The Logic WT_mu

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The Logic WT_mu

Omer Landry Nguena Timo

To cite this version:

Omer Landry Nguena Timo. The Logic WT_mu. 2009. �hal-00383062�

The Logi WT

µ

Omer-Landry Nguena-Timo

Université Bordeaux 1,LaBRI, CNRS

351 ours de laLibération, 33400Talene-FRANCE

omer-landry.nguena-timolabri.fr

Abstrat

Thepowerof Model-hekingdepends ontheexpressivepowerofmodelsof systems

andmodelsofspeiations.ThepaperintroduesWT

µ

^{, areal-timelogiwith theleast}

and thegreatestxpointoperators.WT

µ

^{isaweaktimed extensionofthe}

µ

^-alulus;it

is losed to L

ν

^{. As} Event-reordinglogi, WT

µ

^{desribes properties on}Event-reording automata.

WeshowthatWT

µ

^{ismoreexpressivethan}Eevent-reordinglogi.Inpartiular,with

WT

µ

^{formulas, one an require ourrenes of an event at all the time instants that}

satises atiming onstraint.We providean exponential-time deisionproedure forthe

model-hekingofWT

µ

^.

1 Introdution

ThepowerofModel-hekingdepends ontheexpressivepowerofmodelsof systemsand

modelsofspeiations.Ourgoalistopresentanewexpressivexpointlogifordesribing

propertiesonalassofreal-timesystems.Asigniantpropertythatourlogiisableto

desribethe requirement of the ourrenean event in all the time satisfying a timing

onstraint(neessity modaloperator). Wearguethat suh akindofpropertyannotbe

desribedwithEvent-reordinglogi(ERL)[Sor02℄thathasbeenintroduedbySoreafor

desribingpropertythesamelassofreal-time systems.

Real-timesystemsaremodeledwithtimedproesses.Timedproessesarenothingelse

butevent-reordingautomata[AFH99℄withoutanaeptaneondition.Timedproesses

have loalloks eah assoiatedto anevent andsuh alokgathers thetime elapsed

sine the last ourrene of the orresponding event. A timed proess is a nite state

labelled transition systemwhose transitions (

p −→ ^g,a p ^′

^{)are labelledwith onstraintson}

loks and events. A onstraint onloks isjust aonjuntionof omparisons of values

ofalokwithanintegeronstant.Clokareinterpretedoverrealnumbers.Thevalueof

eahlokgrowsontinuouslyandwiththesamerateasthetimeunlessitisreset.When

theproessisin somestate,thetimeelapses ontinuously(the valuesofthelokstoo)

untilaneventours.Then,theproessinstantaneouslyseletsatransitionlabelledwith

that eventandhekswhethertheonstraint(

g

^{)onthe hosentransitionis satisedby}

the values of loks before it resets the lok assoiated to the eventand movesto the

targetstateofthetransition.Iftheonstraintisnotsatised,theproessdoesnothange

thestate.

The logi that we introdue in this paperis alled WT

µ

^{. The logi WT}

µ

^{is a weak}

timed extensionofthestandard

µ

^{-alulus.FormulasofWT}

µ

^areinterpretedovertimed proesses. Timed proesses are nothing else but event-reording automata without an

events,whilemodalitiesofERLareindexedwithpairsmadeofaonstraintandanevent.

They are of WT

µ

^{are of theform}

hgi

^and

[g]

^{in addition to the lassial modalities of}

the

µ

^{-alulus indexed with event (}

hai

^and

[a]

^). Intuitively, a state of a timed proess

p

^satises

hgiϕ

^{from a given} time-ontext desribed by avaluation

v

^{if by letting time}

elapse in it,itis possibleto reah amomentwhen thevaluesofthelokssatisfy

g

^and

inthatmoment,theformula

ϕ

^{issatised.Astate}

p

^{ofatimedproesssatises}

[g]ϕ

^from

a time-ontext

v

^{ifwhenever startingfrom}

v

^{welet thetime pass and reah amoment}

when

g

^{is satisedthen}

ϕ

^{is satisedin that moment. We onsider the}model-heking problem forWT

µ

^{;that is: Doesatimed proess satisfyaWT}

µ

^{formula.We providean}

exponential-timedeisionproedureforthat problem.

Weompare WT

µ

^{with ERL.ERL is also presented[Sor02℄ asatimed extension of}

the

µ

^{-alulus;andmodelsofERL formulasaretimedproesses.InERL,modalitiesare}

indexed bothwith aneventand aonstraint(

[g, a]

^,

hg, ai

^{).A stateofatimedproess}

p

satises

hg, aiϕ

^{from a given} time-ontext desribed by avaluation

v

^{if by letting time}

elapseinit,itispossiblethattheevent

a

^{oursinamomentwhenthevaluesoftheloks}

satisfy

g

^{andaftertheourreneof}

a

^{,theproessgoestoastatethatsatises}

ϕ

^.Astate

ofatimedproess

p

^satises

[g, a]ϕ

^fromagiventime-ontextdesribedbyavaluation

v

ifaftertheourreneof

a

^{inamomentwhenthevaluesoftheloks(obtainedbyletting}

time elapses in

v

^{) satisfy}

g

^{the proess always goes to a statethat satises}

ϕ

^{. We will}

showthat WT

µ

^{ismoreexpressivethanERL aseveryformulaofERL anbetranslated}

into an equivalent WT

µ

^{formula;and there are someformulasof WT}

µ

^{that annot be}

translated into formulas of ERL. In partiular with WT

µ

^{, it is possible to requirethe}

ourreneaneventinallthetimesatisfyingatimingonstraint;butitisnotwithERL.

Related results:Logis (TML[HLY91℄,

L ^t _µ

^{[SS95℄ L}

ν

^{[LLW95℄)that enableto de-}

sribethethe neessity modal operator hasbeenonsidered fordesribing propertieson

timedautomatabutthedeidabilityofthesatisabilityproblemhasnotbeenestablished.

Laroussinieetal.[LLW95℄haveintroduedthelogi

L ν

^{asamorepowerfullogithanthe}

onein[HLY91,SS95℄butitssatisabilityproblemisstillopenandnodisjuntivenormal

form hasbeenprovided [BCL05℄. Thelogis

L ν

^{and WT}

µ

^are inomparableastheyare not interpretedoverthe samemodel and

L ν

^{doesnotallowthe least xpoint operator.}

But, if we restritthe interpretation of

L ν

^{on timed proesses, we laim that}

hgiϕ

^will

havethesamemeaningasthe

L ν

^formula

hδi(g ∧ ϕ)

^and

[g]ϕ

^{willhavethesamemeaning}

asthe

L ν

^formula

[δ](g → ϕ)

^.

This paper is organisedasfollows:We presentresults forthe model-hekingof the

µ

^{-alulus in the next setion. We present time proesses in Setion 3. In that setion}

wealsopresentwellknownoneptsandresultsonerningregion,onstraint,andtimed

abstrat bisimulation.In Setion 4we presentWT

µ

^{and itssemantis.Weonsider the}

model-hekingproblemforWT

µ

^{inSetion5.InSetion6,wepresentERLandweshow}

that WT

µ

^{is more expressive than ERL. We onlude the paper with future works on}

WT

µ

^.

2 Two Player Parity Game and

µ

^{-alulus Results}

2.1 Two Player Parity Games and Multi-Parity Games

We present a omplexity result for heking a winning strategy in a twoplayer games

withparityondition.Wealsopresentthenotionoftwomulti-paritygame.

Denition 1 A two player parity game(see [Zie98℄) is a tuple

G = hN E , N A , T ⊆ N ² , Acc G i

^where

hN, T i

^{is a graph with the nodes (or positions)}

N = N A ∪ N E

^par-

titionedinto

N E

^and

N A

^.

N E

^{denotesthesetofnodesoftheplayer}

Eve

^and

N A

^denotes

the set of nodes of the player

Adam

^{. The winning ondition}

Acc G ⊆ N ^ω

^{, is a parity}

onditiononthenodes.Thegameisnite if

N

^isnite.

Aplay between

Eve

^and

Adam

^fromsomenode

n ∈ N

^{proeedsasfollows:if}

n ∈ N E

then

Eve

^{makesahoie of asuessorotherwise}

Adam

^{hooses asuessor; from this}

suessor the samerule applies and the play goes on foreverunless one of the parties

annotmakeamove.Aplayisniteifaplayerannotmakeamoveandthenheloosethe

play.Intheasethattheplayisaninnitepath

π = n 0 n 1 n 2 · · ·

^,

Eve

^winsif

π ∈ Acc G

^.

Otherwise

Adam

^{isthe winner.Among winning onditionsintrodued in the}literature, we onsider the parity ondition. A strategy

σ

^for

Eve

^{is afuntion assigning to every}

sequeneofnodes

~n

^{endinginanode}

n

^from

N E

^avertex

σ(~n)

^{whihisasuessorof}

n

^.

A play from

n

^{onsistent with}

σ

^{is anite orinnitesequene}

n 0 n 1 n 2 · · ·

^{suh that}

n i+1 = σ(n i )

^forall

i

^with

n i ∈ N E

^{. Thestrategy}

σ

^{iswinning for}

Eve

^{from thenode}

n

ifandonlyifalltheplaysstartingin

n

^{andonsistentwith}

σ

^{arewinning.Thestrategies}

for

Adam

^{is are dened similarly. A node is winning ifthere exists a strategy winning}

from it.A gameis determined ifeverynodeis winning foroneof theplayer.Astrategy

is positional if itdoesnot depend on the sequenesof nodes that were played till now,

butonlyonthepresentnode.Sosuhastrategyfor

Eve

^anberepresentedasafuntion

σ : N E → N

^{andidentiedwithahoieofedgesinthegraphof thegame.}

Now we state the following results on two player games (see [GH82, EJ91, Jur00,

VJ00℄).

Theorem2 Every paritygame isdetermined. Ina twoplayer parity game aplayer has

a winningpositional strategyfrom eahof his nodes.Thereis aneetive proedurethat

deides whois awinner fromagiven node ina nitegame, andthat proedure worksin

time

O |T | ×

2 × |N | d

⌈d/2⌉ !

where,

d

^{isthe maximalparityindex.}

2.2 The

µ

^-Calulus

The

µ

^{-alulusintroduedbyKozen[Koz82℄(seealso[AN01℄) isanexpressivetemporal}

logithat extendsmodallogiwiththegreatest(

ν

^)andleast(

µ

^{)xpointoperators.We}

presentthesyntaxand thesemantisof the

µ

^{-alulus.Thenwestatesomewell known}

resultsthatinludetheomplexityofthemodel-hekingproblem,theomplexityofthe

satisabilityproblem andadisjuntivenormalform theorem. The omplexityresultfor

themodel-hekingisobtainedbyredutiontohekingifthere isawinning strategyin

atwoplayerparitygame.

2.2.1 Denitionsand Semantis

Denition 3 The syntax of the

µ

^{-alulus is dened over a set}

Var = {X, Y, . . .}

^of

variables, aset

Σ

^{ofevents.Itisgivenbythefollowinggrammar:}

ϕ ::= tt |

| X | ϕ ∨ ψ | ϕ ∧ ψ | haiϕ | [a]ϕ | µX.ϕ(X) | νX.ϕ(X )

In theabove,

X ∈ Var

^,

a ∈ Σ

^;and

tt

^{and denote theformula that are alwaystrue}

and false respetively;

hai

^and

[a]

^{denote the} existential and the universal modalities indexedwiththeevent

a

^{;theyrepresentexists}

a

^{-suessorandall}

a

^{-suessormodalities}

respetively.Theformulas

µX.ϕ(X )

^and

νX.ϕ(X )

^representrespetivelytheleastandthe greatestxpointformula.

Foraformula

ϕ

^{,thelosure[Koz82℄of}

ϕ

^,

sub(ϕ)

^{isdened asfollows:}

Denition 4 Thelosure

sub(ϕ)

^of

ϕ

^{isthesmallestset offormulassuhthat:}

• ϕ ∈ sub(ϕ)

•

^if

ψ 1 ∨ ψ 2 ∈ sub(ϕ)

^theboth

ψ 1 , ψ 2 ∈ sub(ϕ)

•

^if

ψ 1 ∧ ψ 2 ∈ sub(ϕ)

^theboth

ψ 1 , ψ 2 ∈ sub(ϕ)

•

^if

haiψ ∈ sub(ϕ)

^then

ψ ∈ sub(ϕ)

•

^if

[a]ψ ∈ sub(ϕ)

^then

ψ ∈ sub(ϕ)

•

^if

σX.ψ(X ) ∈ sub(ϕ)

^then

ψ(X ) ∈ sub(ϕ)

^,where

σ ∈ {ν, µ}

The formulasin

sub(ϕ)

^{are alled the} subformulas of

ϕ

^{. Fora formula}

ϕ

^,

sub(ϕ)

^is

nite and,bydenition,itisnotlargerthatthenumberofsymbolsusedin

ϕ

^.

Denition 5 Theset

f ree(ϕ)

^{offreevariableofa}

µ

^{-alulusformula}

ϕ

^isdenedindu-

tivelyasfollows:

• f ree(tt ) = f ree(

) = ∅

• f ree(X ) = {X }

• f ree(ϕ ∨ ψ) = f ree(ϕ) ∪ f ree(ψ)

• f ree([a]ϕ) = f ree(haiϕ) = f ree(ϕ)

• f ree(µX.ϕ(X)) = f ree(νX.ϕ(X )) = f ree(ϕ) \ {X}

A variable

X

^{isfree in aformula}

ϕ

^if

X ∈ f ree(ϕ)

^.

Denition 6 Avariable

X

^{isbound inaformula}

ϕ

^{ifthereisasubformula}

σX.ψ(X)

^of

ϕ

^with

σ ∈ {µ, ν}

^.

Denition 7(Wellnamed) Weallaformulawell named iftheexpression

µX.ϕ(X)

(or

νX.ϕ(X )

^{)ours atmostoneforeahvariable}

X

^.

Byrenamingvariablesifneessary,everyformulaanbetranslatedintoanequivalent

wellnamedformula.Inwhatfollows,withoutlossofgenerality,weassumethatformulas

arewellnamed.

Denition 8(Binding) Thebinding denition ofaboundvariable

X

^inawellnamed

formula

ϕ

^,

D ϕ (X )

^{is the unique subformula of}

ϕ

^{of the form}

σX.ψ(X)

^{. We will omit}

subsript

ϕ

^{whenitausesnoambiguity. Weall}

X

^a

µ

^{-variable when}

σ = µ

^{, otherwise}

we all

X

^a

ν

^{-variable. The funtion}

D ϕ

^{assigning to every bound variable its binding}

denition in

ϕ

^{willbealledthebindingfuntion assoiatedwith}

ϕ

^.

Denition 9 A sentene isawellnamedformulawithoutfreevariables.

Denition 10 Thedependenyorder

≤ ϕ

^{overtheboundvariablesofaformula}

ϕ

^,isthe

leastpartialordersuhthatif

X

^oursin

D ϕ (Y )

^(and

D ϕ (Y )

^{isasubformulaof}

D ϕ (X )

⁾

then

X ≤ ϕ Y

^{. When}

X ≤ ϕ Y

^{,itisalsosaidthat}

Y

^dependson

X

^or

X

^isolderthan

Y

^.

Denition 11 Variable

X

ⁱⁿ

µX.ϕ(X)

^{isguarded ifeveryourreneofXin}

ϕ(X)

^isin

the sopeof somemodalityoperator

hi

^or

[]

^{. Wesay thata formula isguarded if every}

boundvariable intheformulaisguarded.

Alternation depth desribes the number of alternations between least and greatest

xpointoperators.

Denition 12 The alternation depth of a formula denoted by

alt(ϕ)

^{is the numberof}

nestingbetween

µ

^and

ν

ⁱⁿ

ϕ

^{;itisreursivelydenedasfollows:}

• alt(tt ) = alt(

) = alt(X ) = 0

• alt(ϕ ∧ ψ) = alt(ϕ ∨ ψ) = max(alt(ϕ), alt(ψ))

• alt(haiϕ) = alt([a]ϕ) = alt(ϕ)

• alt(µX.ϕ(X)) = max({1, alt(ϕ(X )} ∪ {1 + alt(νY.ψ(Y )) | νY.ψ(Y ) ∈ sub(ϕ); X ≤ ϕ

Y })

• alt(νX.ϕ(X )) = max({1, alt(ϕ(X )} ∪ {1 + alt(µY.ψ(Y )) | µY.ψ(Y ) ∈ sub(ϕ); X ≤ ϕ

Y })

Formulas of the

µ

^{-alulus are} interpreted over

Σ

^{-labelled transition systems. The}

semantis of a

µ

^{-alulusformula}

ϕ

^{is aset of states of a}

Σ

^{-labelled transition system}

S = hS, Σ, s ⁰ , ∆ S i

^{where the formula holds under a given valuation of variables}

Val : Var → 2 ^S

^{, and itis denoted by}

[[ϕ]] Val ^S

^{. Givena valuation of variables}

Val

^{and aset of}

states

T ⊆ S

^{, the valuation}

Val [X/T ]

^{is the valuation}

Val

^{with the} substitution that assoiatesthestatesof

T

^{withthevariable}

X

^{.Formally,for}

Y ∈ Var

^,

Val [X/T ](Y ) = T

if

Y = X

^and

Val(Y )

^{otherwise.Wedenetherelation}

^{betweenastate}

s

^{ofatransition}

system

S

^{, avaluation}

Val

^{and aformula}

ϕ

^{. Wewrite}

S, s, Val ϕ

^{when theformula}

ϕ

holdsin

s

^orequivalently

s

^satises

ϕ

^.Therelation

^{isdenedasfollows:}

• S, s, Val X

^if

s ∈ Val (X )

• S, s, Val ϕ 1 ∨ ϕ 2

^if

S, s, Val ϕ 1

^or

S, s, Val ϕ 2

• S, s, Val ϕ 1 ∧ ϕ 2

^if

S, s, Val ϕ 1

^and

S, s, Val ϕ 2

• S, s, Val haiϕ

^ifthereis

s −→ ^a s ^′

^suhthat

S, s ^′ , Val ϕ

• S, s, Val [a]ϕ

^ifforall

s −→ ^a s ^′

^wehave

S, s ^′ , Val ϕ

• S, s, Val µX.ϕ(X)

^if

s ∈ ∩{T ⊆ S | [[ϕ(X)]] ^S Val [X/T] ⊆ T }

^.

• S, s, Val νX.ϕ(X )

^if

s, ∈ ∪{T ⊆ S | T ⊆ [[ϕ(X )]] ^S Val [X/T] }

Thenwedene

[[ϕ]] ^S Val = {s ∈ S | S, s, Val ϕ}

^{.It issaidthata}

Σ

^{-labelledtransition}

system

S

^{is amodel ofaformula}

ϕ

^when

s ⁰ ∈ [[ϕ]] ^S Val

^{;in this ase wewrite}

S, Val ϕ

^.

Thevaluation

Val

^{isomittediftheformuladoesnotontainsfreevariables.}

It is known (see [Eme90℄ for a survey) that properties expressed in temporal logis

LTL,CTL,andCTL

∗

anbeenodedas

µ

^{-alulusformulasandthatthereareformulas}

ofthe

µ

^{-alulus(forinstane}

νX.haihaiX

^{)thatannotbewritteninCTL}

^∗

^.

Given twoformulas

ϕ 1

^and

ϕ 2

^{, we oftenusethe notation}

ϕ 1 ≡ ϕ 2

^{to saythat}

ϕ 1

^is

equivalent to

ϕ 2

^{,meaningthat foreverylabelledtransition system}

S

^andvaluation

Val

^,

[[ϕ 1 ]] ^S Val = [[ϕ 2 ]] ^S Val

^.

It is standard to onsider the negation operator (

¬

^{) on}

µ

^{-alulus sentenes. This}

operatorisdenedasfollows:

• ¬tt ≡

• ¬

≡ tt

• ¬(ϕ 1 ∧ ϕ 2 ) ≡ ¬ϕ 1 ∨ ¬ϕ 2

• ¬(ϕ 1 ∨ ϕ 2 ) ≡ ¬ϕ 1 ∧ ¬ϕ 2

• ¬haiϕ ≡ [a]¬ϕ

• ¬[a]ϕ ≡ hai¬ϕ

• ¬µX.ϕ(X) ≡ νX.¬ϕ(¬X)

• ¬νX.ϕ(X ) ≡ µX.¬ϕ(¬X)

Thefollowingpropositionisstandard.

Proposition13 Given asentene

ϕ

^{, a}

Σ

^{-labelledtransition system}

S

^{and avaluation}

Val

^,

[[¬ϕ]] ^S Val = S \ [[ϕ]] ^S Val

Thanks to the proposition just above, we anuse the negation operator anappear in

µ

^{-alulussentenes.}

Letuspresentsomeresultsonthe

µ

^-alulus.

Proposition14 ([Koz82℄) Everyformulaisequivalentto someguardedformula.

2.2.2 Model-Cheking Results

Informally,thetaskofhekingwhetheranitestatetransitionsystem,

S = hS, Σ, s ⁰ , ∆ S i

is amodelofasentene

ϕ

^{anbeseenastwoplayerparitygamewhosenodesare setof}

tuples oftheform

(s, ψ)

^where

s ∈ S

^and

ψ

^{isasubformulaof}

ϕ

^{. Positions oftheplayer}

Eve

^onstainsubformulasofoneoftheforms

tt , ϕ 1 ∨ϕ 2 , haiψ

^{.Theotherpositionsbelong}

to the player

Adam

^{.The initialposition of thegame is}

(s ⁰ , ϕ)

^{. Theset of movesofthe}

gamesaresuhthat:

•

^{There isnomovefromeither}

(s, tt)

^or

(s,

)

^.

•

^From

(s, ϕ ∧ ψ)

^{aswellas from}

(s, ϕ ∨ ψ)

^{therearemovesto}

(s, ϕ)

^andto

(s, ψ)

^.

•

^From

(s, [a]ϕ)

^{and from}

(s, haiϕ)

^{there are moves to}

(s ^′ , ϕ,

^{for every}

s ^′

^{suh that}

s −→ ^a s ^′

^.

•

^{There isamovefrom}

(s, σX.ϕ(X ))

^to

(s, ϕ(X ))

•

^{There isamovefrom}

X

^to

(s, ϕ(X))

^where

D(X) = σX.ϕ(X)

Theaeptaneonditionisgivenbytheparityfuntion

rank : Q → N

^{dened by:}

rank(ψ) =







0

^if

ψ

^{isnotavariable}

2 × alt(D(X ))

^where

ϕ = X

^and

X

^isa

ν

^-variable

2 × alt(D(X )) + 1

^where

ϕ = X

^and

X

^isa

µ

^-variable

Oneanshowthat

S

^{isamodelofaformulaifplayer}

Eve

^{hasawinning strategyin}

thethegame.Thisgivesanintuitiveideabehindthefollowingresults.

Theorem15 ([EJ91, Tho97,Jur00℄) Let

S = hS, Σ, s ⁰ , ∆ S i

^bea

Σ

^{-labelledtransition}

system and let

ϕ

^{be a}

µ

^{-alulus formula. The} model-heking problem for

ϕ

^and

S

^is

solvable in time

O |∆ S | × |sub(ϕ)| × |S| × |sub(ϕ)|

⌊alt(ϕ)/2⌋

⌈alt(ϕ)/2⌉ !!

3 Timed Proesses

We present timed proesses as event-reording automata without aeptane [AFH99℄.

Werstlypresentthenotionsofregion[ACD

+

92,LY97,AFH99,AD94℄anditsproperty.

All theresultspresentedin thissetionarewell-known.

3.1 Cloks and Valuations

Cloksarevariablesevaluatedoverreal numbers.Therearetwooperationsontime,the

timeelapseoperationthatgivesthevalueofthelokafteradelayandtheresetoperation

that setsthevalueofaloksto

0

^.

Let

R ⁺

^{bethesetof nonnegativereal numbers.Weonsider}

H = {h 1 , h 2 , . . . }

^aset

ofloksvariables(orloksforsimpliity).

Denition 16 Avaluation onasetoflok

H

^{isatotalfuntion}

v : H → R ⁺

^.

The symbol

V

^{represents the set of} valuations. Given a valuation

v ∈ V

^{, and a lok}

h ∈ H

^{,thevaluation}

v + t

^{isdened by}

[v + t](h) = v(h) + t

^{and,thevaluation}

v[h := 0]

isdenedby

v[h := 0](h ^′ ) = 0

^if

h = h ^′

^else

v[h := 0](h ^′ ) = v(h ^′ )

^{.Wesaythatavaluation}

v

^{isasuessor ofavaluation}

v ^′

^if

v = v ^′ + t

^forsome

t ∈ R ⁺

^.

Example:Let

H = {h 1 , h 2 }

^{beasetoftwoloks.InTable1,wepresentsomevaluations}

on

h

^{aresomevaluationon}

H

^.

v 0 (h 1 ) = 0 v 0 (h 2 ) = 0

v 1 (h 1 ) = 0.35 v 1 (h 2 ) = 0.35

v 2 (h 1 ) = 0.35 v 2 (h 2 ) = 0

v 3 (h 1 ) = 0.85 v 3 (h 2 ) = 0.50

v 4 (h 1 ) = 0

v ⁴ (h ² ) = 0.50

v 5 (h 1 ) = 0.35 v ⁵ (h ² ) = 0.85

Table1:Examplesof valuations.

These valuations are suh that

v 1 = v 0 + 0.35

^,

v 2 = v 1 [h 2 := 0]

^,

v 3 = v 2 + 0.50

^,

v 4 = v 3 [h 1 := 0]

^,

v 5 = v 4 + 0.35

^and

v 2 = v 5 [h 2 := 0]

^{. In Figure 1 we give another}

representationsofthesevaluationsin Cartesianreferene.

0 1

0 1 h 1

h ²

v ⁰ v 4

v 3

v 2

v ¹ v 5

Figure1:RepresentationofvaluationsinCartesianreferene.

3.2 Constraints

Constraintsareonjuntions ofsimpleonstraints;andasimpleonstraintisaompar-

ison of alok with aninteger (diagonalfree simpleonstraint)ora omparisonof the

dierenebetweentwolokswithandinteger.Diagonalfreeonstraintsuseonlydiagonal

free simple onstraints.Constraintsare interpreted over valuations. The semantis of a

onstraintisthesetofvaluationssatisfyingit.Wewillalsoonsidertwotypesofatomi

onstraints :retangular onstraints andtriangularonstraints.

Denition 17 A simple onstraint dened on aset of loks

H

^{is an equation of the}

form

h − h ^′ ⊲⊳ n

^or

h ⊲⊳ n

^where

n ∈ N

^,

⊲⊳

^isoneof

{<, ≤, ≥, >}

^and

h, h ^′ ∈ H

^.

A diagonalfree simpleonstraint isasimpleonstraintoftheform

h ⊲⊳ n

^.

Denition 18 A lokonstraint overaset ofloks

H

^{isaonjuntionof simpleon-}

straints.

Φ H

^{,denotesthesetoflokonstraintsover}

H

^.Adiagonal-free lokonstraint isalokonstraintthatusesonlydiagonalfreesimpleonstraints.

Gds H

^{denotestheset}

ofdiagonal-freelokonstraintsover

H

^.

Wewill oftenwrite

h = n

^or

h − h ^′ = n

^asanabbreviationof

h ≤ n ∧ h ≥ n

^.Wealso

write

h − h ^′ = n

^{torepresenttheonstraint}

h − h ^′ ≤ n ∧ h − h ^′ ≥ n

^.

Laterweonsidertwospeial lokonstraints

tt

^{and denedby:}

tt = V

h∈H h ≥ 0

and

= V

h∈H h < 0

^.

The notion of a onstraint satised in a given valuation denoted

v g

^{is dened}

indutivelyasfollows:

• v h ⊲⊳ n

^ifandonlyif

v(h) ⊲⊳ n

• v h − h ^′ ⊲⊳ n

^ifandonlyif

v(h) − v(h ^′ ) ⊲⊳ n

• v g 1 ∧ g 2

^ifandonlyif

v g 1

^and

v g 2

The meaning of a onstraint

g

^{, denoted}

[[g]]

^{, is the set of valuations in whih it is}

satised.Clearly,

[[g]] = {v : v g}

^{.Itbeomesobviousthat}

[[tt]] = H → R ⁺

^and

[[

]] = ∅

^.

Denition 19 Aonstraint

g

^isinonsistentif

[[g]] = ∅

^.

Denition 20 The bound of aonstraint

g

^{, denoted by}

M g

^{, is the maximal onstant}

that appears in it. The bound of a set of onstraintsis the maximal value among the

bounds ofonstraintit ontains.A setof onstraintsis

M

^{-bounded ifeveryonstantin}

itissmallerthan

M

^.

Nowweonsideratomionstraintsandweshowhowtodeomposeaonstraintinto

anequivalentset ofatomionstraints.

Denition 21 Forainteger

M ∈ N

^,a

M

-retangular onstraint isaonjuntionofthe form

V

h∈H g h

^where

g h

^{isaonstraintoftheform}

c < h < c + 1

^or

h = c

^or

h > M

^with

c ∈ N ∩ [0..M [

^.

Thesetofall

M

-retangularonstraintsisdenotedby

Agds _H (M )

^.Thesymbol

Agds _H

will denotetheset

S

M ∈ N Agds _H (M )

Denition 22 A

M

-triangularonstraintisaonjuntionoftheform

V

h∈H g h ∧ V

(h,h ^′ )∈H ² g h,h ^′

where

g h,h ^′

^{isaonstraintofthe forms}

c < h − h ^′ < c + 1

^or

h − h ^′ = c

^or

h − h ^′ > M

and

g h

^isoftheform

c < h < c + 1

^or

h = c

^or

h > M

^with

c ∈ N ∩ [0..M[

^.

Thesymbol

T gds _H (M )

^{denotesthesetofallof}

M

-triangularonstraints.Thesymbol

T gds _H

^{denotestheset}

S

M ∈ N T gds _H (M )

^.

Notation:Weoftenusethesymbol

g ˆ

^{todenoteaonstraintin}

Agds _H (M )

^or

T gds _H (M )

forsome

M

^{.Laterthetermsatomionstraintswilloftenbeusedinplaeofretangular}

onstraintsortriangularonstraints.

Letus rstreallthefollowingfat resultingfrom denitionsofatomionstraints.

Fat23 (atomiity) Let

M ∈ N

^beaonstant.

• ∀ˆ g, g ˆ ^′ ∈ T gds _H (M )

^,if

[[ˆ g]] 6= [[ˆ g ^′ ]]

^then

[[ˆ g]] ∩ [[ˆ g ^′ ]] = ∅

• ∀ˆ g, g ˆ ^′ ∈ Agds _H (M )

^{, if}

[[ˆ g]] 6= [[ˆ g ^′ ]]

^then

[[ˆ g]] ∩ [[ˆ g ^′ ]] = ∅

• ∀(ˆ g, ˆ g ^′ ) ∈ Agds _H (M ) × T gds _H (M )

^{, either}

[[ˆ g ^′ ]] ∩ [[ˆ g]] = ∅

^or

[[ˆ g ^′ ]] ⊆ [[ˆ g]]

The rsttwo items statethat either the semantis of twoatomi onstraintsof the

samenatureareequal,ortheyaredisjoint.Thelastitemoftheabovefatstatesthatthe

semantisof atriangular onstraintis either inluded in thesemantisof aretangular

onstraints,orthetwosemantisaredisjoint.

Example: In Figure 2,we illustrate the onepts of onstraints and diagonal free on-

straints. The onstraints

g 1

^and

g 3

^{are general onstraints while the onstraint}

g 2

^is

diagonal free.Moreover

[[g 3 ]] = [[g 1 ]] ∧ [[g 2 ]]

^{.The onstraint}

g 2

^{isaretangular onstraint}

0 1 2

0 1 2 h _a

h b

g 1 = 0 ≤ h _a ≤ 3 ∧ 0 ≤ h _b ≤ 2 ∧ −1 ≤ h _a − h _b ≤ 1

g 2 = 1 < h a < 2 ∧ 0 < h _b < 1

g 3 = 1 < h _a < 2 ∧ 0 < h _b ≤ 1 ∧ −1 ≤ h _a − h _b ≤ 1

Figure2:Illustrationof onstraintsanddiagonalfreeonstraints.

in

Agds H (2)

^{andtheonstraint}

g 3

^{is atriangularonstraint.}

Normalization andRetangularisation Untiltheendofthissubsetionweon-

siderthedeompositionofdiagonalfreeonstraintintosetofretangularonstraints.We

willneedtoonsideronstraintsthatdonotinvolveonstantsgreaterthanaxedbound.

For that purpose, we present the normalisation operation

norm N

^{that we use later to}

deomposeonstraints.

Denition 24 The

N

-normalizationofasimpleonstraint

C

^{istheonstraint}

norm N (C)

dened by:

• norm N (h ⊲⊳ n) = tt

^if

⊲⊳∈ {<, ≤}

^and

n > N

^.

• norm N (h − h ^′ ⊲⊳ n) = tt

^if

⊲⊳∈ {<, ≤}

^and

n > N

^.

• norm N (h ⊲⊳ n) = h > N

^if

⊲⊳∈ {>, ≥}

^and

n > N

^.

• norm N (h − h ^′ ⊲⊳ n) = h − h ^′ > N

^if

⊲⊳∈ {>, ≥}

^and

n > N

^.

•

^{Intheotherases}

norm N

^{doesnotmodifytheonstraint.}

Givenaonstraint

g

^andaninteger

N

^,the

N

-normalizationof

g

^,

norm N (g)

^isobtained

bynormalizingeahsimpleonstraintourringin

g

^.

Lemma25 Let

C

^,adiagonal-freesimpleonstraint,thereisaonstant

M

^suhthat:

•

^forevery

N ≥ M

^,

[[norm M (C)]] = [[norm N (C)]] = [[C]]

•

^forevery

N < M

^,

[[norm M (C)]] ( [[norm N (C)]]

Proof

1. When

C

^hastheform

h ⊲⊳ n

^with

⊲⊳∈ {<, ≤}

^andonsider

M = n

^,

(a) Let

N ≥ M

^,

norm N (h ⊲⊳ n)

^isequalto

norm M (h ⊲⊳ n)

^{and theyareequalto}

h ⊲⊳ n

^{andwegettheresultthat}

[[norm M (C)]] = [[norm N (C)]] = [[C]]

^.

(b) Let

N < M

^,

norm N (h ⊲⊳ n) = h ≥ 0

^.Clearly

[[norm M (C)]] ( [[norm N (C)]]

^.

2. When

C

^hastheform

h ⊲⊳ n

^with

⊲⊳∈ {>, ≥}

^andonsider

M = n

^,

(a) Let

N ≥ M

^,

norm N (h ⊲⊳ n)

^isequalto

norm M (h ⊲⊳ n)

^{and theyareequalto}

h ⊲⊳ n

^{andwegettheresultthat}

[[norm M (C)]] = [[norm N (C)]] = [[C]]

^.

(b) Let

N < M

^{, then}

norm N (h ⊲⊳ n) = h ⊲⊳ N

^and

[[norm M (C)]] = h ⊲⊳ M

^.

Clearly,

[[norm M (C)]] ( [[norm N (C)]]

^.

Letusreallthatforaonstraint

g

^,

M g

^{denotesthemaximalonstantourringin}

g

^.

Weuse thelemmaabovetoshowthat the

M

-normalisationofaonstraintdoesmodify itssemantiswhen

M

^{isgreaterorequalto}

M g

^.

Proposition26 Let

g ∈ Gds H

^,

•

^forevery

M ≥ M g

^,

[[norm M (g)]] = [[norm N (g)]] = [[g]]

•

^forevery

M < M g

^,

[[norm M (g)]] ( [[norm N (g)]]

Proof

Bydenitions

g = V

i=1..n C i

^and,

[[norm M (g)]] = T

i=1..n [[N orm M (C i )]]

^.As

M g

^isgreater

that theonstant usedin every

C i

^{, we get, using 25 that for}

M ≥ M g

^,

[[norm M (g)]] = [[norm N (g)]] = [[g]]

^andfor

M < M g

^,

[[norm M (g)]] ( [[norm N (g)]]

Example:Consideringtheonstraint

g = 0 ≤ h a ≤ 3 ∧ 0 ≤ h b ≤ 2

^{,wepresentinTable2}

theresultsof

M

-normalisationoperationsdepending onthevalueof

M

^{.It iseasytosee}

M

norm M (g)

0

tt

1

tt

2

0 ≤ h b ≤ 2

3

0 ≤ h a ≤ 3 ∧ 0 ≤ h b ≤ 2

Table2:Illustrationofthenormalisationoperation.

that forevery

M < 2

^,

[[g]] ⊆ [[norm M (g)]]

^andforevery

M ≥ 2

^,

[[g]] = [[norm M (g)]]

To obtain the deomposition of diagonal onstraints,we rstly deompose diagonal

freeonstraintsintoaset(possiblyinnite)ofunboundedretangularonstraints.Then,

weusethenormalisation proedureaboveoneah atomionstraintinthat set to have

anite set of bounded retangular onstraints.The deomposition ofdiagonalfree on-

straints into a set of unbounded retangular onstraints is performed in two steps: in

Lemma27wedeomposesimplediagonalfreeonstraintsandweusethatdeomposition

in Proposition28todeomposediagonalfreeonstraints.

Lemma27 Foreverydiagonalfreesimpleonstraint

C

^,thereisaset

Rect(C)

^ofatomi

diagonalfreesimpleonstraintssuhthat

[[C]] = S

C ^′ ∈Rect(C) [[C ^′ ]]

^.

Proof

Let

C

^{beadiagonalfreeonstraint}

C

^{.Weonstrutaset}

Rect(C)

^{dependingontheform}

of

C

^{; andweshowthatfor every}

v ∈ V

^,

v C

^{ifand onlyifthereis}

C ^′ ∈ Rect(C)

^suh

that

v C ^′

^.

1. if

C

^isoftheform

h < n

^{then set}

Rect(C) = {i < h < i + 1, h = i | i = 0..n − 1}

2. if

C

^isoftheform

h ≤ n

^thenset

Rect(C) = {i < h < i+1, h = i | i = 0..n−1}∪{h = n}

3. if

C

^isoftheform

h > n

^{then set}

Rect(C) = {i < h < i + 1, h = i + 1 | i = n..∞}

4. if

C

^{is of the form}

h ≥ n

^{then set}

Rect(C) = {i < h < i + 1, h = i + 1 | i = n..∞} ∪ {h = n}

Theproofthat ineahase,

[[C]] = ∪ _C ′ ∈Rect(C) [[C ^′ ]]

^,isobvious.

Weobservethatsimpleonstraintsofthe form

h > n

^to

h ≥ n

^{aredeomposed into}

innitesetofonstraints.

Proposition28 Foreverydiagonal-freeonstraint

g

^,thereisaset

Rect(g)

^ofretangular

onstraintssuhthat

[[g]] = S

ˆ

g∈Rect(g) [[ˆ g]]

^.

Proof

The resultis aonsequeneofthe Lemma27 aboveasaonstraintsisa onjuntionof

simpleonstraints.

Wesaythat

Rect(g)

^{istheunboundedretangular}deomposition of

g

^.

Now that wehavedeomposed diagonal freeonstraintsinto sets (possibly innite)

ofunbounded retangularonstraints,wewillapply thenormalisationoperationoneah

retangular onstraint in these sets; the result of the appliation of the normalisation

operationwithrespettoaonstant

M

^{willbenitesetof}

M

-retangularonstraints.But weneedtoshowthatthesemantisoftheonstraintresultingfromtheappliationofthe

M

-normalisationoperationonasimplediagonalfreeonstraintisthesameastheunion ofthesemantisofretangularonstraintsinitsunbounded retangulardeomposition.

Lemma29 Foreverydiagonalfreesimpleonstraint

C

^oftheform

h ≤ n

^or

h ≥ n

^,for

every

M ∈ N

^,

[[norm M (C)]] = ∪ C ^′ ∈Rect(C) [[norm M (C ^′ )]]

^.

Proof

If

C

^isoftheform:

• h ≤ n

^,

If

M ≥ n

^then

norm M (C) = C

^andforevery

C ^′ ∈ Rect(C)

^,

norm M (C ^′ ) = C ^′

^.

Thenwegettheresult.

If

M < n

^then

norm M (C) = tt

^.Let

C ^′ h = n

^.FromLemma27

C ^′ ∈ Rect(C)

^and

norm M (C ^′ ) = tt

^then

∪ _C ′ ∈Rect(C) [[norm M (C ^′ )]] = tt

^and

[[norm M (C)]] =

∪ _C ′ ∈Rect(C) [[norm M (C ^′ )]]

^.

• h ≥ n

^,

Theasewhen

M ≥ n

^{isobviousbeauseeveryonstraintin}

Rect(C) ∪ {C}

^is

notmodiedby

norm M

^.

The ase when

M < n

^{is also obvious beause}

norm(C) = h > M

^and

norm M (C ^′ ) = h > M

^forevery

C ^′ ∈ Rect(C)

Nowweaneasilyextendresultsinthelemmaabovetodiagonalfreeonstraints.

Proposition30 Foreverydiagonal-freeonstraint

g

^{, forevery}

M ∈ N

^,

[[norm M (g)]] = S

ˆ

g∈Rect(g) [[norm M (ˆ g)]]

^.

Proof

It isaonsequeneofLemma 29aboveandProposition28

Denition 31 Givena

g ∈ Gds

^{,andandinteger}

M ∈ N

^wedenetheset

Rect M (g) = {norm M (ˆ g) | g ˆ ∈ Rect(g)}

.

FromProposition26,wegetthateverydiagonal-freeonstraintusingonstantsmaller

thananinteger

M

^{anbedeomposed intoanitesetof}

M

-retangularonstraints.

Proposition32 Foreveryonstraint

g ∈ Gds

^,forevery

M ≥ M g

^,

[[g]] = S

ˆ

g∈Rect M (g) [[ˆ g]]

^.

Proof

FromProposition30

[[norm M (g)]] = S

ˆ

g∈Rect(g) [[norm M (ˆ g)]]

^orequivalently

[[norm M (g)]] = S

ˆ

g∈Rect M (g) [[ˆ g]]

^{. From} Proposition 26 for

M ≥ M g

^,

[[g]] = [[norm M (g)]]

^{and weget the}

result.

Remark:Thesamekindof propertyanbeestablishedforgeneralonstraintsandtri-

angular onstraints. As retangular onstraintsontain triangularonstraints every

M

^-

bounded diagonal free atomi onstraint an be deomposed into a nite union of

M

^-

boundedtriangularonstraints.

Fromtheremarkabovewehavethefollowingorollary.

Corollary 33 Every onstraint or diagonal free onstraint an be deomposed into a

nite equivalentsetoftriangularonstraints.

3.3 Regions

We present a partitioning of the valuations into a nite number of equivalene lasses

alled regions. Valuations in the same region must satisfy the same lok onstraints,

theirtime suessorsmustalso satisfythesamelokonstraints,andtheymustsatisfy

thesamelokonstraintsafter alokisreset.

ThedenitionofaregionwepresentherehasbeenintroduedbyAlurandDill[AD94℄

for analysing timed automata using onlydiagonal -freeonstraints.The equivalene re-

lation between valuations is dened with respet to some integer

M

representing the maximal valueusedin onstraints.Thedenition ofthat relationis somehowrelatedto

the denition of atomi onstraintsas atomi onstraints an not be deomposed into

smaller onstraints. Thus, twoequivalent valuations agree on the integral part of eah

lokwhosevaluesaresmallerthan

M

^{andtheyalsoagreeontheorderonthefrational}

partofthevaluesoftheloks.

Forarealnumber

n

^let

⌊n⌋

^{denotetheintegralpartof}

n

^and

{n}

^{denotethefrational}

partof

n

^.

Let

M

^{beanaturalnumber.Considerthe}parametrisedbinaryrelation

∼ ^M ⊆ V H × V H

overvaluationsdened by,

v ∼ ^M v ^′

^if:

1.

v(h) > M

^ifandonlyif

v ^′ (h) > M

^foreah

h ∈ H

^;

2. if

v(h) ≤ M

^,then

⌊v(h)⌋ = ⌊v ^′ (h)⌋

^forevery

h ∈ H

^;

3. if

v(h) ≤ M

^,then

{v(h)} = 0

^ifandonlyif

{v ^′ (h)} = 0

^forevery

h ∈ H

^,and;

4. if

v(h) ≤ M

^and

v(h ^′ ) ≤ M

^,then

{v(h)} ≤ {v(h ^′ )}

^ifandonlyif

{v ^′ (h)} ≤ {v ^′ (h ^′ )}

forevery

h, h ^′ ∈ H

^.

Proposition34 ([AD94℄) Therelation

∼ ^M

^{is an equivalenerelation over the set of}

valuationswithat most

2 ^3|H|−1 × |H|! × (M + 1) ^|H|

^{equivalenelasses.}

Therelation

∼ ^M

^{isdened asaonjuntionoffour} properties.Eahpropertydenesan equivalene relation;letus denote them by

∼ ^M ₁ , . . . , ∼ ^M ₄

^, respetively. Foreahof these four relationswewillgiveanupperbound onthenumberofits equivalenelasses.The

produtoftheseboundswill giveanupperbound on

∼ ^M

^{asthelateristhe}intersetion ofthefourequivalenerelations.

The relation dened by the rst ondition has

2 ^|H|

^{equivalene lasses, as the only}

thing that ountsis whetherthe valueofalok isbiggerthan

M

^{ornot.Similarlythe}

third relationhas

2 ^|H|

^{equivalenelasses. Thenumberof lassesof theseond relation}

is

(M + 1) ^|H|

^{as thereare}

M + 1

^{possibleintegervaluesofinterest.Finally, thenumber}

of lassesof thefourth relationis bounded by thenumberofpermutationsof theset of

loksmultipliedby

2 ^|H|−1

^{asforeverytwoloksonseutiveina}permutationweneed to deideiftheyareequaloriftheseondisstritlybiggerthantherst.

Summarizing,weget

2 ^3|H|−1 |H!|(M + 1) ^|H|

^.

Weuse

Reg(M )

^(or

Reg

^{forshort) to representtheset of equivalene lassesofthe}

relation

∼ ^M

^.

Denition 35 A region [AD94℄ is an equivalene lass of the relation

∼ ^M ⊆ V H × V H

dened above.

InFigure3weillustrateregionfordiagonalfreeonstraintsforthemaximalonstant

M = 2

^{.InFigure3valuationsearlierpresentedinTable1arenot}equivalent.Aregionin thegureiseitheraornerpoint(forexample

(0, 2)

^{),anopenlinesegment(forexample}

0 < h 1 = h 2 < 1

^{)or anopenbox(forexample}

0 < h 1 < h 2 < 1

^).

0 1 2

0 1 2 h 1

h 2

v 0

v 4

v 3

v 2

v 1

v 5

Figure3:Regionillustration.

Fromthedenitionof

∼ ^M

^{,itomesthatanequivalenelassanbe}representedusing atriangularonstraintin

g

^{.Aordingtothedenitionof}

∼ ^M

^{,twovaluationsthatbelong}

to thesameequivalenelasssatisfyonstraintoftheform:

• h = i h

^or

i h < h < i h + 1

^foreah

h ∈ H

^where

i h ∈ {0, 1, . . . , M }

^andweassume

M + 1 = ∞

^{.Thisisaonsequeneof}

∼ ^M ₁

^,

∼ ^M ₂

^,

∼ ^M ₃

^.

• h − h ^′ = i hh ^′

^or

i hh ^′ < h − h ^′ < i hh ^′ + 1

^{for eah ouple}

(h, h ^′ ) ∈ H ²

^{suh that}

h ⊲⊳ M

^and

h ^′ ⊲⊳ M

^with

⊲⊳∈ {=, <}

^{.Thisisaonsequeneof}

∼ ^M ₄

^.

Given avaluation

v

^,

[v]

^{denotesthe equivalene lass (region)of}

v

^{. We also usethe}

letter

r

^{to represent a region. Given a region}

r

^{, we dene}

r + t = {[v + t] | v ∈ r}

^,

r↑ = {r + t | t ∈ R ≥0 }

^,and

r[h := 0] = {v[h := 0] : v ∈ r}

^{. Wewrite}

r ⊆ g

^for

r ⊆ [[g]]

^.

Proposition36 Let

G

^beasetof

M

^{-boundedonstraintsthen}

Reg(M )

^satises:

P1

∀g ∈ G, r ∈ Reg

^,either

r ⊆ [[g]]

^or

[[g]] ∩ r = ∅

^.

P2

∀r, r ^′ ∈ Reg

^{, ifthere exists some}

v ∈ r

^and

t ∈ R ≥0

^suhthat

v + t ∈ r ^′

^{, thenfor}

every

v ^′ ∈ r

^{thereis some}

t ^′ ∈ R ≥0

^suhthat

v ^′ + t ^′ ∈ r ^′

^.

P3

∀r, r ^′ ∈ Reg, ∀h ∈ H

^,if

r[h := 0] ∩ r ^′ 6= ∅

^,then

r[h := 0] ⊆ r ^′

^.

Proof

WeshowP1intherstitem,P2in theseonditemandP3inthelastitem.

1. Let

g ∈ G

^,fromProposition 32let

[[g]] = S

g _i ∈Rect _M (g) [[ˆ g i ]]

^{. Eah}

ˆ g i

^{isaretangular}

onstraint.

[[g]] ∩ r = S

g i ∈Rect M (g) [[ˆ g i ]] ∩ r)

^{.FromFat23thereisatmostone}

i

^suh

that

r

^intersets

ˆ g i

^{. It followsthat}

r

^{intersets a onstraint}

ˆ g i

^of

Rect M (g)

^{if and}

onlyif

ˆ g i

^ontains

r

^{.Wehavethatif}

v r

^then

v g

^.

2. Let

v, v ^′ ∈ r

^,adding

t

^to

v

^{maymodifytheintegerpartofthevalue(withrespetto}

v

^{)ofsomeloksormaymodifytheorderonthefrationalpartofthevalue(with}

respetto

v

^{)ofloks.Weaimatndatime}

t ^′

^suhthat:

- Theintegerpartof thevalueof eah lokwithrespetto

v ^′ + t ^′

^{is equaltothe}

integerpartofthevalueofeahlokwithrespetto

v + t

-Theorderofthefrationalpartsofloksin

v ^′ + t ^′

^isthesamein

v + t

^.

-Thesetoflokswithzerofrationalpartin

v + t

^isthesamein

v ^′ + t ^′

^.

Let

|H| = n

^andassumeapermutation

π

^of

{1, . . . , n}

^suhthat

{v(h π 1 )} ⊲⊳ 1 {v(h π 2 )} ⊲⊳ 2 , . . . , ⊲⊳ n−1 {v(h π n )}(∗)

with

⊲⊳ i ∈ {<, =}

^.

Let

t ∈ R ≥0

^{. Itis learthat}

{v(h) + t} = {v(h) + {t}}

^{. Onlythefrationalpartof}

t

^{mayaettheorderin}

(∗)

^.

There maybealargestindex

j

^suhthat

{v(h π _j ) + {t}} = {v(h π _j )} + {t}

^.Inase,nosuh

j

^{exists, take}

j = n

^.

Clearly,

{v(h π j ) + {t}} ≥ {v(h π j )}

^and;

∀k > j

^wehave:

{v(h π _k ) + {t}} < {v(h π _k )}

^and

{v(h π _k ) + {t}} < {v(h π j ) + {t}}

^.

Wegetthat:

{v(h π j+1 ) + {t}} ⊲⊳ j . . . ⊲⊳ n−1 {v(h π n ) + {t}} < {v(h π j ) + {t}}

Similarly,weestablishthat

{v(h π j ) + {t}} < {v(h π j− 1 ) + {t}}⊲⊳ j−2 . . . ⊲⊳ 1 {v(h π 1 ) + {t}}

where

⊲⊳ k =>

^if

⊲⊳ j ∈ {<}

^otherwise

⊲⊳ j ∈ {=}

^,

∀k ≤ j