Axiomati Satisability of LTL Formulae

In order to de ide axiomati satisabilityof LTL formulae, we will onstru t an

au-tomatonwhosesu essfulruns orrespondto omputations fortheinput. Noti ethat

a omputation  :N !

P

(P) an be seen also as a unarytree, that is,a tree where every node hasexa tlyone su essor. Morepre isely,ea h node representsone point

in time and the su essor relation in this tree is given by the standard ordering of

natural numbers. Thus, the automaton we onstru t will have the unique unlabeled

unarytree asinput. Thestatesofthisautomatonwillbesets ofLTLformulae,whi h

intuitivelyrepresenttheset ofall formulae thataresatisedata givenpointintime.

In that sense, these states orrespond to the Hintikka sets dened in the previous

subse tion. Noti e nonetheless that this orresponden e willnot bepre ise sin e for

LTLwe willfollow theideasofprevious automata onstru tions(e.g. [WVS83 ℄), and

hen e willnotassumethat theformulaeare innegationnormal form. Givenan LTL

formulaandasetofLTLformulaeR,wedenethe losureof(;R)asthesetofall

subformulae of  and R, and their negations, where doublenegations are an elled.

Thisset isdenoted by l(;R).

The statesofourautomatonareso- alledelementarysets offormulae,whi hplay

theroleoftheHintikkasets oftheprevioussubse tion; thatis,they aremaximaland

onsistent sets of subformulae in l(;R).

Denition 2.21 (Elementary set). Aset H l(;R) is alled an elementaryset

for (;R) if it satises the following onditions:

 :2H i 2= H;

 ^ 2H i f; gH;

 2H implies U 2H;

 if U 2H and 2= H,then 2H

As we have said before,theautomaton forsatisabilityof LTLformulaewilltake

unary trees asinputs; i.e., its runs willbe innite words over the set of states. The

transitionrelationisthusbinary. Thistransitionrelationmakessurethatthetemporal

next formula inthelabelofanode,thenitssu essornodemust ontain . This

is formalisedbythefollowing denition.

Denition 2.22 (Compatible). A tuple (H;H 0

) of elementary sets is alled

om-patible i it satises the following onditions:

 for all 2 l(;R), 2H i 2H 0

; and

 for all

1 U

2

2 l(;R),

1 U

2

2H i either (i)

2

2H or (ii)

1

2H and

1 U

2 2H

0

.

The runs of our automaton will be sequen es of elementary sets where ea h two

onse utiveonesforma ompatibletuple. In ontrastto the aseforSI,thepresen e

ofarunofthisautomatondoesnotimplytheexisten eofa omputation. Thereason

is that one an delay the satisfa tion of an until formula indenitely; that is, every

node inthe run may have theformula

1 U

2

whilenone has

2

, violating this way

thelast onditioninthedenitionofa omputationfortheinput(seeDenition2.9).

In order to rule out these kinds of runs and make sure that ea h until formula is

eventuallysatised,we willimposea generalised Bu hi onditionwhi h introdu esa

set of nalstates forea h untilformulain l(;R). Intuitively,ea h su h set ofnal

states isin harge ofenfor ing theeventual satisfa tionof one spe i untilformula.

Denition 2.23 (Automaton A sat

;R

). Let  and R be an LTL formula and a set

of LTL formulae, respe tively, and let 

1 U

1

;:::;

n U

n

be all the until formulae in

l(;R). Thegeneralised B u hiautomaton A sat

;R

:=(Q;
;I;F

1

;:::;F

n

) is givenby

 Q is the setof all elementary sets for (;R);


onsists of all ompatible pairs (H;H 0

)2QQ;

 I :=fH 2QjR[fgHg;

 for 1in;F

i

:=fH 2Qj

i

2H or 

i U

i

= 2Hg.

The su essfulruns of thisautomaton whoseroot is labelledwithan initial state

orrespondtothe omputationsfortheinput(;R). Fromthis,weobtainthe

follow-ing result[WVS83℄.

Theorem 2.24. Let  be an LTL formula and R a set of LTL formulae. The

au-tomaton A sat

;R

has a su essfulrun r with r(")2I i  isaxiomati satisable w.r.t.

R.

From thistheorem itfollowsthat axiomati satisabilityof LTL formulae an be

de idedbyan emptiness teston theautomaton A sat

;R .

Inthis hapter we have des ribedseveralpreviouslyknownalgorithmsfor

reason-in lusionof more omplex onstru tors and axiomsrestri ting theinterpretationsfor

on epts and roles in DLs. We then left the DL familyto in lude also the temporal

operatorsfor LTL.

Broadly, we showed the main hara teristi s of two dierent approa hes for

on-stru ting de isionpro edures. Onone hand,the tableau-based method,that tries to

onstru t a model whilekeepingtherestri tions imposedby theaxioms(in luded as

expansion rules). On the other hand is the automata-based approa h that tries to

onstru t an automaton forwhi han emptiness test leadsto a orre t de ision.

The parti ularinstan esofde isionpro edurespresentedinthis hapterwillhelp

us formalise the notions of general tableau algorithms (in Chapter 3) and so- alled

axiomati automata (in Chapter 5), respe tively. We will then show how ea h of

these de ision pro edures an be modied to obtain what is alled a pinpointing

pro edure; intuitively, one that will allow us to dedu e how the presen e of ertain

axioms in uen es the property being tested. The output of a pinpointing pro edure

willbe the so- alled pinpointing formula, from whi h all explanationsand diagnoses

Tableaux and Pinpointing

The previous hapter introdu ed pro edures that allow us to de ide if a property,

su h as subsumptionorsatisabilityof on ept names, follows from a set of axioms.

Thesets ofaxiomsused ould take verydierent shapes;namely, on ept denitions,

assertional axioms, or GCIs, in the ase of DLs, or LTL formulae. The de ision

pro edures we presented ame in two avours: the tableau-like and the

automata-based pro edures. It is thegoal of thiswork to show how to extend them insu h a

waythat, on eade isionismade,weareable tojustify itbyretrievingthoseaxioms

thatarerelevantfortheobtainedanswer. Theapproa hfollowedinthiswork onsists

onndingamonotoneBooleanformula,whi hwe allpinpointingformula,fromwhi h

the desired sets of axioms an be dedu ed. The present and following hapters will

deal withthe tableau-like methods, whilewe delay thetreatment of automata-based

pro edures untilChapter 5.

Before we an begin with the task of extending any kind of algorithm, we need

to formally des ribe the problemthat we are trying to solve; namely, theproperties

thatshouldbesatisedbythepinpointingformula. Thisinturnwillrequireaformal

denition of the kinds of properties that the original pro edures de ide. All these

notionsare introdu ed inSe tion3.1.

Afterwards,we pro eedto des ribe extensions of tableau-like de isionpro edures

that ompute the desired pinpointing formula. In order to improve understanding,

thisisdoneintwosteps. We rstfo usinthespe ial aseofgroundtableauxofwhi h

thesubsumptionalgorithmof Se tion2.3.1 isan instan e. We thengeneralise allthe

notions and results to what we all general tableaux in Se tion 3.3. This notion

en- ompassesthepro eduresdes ribedinSe tions2.3.2and2.3.3, butisnotabletodeal

with blo king onditions as des ribed in the last two se tions of the previous

hap-ter. Thepinpointingextensionsofgeneraltableaux areshownto orre tly omputea

pinpointingformulawheneverthey terminate.

Theextensionpresentedinthis hapterfollowstheideasintrodu edbyBaaderand

Hollunder in[BH95 ℄. There, the onsisten y algorithm forALC ABoxes is extended

by a labellingte hnique that ultimately omputes a pinpointing formula. A similar

approa h was followed by S hloba h and Cornet [SC03 ℄ for on ept unsatisability

withrespe tto so- alledunfoldableALC terminologies. The maindieren ebetween

BaaderandHollunder'sapproa handthatbyS hloba handCornetisthatthelatter

tries to nd the sets of axioms that are relevant to unsatisability dire tly, rather

than by usingthe intermediarypinpointing formulaas donein theformer approa h.

Inreality,theresultobtainedusingthemethodin[SC03℄ anbeseenasapinpointing

formulawrittenindisjun tivenormalform. Althoughtheseideashavebeenextended

to in lude additional onstru tors or usedierent kinds of axioms (see, forinstan e,

[PSK05, MLBP06℄), ea h of these extensions has beenmade to work spe i ally for

the language being studied. Nonetheless, ex ept for the ase dealing with blo king

[LMP06 ℄ thatneeds spe ialattention, they all followthesame basi ideas.

Unfortunately, as shown at the end of this hapter, there is no warranty that

the extended algorithm will stop after a nite number of steps, even if the original

tableaudoes. Thisfa tisspe iallyrelevantsin enoneof thepapers itedsofardeals

with termination of the extensions they present. A tually, termination is usually

disregarded astrivially followingfrom thesame ausesof terminationof the original

tableau, givingno furtherinsightinto whi h these ausesare inreality. It willbe the

taskofChapter4tointrodu eaframeworkwhereboth,tableauxandtheirpinpointing

extensions,areguaranteedtoterminate. Itisinthat haptertoothatwewillintrodu e

thenotionof blo kingforgeneraltableaux and theirpinpointingextensions.

3.1 Basi Notions for Pinpointing

Webeginthisse tionbydeningthegeneralform oftheinputsforthede ision

algo-rithms used along thiswork. These inputs, alled axiomatised inputs, onsist of two

parts. Intuitively,one part orrespondsto a knowledge base, that is,a set of axioms

possibly restri tedto satisfy additionalinternal restri tions, and the other expresses

the instan e of theinferen e problemthat needs to be tested againstthis knowledge

base. The internal restri tions in the set of axioms are ne essary for modelling e.g.

a y li - or SI-TBoxes, where not every set of axioms is allowed. Indeed, a y li

TBoxes requireevery on ept name to appear at most one in theleft-hand-side of a

on ept denition, and SI-TBoxes are restri ted to allow the use of ea h role name

in at most one inverse axioms. But noti e that in both ases, if a set of axioms is

allowedtobeusedasaknowledge base,thenanyof itssubsetsisalsoallowed. Inour

general approa hwekeep thisproperty.

The onsequen es in whi h we are interested need to satisfy a monotoni ity

re-stri tion inthe sensethatadding axiomsto theknowledge base an onlymake more

onsequen es true, but not falsify any that already follows from the original set of

axioms. A property is merely a set of axiomatised inputs, and the de ision

prob-lemasso iatedwithsu h property onsist onde iding,fora givenaxiomatised input,

whether itbelongsto theset ornot. A propertythatmodels onsequen es satisfying

themonotoni ity restri tionstatedabove willbe alled onsequen e property.

Denition 3.1 (Axiomatised input, -property). Let I be a set, alled the set

of inputs, T be a set, alled the set of axioms, and let

P

admis

(T) 

P

fin

(T) be a

set of nite subsets of T.

P

admis

(T) is alled admissible if T 2

P

admis

(T) implies

T 0

2

P

(T) for all T 0

T. An axiomatised inputfor I and

P

(T) is of the

form (I;T) where I2I and T 2

P

admis (T).

A onsequen eproperty(or -propertyfor short)isa setP I

P

admis

The idea behind -properties on axiomatised inputs is to model onsequen e

re-lationsin logi , i.e., the -propertyP holdsifthe inputI \follows" from theaxioms

inT. The monotoni ityrequirement on -properties orresponds to thefa t thatwe

want to restri t the attention to onsequen e relations indu ed by monotoni logi s.

Infa t, fornon-monotoni logi s,lookingat minimalsetsof axiomsthathave agiven

onsequen edoesnotmake mu h sense.

To illustrate Denition 3.1, onsiderthe set N

C

of on ept names. Assume that

I is the set of ordered pairsN

C

N

C

and that T onsists of all H L-GCIs over these

on ept names. Then thefollowingis a -propertya ording to the above denition:

P := f((C ;D);T) j C v

T

Dg: This property represents subsumptionw.r.t. general

H L-TBoxes. As a on rete example, onsider := ((A;B);T) where T onsists of

It is easy to see that 2 P. Note that Denition 3.1 is general enough to apture

other variants of the example above, for instan e, where I 0

onsist of tuples of the

form (C ;D;T

1

)2I

P

fin

(T)and the -property isdenedas

P

For example, if we take the axiomatised input 0

Due to the monotoni ity of -properties, it may well be that some axioms are

irrelevant fordedu inga onsequen e. If we areinterested injustifying su h a

onse-quen e,wewouldneedto getridofallthose irrelevantaxiomsandpresent aminimal

knowledge basefromwhi hthe onsequen e stillfollows. If, onthe ontrary,the

on-sequen e isdete tedasan error,wemightwant to removeonlyenoughaxiomstoget

rid ofit butnotmore,sin ethat might alsoremove some desired onsequen es.

Denition 3.2 (MinA,MaNA). Given an axiomatised input =(I;T) and a

-property P, a setof axioms S T is alled a minimalaxiom set(MinA)for w.r.t.

Note that the notionsof MinA and MaNA areonlyinterestingin the ase where

2 P. In fa t, otherwise the monotoni ity property satised by P implies that

MIN

P( )

=; and MAX

P( )

=fTg. In the above example, where we have 2 P, it

is easy to seethat MIN

P( )

gg. In thevariant ofthe

examplewhereonlysubsetsof fax

1

ThesetMAX

P( )

anbeobtainedfromMIN

P( )

by omputingtheminimalhitting

sets of MIN

P( )

,and then omplementing these sets [SC03, LS05 ℄. A set S T is a

hittingset ofMIN

P( )

ifithasanonemptyinterse tionwitheveryelementofMIN

P( ) ,

and is a minimal hitting set if no stri t subset of S is itself a hitting set. In our

example, the minimal hitting sets of MIN

P( )

gg. The intuition behind this

redu tionsisthat,togetasetofaxiomsthatdoesnothavethe onsequen e,wemust

remove from T at leastone axiom forevery MinA,and thus theminimalhitting sets

give ustheminimalsets to be removed.

Theredu tionwehavejustsket hedshowsthatitisenoughtodesignanalgorithm

for omputing all MinAs, sin e the MaNAs an then be obtained by a hitting set

omputation. It should be noted, however, that this redu tion is not polynomial:

there may be exponentially many hitting sets of a given olle tion of sets, and even

de iding whether su h a olle tion has a hitting set of ardinality nis already an

NP- omplete problem[GJ79 ℄. Also note that there is a similar redu tion involving

hitting sets for omputingtheMinAsfrom allMaNAs.

Instead of omputing MinAs or MaNAs, one an also ompute the pinpointing

formula.

10

To dene the pinpointingformula, we assume that every axiom t2 T is

labeled with a unique propositional variable, whi h we denote as lab(t). Let lab(T)

betheset ofall propositional variableslabelinganaxiom inT. A monotone Boolean

formula over lab(T) isa Booleanformulausing(someof) thevariablesinlab(T) and

onlythe onne tives onjun tionanddisjun tion. Wefurtherassumethattheformula

>, whi h is always evaluated as true, is a monotone Boolean formula. As usual, we

identifyapropositionalvaluation withthesetofpropositionalvariablesitmakestrue.

Foravaluation V lab(T),let T

V

:=ft2T jlab(t)2Vg.

Denition 3.3 (Pinpointing formula). Given a -property P and an axiomatised

input =(I;T), a monotone Boolean formula  over lab(T) is alled a pinpointing

formulaforP and ifthefollowingholdsfor everyvaluationV lab(T): (I;T

V

g as the set of propositional

variables. Itis easyto see that(ax

1

isapinpointingformula forP

and .

Valuations have a natural partial order by means of set in lusion, whi h allows

us to speak about minimal and maximal valuations. The following is an immediate

onsequen e ofthe denitionof apinpointingformula[BH95 ℄.

Lemma 3.4. Let P be a -property, =(I;T) an axiomatised input, and  a

pin-pointingformula for P and . Then

MIN

P( )

= fT

V

jV isa minimal valuation satisfying g

MAX

P( )

= fT

V

jV isa maximal valuation falsifyingg

10

This orrespondsto whatwas alled the lashformula in[BH95 ℄. Here, wedistinguishbetween

the pinpointingformula, whi h anbe denedindependentlyof atableau algorithm, and the lash

This lemma shows that itis enough to design an algorithm for omputinga

pin-pointing formula to obtainall MinAsand MaNAs. However, like theprevious

redu -tion for omputingMAX

P( )

from MIN

P( )

,theredu tionsuggested bythelemma is

notpolynomial. Forexample,to obtainMIN

P( )

from,one anbringinto

disjun -tivenormalformandthenremovedisjun tsimplyingotherdisjun ts. Itiswell-known

thatthis an ausean exponentialblowup. Conversely,however,thesetMIN

P( ) an

dire tly be translatedintothe pinpointingformula

_

S2MIN

P( )

^

s2S

lab(s): (3.2)

Returning to our example, the pinpointing formula obtained in this fashion from

MIN

P( )

=ffax

1

;ax

2

;ax

4 g; fax

2

;ax

In document Axiom-Pinpointing in Description Logics and Beyond (Page 35-43)