Marelo Finger Dov Gabbay Mark Reynolds
Draft for the 2nd editionof the Handbook of Philosophial Logi
The expressivity of a language is always measured with respet to some
other language. That is, when talking about expressivity, we are always
omparing two or more languages. When measuring the expressivity of a
large number of languages, it is usually more onvenient to have a single
languagewithrespettowhihallother languagesanbeompared,ifsuh
alanguage isknown to exist.
Intheaseofpropositionalone-dimensionaltemporal languagesdened
bythe preseneof a xednumberoftemporal onnetives (alsoalledtem-
poral modalities), the expressivity of those languages an be all measured
against a fragment of rst-order logi, namely the monadi rst-order lan-
guage. This is the fragment that ontains a binary < (to represent the
underlyingtemporalorder),=(whihwe assumeisalwaysinthelanguage)
and a set of unary prediates Q
1 (x);Q
2
(x);::: (whih aount for the in-
terpretationof thepropositionalletters, that are interpretedasa subset of
thetemporaldomainT). Indeed, anyone-dimensionaltemporalonnetive
an be dened asa well-formed formula in suh a fragment, known as the
onnetive's truthtable; one-dimensionalityforessuh truthtables tohave
asingle freevariable.
In the ase of omparing the expressivity of temporal onnetives, an-
otherparametermust be taken into aount,namelythe underlyingow of
time. Twotemporallanguagesmayhavethesameexpressivityoveroneow
of time (say, the integers) butmay dier in expressivity over another (e.g.
therationals); see thedisussionon theexpressivityof theUS onnetives
below.
Letusexemplifywhatwemeanbythoseterms. Considertheonnetives
sine(S), until(U), future(F), and past(P). Given a ow of time (T;<;h),
the truth value of eah of the above onnetives at a point t2 T is deter-
minedasfollows:
(T;<;h);tj=Fp i (9s>t)(T;<;h);sj=p;
(T;<;h);tj=Pp i (9s<t)(T;<;h);sj=p;
(T;<;h);tj=U(p;q) i (9s>t)((T;<;h);sj=p^
8y(t<y<s!(T;<;h);yj=q));
(T;<;h);tj=S(p;q) i (9s<t)((T;<;h);sj=p^
8y(s<y<t!(T;<;h);yj=q))
If we assume that h(p) represents a rst-order unary prediate that is
interpretedash(p)T,then thesetruth valuesabove an beexpressed as
rst-orderformulas. Thus:
F
(b) (T;<;h);tj=Pq i
P
(t;h(q)) holdsin(T;<),
() (T;<;h);tj=U(q
1
;q
2 ) i
U (t;h(q
1 );h(q
2
))holdsin(T;<),and
(d) (T;<;h);tj=S(q
1
;q
2 ) i
S (t;h(q
1 );h(q
2
))holdsin(T;<).
where
(a)
F
(t;Q)=(9s>t)Q(s);
(b)
P
(t;Q)=(9s<t)Q(s);
()
U (t;Q
1
;Q
2
)=(9s>t)(Q
1
(s)^8y(t<y <s!Q
2 (y)));
(d)
S (t;Q
1
;Q
2
)=(9s<t)(Q
1
(s)^8y(s<y<t!Q
2 (y))):
# (t;Q
1
;:::;Q
n
) is alled the truth table for the onnetive #. The
number n of parameters in the truth table will be the number of plaes
in the onnetive, e.g. F and P are one plae onnetive, and their truth
tables have a single parameter; S and U are two-plae onnetives, with
truthtables havingtwo parameters.
Itislearthatinsuhaway,westartdeninganynumberofonnetives.
For example onsider (t;Q) = 9xy(t < x < y^8s(x <s < y ! Q(s)));
then(t;Q) means`There isan intervalinthefuture oft insidewhih P is
true.' ThisisatableforaonnetiveF
int
: (T;<;h);tj=F
int
(p)i(t;h(p))
holdsin (T;<):
Weareinonditionofpresentingageneraldenitionofwhatatemporal
onnetive is:
Denition 1.1
1. Anyformula(t;Q
1
;:::;Q
m
) withone freevariable t, inthe monadi
rst-order language with prediate variable symbols Q
i
, is alled an
m-plaetruth table (inone dimension).
2. Given a syntati symbol #for an m-plae onnetive, we say it has
atruth table (t;Q
1
;:::;Q
m
) iforanyT;h and t, ()holds:
(): (T;<;h);tj=#(q
1
;:::;q
m
) i(T;<)j=(t;h(q
1
);:::;h(q
m )):
onnetives are denable using other onnetives. For example, it is well
known that F is denable usingU as Fp U(p;>). As anotherexample,
onsidera onnetive thatstates the existene of a \next"timepoint: Æ
U(>;?).
The onnetive Æ is a nie exampleonhow thedenabilityofa onne-
tive by others depends on the lass of ows of time being onsidered. For
example, ina dense ow of time, Æ an be dened in terms of F and P |
atually,sinethere areno \next"time pointsanywhere,Æ ?. Similarly,
inan integer-like ow oftime, Æ is equivalent to >.
On theother hand, onsiderthe ow (T;<) oftime with a singlepoint
withouta \next time": T =f::: 2; 1;0;1;2;:::g[f(1=n)jn=1;2;3:::g,
with<beingtheusualorder;thenÆ isnotdenableusingP and F. To see
that, supposeforontraditionthatÆ isequivalent toA whereA iswritten
withP and F and, maybe, atoms. Replaeall appearanes of atoms by ?
to obtain A 0
. Sine Æ $ A holdsin the struture (T;<;h 0
) with all atoms
always false, in this struture Æ $ A 0
holds. As neither Æ nor A 0
ontain
atoms, Æ $ A 0
holds in all other (T;<;h) as well. Now A 0
ontains only
P and F, >, and ? and the lassial onnetives. Sine F> P> >
and F? P? ?, at every point, A 0
must be equivalent (in (T;<)) to
either>or? andso annotequalÆ whih istrue at 1and falseat 0. As a
onsequene, Æ isnot denableusingP and F over lineartime.
In general, given a family of onnetives, e.g. fF;Pg or fU;Sg, we an
buildnewonnetivesusingthegivenones. Thatthese newonnetivesare
onnetivesinthesenseof Denition1 followsfrom thefollowing.
Lemma1.2 Let #
1 (q
1
;:::;q
m1 );:::;#
n (q
1
;:::;q
mn
) be n temporal onne-
tives with tables
1
;:::;
n
. Let A be any formula built up from atoms
q
1
;:::;q
m
,the lassial onnetives,andthese onnetives. Thenthereexists
a monadi
A (t;Q
1
;:::;Q
m
) suh that for all T andh,
(T;<;h);tj=A i (T;<)j=
A (t;h(q
1
);:::;h(q
m )):
Proof. We onstrut
A
by indutionon A. The simple ases are:
q
j
=
Q
j (t),
:A
=:
A and
A^B
=
A
^
B .
Forthetemporalonnetivease,weonstruttheformula
#
i (A
1
;:::;A
m
i )
=
i (t;
A
1
;:::;
Am
i
); the right-hand sideis a notationfor theformula ob-
tained by substituting
A
j
(x) in
i
wherever Q
j
(x) appears, with the ap-
propriaterenamingof boundvariables to avoidlashes. The indutionhy-
pothesis is applied over
A1
;:::;
Am
i
and the result is simply obtained by
truthtable of theonnetive #
i
.
A
poralformulaA. Anm-paleonnetive#withtruthtable(t;Q
1
;:::;Q
m )
issaid to be denablefrom onnetives #
1
;:::;#
n
in a ow of time(T;<)
ifthereexistsatemporalformulaAbuiltfromthoseonnetiveswhoserst
ordertranslationis
A
suh that
(T;<)j=
A
$:
The expressive power of a family of onnetives over a ow of time is
measuredbyhowmanyonnetivesitanexpressovertheowoftime. Ifit
an expressany oneivable onnetive (given by amonadi formula),then
thatfamilyof onnetivesis expressivelyomplete.
Denition 1.3 A temporallanguagewithone-dimensionalonnetives is
said to be expressively omplete or, equivalently, funtionally omplete, in
one dimension over a lass T of partialorders i forany monadi formula
(t;Q
1
;:::;Q
m
),there exists an A of thelanguage suh that forany(T;<)
inT,foranyinterpretationh forq
1
;:::;q
m ,
(T;<)j=8t( $
A )(t;h(q
1
);:::;h(q
m )):
In the ases where T = f(T;<)g we talk of expressive ompleteness
over (T;<). For example, the language of Sine and Until is expressively
omplete over integer time and real number ow of time, as we are going
to see in Setion 1.2; but they are not expressivelyomplete over rational
numbers time[GPSS80 ℄.
Denition 1.4 A ow of time (T;<) is said to be expressivelyomplete
(or funtionallyomplete) (in one dimension)i there exists a nite set of
(one-dimensional)onnetiveswhihisexpressivelyompleteover(T;<),in
one dimension.
The qualiation of one-dimensionality in the denitions above will be
explainedwhen we introduethe notionofH-dimensionbelow.
These notions parallel the denability and expressive ompleteness of
lassiallogi. We knowthat inlassiallogif:;!g issuÆient to dene
all other onnetives. Furthermore,for anyn-plae truth table :2 n
! 2
there existsanA(q
1
;:::;q
n
) of lassiallogisuh thatforanyh,
h(A)= (h(q
1
);:::;h(q
n )):
Thisisthe expressive ompletenessof f:;!g inlassiallogi.
tions:
(a) Given anitesetofonnetives anda lassof owsof time,arethese
onnetivesexpressivelyomplete?
(b) Inase the answerto (a) is no,we would like to ask: given a lass of
ows, doesthere exist anitesetof one-dimensionalonnetivesthat
isexpressivelyomplete?
These questions oupy us to the rest of this setion. We show that
thenotionofexpressiveompletenessisintimatelyrelatedtotheseparation
property,whihwe investigate inSetion.
The answer to question (b) is related to the notion of H-dimension,
disussedinSetion1.3.
1.1 Separation and Expressive Completeness
The notion of separation involves partitioning a ow of time in disjoint
parts (typially: present, past and future). A formula is separable if it is
equivalentto anotherformulawhosetemporal onnetivesrefer onlyto one
ofthepartitions.
If every formulaina languageis separable, thatmeans thatwe have at
leastoneonnetivethathasenoughexpressivityovereahofthepartitions.
Sowemightexpetthatthatsetofonnetivesisexpressivelyompleteover
a lass of ows that admits suh partitioning, provided the partitioning is
also expressiblebytheonnetives.
The notionof separation wasinitiallyanalysedin termsof linear ows,
where the notion of present, past and future most naturally applies. So
we start our disussion with separation over linear time. We later extend
separationto generiows.
1.1.1 Separation over linear time
Consider a linear ow of time (T;<). Let h;h 0
be two assignments and
t2T. Wesaythat h;h 0
agree on the past of t, h=
<t h
0
,i foranyatom q
andanys<t,
s2h(q) i s2h 0
(q):
We deneh 0
=
=t
h foragreement of the present,i forany atomq
t2h(q) i t2h 0
(q):
andh =
>t
h,foragreement on the future,iforanyatomq and anys>t,
s2h(q) i s2h 0
(q):
LetT bealassoflinearowsoftimeand Abeaformulainatemporal
languageover (T;<). We say that A is a pure past formula over T, i for
all(T;<)in T,forallt2T,
8h;h 0
;(h=
<t h
0
)impliesthat(T;<;h);tj=Ai (T;<;h 0
);tj=A:
Similarly,we denepure futureand purepresent formulas.
Suh a denition of purity is a semanti one. In a temporal language
ontaining only S and U there is also have a notion of syntati purity as
follows. Aformulais aboolean ombination of
1
,...,
n
ifitisbuilt from
1
, ...,
n
using only boolean onnetives. A syntatially pure present
formula is a boolean ombination of atoms only. A syntatially pure past
formula isa booleanombination offormulas of theform S(A;B) whereA
and B are either pure present or pure past. Similarly,a syntatially pure
future formula is a boolean ombination of formulas of the form U(A;B)
whereA andB are eitherpurepresent orpure future.
It is lear that if A is a syntatially pure past formula, then A is a
purepastformula; similarlyforpurepresentand purefutureformulas. The
onverse,however, isnottrue. Forexample,from thesemantialdenition,
alltemporal temporallyvalidformulas arepurefuture (and purepast,and
purepresent), inludingthose involving S.
We are now inapositionto denetheseparationproperty.
Denition 1.5 LetT bealassoflinearowsoftimeandAbeaformula
inatemporallanguageL.WesayA isseparableinLoverT ithere exists
a formula in L whih is a boolean ombination of pure past, pure future,
and atomi formulas and is equivalent to A everywhere in any (T;<) from
T.
A set of temporal onnetives is said to have the separation property
over T i every formula in the temporal language of these onnetives is
separableinthelanguage(over T).
We nowshowthatseparation impliesexpressiveompleteness.
Theorem 1.6 Let L be a temporal language built from any number (nite
or innite) of onnetives in whih P and F are denable over a lass T
of linear ows of time. If L has the separation property over T then L is
expressively omplete over T.
We have to showthatforany'(t;Q ) inthemonadi theoryof linearorder
with prediate variable symbols Q = (Q
1
;:::;Q
n
), there exists a formula
A = A(q
1
;:::;q
n
) in the temporal language suh that for all ows of time
(T;<) fromT,forallh;t, (T;<;h);tj=A i(T;<)j='(t;h(q
1
);:::;h(q
n )).
We denote thisformulabyA['℄and proeedbyindutionon thedepth
mofnestedquantiersin'. Form=0,'(t) isquantierfree. Just replae
eah appearane of t=t by >, t<t by ?, and eah Q
j
(t)byq
j
to obtain
A['℄.
Form>0,weanassume'=9x (t;x;Q )where hasquantierdepth
m (the8 quantieris treatedas derived).
Assuming that we do not use t as a bound variable symbol in and
that we have replaed all appearanes of t =t by > and t < t by ? then
the atomi formulas in whih involve t have one of the following forms:
Q
i
(t),t<y,t=y,ory<t,wherey ould be xoranyothervariable letter
ourringin .
If we regard t as xed, the relations t < y;t = y;t > y beome unary
andanrewritten, respetively,asR
<
(y),R
=
(y)andR
>
(y),whereR
<
,R
=
and R
>
arenew unaryprediatesymbols.
Then anbe rewrittenequivalentlyas
t
0
(x;Q ;R
=
;R
>
;R
<
);
in whih t appears only in the form Q
i
(t). Sine t is free in , we an go
furtherand prove (by indutionon the quantier depth of ) that t
0 an
be equivalentlyrewrittenas
t
= _
j [
j (t)^
t
j
(x;Q;R
=
;R
>
;R
<
)℄;
where
j
(t)isquantierfree,
Q
i
(t)appearonlyin
j
(t)and notat all in t
j ,
andeah t
j
hasquantierdepthm.
Bytheindutionhypothesis,thereisaformulaA
j
=A
j (q;r
=
;r
>
;r
<
)in
thetemporal languagesuhthat, foranyh;x,
(T;<;h);xj=A
j
i(T;<)j= t
j (x;h(q
1
);:::;h(q
n );h(r
= );h(r
>
);h(r
<
)):
T) to Pq_q_Fq whoseexistene inL isguaranteed byhypothesis. Then
let B(q;r
=
;r
>
;r
<
) = W
j (A[
j
℄^3A
j
). A[
j
℄ an be obtained from the
quantierfree ase.
Inanystruture(T;<)fromT foranyhinterpretingtheatomsq,r
=
;r
>
andr
<
, thefollowingarestraightforwardequivalenes
(T;<;h);tj=B
(T;<;h);tj= W
j (A[
j
℄^3A
j )
W
j
((T;<;h);tj=A[
j
℄^(T;<;h);tj=3A
j )
W
j (
j
(t)^9x((T;<;h);xj=A
j ))
W
j (
j
(t)^9x t
j (x;h(q
1
);:::;h(q
n );h(r
= );h(r
>
);h(r
<
)))
9x W
j (
j (t)^
t
j (x;h(q
1
);:::;h(q
n );h(r
= );h(r
>
);h(r
<
)))
9x t
0 (x;h(q
1
);:::;h(q
n );h(r
= );h(r
>
);h(r
<
)):
NowprovidedweinterprettheratomsastheappropriateRprediates,
i.e.:
h
(r
=
)=ftg,
h
(r
<
)=fsjt<sg, and
h
(r
>
)=fsjs<tg,
we obtain
(T;<;h
);tj=B i 9x (t;x;h
(q
1 );:::;h
(q
n
))i '(t;h
(q
1 );:::;h
(q
n )):
B is almost the A['℄ we need exept for one problem. B ontains,
besides the q
i
, also three other atoms, r
=
;r
>
, and r
<
, and equation ()
from Denition 9.1.1 above is valid for any h
whih is arbitrary on the
q
i
but very speial on r
=
;r
>
;r
<
. We are now ready to use the separation
property (whih we haven't used so far in the proof). We use separation
to eliminate r
=
;r
>
;r
<
from B. Sine we have separation B is equivalent
to a boolean ombination of atoms, pure past formulas, and pure future
formulas.
So there isa boolean ombination =(p
+
;p ;p
0
) suh that
B$ (B
+
;B ;B
0 );
whereB
0 (q;r
>
;r
=
;r
<
)isaombinationofatoms,B
+ (q;r
>
;r
=
;r
<
)arepure
future,and B (q;r
>
;r
=
;r
<
) arepurepast formulas.
Finally,B
=(B
+
;B
;B
0
)where
B
0
=B
0
(q;?;>;?);
B
+
=B
+
(q;>;?;?);
B
=B (q;?;?;>).
Thenwe obtainfor anyh
,
(T;<;h
);tj=B i(T;<;h
);tj=(B
+
;B ;B
0 )
i(T;<;h
);tj=(B
+
;B
;B
0 )
i(T;<;h
);tj=B
:
Hene
(T;<;h
);tj=B
i (T;<)j='(t;h
(q)):
Thisequationholdsforanyh
arbitraryonq,butrestritedonr
<
;r
>
;r
= .
But r
<
;r
>
;r
=
do notappear in it at all and hene we obtain that forany
h,(T;<;h);tj=B
i (T;<)j='(t;h
(q)). Somake A['℄=B
and we are
done.
The onverse is also true: expressive ompleteness implies separation
over linear time. The proof involves using the rst-order theory of lin-
eartime to rst separate a rst-order formula overlinear time; a temporal
formulaistranslatedintotherst-orderlanguage,whereitisseparated;ex-
pressive ompletenessis neededthento translateeah separated rst-order
subformulainto atemporal formula. Detailsare omitted,butan befound
in[GHR94℄.
1.1.2 Generalized Separation
The separation property is not restrited to linear ows of time. In this
setionwe generalizetheseparationpropertyoveranylassofows oftime
andsee thatTheorem 1.6hasa generalised version.
Thebasiidea istohave some relationsthatwillpartitionevery ow of
timeinT,playingthe roleof <, >and =inthelinear ase.
Denition 1.7 Let'
i
(x;y);i=1;:::;nbengivenformulasinthemonadi
language with < and let T be a lass of ows of time. Suppose '
i (x;y)
partition T,that is, for every t in eah (T;<) in T the sets T(i;t) =fs 2
T j'
i
(s;t)g fori=1;:::;nare mutuallyexlusiveand S
i
T(i;t)=T.
In analogy to theway that F and P represented < and >, we assume
thatfor eah ithere isa formula
i
(t;x)suh that'
i
(t;x)and
i
(t;x) are
equivalent over T and
i
is a boolean ombination of some '
j
(x;t). Also
ofthe'
i :
Then we have thefollowingseriesofdenitions:
Foranytfrom any(T;<) inT,foranyi=1;:::;n, we saythattruth
funtionshand h 0
agreeon T(i;t)ifand onlyifh(q)(s)=h 0
(q)(s)for
alls inT(i;t)and all atomsq.
We saythata formulaAis pure'
i
overT ifforany(T;<)inT,any
t2T andanytwotruth funtionsh and h 0
whihagreeon T(i;t),we
have
(T;<;h);tj=A i(T;<;h 0
);tj=A:
The logi L has the generalized separation property over T i every
formulaA of Lisequivalent overT to a booleanombinationof pure
formula.
Theorem 1.8 (generalized separation theorem) Supposethelanguage
L an express over T the 1-plae onnetives #
i
, i=1;:::;n, dened by:
(T;<;h);tj=#
i
(p) i 9s '
i
(s;t) holds in (T;<)
and (T;<;h);sj=p:
If has thegeneralized separation propertyover a lassT of ows oftime
then L is expressively omplete over T.
A proof ofthisresult appearsin[Ami85 ℄. Seealso [GHR94 ℄.
Theonverse doesnotalwaysholdinthegeneralase, foritdependson
thetheoryof theunderlyinglassT.
Asimpleappliationofthegeneralisedseparationtheoremisthefollow-
ing. Supposewehavearstorderlanguagewiththebinaryorderprediates
<,>, =withtheirusualinterpretation,andsupposeitalso ontainsapar-
alleloperatorjdened by:
xjy=
def
:[(x=y)_(x<y)_(y<x)℄:
Supposewehave a newtemporal onnetive D,denedby
(T;<;h);tj=Dq i9xjtsuhthat (T;<;h);xj=q:
Finally,A issaidto bepureparalleloveralassT ofowsof timeiforall
tfrom any (T;<) fromT, forallh=
jt h
0
,
(T;<;h);tj=Ai(T;<;h 0
);tj=A;
whereh=
jt
h i 8xjt8q(x2h(q)$x2h(q)):
It is lear what separation means in the ontext of pure present, past,
future, and parallel. It is simple to hek that the <;>;=;j satisfy the
generalseparationpropertyandotherpreonditionsforusingthegeneralized
separationtheorem. Thusthat theorem gives immediatelythefollowing.
Corollary 1.9 Let L be a language withF;P;D over any lass of ows of
time. If L has a separation then L is expressively omplete.
1.2 Expressive CompletenessofSine and Untilover Integer
Time
Asanexampleoftheappliationsofseparationto theexpressive omplete-
ness of temporal language, we are going to sketh the proof of separation
of the Sine and Until-temporal logi ontaining over linear time. The full
proof an be foundin[Gab89,GHR94 ℄. With separationand Theorem 1.6
we immediatelyobtainthat theonnetivesS and U are expressivelyom-
pleteoverthe integers; theoriginal proof of theexpressive ompleteness of
S and U overthe integersis dueto Kamp[Kam68 ℄.
The basi idea of the separation proess is to start with a formula in
whih S and U may be nestedinside eahother and throughseveral trans-
formationstepswe aregoingto systematiallyremove U from insideS and
vie-versa. This givesus a syntatial separationwhih,obviously, implies
separation.
As we shall see there are eight ases of nestedourrenes of U within
an S to worry about. It should be noted that all the results inthe rest of
this setion have dual results for the mirror images of the formulas. The
mirror imageof a formula isthe formula obtainedbyinterhangingU and
S;forexample,the mirrorimage of U(p^S(q;r);u)is S(p^U(q;r);u).
We start dealing withbooleanonnetivesinside thesope of temporal
operators, with some equivalenes over integer ows of time. We say that
a formula A is valid over a ow of time (T;<) if it is true at all t 2 T;
notation: (T;<)j=A
Lemma1.10 The following formulas (and their mirror images) are valid
over integer time:
U(A_B;C)$U(A;C)_U(B;C);
U(A;B^C)$U(A;B)^U(A;C);
:U(A;B)$G(:A)_U(:A^:B;:A);
Proof. Simplyapplythesemantial denitions.
We now show the eight separation ases involving simple nesting and
atomiformulas only.
Lemma1.11 Letp;q;A,andB beatoms. Theneahoftheformulas below
isequivalent,overintegertime,toanotherformulainwhihtheonlyappear-
anes of theuntil onnetiveareas theformula U(A;B) andno appearane
of that formula is in the sope of an S:
1. S(p^U(A;B);q),
2. S(p^:U(A;B);q),
3. S(p;q_U(A;B)),
4. S(p;q_:U(A;B)),
5. S(p^U(A;B);q_U(A;B)),
6. S(p^:U(A;B);q_U(A;B)),
7. S(p^U(A;B);q_:U(A;B)), and
8. S(p^:U(A;B);q_:U(A;B)):
Proof. We prove therst ase only; omitting theothers. Note thatS(p^
U(A;B);q) is equivalentto
S(p;q)^S(p;B)^B^U(A;B)
_ [A^S(p;B)^S(p;q)℄
_ S(A^q^S(p;B)^S(p;q);q):
Indeed, the original formula holds at t i there is s < t and u > s suh
thatpholdsats, Aat u,B everywherebetweensandu,and q everywhere
between sand t. The three disjuntsorrespond to the ases u >t,u =t,
and u <t respetively. Note that we make essential useof thelinearity of
time.
Wenowknowthebasistepsinourproofofseparation. Wesimplykeep
pullingout Us from under the sopes of Ss and vie versa until there are
no more. Given a formula A, this proess will eventually leave us with a
nationofatoms, formulas U(E;F)withE andF builtwithoutusingS and
formulas S(E;F) withE andF builtwithoutusingU. Clearly,suhaB is
separated.
We startdealingwith morethanone U insidean S. In thisontext, we
allaformulainwhihU and S do notappearpure.
Lemma1.12 Suppose that A and B are pure formulas and that C and D
are suh that any appearane of U is as U(A;B) and is not nested under
any Ss. Then S(C ;D) is equivalent to a syntatially separated formula in
whih U only appears as the formula U(A;B).
Proof. If U(A;B) does notappear thenwe are done. Otherwise, by rear-
rangementofCandDintodisjuntiveandonjuntivenormalform,respe-
tively,and repeateduseofLemma1.10wean rewriteS(C ;D)equivalently
asabooleanombinationofformulasS(C
1
;D
1
)withnoU appearing. Then
thepreedinglemmashowsthateahsuhbooleanonstituentisequivalent
to abooleanombinationof separated formulas. Thuswe have aseparated
equivalent.
Next let usbegintheindutiveproess ofremovingUs from more than
oneS. Wepresenttheseparationinaresendo. Eahstepintroduesextra
omplexity inthe formulabeing separated and uses the previous ase as a
startingpoint.
Lemma1.13 Suppose that A;B, possibly subsripted, are pure formulas.
SupposeC ;D,possiblysubsripted,ontainnoS. ThenEhasasyntatially
separated equivalent if:
the onlyappearane of U in E is as U(A;B);
the onlyappearanes of U in E are as U(A
i
;B
i );
the onlyappearanes of U in E are as U(C
i
;D
i );
E is any U;S formula.
We omitthe proof, referringto [GHR94,Chapter 10℄ fora detaileda-
ount. But note that sine eah ase above uses the previous one as an
indution basis, this proess of separation tends to be highly exponential.
Indeed, the separated version of a formula an be many times larger than
theinitialone. We nallyhave themainresults.
formulain the fU;Sg-language isequivalentto a separated formula.
Proof. Thisfollowsdiretlyfromthepreedinglemmabeause,aswehave
alreadynoted, syntati separationimpliesseparation.
Theorem 1.15 The language fU;Sg is expressively omplete over integer
time.
Proof. Thisfollows from theseparationtheorem and Theorem1.6.
Otherknownseparationand expressiveompleteness resultsover linear
timeare[GHR94℄:
ThelanguagefU;Sgisseparableoverrealtime. Indeed,itisseparable
over anyDedekindompletelinear ow of time. Asa onsequene, it
isalso expressivelyompleteoversuhows.
ThelanguagefU;Sgisnotseparableovertherationals;asaresult,itis
notseparableoverthelassoflinearowsoftime,norisitexpressively
ompleteover suh ows.
The problem of fU;Sg over generi linear ows of time is that they
mayontaingaps. Itispossibletodenea rstorder formulathatmakes a
propositiontrueupuntilagapandfalseafterwards. Suhformula,however,
annotbeexpressedintermsoffU;Sg. Soisthereanextrasetofonnetives
thatis expressivelyomplete over therationals? The answer inthisase is
yes,and theyarealledtheStavi onnetives. Theseareonnetiveswhose
truthvaluedependsontheexisteneofgapsintheowoftime,andtherefore
arealwaysfalseoverintegersorreals. Foradetaileddisussiononseparation
inthepresene ofgaps, pleaserefer to [GHR94,Chapters11 and 12℄.
We remainwiththefollowinggeneriquestion: givenaowoftime,an
we nd a set of onnetives that is expressively omplete over it? This is
thequestionthatweinvestigate next.
1.3 H-dimension
ThenotionofHenkin-orH-dimensioninvolveslimitingthenumberofbound
variables employed in rst-order formulas. We will see that a neessary
onditionfor there to exist a niteset of onnetives whih is expressively
omplete over a ow of time is that suh ow of time have a nite H-
dimension.
onnetives, we will see that if many-dimensional onnetives are allowed,
thanniteH-dimensionimpliestheexisteneofsuhnitesetofonnetives.
However, when we onsiderone-dimensional onnetivessuh asSine and
Until,niteH-dimensionisno longera suÆient ondition.
In fat our approah in this disussionwill be based on a weak many-
dimensional logi. It is many dimensional beause the truth value of a
formulaisevaluatesat morethanone time-point. Itisweakbeauseatomi
formulas are evaluated only at a single time point (alled the evaluation
point),whilealltheotherpointsarethereferene points). Suhweak many
dimensionalityallowsus to denethetruth tableof many dimensionalsys-
tems as formulas in the monadi rst-order language, as opposed to a full
m-dimensional system (in whih atoms are evaluated at m time points)
whihwouldrequirean m-adilanguage.
Anm-dimensionaltableforann-plaeonnetiveisaformulaoftheform
(x
1
;:::;x
m
;R
1
;:::;R
n
), where is a formula of the rst-order prediate
language,writtenwithsymbolsfromf<g[fR
1
;:::;R
n
g,whereR
1
;:::;R
n
are speial m-plae relation symbols. Without loss of expressivity,we will
further assume that eah term y
j
ourring in R
i (y
1
;:::;y
m
) is a always a
variable.
Fix atemporal systemT whoselanguageontainsatoms q
1
;q
2
;:::;the
lassialonnetives, and the speial symbols #
1
;:::;#
j
, standing for n
1 -
,:::;n
j
-plaeonnetivesrespetively. Let
1
;:::;
j
betheirm-dimensional
n
1 -,:::;n
j
-plaetables respetively.
Remark 1.16 Sinethere arenitelymany
i
toonsider,we anfurther
assume that there is b m suh that eah
i
is written with variables
x
1
;:::;x
b only.
The semantisof m-dimensionalformulas is given by:
Denition 1.17 Let(T;<)beaowoftime. Lethbeanassignmentinto
T,i.e.foranyatom q,h(q)T. We denethe truthvalue ofeah formula
Aofthelanguageof T at mindiesa
1
;:::;a
m 1
;t2T underh,asfollows:
1. (T;<;h);a
1
;:::;a
m 1
;tj=q i t2h(q), q atomi.
2. (T;<;h);a
1
;:::;a
m 1
;tj=A^B i(T;<;h);a
1
;:::;a
m 1
;tj=Aand
(T;<;h);a
1
;:::;a
m 1
;tj=B.
3. (T;<;h);a
1
;:::;a
m 1
;tj=:A i(T;<;h);a
1
;:::;a
m 1
;t6j=A.
4. Foreah i(1ij), (T;<;h);a
1
;:::;a
m 1
;tj=#
i (A
1
;:::;A
n
i ) i
i 1 m 1 1 ni
h(A
k )=
def:
f(t
1
;:::;t
m )2T
m
j(T;<;h);t
1
;:::;t
m j=A
k g:
LetL M
denotethemonadilanguagewith<,rst-orderquantiersover
elements, and an arbitrary number of monadi prediate symbols Q
i for
subsetsofT. WewillregardtheQ
i
asprediate(subset)variables,impliitly
assoiatedwiththeatomsq
i
. Wedenethetranslationofanm-dimensional
temporalformulaA into a monadiformulaÆA:
1. If A is an atom q
i
, we set ÆA = (x
1
= x
1
)^:::^(x
m 1
= x
m 1 )^
Q
i (x
m ).
2. Æ(A^B)=ÆA^ÆB,and Æ(:A)=:ÆA.
3. LetA=#
i (A
1
;:::;A
n
i
), where
i (x
1
;:::;x
m
;R
1
;:::;R
n
i
) is the ta-
bleof #
i
. Sine we an always rewrite suhthat all ourrenes of
R
k (y
1
;:::;y
m
) in are suh that the terms y
i
are variables, after a
suitablevariablereplaementwean writeÆAusingonlythevariables
x
1
;:::;x
b as:
ÆA=
i (x
1
;:::;x
m
;ÆA
1
;:::;ÆA
n
i ):
Clearly,a simpleindutiongives usthat:
(T;<;h);a
1
;:::;a
m
j=B i T j=ÆB(a
1
;:::;a
m
;h(q
1
);:::;h(q
k )):
suh thatÆB(a
1
;:::;a
m
;h(q
1
);:::;h(q
k
))uses onlythevariablesx
1
;:::;x
b .
Supposethat K is a lassof ows of time, x=x
1
;:::;x
m
are variables,
and
Q=Q
1
;:::;Q
r
aremonadiprediates. If(x ;
Q),(x ;
Q)areformulas
inL M
with free variablesx and free monadi prediates
Q, we say that
and areK -equivalentifforallT 2K andall subsets S
1
;:::;S
r T,
T j=8x
(x ;S
1
;:::S
r
)$(x ;S
1
;:::;S
r )
:
We say the temporal system T is expressively omplete over K in n
dimensions(1nm)ifforany(x
1
;:::;x
n
;
Q)ofL M
withfreevariables
x
1
;:::;x
n
, there exists a temporal formula B(q) of T built up from the
atoms q=q
1
;:::;q
r
, suh that ^ V
n<im x
i
=x
i
and ÆB are equivalent
inK . Inthisase, K issaid to bem-funtionally omplete inndimensions
(symbolially,FC m
n
);K is funtionally omplete ifitisFC m
1
forsome m.
Finally,we denetheHenkin or H-dimensiond of alass K of ows as
thesmallestd suh that:
For any monadi formula (x
1
;:::;x
n
;Q
1
;:::;Q
r
) in L with free
variables among x
1
;:::;x
n
and monadi prediates Q
1
;:::;Q
r (with
n;r arbitrary), there existsan L M
-formula 0
(x
1
;:::,x
n ,Q
1
;:::;Q
r )
that is K -equivalent to and uses no more than d dierent bound
variable letters.
We now show that for any lass of ows, nite Henkin dimension is
equivalentto funtionalompleteness (FC m
1
forsome m).
Theorem 1.18 For any lass K of ows of time, if K is funtionally om-
plete then K has nite H-dimension.
Proof. Let(
Q ) beanysenteneofL M
. Byfuntionalompleteness,there
exists a B(q) of T suh that the formulas x
1
=x
1
^:::^x
m
= x
m
^(
Q )
andÆB(x
1
;:::;x
m
;
Q) areK -equivalent. WeknowthatÆB iswrittenusing
variables x
1
;:::;x
b
only. Hene the sentene
= 9x
1 :::9x
m ÆB(x
1
;:::;
x
m
;
Q) has at mostb variables, and is learly K -equivalent to . So every
senteneof L M
isK -equivalent toone withatmostbvariables. Thismeans
thatK hasH-dimension at mostb,soitis nite.
We now show the onverse. That is, we assume that the lass K of
ows of time has nite H-dimension m. Then we are going to onstrut a
temporallogithatisexpressivelyompleteoverKandthatisweaklym+1-
dimensional (and that is why suh proof does not work for 1-dimensional
systems: italways onstrutsa logiofdimension at least2).
Let us all this logi system d. Besides atomi propositions q
1
;q
2
;:::
andtheusualbooleanoperators,thissystemhasasetofonstants(0-plae
operators) C
<
i;j
and C
=
i;j
and unary temporal onnetives
i
and
i , for
0 i;j m. If h is an assignment suh that (h(q) T for atomi q, the
semantisof d-formulas is given by:
1. (T;<;h);x
0
;:::;x
m
j=q i x
0
2h(q) forq atomi.
2. Thetables for:;^arethe usualones.
3. (T;<;h);x
0
;:::;x
m j=C
<
i;j ix
i
<x
j
. Similarlywedenetheseman-
tisofC
=
i;j . C
=
i;j
arethusalled diagonalonstants.
4. (T;<;h);x
0
;:::;x
m j=
i
Ai(T;<;h);x
i
;:::;x
i
j=A. So
i
\projets"
thetruth value on thei-th dimension.
5. (T;<;h);x
0
;:::;x
m j=
i
A i (T;<;h);x
0
;:::;x
i 1
;y;x
i+1
;:::;x
m j=
Aforally 2T. So
i
is an\always" operator forthei-th dimension.
Lemma1.19 Let be a formula of L written only using the variable
letters u
0
;:::;u
m
, and having u
i
1
;::;u
i
k
free for arbitrary k m. Then
thereexists a temporal formula A of d suh that for all h;t
0
;:::;t
m 2T,
(T;<;h);t
0
;:::;t
m
j=A iK ;hj=(t
i
1
;:::;t
i
k ):
Proof. By indutionon . Assume rst that is atomi. If is u
i
< u
j
let A=C
<
i;j
ifi6=j,and ? otherwise. Similarlyforu
i
=u
j
. If is Q(u
i ),
letA be
i (q).
The lassial onnetives present no diÆulties. We turn to the ase
where is 8u
i (u
i1
;::;u
i
k
). By indutionhypothesis, let A bethe formula
orresponding ;then
i
Ais theformulasuitablefor.
We are now inapositionto prove theonverse of Theorem1.18.
Theorem 1.20 For any lass K of ows of time, if K has nite H-dimen-
sion then K isfuntionally omplete.
Proof. Let(u
0
)beanyformulaofL M
withonefreevariableu
0
. AsKhas
H-dimensionm, we an supposethat is written withvariablesu
0
;:::;u
m .
ByLemma 1.19there existsanA ofT suhthat foranyT 2K , t2T,and
assignment h into T;(T;<;h);t;:::;tj=A iK ;hj=(t).
Asan appliationof theresultsabove, we showthat thelassofpartial
ordersisnotfuntionallyomplete. Foronsidertheformulaorresponding
to thestatement there areat least n elements in the order:
n
=9x
1
;:::;x
n
^
i6=j [(x
i 6=x
j
)^:(x
i
<x
j )℄:
Itan be shown thatsuh formulaannot be writtenwithless thennvari-
ables (e.g. [GHR94℄). Sine we are able to say that there are at least n
elements in the order forany niten, thelass partial ordershave innite
H-dimensionand byTheorem 1.18itis notfuntionallyomplete.
Ontheotherhand,therealsandtheintegersmusthaveniteH-dimen-
sion, for the fU;Sg temporal logi is expressively omplete over both. In-
deed, [GHR94℄ shows that it has H-dimension at most 3, and so does the
theoryof linearorder.
There is a profusion of logis proposed in the literature for the modelling
of a variety of phenomena, and many more willsurely be proposed in the
future. Agreat partofthose logisdeal onlywith\stati" aspets,and the
temporal evolution is left out. But eventually, the need to deal with the
temporal evolution of a model appears. What we want to avoid is the so
alled reinvention of the wheel, that is, reworking from srath the whole
logi,its language, inferenesystem and models,and reprovingallits basi
properties,when thetemporal dimensionis added.
We therefore show here several methods for ombining logi systems
andwe studyiftheproperties of theomponent systemsare transferred to
theirombination. Weunderstand alogi systemL
L
asomposed ofthree
elements:
(a) alanguageL
L
,normallygiven bya setof formationrulesgenerating
wellformedformulasoverasignatureand asetof logialonnetives.
(b) Aninferene system,i.e.a relation,`
L
,betweensets of formulas,rep-
resentedby `
L
A. Asusual, `
L
Aindiates that?`
L A.
() ThesemantisofformulasoveralassKofmodelstrutures. Thefat
that a formulas A is true of or holdsat a model M2K is indiated
byMj=A.
Eah method for ombining logi systems proposes a way of generat-
ing thelanguage, inferene system and model strutures from those of the
omponent system.
Therst methodpresentedhereaddsatemporaldimensionTto alogi
system L, alled the temporalisation of a logi system T(L) , with an auto-
matiwayof onstruting:
thelanguageof T(L) ;
theinferene systemofT(L) ; and
thelassof temporalmodelsofT(L) .
We do that in a way that the basi properties of soundness, ompleteness
and deidability are transfered from the omponent logis T and L to the
ombined systemT(L) .
Ifthetemporalisedlogiisitself atemporallogi, wehave atwo dimen-
sionaltemporal logiT(T 0
). Suh a logiis too weak, however, beause,by
onstrution, thetemporal logiT 0
annot refer to the thelogi system T.
Wethereforepresenttheindependent ombination TT inwhihtwotem-
poral logisaresymmetriallyombined. Asbefore,thelanguage,inferene
systems and modelsof TT 0
, and show that the properties of soundness,
ompletenessand deidabilityaretransferred form Tand T 0
to TT 0
.
Theindependentombinationisnotthestrongestwaytoombinelogis;
inpartiular,theindependentombinationoftwolineartemporallogidoes
not neessarily produe a two-dimensional grid model. So we show how
to produe the full join of two linear temporal logis TT 0
, suh that
all modelswill be two-dimensional grids. However, in this ase we annot
guarantee that the basi properties of T and T 0
are transferred to TT 0
.
Inthissense,theindependentombination TT 0
isaminimal symmetrial
ombination oflogisthatautomatiallytransfersthebasiproperties. Any
furtherinterationbetweenthelogis hasto be separatelyinvestigated.
As a nal wayof ombining logis,we present methods of ombination
thataremotivatedbythestudyofLabelledDedutiveSystems(LDS)[Gab96 ℄.
All temporal logis onsidered for ombination here are assumed to be
linear.
2.1 Temporalising a Logi
Therstoftheombinationmethods,knownas\addinga temporal dimen-
sion to a logi system" or simply\temporalising a logisystem", has been
initiallypresented in[FG92℄.
Temporalisation is a methodology wherebyan arbitrary logi system L
an be enrihed with temporal features from a linear temporal logi T to
reate a new,temporalised systemT(L) .
We assume thatthe languageoftemporal systemT istheUS language
andits inferene systemisan extensionsof thatof US=K
l in
,withits orre-
spondinglass oftemporal linearmodelsK K
l in .
WithrespettothelogiLweassumeitisanextensionoflassiallogi,
thatis,allpropositionaltautologiesarevalidinit. ThesetL
L
ispartitioned
intwosets,BC
L
and ML
L
. AformulaA2L
L
belongsto thesetof boolean
ombinations, BC
L
, i itis built up from other formulas by theuse of one
of the boolean onnetives : or ^ or anyother onnetive dened only in
termsof those;it belongsto the setof monolithiformula ML
L
otherwise.
IfLisnotanextensionoflassiallogi,we an simply\enapsulate" it
inL 0
withaone-plaesymbol#notourringineitherLorT,suhthatfor
eah formula A2L
L
, #A2L
L 0
, `
L Ai `
L 0
#Aand themodel strutures
of#A arethose ofA. Notethat ML
L 0
= L
L 0
,BC
L 0
=?.
The alphabet of the temporalisedlanguage uses the alphabetof Lplus
otherwise,we useS and U oranyother properrenaming.
Denition 2.1 Temporalised formulas The set L
T(L)
of formulas of the
logisystemL isthesmallestset suhthat:
1. IfA2ML
L
,thenA2L
T(L)
;
2. IfA;B2L
T(L)
then:A2L
T(L)
and (A^B)2L
T(L)
;
3. IfA;B 2L
T(L)
thenS(A;B)2L
T(L)
and U(A;B)2L
T(L) .
Note that, for instane, if 2 is an operator of the alphabet of L and A
and B are two formulas in L
L
,the formula2U(A;B) is not in L
T(L) . The
languageof T(L) is independent of theunderlyingowof time, butnotits
semantis and inferene system, so we must x a lass K of ows of time
over whih thetemporalisation is dened; ifM
L
is a model in the lass of
models of L, K
L
, for every formula A 2 L
L
we must have either M
L j=A
or M
L
j= :A. In the ase that L is a temporal logi we must onsider a
\urrent time"o aspartof its modelto ahieve that ondition.
Denition 2.2 Semantis of the temporalised logi.
1
Let (T;<) 2K be
a owof time and let g :T ! K
L
be a funtionmapping every timepoint
in T to a model in the lass of models of L. A model of T(L) is a triple
M
T(L)
=(T;<;g) and thefatthat Ais true inM
T(L)
at time tis written
asM
T(L)
;tj=A and denedas:
M
T(L)
;tj=A,A2ML
L
ig(t)=M
L
and M
L j=A.
M
T(L)
;tj=:A iit isnotthease thatM
T(L)
;tj=A.
M
T(L)
;tj=(A^B) iM
T(L)
;tj=A and M
T(L)
;tj=B.
M
T(L)
;tj=S(A;B) ithere exists s 2T suh that s < t and
M
T(L)
;s j= A and for every u 2 T, if
s<u<tthen M
T(L)
;uj=B.
M
T(L)
;tj=U(A;B) ithere exists s 2T suh that t < s and
M
T(L)
;s j= A and for every u 2 T, if
t<u<sthen M
T(L)
;uj=B.
The inferenesystem ofT(L)=K is given bythe following:
Denition 2.3 Axiomatisation for T(L) An axiomatisation for the tem-
poralisedlogi T(L)isomposedof:
1
WeassumethattheamodelofTisgivenby(T;<;h)wherehmapstimepointsinto
setsofpropositions(insteadofthemoreommon,butequivalent,mappingofpropositions
intosetsoftimepoints);suhnotationhighlightsthatinthetemporalisedmodeleahtime
pointisassoiatedtoamodelofL.
Theinferene rulesof T=K;
Forevery formulaA inL
L ,if`
L
A then`
T(L)
A,i.e.alltheoremsof L
aretheoremsof T(L). This infereneruleis alledPersist.
Example 2.4 Consider lassialpropositional logi PL =hL
PL
;`
PL
;j=
PL
i. Its temporalisation generates the logi system T(PL) = hL
T(PL)
;`
T(PL)
;j=
T(PL)
i. It is notdiÆultto see thatthe temporalisedversion of PL over
any K isatuallythe temporallogi T=US=K .
If we temporalise over K the one-dimensional logi system US=K we
obtain the two-dimensional logi system T(US) = hL
T(US)
;`
T(US)
;j=
T(US) i
=T 2
(PL)=K. Inthisase wehave torenamethetwo-plaeoperatorsS and
U ofthetemporalisedalphabetto,say,S andU. Note,however, howweak
thislogiis,forS and U annotour withinthe sope ofU and S.
In order to obtain a model for T(US), we must x a \urrent time",
o
1
, inM
US
= (T
1
;<
1
;g
1
) , so that we an onstrut the model M
T(US)
=
(T
2
;<
2
;g
2
)aspreviouslydesribed. Notethat,inthisase,theowsoftime
(T
1
;<
1
)and(T
2
;<
2
)neednottobethesame. (T
2
;<
2
)istheowoftimeof
theupper-leveltemporal system whereas(T
1
;<
1
) isthe ow of timeof the
underlyinglogi whih, in this ase, happens to be a temporal logi. The
satisabilityof aformulain amodelof T(US) needs two evaluationpoints,
o
1 and o
2
;therefore itis atwo-dimensional temporal logi.
The logi system we obtain by temporalising US-temporal logi is the
two-dimensionaltemporallogidesribed in[Fin92℄.
This temporalisation proess an be repeated n times, generating an n
dimensionaltemporallogiwithonnetivesU
i
;S
i
,1in, suh thatfor
i<j U
j
;S
j
annot ourwithinthesopeof U
i
;S
i .
Weanalysenowthetransferofsoundness,ompletenessanddeidability
fromTandLtoT(L) ;thatis,weareasumingthelogisTandLhavesound,
ompleteanddeidableaxiomatisationswithrespettotheirsemantis,and
wewillanalysehowsuhpropertiestransfertotheombinedsystemT(L) . It
isaroutinetasktoanalysethatiftheinferenesystemsofTandLaresound,
soisT(L) . Sowe onentrate on theproofof transfereneof ompleteness.
Completeness
WeprovetheompletenessofT(L)=Kindiretlybytransformingaonsistent
formula A ofT(L) into "(A) and then mappingit into a onsistent formula
ofT. Completenessof T=Kisusedto ndaT-modelforA that isusedto
onstrut amodelfortheoriginalT(L)formula A.
We rst dene the transformation and mapping. Given a formula A 2
L
T(L)
,onsiderthefollowingsets:
Lit(A) = Mon(A)[f:B jB 2Mon(A)g
In(A) = f
^
j Lit(A)and `
L
?g
where Mon(A) is theset of maximal monolithisubformulae of A. Lit(A)
is the set of literals ourring in A and In(A) is the set of inonsistent
formulas that an be builtwiththose. We transformAinto Aas: "(A):
"(A) = A^ V
B2In(A)
(:B ^ G:B ^ H:B)
Thebigonjuntionin"(A)isatheoremofT(L) ,sowehavethefollowing
lemma.
Lemma2.5 `
T(L)
"(A) $A
If K is a sublass of linear ows of time, we also have the following
property(thisiswhere linearityis usedintheproof).
Lemma2.6 LetM
T
beatemporal modeloverKK
l in
suhthatfor some
o2T, M
T
;oj=(A). Then, for every t2T, M
T
;tj=(A).
Therefore, ifsome subsetof Lit(A)isinonsistent,thetransformed for-
mula "(A) puts that fat in evidene so that, when it id mapped into T,
inonsistentsubformulaewillbe mapped into falsity.
NowwewanttomapaT(L) -formulaintoaT-formula. Forthat,onsider
anenumerationp
1 ,p
2
,::: ,ofelementsofP andonsideranenumerationA
1 ,
A
2
, ::: , of formulae inML
L
. The orrespondene mapping :L
T(L)
!L
T
isgiven by:
(A
i
) = p
i
forevery A
i 2ML
L
;i=1;2:::
(:A) = :(A)
(A^B) = (A)^(B)
(S(A;B)) = S((A);(B))
(U(A;B)) = U((A);(B))
The followingistheorrespondene lemma.
Lemma2.7 Theorrespondene mapping isa bijetion. Furthermore ifA
isT(L) -onsistent then (A) is T-onsistent.
A
Lit(A)jM
T
;tj=(B)g isnite andL-onsistent.
Proof. Sine Lit(A) is nite, G
A
(t) is nite for every t. Suppose G
A (t)
isinonsistentfor some t, thenthere existfB
1
;:::;B
n g G
A
(t) suh that
`
L V
B
i
! ?. So V
B
i
2 In(A) and :(
V
B
i
) is one of the onjunts of
"(A). ApplyingLemma2.6toM
T
;oj=("(A))wegetthatforeveryt2T,
M
T
;tj=:(
V
(B
i
)) butby,the denitionof G
A ,M
T
;tj= V
(B
i
),whih
isa ontradition.
We are nallyreadyto prove theompleteness ofT(L)=K.
Theorem 2.9 (Completeness transfer for T(L) ) If thelogial systemL
is omplete and T is omplete over a sublass of linear ows of time K
K
l in
, then the logial systemT(L) isomplete over K.
Proof. AssumethatAisT(L) -onsistent. ByLemma2.8,wehave (T;<)2
K andassoiatedto every timepointinT we havea niteand L-onsistent
set G
A
(t). By (weak) ompleteness of L, every G
A
(t) has a model, so we
denethetemporalisedvaluation funtiong:
g(t) = fM t
L jM
t
L
isa model ofG
A (t)g
Consider the model M
T(L)
= (T;<;g) over K. By strutural indution
over B, we show that for every B that is a subformula of A and for every
timepoint t,
M
T
;tj=(B) iM
T(L)
;tj=B
We show only the basiase, B 2Mon(A). Suppose M
T
;tj=(B); then
B 2G
A
(t)and M t
L
j=B,and hene M
T(L)
;tj=B. SupposeM
T(L)
;tj=B
and assume M
T
;t j= :(B); then :B 2 G
A
(t) and M t
L
j= :B, whih
ontradits M
T(L)
;t j= B; hene M
T
;t j= (B). The indutive ases are
straightforwardand omitted.
So,M
T(L)
isa modelfor AoverKand theproofis nished.
Theorem2.9givesussoundand ompleteaxiomatisationsforT(L)over
manyinterestinglassesofows oftime,suhasthelassofalllinear ows
oftime, K
l in
,theintegers, Z,and thereals, R. Theselassesare, intheirT
versions,deidableand theorrespondingdeidabilityofT(L)isdealt next.
Notethattheonstrutionaboveisnitisti,andthereforedoesnotitself
guaranteethatompatnessistransferred. However,animportantorollary
of theonstrution above isthat the temporalised systemis a onservative
of anoriginal systemis provable inthe ombined system. Formally,L
1 is a
onservative extension ofL
2
ifitis anextension ofL
2
suh thatifA2L
L
2 ,
then`
L
1
Aonlyif`
L
2 A.
Corollary 2.10 LetLbeasoundandompletelogisystemandTbesound
and omplete over K K
l
in. The logi system T(L) is a onservative ex-
tension of both L andT.
Proof. Let A 2 L
L
suh that `
T(L)
A. Suppose by ontradition that 6`
l
ogiLA,sobyompletenessofL, there existsa modelM
L
suhthatM
L j=
:A. We onstrut a temporalised model M
T(L)
= (T;<;g) by making
g(t) =M
L
for all t 2 T. M
T(L)
learly ontradits the soundness of T(L)
and therefore that of T, so `
L
A. This shows that T(L) is a onservative
extensionof L; theproof ofextensionof T issimilar.
Deidability
Thetransfer of deidabilityis also doneusingtheorrespondene mapping
and thetransformation. Suh atransformation isatuallyomputable,
asthefollowingtwo lemmasstate.
Lemma2.11 For any A 2 L
T(L)
, if the logi system L is deidable then
thereexists an algorithm for onstruting "(A).
Lemma2.12 Overa linear ow of time, for every A2L
T(L) ,
`
T(L)
A i `
T
("(A)):
Deidabilityisa diretonsequeneof these two lemmas.
Theorem 2.13 If L is a deidable logi system, and T is deidable over
KK
l in
, then the logi system T(L) isalso deidable over K .
Proof. ConsiderA2L
T(L)
. SineLisdeidable,byLemma2.11thereisan
algorithmiproeduretobuild"(A). Sineisareursivefuntion,wehave
analgorithmto onstrut ("(A)), and dueto thedeidabilityofT overK ,
we have an eetive proedure to deide ifit is a theorem or not. Sine K
islinear, by Lemma2.12 thisis also a proedure for deiding whether A is
atheorem ornot.
We now deal with the ombination of two temporal logi systems. One of
the will be alled the horizontal temporal logi US, while the other will
be the vertial temporal logi
U
S. If we temporalise the horizontal logi
withthe vertial logi, we obtain a very weakly expressive system; if USis
theinternal(horizontal)temporal logiinthetemporalisationproess (F is
derived in US), and
U
S is the external (vertial) one (F is dened in
U
S),
we annotexpressthat vertialand horizontal future operators ommute,
FFA$FFA:
In fat, the subformula FFA is not even in the temporalised language of
U
S(US), nor is the whole formula. In other words, the interplay between
the two-dimensions is not expressible in the language of the temporalised
U
S(US).
The idea isthen to denea method forombiningtemporal logis that
issymmetrial. Asusual,we ombine thelanguages,inferene systemsand
lassesofmodels.
Denition 2.14 LetOp(L)bethesetofnon-booleanoperatorsofageneri
logiL. Let T and T be logisystems suh thatOp(T)\Op(T) =?. The
fully ombined language of logi systems T and T over the set of atomi
propositionsP,isobtainedbytheunionoftherespetivesetof onnetives
andtheunionoftheformationrulesof thelanguages ofbothlogisystems.
LettheoperatorsU andS beinthelanguageofUS andU andS bein
thatof
U
S. Theirfullyombined languageoverasetofatomi propositions
P isgiven by
every atomi propositionisinit;
ifA;B areinit, soare :Aand A^B;
ifA;B areinit, soare U(A;B) andS(A;B).
ifA;B areinit, soare U(A;B) andS(A;B).
Not onlyare the two languages taken to be independent of eah other,
buttheset ofaxiomsof thetwo systems aresupposedto bedisjoint;sowe
allthefollowing ombination method theindependent ombination of two
temporallogis.
Denition 2.15 Let US and US be two US-temporal logi systems de-
nedoverthe same set P ofpropositionalatoms suh that their languages
are independent. The independent ombination US
U
S is given by the
following:
Thefully ombined languageofUS and
U
S.
If(;I) is an axiomatisation forUS and (;I) is an axiomatisation
for
U
S, then ([;I[I) is an axiomatisation for US
U
S. Note
that, apart from the lassialtautologies, theset of axioms and
aresupposed to be disjoint,butnottheinferene rules.
ThelassofindependentlyombinedowsoftimeisKKomposedof
biorderedowsoftheform(
~
T;<; <)wheretheonnetedomponents
of(
~
T;<)arein K and theonnetedomponentsof (
~
T;<) areinK,
and
~
T is the(notneessarilydisjoint)unionof thesetsof timepoints
T and T thatonstitute eah onnetedomponent.
A model struturefor US
U
S over KK is a 4-tuple (
~
T;<;<;g),
where(
~
T;<; <)2KKand gisan assignmentfuntiong:
~
T !2 {
.
Thesemantisofa formulaAina modelM=(
~
T;<; <;g) isdened
asthe union of therules dening thesemantis of US=K and
U
S=K.
The expression M;t j= A reads that the formula A is true in the
(ombined) model M at thepoint t2
~
T. The semantisof formulas
isgiven by indutioninthestandard way:
M;tj=p i p2g(t)and p2P:
M;tj=:A i itisnotthe asethat M;tj=A.
M;tj=A^B i M;tj=A and M;tj=B.
M;tj=S(A;B) i there existsan s2
~
T withs<t and M;sj=A
and for every u 2
~
T, if s < u < t then
M;uj=B.
M;tj=U(A;B) i there existsan s2
~
T witht<s and M;sj=A
and for every u 2
~
T, if t < u < s then
M;uj=B.
M;tj=S(A;B) i there exists an s2
~
T with s<t and M;sj=A
andforeveryu2
~
T,ifs<u<tthenM;uj=B.
M;tj=U(A;B) i there exists an s2
~
T with t<sand M;sj=A
andforeveryu2
~
T,ift<u<sthenM;uj=B.
Thealsoindependentombinationoftwologisappearsintheliterature
underthenamesof fusion orjoin.
l in
irreexive and total orders; similarly, we assume that the axiomatisations
areextensions of US/K
l in .
The temporalisation proess willbe used as an indutive step to prove
the transferene of soundness, ompleteness and deidability for US
U
S
over KK. We denethedegree alternation of a(US
U
S)-formula Afor
US,dg(A):
dg(p)=0
dg(:A)=dg(A)
dg(A^B)=dg(S(A;B))=dg(U(A;B))=maxfdg(A);dg(B)g
dg(U(A;B))=dg(S(A;B))=1+maxfdg(A); dg(B)g
andsimilarlydene dg(A)for
U
S.
Anyformulaofthefullyombinedlanguagean beseenasa formula of
somenitenumberofalternatingtemporalisationsoftheformUS(
U
S(US(::: ))) ;
more preisely, A an be seen as a formula of US(L
n
), where dg(A) = n,
US(L
0
)=US,
U
S(L
0 )=
U
S,andL
n 2i
=
U
S(L
n 2i 1 ),L
n 2i 1
=US(L
n 2i 2 ),
fori=0;1;:::;d n
2 e 1.
Indeed, not only the language of US
U
S is deomposable in a nite
numberoftemporalisation,butalsoitsinferenes,asthefollowingimportant
lemmaindiates.
Lemma2.16 Let US and
U
S betwo omplete logi systems. Then, A isa
theorem of US
U
S i it is a theorem of US(L
n
), where dg(A)=n.
Proof. If A is a theorem of US(L
n
) ,all theinferenes inits dedution an
be repeated inUS
U
S,soit isa theoremof US
U
S.
SupposeAisa theoremofUS
U
S;letB
1
;:::;B
m
=A bea dedution
of A in US
U
S and let n 0
= maxfdg(B
i )g, n
0
n. We laim that eah
B
i
is a theorem of US(L
n 0
). In fat, by indution on m, if B
i
is obtained
inthe dedution by substituting into an axiom, the same substitution an
be donein US(L
n 0) ;
if B
i
is obtained byTemporal Generalisation from B
j ,
j<i, then by theindutionhypothesis, B
j
is a theorem of US(L
n 0
) and so
isB
i
;ifB
i
isobtainedbyModus Ponens fromB
j
and B
k
,j;k <i, thenby
theindutionhypothesis,B
j
and B
k
aretheoremsof US(L
n 0
)and sois B
i .
So A is a theorem of US(L
n 0)
and, sine US and
U
S are two omplete
logi systems, by Theorem 2.9, eah of the alternating temporalisations in
US(L
n 0
)isaonservativeextensionof theunderlyinglogi;itfollowsthatA
isa theoremof US(L
n
),as desired.
transfereneofsoundness,ompletenessanddeidabilityalsofollowsdiretly
fromthisresult.
Theorem 2.17 (Independent Combination) LetUSand
U
Sbetwosound
and omplete logi systems over the lasses K and K , respetively. Then
their independent ombination US
U
S is sound and omplete over the
lass KK. If US and
U
S are omplete and deidable, so is US
U
S.
Proof. Soundness follows immediately from the validity of axioms and
inferenerules.
Weonlyskeththeproofofompletesshere. GivenaUS
U
S-onsistent
formula A, Lemma 2.16 is used to see that it is also onsistent in US(L
n ),
so a temporalised US(L
n
)-model is builtfor it. Then, by indutionon the
degreeofalternationofA,thisUS(L
n
)isusedtoonstrutaUS
U
S-model;
eah step of suh onstrution preserves the satisfatibility of formulas of a
limited degree of alternation, so in the nal model, A, is satisable; and
ompletenessisproved. Fordetails, see[FG96℄.
For deidability, suppose we want to deide whether a formula A 2
US
U
Sisatheorem. ByLemma2.16,thisisequivalenttodeidingwhether
A2US(L
n
)isatheorem,wheren=dg(A). SineUS/Kand
U
S/Kareboth
ompleteanddeidable,bysuessiveappliationsofTheorems2.9and2.13,
it follows that the following logis are deidable: US(
U
S),
U
S(US(
U
S)) =
U
S(L
2 ) ,::: ,
U
S(L
n 1 )=L
n
;soathelastappliationofTheorems2.9and2.13
yieldsthatUS(L
n
) isdeidable.
2.2.1 The minimalityof the independent ombination
The logi US
U
S is the minimal logi that onservatively extends both
US and
U
S. Thisresultwasrstshownfortheindependentombination of
monomodallogis independentlyby[KW91℄and [FS91 ℄.
Indeed, suppose there is another logi T
1
that onservatively extends
both US and
U
S but some theorem A of US
U
S is not a theorem of T
1 .
But A an be obtained by a nite number of inferenes A
1
;:::;A
n
= A
using only the axioms of US and
U
S. But any onservative extension of
US and
U
S must be able to derive A
i
, 1 i n, from A
1
;:::;A
i 1 , and
thereforeit mustbe ableto deriveA; ontradition.
One wehave thisminimalombination betweentwologisystems,any
other interation between the logis must be onsidered on its own. As
an example, onsider the following formulas expressing the ommutativity
validovera model ofUS
U
S:
I
1
FFA$FFA
I
2
FPA$PFA
I
3
PFA$FPA
I
4
PPA$PPA
Nowonsidertheprodut oftwolineartemporalmodels,givenasfollows.
Denition 2.18 Let (T;<) 2 K and (T;<) 2 K be two linear ows of
time. The produt of those ows of timeis (T T;<;<). A produt model
over KK is a 4-tuple M= (T T;<;<;g), where g : T T ! 2 {
is a
two-dimensional assignment. The semantis of the horizontal and vertial
operators are independentof eah other:
M;t;xj=S(A;B) i there exists s<t suh that M;s;x j=A and
forall u,s<u<t, M;u;x j=B.
M;t;xj=S(A;B) i there exists y<x suh that M;t;y j= A and
forall z,y<z<x,M;t;zj=B.
SimilarlyforU andU,thesemantisof atoms and boolean onnetives
remainingthestandardone. AformulaAisvalidoverKKifforallmodels
M=(T;<;T;<;g),forall t2T and x2T we have M;t;xj=A.
It is easy to verify that the formulas I
1 {I
4
are valid over produt mod-
els. We wonder if suh produt of logis transfers the properties we have
investigatedforthepreviouslogis. Theansweris: itdepends. Wehavethe
followingresults.
Proposition 2.19 (a) There isa sound andompleteaxiomatisation for
US
U
S over the lasses of produt models K
l in K
l in , K
dis K
dis ,
Q Q, K
l in K
dis , K
l in
Q and Q K
dis
[Fin94 ℄.
(b) Thereareno nite axiomatisations for the valid two-dimensional for-
mulas over the lasses ZZ,N N and RR [Ven90℄.
Notethatthealltheomponentone-dimensionalmentionedabove logi
systems are omplete and deidable, but their produt sometimes is om-
plete, sometimes not. Also, the logis in (a) are all deidableand those in
(b)areundeidable.
Thisistoillustratethefollowingidea: givenanindependentombination
of two temporal logis, the addition of extra axioms, inferene rules or an
