Prefixed Tableaux Systems for Modal Logics with Enriched Languages

HAL Id: hal-03195295

https://hal.archives-ouvertes.fr/hal-03195295

Submitted on 10 Apr 2021

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Prefixed Tableaux Systems for Modal Logics with Enriched Languages

Philippe Balbiani, Stéphane Demri

To cite this version:

Philippe Balbiani, Stéphane Demri. Prefixed Tableaux Systems for Modal Logics with Enriched Lan- guages. 15th International Joint Conference on Artificial Intelligence (IJCAI’97), Aug 1997, Nagoya, Japan. pp.190-195. �hal-03195295�

Prexed Tableaux Systems for Modal Logis with Enrihed Languages

Philippe Balbiani

Laboratoired'informatique deParis-Nord,

AvenueJean-Baptiste Clement,

93430 Villetaneuse, Frane.

Stephane Demri

LaboratoireLEIBNIZ,

46Avenue Felix Viallet,

38031 Grenoble, Frane.

Abstrat

We present sound and omplete prexed

tableauxsystemsforvariousmodallogiswith

enrihed languages inluding the \dierene"

modaloperator[6=℄andthe\onlyif"modalop-

erator[ R ℄. Theselogisareofspeialinterest

in Artiial Intelligene sine their expressive

powerishigherthanthestandardmodallogis

andformostofthemthesatisabilityproblem

remainsdeidable. Wealsoinludeinthepaper

deisionproeduresbasedonthesesystems. In

theonlusion,werelateourworkwithsimilar

onesfromtheliteratureandweproposeexten-

sionstootherlogis.

1 Introdution

Thedenitionoflogialformalismsthatmodelognitive

and reasoning proesseshas beenalways onfronted to

two issues: how toderease theexpressivepowerofex-

istinguntratablelogisinordertoobtaintratablefrag-

ments and how to inrease the expressive power of de-

idablelogiswhilepreservingdeidability-thisinludes

forinstanetheextensionofknowndeidablefragments

of the lassial logi. These fragments inlude various

modal logis (see e.g.

[

Hughes and Cresswell, 1984

℄

)

if one translates them in the standard way to lassial

logi. ThemodallogishavebeenreognizedintheAr-

tiial Intelligene ommunity as serious andidates to

apture dierent aspets of reasoning aboutknowledge

(see e.g.

[Fagin

et al.,

1995℄).

However the standard

modal logishavea restrited expressive power(forin-

stanethelass ofirreexiveframes isnotdenable by

amodalformulaofthelogiK).

Thatiswhyintheliteraturevariousmodallogiswith

enrihedlanguageshavebeendened. Mostofthework

done forthese logis hasbeen dediated to study their

expressive power (see e.g.

[Goranko

and Passy, 1992;

Rijke,

1993℄).

In the paper our aim is to analyze var-

ious features relatedto the mehanization of numerous

modallogiswith enrihedlanguages. To doso, wede-

ne prexed tableaux whih are known to be lose to

thesemantisofthelogisandtheyallowauser-friendly

presentationoftheproofs. Moreover,theuseofprexes

(see e.g.

[Fitting,

1983; Wallen, 1990; Massai, 1994;

Governatori, 1995

℄

) is known to take advantage of the

omputationalfeaturesofthe logis. Namely,eahpre-

x ourring at some stage of the proof ontains some

information about part of the urrent proof. However

weignorewhether a matrixharaterizationof thelog-

istreatedhereinexistinordertoavoidsomeredundan-

iesinthetableauxproofsearh-notationalredundany,

irrelevaneandnon-permutability

[Wallen,1990℄.

The logistreatedinthepaperontain variousoper-

ators thatdierfrom thestandardneessityoperator2

(alsonoted[R ℄):

the diereneoperator [6=℄that allowsto aess to

the worlds dierent from the urrent world (see

e.g. appliations of its use in [

Segerberg, 1981;

Sain, 1988;Koymans, 1992;Rijke,1993

℄

)

theomplementoperator[ R ℄thatallowstoaess

to theworldsnotaessiblefromtheurrentworld

(see e.g.

[Humberstone,

1983; Goranko, 1990a;

Levesque,1990;Lakemeyer, 1993℄)

and by aside-eettheuniversaloperator[U℄ that

allowstoaessto anyworldofthemodel (seee.g.

[Goranko

and Passy,

1992℄).

[U℄A an be dened

invarious ways: forinstane [U℄A=

def

A^[6=℄A or

[U℄A=

def

[R ℄A^[ R ℄A.

Adding theseoperatorsto standard modallogis an

signiantly inrease their expressive power. For in-

stane every nite ardinality is denable in a modal

logi whose language ontains [6=℄

[

Koymans, 1992

℄

.

Mostofthelogisdealt withinthepaperhavea deid-

ablesatisabilityproblemandweshallprovidedeision

proedures based on our systems. Howeverbeause of

the expressive power of the logis ouraluli have two

original features: a urrent information C is assoiated

to eah branh of a tableau and a restrited ut rule

is inluded in various aluli that an be viewed as a

modalvariantof theutruleinthed'Agostino's aluli

[d'Agostino,1993℄.

The rest of the paper is strutured as follows. Se-

tion2 presentsthelogis onsidered inthepaper. The

setions3,4,5and 6present thealuliforthevarious

logisaswellasthedeisionproedures. Beauseoflak

of spae we have omitted part of the proofs as well as

thepossibleextensions wherethe aessibilityrelations

satisfystandardonditions(reexivity,symmetry,tran-

sitivity,:::). Setion7omparesouraluliwithexisting

ones forothermodallogisand onludesthepaperby

presentingpossibleextensions.

2 Enrihed multi-modal logis

2.1 Syntax and semantis

AmodallanguageLisdeterminedbythreesetsthatare

supposedtobepairwisedisjoint: asetFor

0

=fp;q;:::g

of propositional variables, a set f:;^g of propositional

operators(theonnetives_;);,aredenedasforthe

propositionalalulus)anda(possiblynite)ountable

set OP = f[i℄ : i 2 Ig of modal operators. The set of

formulaeForofthelanguageLisdenedbythefollowing

grammar: A ::= pj :Aj A^Bj A where p2For

0 ,

A;B 2For and 2OP. Inthe sequelwe assumethat

OP is nite and as usual hiiA =

def

:[i℄:A. A frame

isa struture (W;(R

i )

i2I

)where W isa non-emptyset

of worlds (sometimes also alledknowledge states) and

(R

i )

i2I

is a family of binary relations onW. A model

M is a struture (W;(R

i )

i2I

;V) where (W;(R

i )

i2I ) is

a frame and V is mapping For

0

! P(W), the power

set of W. For eah set W, wewrite id

W

(resp. dif

W )

to denote the binary relation fhw;wi : w 2 Wg (resp.

W W nid

W

). Let M =(W;(R

i )

i2I

;V) bea model.

As usual, we say that a formula A is satised by the

worldw2W (denotedbyM;wj=A)whenthefollowing

onditionsaresatised:

M;wj=piw2V(p)forallp2For

0 ,

M;wj=:AinotM;wj=A,

M;wj=A^BiM;wj=AandM;wj=B,

M;w j=[i℄A i forall w 0

2W suh that (w;w 0

)2

R

i

,wehaveM;w 0

j=A.

InthesequelbyalogiLweunderstandapairhFor;Si

suhthatFor isa setofformulaefromagivenlanguage

andSisasetofmodels. AformulaAissaidtobeL-valid

ifor allmodels M2S andall w2W,M;wj=A. A

formulaAissaidtobeL-satisablei:AisnotL-valid.

2.2 Logis in the paper

Inthepaperweshallonsidernumerouslogisthat ad-

mitinterationsbetweenthemodaloperators:

1. K

I

= hFor;Si is the logi suh that S is the set

of allthe models. TheK

I

-satisabilityproblem is

[F 1995℄).

2. L([R ℄;[ R ℄)=hFor;Si(seee.g.

is the logi suh that I = f1;2g and M =

(W;R

1

;R

2

;V)2S iR

1

=WWnR

2

. Thesatis-

abilityproblemisdeidableandEXPTIME-hard

[

Spaan,1993

℄

. Similar modal logisare onsidered

intheontextofknowledgerepresentationandrea-

soning(seee.g.

[

Lakemeyer,1993

℄

).

3. L([6=℄)=hFor;Si(seee.g.

[Segerberg,1981℄)

isthe

logi suh that I =f1g and M= (W;R

1

;V)2 S

i R

1

=dif

W

. TheL([6=℄)-satisabilityproblem is

NP-omplete whenFor

0

isinniteandinPother-

wise(seee.g.

[Spaan,

1993;Demri, 1996℄).

4. K

I

([6=℄) = hFor;Si is the logi suh that 1 2 I

(a distinguished element of I), ard(I) 2 and

M=(W;(R

i )

i2I

;V)2S iR

1

=dif

W

. Axiomati-

zation ofK

I

([6=℄)hasbeenstudiedin [Rijke,

1993;

Balbiani, 1997℄.

For I = f1;2g, the K

I ([6=℄)-

satisabilityproblemisdeidableandEXPTIME-

omplete

[Rijke,1993℄.

ThemodelsforL([R ℄;[ R ℄)satisfy(?)R

1

=WWn

R

2

. Ifwerequire (??)R

1

=dif

W

then[2℄A,Aisvalid

in this new logi. L([6=℄) an be seen as L([R ℄;[ R ℄)

exeptthat themodelssatisfy(?)and(??)andonly[1℄

isinthelanguage. Moreover,K

I

([6=℄)is obtainedfrom

L([6=℄)by adding the operators f[i℄ : i 2 Inf1gg that

behaveasinK

I

. Thenotionofomplementaryrelations

isthereforeruial inthesemantisofthelogis.

It is not the purpose of this setion to reall all

the features of the expressive power of the abovemen-

tionedlogis(seee.g.

[

Goranko,1990a;Koymans,1992;

Rijke,1993

℄

). Bywayof exampleweonsider thelogi

K

I

([6=℄)with I =f1;2g. As usual, a lass F of frames

(W;R

1

;R

2

)issaidtobeK

I

([6=℄)-denableithereexists

aK

I

([6=℄)-formulaAsuhthatforallframes(W;R

1

;R

2 ),

(W;R

1

;R

2

) 2F i (W;R

1

;R

2

)j=A (i.e. for all valua-

tionsV andallw2W, (W;R

1

;R

2

;V);wj=A). A sim-

ilar notion of denability an be naturally dened for

otherlogis.

Fat2.1.

[Goranko,

1990b;Koymans, 1992℄

All universal rst-order onditions on R ;= are

K

I

([6=℄)-denable.

Everynite ardinalityisL([6=℄)-denable.

Eah universal rst-order formula on R is

L([R ℄;[ R ℄)-denable.

The statementsof Fat 2.1 do nothold forthe logi

K

I

: for example the lass of irreexive frames is not

K

I

-denable.

3 Tableaux for K

I

Thealulus dened for K

I

in this setionan be eas-

:[C℄

:

1 [C℄

rule

:2 [C℄

:[C℄

:1 [C℄j:2 [C℄

rule

: i

[C℄

k i

: i

0 [C℄

i

rule; newk2!onthebranh

: i

[C℄

0

: i

0 [C℄

i

rule

if 0

isalreadyonthebranhandforsomek2!, 0

=k i

.

Figure1: TableauxsystemforK

I

[Fitting, 1983℄)

but it will bethe opportunityto intro-

duevariousdenitionssmoothly.

We shall dene prexed tableaux following the

methodologydesribedin

[Fitting,1983℄.

Wemakesub-

stantialuseoftheuniformnotationfor modalformulae

dened in

[Fitting, 1983℄.

Four types of formulae are

usually distinguished: (neessity), (possibility),

(onjuntion)and (disjuntion). For i 2I, weintro-

due the types i

and i

. For instane, :hiiA and [i℄A

areoftype i

( i

0

denotestheformulae:AandArespe-

tively)and:[i℄AandhiiAareoftype i

( i

0

denotesthe

formulae:AandArespetively).

A prexed formula is a triple of the form : A [C℄

where isa prex, i.e. is anite sequeneofnatural

numbers possibly supersripted by some i 2 I, A is a

formulaandCisaouplehC

1

;C

2

i. EahC

i

isasetofpairs

ofprexes. Whentheontext islearweomit or[C℄.

TheonditionCistheurrentinformationonthebranh

that is stored during its development. At eah stepof

the development of a branh, C is idential for all the

prexedformulaeon that branh, i.e. C is anattribute

for branhes. Werefer to a prexed formulaas atomi

if it isof the form : p [C℄ or ::p [C℄ when p is an

atomi formula. Figure1 presentsthe prexedtableau

systemforthelogiK

I

. Observethat theondition[C℄

isofnouseinthisalulus.

In the sequel we omit the presentation of the -rule

(deompositionofonjuntions)andthe-rule(deom-

positionof disjuntions)but theserulesare inludedin

any forthoming alulus. A branh is losed ifit on-

tainsontraditoryprexedformulae(foranyformulaA,

:Aand::Aareontraditory). Atableau islosed

ifevery branh is losed. A formulaA issaid tohave a

losedtableau ithere isa losed tableau whih rootis

0 : :A [h;;;i℄. Termination ours when no operation

is possible. A branh is open if it is not losed and a

tableauisopenifatleastonebranh issuh.

Theorem 3.1. AformulaAisK -validiAhasalosed

: [C℄

0

: i

0 [C℄

i

rule;i2f1;2g

ifC

i (h;

0

i;C)holdsand 0

alreadyoursonthebranh.

: i

[C℄

k i

: i

0 [C℄

i

rule; newk2!onthebranh

if thereis no 0

suhthat 0

: i

0

onthebranhandeither

Ci(h;

0

i;C)or(forall: i

onthebranh, 0

: i

0

isonthe

branh).

00

:A[C℄

00

:A[C 0

℄j 00

:A[C 00

℄j 00

:A[C 000

℄

; 0

notalreadyappliedwiththisrule

Figure2: TableauxsystemforK

1;2

tableaubuiltwiththerulespresentedinFigure1.

The proof of Theorem 3.1 an be easily obtained from

existingonesfromtheliterature [

Fitting,1983

℄

.

4 Tableaux for L([R℄;[ R℄)

Instead of dening a sound and omplete alulus for

the logi L([R ℄;[ R ℄) we dene a sound and omplete

alulusfor thelogiK

1;2

(I =f1;2g)haraterizedby

the models (W;R

1

;R

2

;V) where R

1 [R

2

= W W

(we do not require R

1

\R

2

= ;). It is known that

L([R ℄;[ R ℄) and K

1;2

havethe same lass of valid for-

mulae

[Goranko,1990a℄

andweshallprovide adeision

proedure for the set of K

1;2

-valid formulae based on

ourtableaux approah. Atually from the alulus for

K

1;2

theareful reader will observe that a alulus for

L([R ℄;[ R ℄)anbeeasilydened. Howeverthealulus

forK

1;2

ismoreadequatetodeneadeisionproedure.

TherulesforthelogiK

1;2

arethoseinFigure2where

C 0

=hC

1

[fh;

0

ig;C

2 i,C

00

=hC

1

;C

2

[fh;

0

igi,

C 000

=hC

1

[fh;

0

ig;C

2

[fh;

0

igi.

For thelogiK

1;2 , C

i (h;

0

i;C)holds(i2f1;2g)iei-

therh;

0

i2C

i or

0

=k i

forsomek2!. Intuitively,

C

i

enodes the aessibility relation R

i

. Theondition

Couldbedeletedinthedenitionofthealulussine

itonlystoressomeinformationaboutthewaytherules

havebeenappliedonthebranh. However,ifonewishes

toimplementouraluli,theatualpresentationiswell-

suitedforthis purpose. Forinstane the i

-rulean be

read as follows. If the formula : i

ours on the

branh and ifthe urrent informationon the branh is

C then add 0

: i

0

on the branh and C remains un-

hanged. Itisworthobservingthat theutruleannot

be deleted unless ompleteness is lost. This property

is also shared by the ut rule inthe aluli dened in