icm
odel
theo
ryof
modal
logi
c(s
ee[4
,2]
for
mor
ein
form
atio
n)
usi
ng
Kri
pke
,or
rela
tion
al,
stru
c-tu
res.
Thes
enot
esar
eor
ganiz
edas
follow
s.W
ebeg
inw
ith
ash
ort
revie
wof
modal
logi
can
dit
sse
man
tics
inte
rms
ofK
ripke
stru
cture
s.A
fter
that
,w
ew
ill
pre
senttw
oex
ample
sw
hic
hw
illm
otiv
ate
the
rest
ofou
rst
udy.
Sec
tion
2co
nta
ins
som
ebas
icte
rmin
olog
yan
dre
sult
sab
out
nei
ghbor
hood
fram
es.
Inth
ere
mai
nin
gse
ctio
nsw
ew
illsy
stem
atic
ally
study
the
model
theo
ryof
modal
logi
cw
ith
resp
ect
tonei
ghbor
hood
model
s.T
he
bas
icm
odal
langu
age,
den
oted
byL
(At)
,w
her
eAt
isa
set
ofat
omic
sente
nce
sis
gener
ated
by
the
follow
ing
gram
mar
1:
p|¬
!|!
!!|!
!|"
!
wher
ep"
At.
Are
lati
onalfr
am
eis
apai
r#W
,R$
wher
eR
isa
rela
tion
onW
(i.e
.,R%
W&
W).
Are
lati
onal
model
bas
edon
afr
ame
Fis
apai
r#F
,V$
wher
eV
:At'
2Wis
ava
luat
ion
funct
ion.
For
mula
sfr
omL
are
inte
rpre
ted
at
1T
ypic
ally
only
one
of!
and
"is
take
nas
prim
itiv
ean
dth
eot
her
isde
fined
tobe
the
dual
,fo
rex
ampl
e"!
isso
met
imes
defin
edto
be¬
!¬
!.
We
have
opte
dto
take
both
!an
d"
aspr
imit
ive
for
reas
ons
whi
chw
illbe
com
ecl
ear
late
rin
Sect
ion
2.
1
l
root6|=
�A(p∧q)
Review:(Propositional)ModalLogic
BasicModalLanguage:φ
:=p|¬φ|φ
∧φ|�
φ,wherep∈
Φ0.
RelationalSemantics:
AKripkemodelisatuple〈W
,R,V〉
whereR⊆W×W
andV
:Φ0→
2W.
Review:(Propositional)ModalLogic
BasicModalLanguage:φ
:=p|¬φ|φ
∧φ|�
φ,wherep∈
Φ0.
RelationalSemantics:
AKripkemodelisatuple〈W
,R,V〉
whereR⊆W×W
andV
:Φ0→
2W.
LetR
(w)
={v
|wRv}
Truth
atastate:
LetM
=〈W
,R,V〉beamodelwithw∈W,
M,w
|=�φi�
R(w
)⊆
(φ)M
where
(φ)M
isthetruth
setofφ.
Review:(Propositional)ModalLogic
Given
arelationalframe〈W
,R〉
Review:(Propositional)ModalLogic
Given
arelationalframe〈W
,R〉
Thesetofnecessary
propositions:N
w={X
|R(w
)⊆X}
Review:(Propositional)ModalLogic
Given
arelationalframe〈W
,R〉
Thesetofnecessary
propositions:N
w={X
|R(w
)⊆X}
Fact:
Nwisclosedunder
intersections,supersets
andcontainsthe
unit
andcontainsaminim
alelem
ent:∀w
,∩N
w∈N
w
OtherExamples
Fixt∈
[0,1
]:
IntendedInterpretationof
�φ:φis
assigned
(subjective)
probability>t
Fact:
�φ∧
�ψ→
�(φ∧ψ
)isnotvalidunder
thisinterpretation.
OtherExamples
Fixt∈
[0,1
]:
IntendedInterpretationof
�φ:φis
assigned
(subjective)
probability>t
Fact:
�φ∧
�ψ→
�(φ∧ψ
)isnotvalidunder
thisinterpretation.
Other
examples:
ConcurrentPropositionalDynamicLogic,
Parikh'sGameLogic,orPauly'sCoalitionLogic,Alternating-tim
e
Tem
poralLogic
Extensiveliterature
onthe�logicalomniscience�problem,Deontic
Logics
NeighborhoodSemanticsforPropositionalModalLogic
ANeighborhoodFrameisatuple〈W
,N〉whereN
:W→
22W
ANeighborhoodModelisatuple〈W
,N,V〉whereV
:Φ0→
2W
Truth
inamodelisde�ned
asfollow
s
•M,w
|=pi�w∈V
(p)
•M,w
|=¬φ
i�M,w
6|=φ
•M,w
|=φ∧ψi�M,w
|=φandM,w
|=ψ
•M,w
|=�φi�
(φ)M
∈N
(w)
SomeHistory
NeighborhoodModelswere�rstdiscussed
in(Scott1970,Montague
1970)�
perhaps(M
cKinseyandTarski1944)should
becited?See
(Segerberg1971)and(C
hellas1980)fordiscussionsof
neighborhoodsemantics
forpropositionalmodallogics.
Non-norm
alModalLogics
E�φ↔¬♦¬φ
RE
φ↔ψ
�φ↔
�ψ
M�
(φ∧ψ
)→
(�φ∧
�ψ
)
C(�φ∧
�ψ
)→
�(φ∧ψ
)
N�>
K�
(φ→ψ
)→
(�φ→
�ψ
)
Non-norm
alModalLogics
E�φ↔¬♦¬φ
RE
φ↔ψ
�φ↔
�ψ
M�
(φ∧ψ
)→
(�φ∧
�ψ
)
C(�φ∧
�ψ
)→
�(φ∧ψ
)
N�>
K�
(φ→ψ
)→
(�φ→
�ψ
)
ClassicalModalLogics:
E=E
+RE
+PC,EM
=E
+M,
EC
=E
+C,etc.
Non-norm
alModalLogics
E�φ↔¬♦¬φ
RE
φ↔ψ
�φ↔
�ψ
M�
(φ∧ψ
)→
(�φ∧
�ψ
)
C(�φ∧
�ψ
)→
�(φ∧ψ
)
N�>
K�
(φ→ψ
)→
(�φ→
�ψ
)
ClassicalModalLogics:
E=E
+RE
+PC,EM
=E
+M,
EC
=E
+C,etc.
Fact:
K=
EM
CN
Constraints
onneighborhoodframes
•MonotonicorSupplemented:
IfX∩Y∈N
(w),then
X∈N
(w)andY∈N
(w)
•Closedunder�niteintersections:
IfX∈N
(w)and
Y∈N
(w),then
X∩Y∈N
(w)
•Containstheunit:W
∈N
(w)
•Augmented:
Supplementedplusforeach
w∈W,
⋂ N(w
)∈N
(w)
De�nabilityResults
1.F|=
�(φ∧ψ
)→
�φ∧
�ψi�F
isclosedunder
supersets
(monotonicframes).
2.F|=
�φ∧
�ψ→
�(φ∧
�ψ
)i�F
isclosedunder
�nite
intersections.
3.F|=
�>
i�F
containstheunit
4.F|=
EM
CN
i�F
isa�lter
5.F|=
�φ→φi�
foreach
w∈W,w∈∩N
(w)
6.Andso
on...
Completeness
Results
•Eissoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
•EM
issoundandstrongly
complete
withrespectto
theclass
of
allmonotonic
neighborhoodframes
•EC
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatareclosedunder�nite
intersections
•EN
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatcontain
theunit
•K
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatare�lters
•K
issoundandstrongly
complete
withrespectto
theclass
of
allaugmented
neighborhoodframes
Completeness
Results
•Eissoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
•EM
issoundandstrongly
complete
withrespectto
theclass
of
allmonotonic
neighborhoodframes
•EC
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatareclosedunder�nite
intersections
•EN
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatcontain
theunit
•K
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatare�lters
•K
issoundandstrongly
complete
withrespectto
theclass
of
allaugmented
neighborhoodframes
Completeness
Results
•Eissoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
•EM
issoundandstrongly
complete
withrespectto
theclass
of
allmonotonic
neighborhoodframes
•EC
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatareclosedunder�nite
intersections
•EN
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatcontain
theunit
•K
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatare�lters
•K
issoundandstrongly
complete
withrespectto
theclass
of
allaugmented
neighborhoodframes
Completeness
Results
•Eissoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
•EM
issoundandstrongly
complete
withrespectto
theclass
of
allmonotonic
neighborhoodframes
•EC
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatareclosedunder�nite
intersections
•EN
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatcontain
theunit
•K
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatare�lters
•K
issoundandstrongly
complete
withrespectto
theclass
of
allaugmented
neighborhoodframes
Completeness
Results
•Eissoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
•EM
issoundandstrongly
complete
withrespectto
theclass
of
allmonotonic
neighborhoodframes
•EC
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatareclosedunder�nite
intersections
•EN
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatcontain
theunit
•K
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatare�lters
•K
issoundandstrongly
complete
withrespectto
theclass
of
allaugmented
neighborhoodframes
Completeness
Results
•Eissoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
•EM
issoundandstrongly
complete
withrespectto
theclass
of
allmonotonic
neighborhoodframes
•EC
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatareclosedunder�nite
intersections
•EN
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatcontain
theunit
•K
issoundandstrongly
complete
withrespectto
theclass
of
allneighborhoodframes
thatare�lters
•K
issoundandstrongly
complete
withrespectto
theclass
of
allaugmented
neighborhoodframes
Questions
Whatistheprecise
connectionbetweenneighborhoodsemantics
for
modallogicandrelationalsemantics
formodallogic?
Whatistheexpressivepow
erofthebasicmodallanguageover
neighborhoodframes?
SomeResults
•Foreach
Kripkemodel〈W
,R,V〉,thereisanpointwise
equivalentaugm
ented
neighborhoodmodel〈W
,N,V〉,andvice
versa
(see
(Chellas,1980)formore
inform
ation).
•Thereare
logicswhichare
complete
withrespectto
aclass
of
neighborhoodframes
butnotcomplete
withrespectto
relationalframes
(Gabbay
1975,Gerson1975,Gerson1976).
•Thereare
logicsincomplete
withrespectto
neighborhood
frames
(Martin
Gerson,1975;Litak,200?).
SomeResults
•Foreach
Kripkemodel〈W
,R,V〉,thereisanpointwise
equivalentaugm
ented
neighborhoodmodel〈W
,N,V〉,andvice
versa
(see
(Chellas,1980)formore
inform
ation).
•Thereare
logicswhichare
complete
withrespectto
aclass
of
neighborhoodframes
butnotcomplete
withrespectto
relationalframes
(Gabbay
1975,Gerson1975,Gerson1976).
•Thereare
logicsincomplete
withrespectto
neighborhood
frames
(Martin
Gerson,1975;Litak,200?).
SomeResults
•Foreach
Kripkemodel〈W
,R,V〉,thereisanpointwise
equivalentaugm
ented
neighborhoodmodel〈W
,N,V〉,andvice
versa
(see
(Chellas,1980)formore
inform
ation).
•Thereare
logicswhichare
complete
withrespectto
aclass
of
neighborhoodframes
butnotcomplete
withrespectto
relationalframes
(Gabbay
1975,Gerson1975,Gerson1976).
•Thereare
logicsincomplete
withrespectto
neighborhood
frames
(Martin
Gerson,1975;Litak,200?).
Kracht-WolterTranslation
Given
aneighborhoodmodelM
=〈W
,ν,V〉,de�neaKripkemodel
M◦
=〈V,R
3,R
63,R
ν,Pt,V〉asfollow
s:
•V
=W∪
2W
•R3
={(v,w
)|w∈W,v∈
2W,v∈w}
•R63
={(v,w
)|w∈W,v∈
2W,v6∈w}
•R
ν={(w,v
)|w
∈W,v∈
2W,v∈ν(w
)}
•Pt=W
Kracht-WolterTranslation
LetL′bethelanguage
φ:=
p|¬φ|φ
∧ψ|[3]φ|[63]φ|[ν]φ|P
t
wherep∈
Atand
Ptisaunary
modaloperator.
De�neST
:LN
ML→L′asfollow
s
•ST
(p)
=p
•ST
(¬φ)
=¬S
T(φ
)
•ST
(φ∧ψ
)=ST
(φ)∧ST
(φ)
•ST
(�φ)
=〈ν〉(
[3]ST
(φ)∧
[63]¬ST
(φ))
Kracht-WolterTranslation
Theorem
Foreach
neighborhoodmodelM
=〈W
,ν,V〉andeach
form
ualaφ∈L
NM
L,foranyw∈W,
M,w
|=φ
i�M
◦ ,w|=
Pt→ST
(φ)
(Thetranslationissimplerifmonotonicityisassumed)
ExpressivePowerofModalLogic(w.r.t.RelationalFrames)
VanBenthem
CharacterizationTheorem
Ontheclass
of
KripkeStructures,ModalLogicisthebisimulationinvariant
fragmentof�rst-order
logic.
BisimulationsforNeighborhoodStructures
FirstAttempt:
LetM
=〈W
,N,V〉andM
′=〈W
′ ,N′ ,V′ 〉be
twoneighborhoodstructuresands∈W
andt∈W′ .Anon-empty
relationZ⊆W×W′isabisimulationbetweenM
andM
′if
•(prop)IfwZw′then
wandw′satisfythesameform
ulas
•(back)IfwZw′andX∈N
(w)then
thereisaX′⊆W′such
thatX′∈N′ (w′ )and∀x
′∈X′∃x
∈X
such
thatxZx′
•(forth)IfwZw′andX′∈N′ (w′ )then
thereisaX⊆W
such
thatX∈N
(w)and∀x
∈X∃x
′∈X′such
thatxZx′
BisimulationsforNeighborhoodStructures
FirstAttempt:
LetM
=〈W
,N,V〉andM
′=〈W
′ ,N′ ,V′ 〉be
twoneighborhoodstructuresands∈W
andt∈W′ .Anon-empty
relationZ⊆W×W′isabisimulationbetweenM
andM
′if
•(prop)IfwZw′then
wandw′satisfythesameform
ulas
•(back)IfwZw′andX∈N
(w)then
thereisaX′⊆W′such
thatX′∈N′ (w′ )and∀x
′∈X′∃x
∈X
such
thatxZx′
•(forth)IfwZw′andX′∈N′ (w′ )then
thereisaX⊆W
such
thatX∈N
(w)and∀x
∈X∃x
′∈X′such
thatxZx′
BisimulationsforNeighborhoodStructures
FirstAttempt:
LetM
=〈W
,N,V〉andM
′=〈W
′ ,N′ ,V′ 〉be
twoneighborhoodstructuresands∈W
andt∈W′ .Anon-empty
relationZ⊆W×W′isabisimulationbetweenM
andM
′if
•(prop)IfwZw′then
wandw′satisfythesameform
ulas
•(back)IfwZw′andX∈N
(w)then
thereisaX′⊆W′such
thatX′∈N′ (w′ )and∀x
′∈X′∃x
∈X
such
thatxZx′
•(forth)IfwZw′andX′∈N′ (w′ )then
thereisaX⊆W
such
thatX∈N
(w)and∀x
∈X∃x
′∈X′such
thatxZx′
Only
worksformonotonicmodallogics
BoundedMorphism
LetM
1=〈W
1,N
1,V
1〉andM
2=〈W
2,N
2,V
2〉betwo
neighborhoodmodels.
Aboundedmorphism
fromM
1toM
2is
amapf
:W1→W
2such
thatforallX⊆W
2andw∈W
1,
f−
1[X
]∈N
1(w
)i�
X∈N
2(f
(w))
andforallp,w∈V
1(p
)i�f(w
)∈V
2(p
)
BoundedMorphism
LetM
1=〈W
1,N
1,V
1〉andM
2=〈W
2,N
2,V
2〉betwo
neighborhoodmodels.
Aboundedmorphism
fromM
1toM
2is
amapf
:W1→W
2such
thatforallX⊆W
2andw∈W
1,
f−
1[X
]∈N
1(w
)i�
X∈N
2(f
(w))
andforallp,w∈V
1(p
)i�f(w
)∈V
2(p
)
LemmaLetM
1=〈W
1,N
1,V
1〉andM
2=〈W
2,N
2,V
2〉betwo
neighborhoodmodelsandf
:W1→W
2abounded
morphism.
Then
foreach
modalform
ulaφ∈L
NM
Landstatew∈W
1,
M1,w
|=φi�
M2,f
(w)|=φ
BehavorialEquivalence
Twomodel-state
pairsM
1,w
1andM
2,w
2arebehavorially
equivalentprovided
thereisaneighborhoodmodelN
=〈W
,N,V〉
such
thatthereare
boudned
morphismsfromffromM
1toN
and
gfromM
2toN
andf(w
1)
=g(w
2).
Two-SortedFirst-OrderLanguageforNeighborhoodStructures
ViewM
◦=〈V,R
3,R
63,R
ν,Pt,V〉asa2-sorted�rst-order
structure.
LetL
2beatwo-sorted
�rst-order
language(pointvariablesand
�set�variables)
Fact:
First-order
structuresthatare
generatedbyneighborhood
structures(i.e.,oftheform
M◦forsomeneighborhoodstructure
M)canbeaxiomatized(inL
2).
Standard
Translation
Wemapform
ulasofthebasicmodallanguagetoL
2:
•st
x(p
)=Px
•st
x(¬φ)
=¬st x
(φ)
•st
x(φ∧ψ
)=st
x(φ
)∧st
x(ψ
).
•st
x(�φ)
=∃u
(xR
νu∧
(∀z(uR3z→st
z(φ
))∧∀z′ (uR63z′→
¬st z
′ (φ))
ST
x(φ
)=Wx→st
x(φ
)
CharacterizationTheorem
forClassicalModalLogic
Theorem
(Pauly)Ontheclass
ofneighborhoodmodels,monotonic
modallogicisthemonotonicbisimulationinvariantfragmentofL
2
CharacterizationTheorem
forClassicalModalLogic
Theorem
(Pauly)Ontheclass
ofneighborhoodmodels,monotonic
modallogicisthemonotonicbisimulationinvariantfragmentofL
2
Theorem
Ontheclass
ofneighbourhoodmodels,modallogicis
thebehaviouralequivalence-invariantfragmentofL
2.
Jointwork
withHelleHvid
HansenandClemensKupke
Conclusions
•Modeltheory
ofmodallogicwithrespectto
neighborhood
structures
•MonotonicModalLogicshavebeenstudied(eg.Hansen,2003)
•Largeliterature
onthetopologicalinterpretationofmodallogic
ModalLangu
age
forTopology:ExpressivityandDe�
nability,B.
tenCate,D.Gabelaia
andV.Sustretov,2006.
Thankyou.