PP 2008 19: Modal Fixed Point Logic and Changing Models

JohanvanBenthem 1

andDaisukeIkegami 2

1

UniversiteitvanAmsterdamPlantageMuidergraht24, 1018TV,Amsterdam,The

Netherlands,johansiene.uva.nl, http://staff.siene.uva.nl/

johan/

2

UniversiteitvanAmsterdamPlantageMuidergraht24, 1018TV,Amsterdam,The

Netherlands,ikegamisiene.uva.nl

ThispaperisdediatedtoProfessorBorisTrakhtenbrot,whosework

andspirithaveinspiredusandsomanyothersinoureld.

Abstrat. We show that propositional dynami logi and the modal

-alulus are losed under produt modalities, as dened in urrent

dynami-epistemilogis.Ouranalysislariesthelattersystems,while

alsoraisingsomenewquestionsaboutxed-pointlogis.

1 Basi Closure Properties of Logis

Standard rst-order logi has some simple but important losure properties.

First,itis losedunder relativization: foreveryformula andunaryprediate

letter P, there is a formula () P

whih says that holds in the sub-model

onsisting of all objets satisfying P. One usually thinks of relativization as

a syntatioperation whih transforms the given formula by relativizing eah

quantier9xto9x(Px^ andeahquantier8xto8x(Px!.Butoneanalso

think of evaluating theoriginal formula itself, but then in a hangedsemanti

model. Theonnetionbetweenthetwoviewpointsisstatedin

Fat 1 (RelativizationLemma).

M;s() P

() MjP;s:

where M j P is the restrition of the model M to its sub-model dened by

the prediate (or formula with one free variable) P. Relativization is a useful

property of abstrat logis, and it is used extensively in proofs of Lindstrom

theorems.Alsousefulislosureunderprediatesubstitutions[ =P℄,whihmay

again be read aseither a syntatioperation, oras a shift to evaluation in a

suitablyhangedmodel,viathefollowingwell-known

Fat 2 (SubstitutionLemma).

M;s[ =P℄ () M[P:= M

℄;s:

where M[P:= M

℄isthemodelMwith thedenotationoftheprediateletter

interpretationbetweentheories.E.g.,embeddingtherst-orderorderingtheory

of the rational numbers into that of the integers requires taking rationals as

orderedpairsofrelativelyprimeintegers(adenablesubsetofthefullCartesian

produtZZ),andredeningtheirorder<aordingly.Thus,wenowalsohave

aprodut onstrution where ertaindenable tuples beome thenew objets.

As is easy to see, the rst-order language is also losed under suh produt

onstrutions-inasensewhihwewillnotspellout.Forourpurposehere,we

will dene a preise sense of `produt losure' in terms of modal logi below,

returningtothegeneralsituation attheend.

Thethreementionedpropertiesalsoholdofmanylanguagesextending

rst-orderlogi,suhasLFP(FO),rst-orderlogiwithaddedxed-pointoperators.

But as we just said, ourfous in this note will be on modal languages, whih

are rather fragmentsof a full rst-order logioverdireted graphswith unary

prediates,althoughwealsoaddxed-pointoperatorslateron.Forsuhmodal

languages, and espeially vividly, in their epistemi interpretation as logis of

knowledgeandinformationow,theabovepropertiesaquirespeialmeanings

ofindependentinterest.

2 Closure Properties of Modal Languages

2.1 Epistemi Logi

Take a modal language with proposition letters, Boolean operators, and

uni-versal modalities [i℄ whih we read as stating what agent i knows, or maybe

better: whatis trueto thebest of i'sinformation. Morepreisely,in epistemi

pointedgraphmodelsM withatualworlds,representingtheinformationofa

groupof agents:

M;s[i℄ () forallt;ifsR

i

t;thenM;t:

2.2 Publi Announementand Denable Submodels

Inthisepistemisetting, takingtherelativizationoftheurrentmodelM;sto

its sub-model MjP;s onsisting of all points satisfying the formula P is the

naturalrenderingofaninformationalevent!P ofpubliannounementthatP is

urrentlytrue.Thus,modelhangereetsinformationupdate.Thelanguageof

publi announement logi PALextends epistemilogi, making these updates

expliitbyaddingmodaloperators[!P℄fortruthfulannounementations:

ators[i℄plusfour redutionaxioms:

[!P℄q $ P !q for atomifats q;

[!P℄: $ P !:[!P℄;

[!P℄^ $ [!P℄^[!P℄ ;

[!P℄[i℄ $ P ![i℄(P ![!P℄):

Weanreadthesepriniplesasaompletereursiveanalysisofwhatagents

knowaftertheyhavereeivednewinformation. Butaswaspointedoutin van

Benthem2000[4℄,thisompletenesstheoremduetoPlazaandGerbrandyreally

juststatesthestandardreursivelausesforperformingsyntatirelativization

ofmodalformulas.Thusthetehnialquestionbeomeswhihmodallanguages

arelosedunder relativization.

This is not always the ase. E.g., onsider an epistemi languagewith an

operatorofommonknowledge(everyoneknowsthat everyoneknowsthat, and

soon.),orsemantially:

M;sC

G

() forallworldst reahablefromsbysomenite

sequeneof

i

steps(i2G);M;t:

Thisamountstoaddinganoperatorofreexive-transitivelosureovertheunion

of all individual aessibilityrelations. This innitary operationtakesus from

the basi modal language into a fragment of so-alled propositional dynami

logi(PDL).Itanbeshownthatthisfragmentdoesnothavetherelativization

property:indeed, the formula [!p℄C

G

q is not denable without modalities [!p℄.

VanBenthem,vanEijk&Kooi2006[5℄provedthisandgoontoproposeriher

epistemilanguages,usingriherfragmentsofPDLwhihdohaverelativization

losure, using so-alled `onditional ommon knowledge' C

G

(; ) whih says

that is trueineveryworldreahablewithstepsstayinginside the -worlds.

Remark 1. These observations are reminisent of the fat that languageswith

generalizedquantiersmaylakrelativizationlosure.Anexampleisrst-order

logiwiththeaddedquantier\formostobjets".Togetthelosure,oneneeds

toaddatrulybinaryquantier\Mostare ".

2.3 General Observation and Produt Update

Publi announement is just one mehanism of information ow. In real-life

senarios,dierentagentsoftenhavedierentpowersofobservation.Tomodel

this,dynami-epistemi logi(DEL) workswithevent models

A=(E;fR

i g

i ;PRE):

Here the preondition funtion maps events e to preondition formulas PRE

i

`produtupdate'turnsaurrentmodelM;sintoamodelMA;(s;e)reording

the information of dierent agents after some event e has taken plae in the

epistemi setting represented by A. Produt update redenes the universe of

relevantpossibleworlds,andtheepistemiaessibilityrelationsbetweenthem:

MAhasdomainf(s;a)jsaworldinM;aaneventin A;(M;s)PRE

a g:

Thenewunertaintiessatisfy(s;a)R

i

(t;b) ifbothsR

i

t andaR

i b:

Thevaluation forpropositionletterson(s;e)isjustasthatforsinM:

Here unertainty among new worlds (s;a);(t;b) an only ome from old

un-ertainty among s;t via indistinguishable events a;b. In general, this produt

onstrution an blow upthe size of theinput model M - it does notjust go

toadenablesub-model.Inwhatfollows,wewillassumethattheeventmodels

arenite,thoughinnitaryversionsarepossible.

Despitetheapparentomplexityofthisprodutonstrution,thereisa

nat-uralmathingdynamiepistemilanguageDELwithanewmodality[A;e℄:

M;s[A;e℄ () ifM;sPRE

e

; thenMA;(s;e):

Theorem2. DELisompletely axiomatizable.

Proof. Theargument,duetoBaltag,Moss&Soleki1998[2℄,isasfollows.The

atomiandBooleanredutionaxiomsinvolvedareliketheearlieronesforpubli

announement,but hereistheessentiallausefortheknowledgemodality:

[A;e℄[i℄ () PRE

e !

^

eRif inA

[i℄[A;f℄:

Bysuessiveappliationofsuhpriniples,alldynamimodalitiesanbe

elim-inatedtoobtainastandardepistemiformula. ut

Wesumthisup,somewhatloosely,bystatingthefollowing:

Fat 3. Basi epistemi logiisprodut-losed.

Butagain,thesituationgetsmoreompliatedwhenweaddommon

knowl-edge.Inthisase,noredutiontothelanguagewithout[A;e℄modalitiesis

pos-sible. Van Benthem, van Eijk &Kooi 2006 [5℄ solve this problem by moving

tothelanguageE-PDLwhihisjust thepropositionaldynamilogiversionof

epistemi logi, but now allowing the formation of arbitrary `omplex agents'

usingthestandardPDLprogramvoabulary:

basiagentsi,tests?onarbitraryformulasofthelanguage,

unions,ompositions,andKleeneiteration.

Theyprovide an expliitaxiomatization for the dynami-epistemi version

`E-PDL is losed under the produt onstrution'. In what follows, for

onve-niene,weuseobviousexistentialounterpartstotheearlieruniversalmodalities.

Hereis theentralobservationoftheabovepaper:

Theorem3. For all 2E-PDL, and all ation models A with event a, the

formula hA;ai has anequivalent formulain E-PDL.

Publiannounements!P arespeialationmodelswithjustoneeventwith

preonditionP, equallyvisible to all agents. Thus, thetheorem also says that

E-PDL,orPDL,islosedunder relativization-asobservedearlierinvan

Ben-them2000[4℄.Inaddition,E-PDLhasbeenshowntobelosedunderprediate

substitutionsin Kooi2007[11℄.

The point of the urrent paper is to analyze this situation more formally,

in termsofgenerallosurepropertiesofmodallanguages,andtheirxed-point

extensions. In partiular, we provide a new proof of Theorem 3 larifying its

bakgroundinmodalxed-pointlogi.

3 Closure under Relativization for Modal Standard

Languages

Itiseasytoseethatthebasimodallanguageislosedunderrelativization.The

proedure relativizesmodalities,just asonedoeswithquantiersin rst-order

logi.Likewise,wealreadymentionedthatpropositionaldynamilogiislosed

under relativization.Thisrequiresanoperationwhihalsotransformsprogram

expressions,asfollows:

([℄) P

=[jP℄() P

:

Hereonemustalsorelativizeprograms toprogramsjP,asfollows:

ijP =?P;i;?P

?jP =?(^P)

([)jP = jP[jP

(;)jP = jP;jP

(

)jP = (jP)

:

Finally,onsiderthemostelaboratemodalxed-pointlanguage,theso-alled

-alulus.Formulas(q)withonlypositiveourrenesofthepropositionletter

qdene amonotonisettransformationin anymodelM:

F M

(X)=fs2Mj(M[q:=X℄;s):g

Theformulaq(q)denesthesmallestxedpointofthistransformation,

whihanbeomputedin ordinalstagesstartingfrom theemptysetasarst

approximation.Likewise,q(q)denesthegreatestxedpointofF M

Both exist formonotone maps, by the Tarski-Knaster theorem (Bradeld and

Stirling 2006[8℄). Foronveniene, we assumethat eah ourreneof a

xed-pointoperatorbindsauniquepropositionletter.Hereisourrstobservation.

Fat 4. Themodal -alulus islosed underrelativization.

Proof. Weshowtheuniversalvalidityofthefollowinginterhangelaw:

h!Piq(q)$P^qh!Pi(q): (1)

Here the ourrenes of q are still syntatially positive in h!Pi(q) - in an

obvious sense.Now to prove (1), ompare the following identities, for all sets

X P M

:

F M

h!Pi

(X)=fs2MjM[q:=X℄;sh!Pi(q)g

=fs2MjP j(MjP)[q:=X℄;s(q)g

=F MjP

(X):

Itshould belear that theapproximationmaps onbothsides nowworkin

exatlythesameway. ut

Still,thereisadierenewithstandardxed-pointlogi.Oneusuallythinks

of,e.g.,asmallestxed-pointformulaq(q)asdeningthelimitofasequene

of ordinalapproximationsstartingfrom theemptyset, whose suessorstages

areomputedbysubstitutionofearlierones:

0

=?; +1

=(

=q):

Butthisanalogybreaks downbetweenthetwosidesoftheaboveequation (2).

Theapproximationsequenesdenedin adiretmannerwill diverge.Consider

themodalformulas

(q)=q; P =}>

inamodelonsistingofthenumbers1;2;3intheirnaturalorder.Bothsequenes

in equation(2)startwiththeemptyset,dened by?,but thentheydiverge:

forh!}>iqq: for}>^qh!}>iq:

h!}>i?; onlytrueat 2 }>^h!}>i?; onlytrueat2

h!}>i?; trueat 1;2 }>^h!}>ih!}>i?; onlytrueat2:

Thereasonfor thedivergene isthat theformula ontheright-handside keeps

prexingformulaswithdynamimodel-hangingmodalities,sothatwearenow

evaluatinginmodelsoftheform (MjP)jP,et.

The general observation explaining this divergene involves another basi

announement modalitiesh!Piareadded.

E.g., onsider againour three-pointmodelM, withaproposition letterptrue

at 2 only, and let be the formula h!}>i}p. Now onsider the substitution

[(h!}>i>)=p℄.Firstonsiderthemodelafterperformingthissubstitution: itwill

assignpto f1;2g.Hene[p:=(h!}>i>) M

℄h!}>i}pwillbetruein1.Next

per-formthesubstitutionsyntatiallytoobtaintheformulah!}>i} h!}>i>

:this

istruenowherein themodelM.

Sinethemodallanguageissimplytranslatableintorst-orderlogi,asimilar

observationholds for rst-order logiwith relativization operators () P

added

aspartofitssyntax.Theresultinglanguagedoesnotsatisfytheusual

Substitu-tionLemma,sinethemodel-hangingoperators() P

reatenewontextswhere

formulas an hange their truth values. So,model-hangingoperators are nie

devies,but theyexataprie.

Remark 2 (Alternativedynami denitionsof substitution).

Fat 5 holds for the straightforwardoperational denition of substitutions

[=p℄ as syntatially replaingeah ourreneofpin byanourreneof

. However,thereis analternative.Inlinewith earlierapproahesin `dynami

semantis'ofrst-order logi(f.van Benthem1996 [3℄),Kooi2007 [11℄treats

substitutions[=p℄asmodalitieshangingtheurrentmodelinitsdenotationfor

p.Thesenewmodalitiessatisfyobviousreursiveaxiomspushingthemthrough

Booleansandstandardmodaloperators.Topushthem alsothroughpubli

an-nounementmodalities,oneanrstrewrite thelatterviatheirPALreursion

axioms, and only then apply the substitution to the omponents. Van Eijk

2007[9℄showshowthisprovidesanalternativesyntatioperationaldenition

of substitution, working inside out. One rst redues innermost PAL or DEL

formulas totheir basimodalequivalents,and thenperformsstandard

synta-tisubstitutionin these. Thoughnotompositional,thisproedure iseetive.

When applied to the two approximation sequenes in our earlier problemati

example,thesewouldnowomeoutbeingthesameafterall.

Thus, dynami modal languages are losed under semanti substitutions,

but nding the preise orresponding syntati operation in their stati base

languagerequires someare.

4 Closure of Dynami Logi under Produts

Theorem 3said that the languageE-PDL is losed under the produt

opera-tionhA;ei.TheproofinvanBenthem,vanEijk&Kooi2006[5℄usesspeial

argumentsinvolvingKleene'sTheorem fornite automata andprogram

trans-formations.Weprovideanewproofwhih providesfurtherinsightbyrestating

thesituationwithin modalxed-pointlogi.

Re-dardDELredutionaxioms.TheremainingaseisthatofformulashA;aihi

withanE-PDLmodalityinvolvingaomplexepistemiprogram.Toproeed,

we need a deeper analysis of program struture. The following result an be

provedtogetherwithTheorem3byasimultaneousindution:

Theorem4. Forall A;a, and programs 0

2 E-PDL,there exist E-PDL

pro-gramsT

0

a;b

(for eah b2A) suhthat,for allE-PDL formulas ,

M;shA;aih 0

i () M;s _

b2A hT

0

a;b

ihA;bi :

Proof. Weuseindutionontheonstrutionoftheprogram 0

.

Case1: 0

=i.

hA;aihii () PRE

a and

_

aR

i bin A

hiihA;bi :

Thisanbebroughtintoourspeialformbysetting

T i

a;b

=?PRE

a

;iifaR

i

bin A; andT i

a;b

=?; otherwise.

Case2: 0

=?forsomeformula.

hA;aih?i () hA;ai(^ ) ()

hA;ai^hA;ai () h?(hA;ai)ihA;ai :

Here the less omplex formula hA;ai anbe taken to be in the language of

E-PDLalready,bythesimultaneousindutionprovingTheorem3.Itiseasyto

thendenetheorrettransitionprediatesT ?

a;b

foralleventsb2A.

Case3: 0

=[ forsomeformulas,.

hA;aih[i () hA;ai(hi _hi ) ()

hA;aihi _hA;aihi ind:hyp:

() _

b2A hT

a;b

ihA;bi _ _

b2A hT

a;b

ihA;bi

and, byreombiningpartsof this disjuntion,usingthe validPDL-equivalene

hi _hi $h[i ,wegettherequirednormalform.

Case4: 0

=; forsomeformulas,.

hA;aih;i () hA;aihihi

ind.hyp.1

()

_

b2A hT

a;b

ihA;bihi

ind.hyp.2

() _

b2A (hT

a;b i

_

2A hT

b;

ihA;i )

and here,using the minimallogi of PDLagain, substituting onespeialform

Case5: = forsomeprogram.

Theruxliesinthisnal ase:ombinationswithKleeneiterations

hA;aih

i donotredue asbefore. Butevenso,weananalyzethem in the

samestyle, using asimultaneous xed-point operator q

b

dening the

propo-sitionshA;bih

i for alleventsb2A in onefell swoop.Theneed forthis

si-multaneousreursionexplainsearlierdiÆultiesintheliteraturewithredution

axiomsforommonknowledgewithprodut update.Tondtherightshema,

rstreallthePDLxed-pointequationforKleeneiteration:

hA;aih

i () hA;ai( _hih

i ) ()

hA;ai _hA;aihih

i

ind.hyp.

() hA;ai _ _

b2A hT

a;b

ihA;bih

i :

Here, again beause of the simultaneous indutive proof with Theorem 3, we

anthink of therst disjuntasbeingsomeformula

a

of E-PDL. Theresult

ofthisunpakingaresimultaneousequivalenesoftheform(withpropositional

variablesq

a

foreaha2A):

q

a $

a _

_

b2A hT

a;b iq

b

: ()

Lemma1. The denotations of the modal formulas hA;aih

i in amodel M

arepreisely thea-projetionsofthesmallestxed-pointsolutiontothe

simulta-neousequations().

Proof (Lemma1). Here,smallestxed-pointsforsimultaneousequationsinthe

alulusare omputedjust asthose for singlexed-pointequations. lemma

1 follows by a simple indution, showing that the standard meanings of the

modalformulashA;aih

i inamodelMareontainedinanysolutionforthe

simultaneous xed-pointequation.

Wealulatethemeaningoftheleastxed-pointof()throughthe

approx-imationproedureandshowitisequaltothatof hA;aih

i (a2A).

From now on, weidentify formulas by their truth sets in M,reading as

fm2Mj(M;m)g.Forsimpliity,werewrite()asfollows:

q

i =

i _

_

1jn hT

i;j iq

j

(1in)

LetF bethemonotoneoperatorfrom P(M) n

to itselfinduedbytheright

hand side of (), where n is the numberof elements in A. More preisely, for

X =(X

1 ;;X

n

)2P(M) n

,F(X)=(Y

1 ;;Y

n

)whereforeah1in,

Y

i =

m2Mj(M[fq

j :=X

j g

j=1;;n

℄;m)

i _

_

hT

i;j iq

Next,forX 2P(M) , denehF (X)j2Oniasfollows:

F 0

(X)=X

F +1

(X)=F(F (X)) F (X)= [ < F

(X)if isalimitordinal.

Foranym2!,weanprovethefollowingequationbyindution onm.

F m

(?)=

_

1j1;j2;;jm 1n [ i _hT i;j1 i j 1 _hT i;j1 ihT j1;j2 i j 2

__hT

i;j1 ihT

j1;j2 ihT

jm 2;jm 1 i jm 1 ℄ 1in : Hene F ! (?)= n

(9m<!)(9j

1 ; ;j

m 1 )hT

i;j

1 ihT

j m 2 ;j m 1 i jm 1 o 1in ;

whihimpliesF !

(? )=F !+1

(?):theleastxed-pointisreahedin ! steps.

ThereforeweonlyhavetoshowthatfhA;a

i ih i g 1in =F ! (? ).

Reallthat,bythedeningpropertyofT

i;j

,foranyE-PDLformula 0

,any

1inandanystatesin M,

M;shA;a

i ihi

0

() M;s _ 1jn hT i;j ihA;a j i 0 :

By using this onditionrepeatedly, we get the following equivalene: for any

n-tuplesofelementsin M andanyiwith1in,

M;s i hA;a i ih i

()(9m2!)M;s

i

hA;a

i ihi

m

()(9m2!)(9j

1 )M;s

i hT i;j 1 ihA;a j1 ihi m 1

()(9m2!)(9j

1 ;j

2 )M;s

i hT i;j1 ihT j1;j2 ihA;a j 2 ihi m 2 ()

()(9m2!)(9j

1 ;;j

m )M;s

i hT i;j1 ihT j1;j2 ihT

jm 1;jm ihA;a

jm i

()(9m2!)(9j

1 ;;j

m )M;s

i hT i;j1 ihT j1;j2 ihT

jm 1;jm i

jm

()s

i 2 F ! (? ) i where F ! (? ) i

isthei-thoordinateofF !

(? ).Hene

(8i) M;s

i hA;a i ih i

() s2F !

iates q

i

after ! steps, one gets the ountable disjuntion over allnite `path

formulas' ofthe formhT

i;j

i ;T

j

1 ;j

2 ; ;T

jn;k i

k

.And thelatterare exatlythe

meaningsoftheoriginalpropositionshA;aih

i .

Butwe arenotdone yet. Whatweneedto shownextis that thesolutions

obtainedinthiswayareatuallyinthelanguageE-PDL! Thefollowinglemma

tellsustherelevantfataboutthe-alulus.Simultaneousxed-pointequations

of the above speial disjuntive shape () an be solved one by one, and the

solutionslieinsidedynamilogi.

Lemma2. Anysystemof simultaneousxed-pointequations of ()has an

ex-pliitminimal solutionforeahq

a

inE-PDL.Moreover,the solutionsretainthe

speialdisjuntive formdesribedinTheorem4.

Proof (Lemma2). TheindutiveproedureproduingexpliitE-PDLsolutions

workslinebyline-likeGaussianEliminationinasystemoflinearequations.

{ Case1.Thereisonlyoneq-variable,aswithpubliannounements.

The line reads q

1

$

1 _h

1;1 iq

1

. The expliit solution works just as in

standarddynamilogi,in the

q

1 =h

1;1 i

1 :

{ Case2.Therearenlinesin thereursionshema,withn>1.

Werstsolveforthevariableq

1

asinCase1-obtaininganexpliitE-PDL

formula

1 (q

2 ;;q

n

)in the otherreursion variables. Wethen substitute

thissolutionin theremaining n 1equations, andsolvethese indutively.

Finally, the solutions thus obtained for the q

2 ; ;q

n

are substituted in

1 (q

2 ;;q

n

)toalso solveforq

1 .

Somesyntatihekingwill showthat these solutionsremainin the syntati

formatdesribedin Theorem4.Butofourse,wealsoneedtoshowthatthisis

reallyasolutionfortheabovexed-pointequations(),andindeedthesmallest

one.Toprovethat,we formulate thealgorithm moreformallyin the following

way(f. Arnold&Niwinski[1℄foramoreextensivetreatment).

For any monotone operator G, let G

denote the least xed point of G.

Let F:P(M) n

! P(M) n

be the monotone operator indued by the n

equa-tions in (). Now take any X

2 ;;X

n

2 P(M) and x them. Next, dene

F

X2;;Xn

:P(M)!P(M)asfollows:

F

X

2 ;;X

n (X

1

)= F(X

1 ;;X

n )

1 ;

where (X)

i

isthei-thoordinateofX.Sine F ismonotone, F

X

2 ;;X

n isalso

monotone.ThendeneF

X

3 ;;X

n

:P(M)!P(M):

F

X3;;Xn (X

2

)= F((F

X2;;Xn )

;X

2 ;;X

n )

2 n X2;;Xn

arebothmonotone.ContinuethisproessuntilwedeneF

;

.Thenthesolution

oftheearlier`Gaussian'algorithmistheuniqueF 0

suh that

(F 0

)

i = F

(F 0

)i+1;;(F 0

)n

(1in):

Note howweomputetherightmostxed-pointrsthere,and thensubstitute

leftward.Hene allwehavetoshowisthefollowing:

Claim 1. F

=F

0

.

TheproofisinArnold&Niwinski[1℄(seeSetion1.4.inthisbook).Tomake

ourpaperself-ontained,wewillputaproofinanAppendix below. ut

Thisonludes theproofsofTheorems3and4. ut

Illustration 1. Weompute the solutionsfor the updatemodel A=

GFED ABC

a 1

2

//

GFED ABC

b 1

2

rr

PRE

a

=p; PRE

b =>

This desribes aseurity senario whereagent1orretly observes that eventa

istakingplae,whileagent2mistakenlybelievesthatbours.Hereisa

desrip-tionof thenon-trivial ommon knowledge for 1;2arisingfrom thissenario, by

writingoutthe xedpointequation for hA;aih(1[2)

ir.

Bystep1inthe proof,

T 1

a;a

=?PRE

a

;1=?p;1;T 1

a;b =?

T 1

b;a

=?; T

1

b;b

=?PRE

b ;1=1

T 2

a;a

=?; T

2

a;b

=?PRE

a

;2=?p;2

T 2

b;a

=?; T

2

b;b

=?PRE

b ;2=2:

Then by step3,

T 1[2

a;a =T

1

a;a [T

2

a;a

=?p;1;T 1[2

a;b =T

1

a;b [T

2

a;b =?p;2

T 1[2

b;a =T

1

b;a [T

2

b;a

=?; T 1[2

b;b =T

1

b;b [T

2

b;b

=1[2:

Nowput

q =hA;aih(1[2)

ir; q =hA;bih(1[2)

q

a

=hA;air_hT 1[2

a;a iq

a _hT

1[2

a;b iq

b

=(PRE

a

^r)_h?p;2iq

b

_h?p;1iq

a

=((p^r)_h?p;2iq

b

)_h?p;1iq

a

and

q

b

=hA;bir_hT 1[2

b;a iq

a _hT

1[2

b;b iq

b

=(PRE

b

^r)_h1[2iq

b

=r_h1[2iq

b :

Sinethe orderofeliminatingvariablesdoesnotinuenethe solutions,werst

solve q

b

asfollows:

q

b

=h(1[2)

ir:

Bysubstitutingthis solutionin theabove equation forq

a ,

q

a

= (p^r)_h?p;2ih(1[2)

ir

_h?p;1iq

a

= (p^r)_h?p;2;(1[2)

ir

_h?p;1iq

a :

Hene

q

a

=h(?p;1)

i (p^r)_h?p;2;(1[2)

ir

=h(?p;1)

i(p^r)_h(?p;1)

;?p;2;(1[2)

ir

We an easily hek that these q

a ;q

b

satisfy the equations we gave by an

independent semantiargument.

Remark 3. Thealulationinthisexampleisreallyjustthefollowingwell-known

fat aboutthemodal-alulus:

Let (q

1 ;q

2 ), (q

1 ;q

2

) be positive formulas in the modal -alulus. Then

thesimultaneousleastxedpointsofthese formulasis

q

1 :(q

1 ;q

2 : (q

1 ;q

2 ));q

2 : (q

1 :(q

1 ;q

2 );q

2 )

:

IntheproofofClaim1(f.theAppendix),weonlyusetheonditionthatF

ismonotone.Thismeansweangeneralizetheresultasfollows:

Corollary 1. Themodal-alulusislosedundertheformationof

simultane-ousxed-pointoperators.

5 Closure of the -alulus under Produts

Proof. Weprovethestatementbyindutionontheomplexityofformulas.We

onlyonsiderthexedpointase,astheothersgolikebefore.

Our main task is to analyze xed-point omputations in produt models

MAintermsofsimilaromputationsintheoriginalmodelM.Thefollowing

idea turns out to work here.Let X be a subset of MA. Modulo the event

preonditions possibly ruling out some pairs, we andesribe X, without loss

of information, in termsof the sequene of its projetionsto the eventsin A,

viewedasanitesetofindies.Thus,weandesribetheomputationinMA

bymeans ofanite set ofomputations in M.The followingset of denitions

andobservationsmakesthis preise.

Take any Kripke model M and any event model A. Let n be the

num-ber of elements of A and let A = fa

j g

1jn

. There are anonial mappings

:P(M) n

!P(MA)and:P(MA)!P(M) n

withÆ=id:

(X)= [

1jn (X

j fa

j

g)\(MA);

(Y)=fY

j g

1jn ;

whereY

j

=fx2Mj(x;a

j )2Yg:

Givenapositiveformula(q)inthemodal-alulus,letF MA

:P(MA )

!P(MA)bethemonotonefuntioninduedby(q).DeneF (q)

:P(M) n

!

P(M) n

asfollows:

F (q)

=ÆF MA

Æ:

WelaimthatF MA

ismonotoneifandonlyifF (q)

ismonotone.Suppose

F MA

is monotone. Sine ; are monotone and ompositions of monotone

funtionsaremonotone,F (q)

isalsomonotone.Toprovetheonverse,suppose

F (q)

is monotone. Pikany X;Y 2 P(MA) with X Y. First note that

F MA

(X)F MA

(Y)holdsifandonlyifÆF MA

(X)ÆF MA

(Y)holds.

Hene allwehavetohekisÆF MA

(X)ÆF MA

(Y).But

ÆF MA

(X)=ÆF MA

Æ(X)

=ÆF MA

Æ (X)

=F (q)

(X)

F (q)

(Y)

=ÆF MA

Æ (Y)

=ÆF MA

Æ(Y)

=ÆF MA

(Y);

wheretheaboveinlusion followsfromthemonotoniityofF (q)

and.

Moreover,thereisafurtheranonialorrespondene:ifXisanF (q)

-xed

point, then (X) is an F MA

-xed-point, and if Y is an F MA

-xed-point,

then(Y)isanF (q)

-xed-point.Hene theleastF (q)

-xed-pointorresponds

to theleastF MA

abovemaybeseenasaspeialaseofthe\TransferLemma"(Theorem1.2.15)

in Arnold&Niwinski[1℄. Thislemmaonlyusesour funtion,whileweadded

thefuntion forlarity,torestritaninputto theinverseimage of{whih

is why the equation Æ = id holds. For further bakground to this kind of

argument,f. Bloomand

Esik[7℄.

Sofar,wehaveseenthattheleastF MA

-xed-pointanbeorrelatedwith

the least F (q)

-xed-point in a natural way. Our next task is to show that

hA;aiq:(q) is atually denable in the modal -alulus. For that purpose,

rstnote thathA;a

j

iq(q)denesthej-thoordinateoftheleastF MA

-xed-point. By the denition of , it is also the j-th oordinate of the least

F (q)

-xed-point.Now,sinethemodal-alulusislosedundersimultaneous

xed-pointoperatorsbyCorollary1,ifweanexpressF (q)

byaformulaofthe

modal-aluluswithpositivevariables,wearedone.

Toprovethis, wegeneralizethesyntatianalysisemployedinSetion 4to

formulaswithmanyvariablesq=q

1 ;;q

m

.Foranyformula(q)inthemodal

-alulus,deneF MA

(q)

:P(MA) m

!P(MA)asfollows:

F MA

(q)

(Y)=f(s;a)j (MA)[q

k :=Y

k ℄;(s;a)

(q)g;

whereY 2(MA) m

.

Claim 2. Foranyformula(q)inthe modal-alulus,thereareformulas

suhthat F (q )

=F M

where F (q)

:P(M) mn

!P(M) n

and

() For any 1km,if all the ourrenes of q

k

in are positive (negative

resp.),then for eah1j;j 0

n,all the ourrenes of p

k ;j in(

)

j 0

are

positive (negativeresp.),

Proof (Claim2).Inthefollowingdenitions,weonlydisplaytheessential

argu-mentvariablesneededtounderstandthefuntionvalues.Weprovethestatement

by indution onthe omplexity of. As in the proofof Lemma 1, weidentify

formulaswiththeirtruthsets.Also,if isasequeneofformulas,

j

isthej-th

oordinateof .

{ Case1:=p(pisnotinq).

F (q)

= p^PRE

a

1

;;p^PRE

a

n

:

Hene(

(q) )

j

=p^PRE

aj

.Itiseasytohek().

{ Case2:=q

k (q

k

isthek-thoordinateofq).

F (q )

(X)=fX

k ;j ^PRE

aj g

1jn :

Hene (

(q) )

j =p

k ;j ^PRE

aj

, where p

k ;j

is the j-th variable in the k-th

1 2 F (q ) = 1 ^ 2 : Hene (q) = 1 ^ 2

.Itiseasytohek().

{ Case4:=: 0 . F (q ) =f:( 0 ) j ^PRE a j g 1jn : Hene ( (q) ) j = :( 0 ) j ^PRE a j

. It is easy to hek () by our

indu-tive hypothesis, and the simultaneous denition for positive and negative

ourrenes.

{ Case5:=hii 0

:

Forany1jnandx2M,

x2 F (q)

(X)

j ()

(19j 0

n)(9y2M)

xR

i y^a

j R i a j 0

^y2 F 0 (q) (X) j 0 :

Toseethatthisistrue,observethattheonditiony2 F 0 (q) (X) j 0 implies (y;a j

0)2MA.Therefore,weanput

(q) j = _ ajRia j 0 hii 0 (q) j 0 :

{ Case6:=q 0

0

,wherealltheourrenesofq 0

arepositivein 0 . F (q) (X)= n F MA q 0 0 (q 0 ;q ) ((X)) j o 1jn = n (F MA 0 (q 0 ;q) ((X))) j o 1jn = X 0 7!F M 0 (X 0 ;X) ;

where (F())

is the least F-xed-point. By indution hypothesis, all the

ourrenes of p 0

j

are positive in (

0)

j

0 for any 1 j;j 0

n, where p 0

orrespondsto q 0

. Sine themodal -alulusis losedunder simultaneous

xed-pointoperators,weanput

(q) =p 0 0

(q),that arealsointhe

modal-alulus.Sine-operatorsdonothangethepositivity(negativity)

ofvariablesnotboundedbythem,()alsoholdsin thisase.

u t

Theproofofthelastaseexplainswhyweneededto`blow-up'inthenumber

of variables in Claim 2. Also, we proved thelaim for arbitrary formulas (not

onlyforpositiveones)beauseotherwiseweannotusetheindutionhypothesis

in Case4(ifispositive,then 0

mustbenegative). ut

Remark 5 (Eetive redution axioms).

AsinFat4,weouldalsoanexpliitredutionaxiomforhA;a

j

iq:(q)by

takingthej-thoordinateofthesimultaneous xed-pointexpressionq:

bothE-PDLandthemodal-alulusarelosedundersimultaneousxedpoint

operators (in thease of E-PDL, suh operators havethe speial form of ()).

Theproofofthatfatisessentiallythesame(itisthatofClaim1)butthease

of the full -alulusis easier beausewehave arbitrary-operators, whilein

E-PDL,wehavetohekifthesolutionisalsoin E-PDL.

6 Conlusions and Further Diretions

T hepreedingresultsplaeurrentmodallogisofinformationupdateinamore

generallight,relatingtheir`redutionaxiom'approahforobtainingonservative

dynamiextensionsofexistingstatilogistoabstratlosurepropertiesof

xed-point logis.Ourobservationsalsosuggestanumberof moregeneralissues, of

whihwementionafew.

Fine-strutureofthe-alulus Ourresultsshowthatprodutlosureholds

forbasimodallogi,propositionaldynamilogiPDL,andthe-alulusitself.

Wethinkthat therearefurthernaturalfragmentswiththisproperty,inluding

the -!-alulus,whih onlyallowsxed-pointswhose omputations stop

uni-formly bystage !.Anotheraseto look forprodutlosure isthehierarhyof

nestedxed-pointalternation. Ourproof removesmodalprodutoperators by

meansofsimultaneousxed-points,whihanthenberemovedbynestedsingle

ones,butwehavenotyet analyzeditspreisesyntatidetails.

On another matter, our proof method in Setion 4 suggests that PDL is

distinguishedinside the -alulusasthesmallestfragmentlosedunder some

verysimple`additive'xed-pointequations.This seemsrelatedtothefat that

the semantis of dynami logi only desribes linear omputation traes, and

nomoreomplexonstruts,suhasarbitrarynite trees.Canthisequational

observationbeturnedintoaharaterizationofPDL?

Connetions with automata theory The rstproof of produtlosure for

PDL in van Benthem & Kooi 2004 [6℄ used nite automata to serveas

`on-trollers'restritingstatesequenesinprodutmodelsMA.Theseond,

dier-entproofin VanBenthem,vanEijk&Kooi2006[5℄involvedanon-trivialuse

of Kleene's Theoremfor regularlanguages,and heneagaina onnetionwith

nite automata. What is the exat onnetion of this proof with our speial

unwinding ofsimultaneous `disjuntive'xed-pointequationsinside PDL?Can

Kleene'sTheorembeinterpretedasanormal-formresultinxed-pointlogi?

There may alsobe amoregeneralautomata-based takeon ourarguments,

giventhestrongonnetionbetweenautomatatheoryand-alulus. 3

3

Addedinprint.MartinOtto(p..)hasproposedusingtheprodutlosureofMSOL

and thebisimulation invarianeofthe mu-alulus withaddedprodutmodalities

that manymodal languagesareprodut-losed.Whataboutlogialsystemsin

general?Wewouldliketohaveanabstratformulationwhihappliestoawider

lass of logialsystems, suh asrst-order logi and itsextensions in abstrat

modeltheory.Wefeelthatprodutlosureisanaturalrequirementonexpressive

power,espeiallygivenitsearliermotivationintermsofrelativeinterpretability.

But the orret formulation may have to be stronger than our notion in this

paper.Eveninthemodalase,ourproofswouldalsogothroughifweallowed,

say, denable substitutions for atomi proposition letters in produt models.

Also,onemightalsotrytosplitourmodalnotionintofullprodutlosureplus

prediatesubstitutions, treatingouruseof preonditions asaaseof denable

domainrelativization.

There mayalso be a onnetionhere with the Feferman-Vaught Theorem,

and produt onstrutions reduing truth in the produt to truth of related

statementsintheomponentmodels.Afterall,ourproofofuniformdenability

ofdynamimodaloperatorshA;aiinduesanobvioustranslationrelatingtruth

ofin aprodutmodelMAto thatofsomeeetivetranslationof inthe

omponentmodelM.

Finally, onewayofseeing how strongprodut losurereally is would be to

askaonversequestion.Forinstane,assumethatafragmentofthe-alulusis

produt-losed.Doesitfollowthatitislosedunder simultaneousxed-points?

Inall,ourresults,thoughsomewhattehnialand limitedin sope,seemto

provideavantagepointforraisingmanyinterestingnewquestions.

Aknowledgements

WethankBaldertenCate,JanvanEijk,MartinOtto,OlivierRoy,Luigi

San-toanale,andYdeVenemafortheirremarksandsuggestionsonthisdraft.

Referenes

1. AndreArnoldandDamianNiwinski.Rudimentsof-alulus,volume146.

North-Holland,2001. StudiesinLogiandtheFoundationsofMathematis.

2. Alexandru Baltag, Lawrene S.Moss,and Slawomir Soleki. Thelogi of publi

announements,ommonknowledge,andprivatesuspiions,301999.

3. Johan vanBenthem. Exploring LogialDynamis. CSLI Publiations, Stanford

University,1996.

4. JohanvanBenthem.Informationupdateasrelativization. Tehnialreport,

Insti-tutefor Logi,LanguageandComputation,UniversiteitvanAmsterdam,2000.

5. Johan vanBenthem,JanvanEijk,and BarteldKooi. Logisofommuniation

andhange. Inform.andComput.,204(11):1620{1662,2006.

6. Johan vanBenthemand Barteld Kooi. Redutionaxioms for epistemi ations.

Proeedings Advanes in Modal Logi, Department of Computer Siene, pages

7. Stephen L. Bloom and Zoltan Esik. Iteration theories : the equational logi of

iterativepresses. Springer-Verlag,1993.

8. JulianBradeldandColinStirling. Modalmu-aluli.InHandbookofModalLogi

(Studies inLogiandPratialReasoning,vol.3).ElsevierSiene,2006.

9. JanvanEijk. Theproperdenitionof substitutionindynami logis. working

note,CWI,Amsterdam,2007.

10. GaelleFontaine. Continuousfragmentofthe-alulus. 2008. Submitted.

11. BarteldKooi.Expressivityandompletenessforpubliupdatelogisviaredution

axioms. J.Appl.Non-ClassialLogi,17(2):231{253,2007.

7 Appendix

7.1 Proof ofClaim 1

Proof (Claim 1).BythepropertyofF 0

,itsuÆestoshowthefollowing:

(F ) i = F (F)i+1;;(F)n

(1in):

Weprovethat byindution oni.

{ Case1:i=1.

Sine F (F)2;;(F)n ((F ) 1

)=(F(F

)) 1 =(F ) 1 ; (F ) 1

is a xed point of F

(F)2;;(F)n

. Sine F

(F)2;;(F)n

is the least

xedpointofF

(F)2;;(F)n , F (F)2;;(F)n (F ) 1 . Sine F (F)2;;(F)n (F ) 1

andF ismonotone,

F (F (F)2;;(F)n ) ;(F ) 2

;;(F

) n F((F ) 1 ;(F ) 2

;;(F

) n ) =F : Hene F (F (F ) 2 ;;(F ) n ) ;(F ) 2

;;(F

) n j (F ) j

(2jn):

Combiningthis with

F (F (F ) 2 ;;(F ) n ) ;(F ) 2

;;(F

) n 1 =F (F)2;;(F)n (F (F)2;;(F)n ) =(F (F)2;;(F)n ) ; weget F (F (F ) 2 ;;(F ) n ) ;(F ) 2

;;(F

) n j (F (F ) 2 ;;(F ) n ;(F ) 2

;;(F

) n j

forany1jn,whihmeansthat (F

(F ) 2 ;;(F ) n ) ;(F ) 2

;;(F

)

n

is

anF-prexedpoint.

SineF

istheleastF-prexedpoint,F

isasubsetof

(F (F)2;;(F)n ) ;(F ) 2

;;(F

)

n

,whihimplies(F

Bytheindutionhypothesis, F (F ) i+1 ;;(F ) n ((F ) i

)=(F(F

)) i =(F ) i : Therefore,(F ) i

isaxedpointofF

(F)i+1;;(F)n

.Sine F

(F)i+1;;(F)n

istheleastxedpointofF

(F)i+1;;(F)n , F (F)i+1;;(F)n (F ) i . Let F j

(1 j i) be the ones uniquely determined by the following

equations:

F

j = F

Fj+1;;Fi;(F)i+1;;(F)n

; (1ji 1)

F i = F (F ) i+1 ;;(F ) n :

Bythesameargumentasbefore,weanprove

F(F

1 ;;F

i ;(F

)

i+1

;;(F

) n ) j F j

(1ji);

F j (F ) j

(1ji):

Hene

F F

1 ;;F

i 1 ;(F (F)i+1;;(F)n ) ;(F ) i+1

;;(F

) n F 1 ;;F

i 1 ;(F (F ) i+1 ;;(F ) n ) ;(F ) i+1

;;(F

)

n

;

whihmeans F

1 ;;F

i 1 ;(F (F)i+1;;(F)n ) ;(F ) i+1

;;(F

)

n

isanF

-prexedpoint.SineF

istheleastF-prexedpoint,

F

F

1 ;;F

i 1 ;(F (F)i+1;;(F)n ) ;(F ) i+1

;;(F

)

n

, whih implies

(F ) i F (F ) i+1 ;;(F ) n

. ut

7.2 Produt losure of CF(P)

Inthissubsetion,weshowhowourmethodsapplytotheso-alled`ontinuous

fragment'ofthe modalmu-alulus,where theoperators orrespondingto

for-mulas are Sott ontinuous. (Hene, in partiular, all xed-points are reahed

uniformlyinallmodelsbystageomega.)Thisfragmentwasreently

harater-izedsyntatiallyinFontaine[10℄.

Denition1. LetPROPbethesetof allpropositionletters, andP any subset

of PROP. Let I be the set of all agents. We dene the ontinuous fragments

CF (P) by indution by indution on the omplexity of formulas in the modal

-alulusasfollows:

CF (P):::=p2P j j_j^jhiijx(x)

where is any formulain the modal -alulus withoutany free variable in P,

thefollowingsense:LetP,fx

1 ;;x

n

gbesetsofpropositionalletterswhihare

disjoint.Thenif

1 (x

1 ;;x

n

);;

n (x

1 ;;x

n

)areinCF(P[fx

1 ;;x

n g),

thenfollowingformulaisin CF(P):

0 B B B x 1 x 2 . . . x n 1 C C C A 0 B B B 1 (x 1 ;;x

n ) 2 (x 1 ;;x

n ) . . . n (x 1 ;;x

n ) 1 C C C A

Proposition1. ForanyP PROP,CF(P)isprodutlosed.

Proof. Weprovethestatementbyindutionontheomplexityofformulas.We

onlyonsiderthexed pointase,asotherasesgoinastandardway.Wewill

usethesamenotationsasin theproofofTheorem5.

Theproof isalmost all thesameasthe asefor themodal -alulus.The

diereneisthatCF(P)isnotlosedunderxedpoints.Butbyusingtheabove

Remark,weandealwiththisproblem.

Bythesameargumentin Theorem 5,ifweanexpressF (q)

byaformula

in CF(P[fx

1 ;;x

n

g) forsomefreshvariablesx

1 ;;x

n

,wearedone.

Toprovethis,weneedthefollowingClaim:

Claim 3. Forany setof propositional lettersQ, the followingistrue:

Let(q)beaformulainCF (Q)whereq isasequeneoffreevariables

(pos-sibly notin )withlengthm.Takefreevariables x

k ;j

(1km;1jn)so

that theydonot appearinany preonditionformulas inAor inq orinQ. We

mayassumethissituationinanysubformulaof (q)byhoosing freshvariables

properly. Then there is a sequene of formulas

(q)

in CF(Q[fx

k ;j g) with

lengthnsuhthat F (q)

=F M

(q) .

Proof. Inthefollowingdenitions,weonlydisplaytheessentialargument

vari-ables needed to understand the funtion values. We prove the statement by

indution on the omplexity of . We identify formulas with their truth sets.

Also,if isasequeneofformulas,

j

isthej-thoordinateof .

{ Case1:=p(pisnotinq).

F (q)

= p^PRE

a

1

;;p^PRE

a n : Hene( (q) ) j

=p^PRE

a

j

andthisis inCF (Q).Sine eahx

k ;j

doesnot

appearinanypreonditionformulasinA,(

(q) )

j

isalsoinCF(Q[fx

k ;j g).

{ Case2:=q

k 0 (q

k

0 isthek 0

-thoordinateofq).

F (q )

(X)=fX

k 0 ;j ^PRE aj g 1jn : Hene( (q) ) j =x k 0 ;j ^PRE aj

.BythesamereasoningasinCase1,PRE

aj

isinCF(Q[fx

k ;j

g) andhenex

k 0

;j

^PRE

a

isalsoinCF(Q[fx

1 2 F (q ) = 1 _ 2 : Hene (q) = 1 _ 2

isinCF(Q[fx

k ;j

g) byindution hypothesis.

{ Case4:=

1 ^ 2 . F (q ) = 1 ^ 2 : Hene (q) = 1 ^ 2

isinCF(Q[fx

k ;j

g) byindution hypothesis.

{ Case5:=hii 0

:

Forany1jnandx2M,

x2 F (q)

(X)

j ()

(19j 0

n)(9y2M)

xR

i y^a

j R i a j 0

^y2 F 0 (q) (X) j 0 :

Toseethatthisistrue,observethattheonditiony2 F 0 (q) (X) j 0 implies (y;a j 0

)2MA.Therefore,weanput

(q) j = _ a j R i a j 0 hii 0 (q) j 0 ;

whihisin CF(Q[fx

k ;j g).

{ Case6:=q 0

0

,where 0

is inCF(Q[fq 0 g). F (q) (X)= n F MA q 0 0 (q 0 ;q ) ((X)) j o 1jn = n (F MA 0 (q 0 ;q) ((X))) j o 1jn = X 0 7!F M 0 (X 0 ;X) ;

where(F())

istheleastF-xed-point.Byindutionhypothesis,

0

arein

CF(Q[fx

k ;j g[fy

j

g) where fy

j

gorrespondsto X 0

in theaboveformula

andsatisestheonditionrequiredin theClaim.ThenbyRemark,wean

put (q) =y j 0,

thatareinCF(Q[fx

k ;j

g). ut