Toward a Compositional Theory of Leftist Grammars and Transformations

0.2 MB·English

by  Pierre Chambart

Toward a Compositional Theory of Leftist Grammars
and Transformations⋆
P.ChambartandPh.Schnoebelen
LSV,ENSCachan,CNRS
0 61,av.Pdt.Wilson,F-94230Cachan,France
1
0
2
Abstract. Leftistgrammars[Motwanietal.,STOC2000]arespecialsemi-Thue
n
a systemswheresymbolscanonlyinsertorerasetotheirleft.Wedevelopatheory
J ofleftistgrammarsseenaswordtransformersasatooltowardrigorousanalyses
7 of their computational power. Our maincontributions inthisfirstpaper are(1)
2 constructionsprovingthatleftisttransformationsareclosedundercompositions
andtransitiveclosures,and(2)aproofthatboundedreachabilityisNP-complete

] evenforleftistgrammarswithacyclicrules.
L
F
.
s 1 Introduction
c
[
Leftist grammarswere introducedby Motwaniet al. to study accessibility and safety

1
inprotectionsystems[MPSV00].Inthisframework,leftistgrammarsareusedtoshow
v
thatrestrictedaccessibilitygrammarshavedecidableaccessibilityproblems(unlikethe
7
4 moregeneralaccess-matrixmodel).
0 Leftistgrammarsarebothsurprisinglysimpleandsurprisinglycomplex.Simplicity
5 comes from the fact that they only allow rules of the form “a→ba” and “cd →d”
.
1 where a symbol inserts, resp. erases, another symbol to its left while remaining un-
0 changed. But the combination of insertion and deletion rules makes leftist grammars
0 go beyondcontext-sensitive grammars, and the decidability result comes with a high
1
complexity-theoreticalprice [Jur08]. Mostof all, what is surprisingis thatapparently
:
v leftist grammars had not been identified as a relevant computational formalism until
Xi 2000.
Theknownfactsonleftistgrammarsandtheircomputationalandexpressivepower
r
a are rather scarce. Motwani et al. show that it is decidable whether a given word can
be derived (accessibility) and whether all derivable words belong to a given regular
language (safety) [MPSV00]. Jurdzin´ski and Lorys´ showed that leftist grammars can
define languages that are not context-free [JL07] while leftist grammars restricted to
acyclicrulesarelessexpressivesincetheycanonlyrecognizeregularlanguages.Then
Jurdzin´skishowedaPSPACElowerboundforaccessibilityinleftistgrammars[Jur07],
beforeimprovingthistoanonprimitive-recursivelowerbound[Jur08].
Jurdzin´ski’sresultsrelyonencodingclassicalcomputationalstructures(linear-boun-
dedautomata[Jur07]andAckermann’sfunction[Jur08])inleftistgrammars.Devising
such encodings is difficult because leftist grammars are very hard to control. Thus,
⋆WorksupportedbytheAgenceNationaledelaRecherche,grantANR-06-SETIN-001.
2 P.ChambartandPh.Schnoebelen
forcomputingAckermann’sfunction,devisingtheencodingisactuallynotthehardest
part:the hardertask is to provethatthe constructedleftist grammarcannotbehavein
unexpectedways.Inthisregard,thepublishedproofsarenecessarilyincomplete,hard
to follow, and hard to fully acknowledge. The final results and intermediary lemmas
cannoteasilybeadaptedorreused.
Our Contribution. We developa compositionaltheoryof leftist grammarsand leftist
transformations(i.e.,operationsonstringsthatarecomputedbyleftistgrammars)that
provides fundamental tools for the analysis of their computational power. Our main
contributionsareeffectiveconstructionsforthecompositionandthetransitiveclosure
of leftist transformations.Thecorrectnessproofsfor these constructionsare based on
newdefinitions(e.g.,forgreedyderivations)andassociatedlemmas.
AfirstapplicationofthecompositionaltheoryisgiveninSection6whereweprove
theNP-completenessofboundedreachabilityquestions,evenwhenrestrictedtoacyclic
leftistgrammars.
A secondapplication,andthe main reasonforthispaper,isour forthcomingcon-
structionprovingthatleftistgrammarscansimulate lossychannelsystemsand“com-
pute”allmultiply-recursivetransformationsandnothingmore(basedon[CS08b]),thus
providingaprecisemeasureoftheircomputationalpower.Finally,afterourintroduction
ofPost’sEmbeddingProblem[CS07,CS08a],leftistgrammarsareanotherbasiccom-
putationalmodelthatwillhavebeenshownto captureexactlythe notionofmultiply-
recursivecomputation.
As furthercomparisonwith earlier work,we observethat, of course,the complex
constructionsin[Jur07,Jur08]arebuiltmodularly.However,themodularityisnotmade
fullyexplicitintheseworks,theinterfacingassumptionsareincompletelystated,orare
mixedwith thedetailsofthe constructions,andcorrectnessproofscannotbegivenin
full.
OutlineofthePaper. BasicnotationsanddefinitionsarerecalledinSection2.Section3
definesleftistgrammarsandprovesageneralizedversionofthecompletenessofgreedy
derivations.Sections4introducesleftisttransformersandtheirsequentialcompositions.
Section 5 specializes on the “simple” transformers that we use in Section 6 for our
encodingof3SAT.FinallySection7showsthatso-called“anchored”transformersare
closedunderthetransitiveclosureoperation,thisinaneffectiveway.
2 BasicDefinitions andNotations
Words. Weusex,y,u,v,w,a ,b ,...todenotewords,i.e.,finitestringsofsymbolstaken
fromsomealphabet.Concatenationisdenotedmultiplicativelywithe (theemptyword)
asneutralelement,andthelengthofxisdenoted|x|.
Thecongruenceonwordsgeneratedbytheequivalencesa≈aa(forallsymbolsa
inthealphabet)iscalledthestutteringequivalenceandisalsodenoted≈:everywordx
hasaminimalandcanonicalstuttering-equivalentx′obtainedbyrepeatedlyeliminating
symbolsinxthatareadjacenttoacopyofthemselves.
Wesaythatxisasubwordofy,denotedx⊑y,ifxcanbeobtainedbydeletingsome
symbols (an arbitrarynumber,at arbitrary positions) from y. We further write x⊑S y
TowardaCompositionalTheoryofLeftistGrammarsandTransformations 3
whenallthesymbolsdeletedfromybelongtoS (NB:we donotrequirey∈S ∗),and
let⊒denotetheinverserelation⊑−1.
RelationsandRelationAlgebra. WeseearelationRbetweentwosetsX andY asaset
ofpairs,i.e.,someR⊆X×Y.WewritexRyratherthan(x,y)∈R.TworelationsRand
R′ can be composed,denotedmultiplicativelywith R.R′, and definedby x(R.R′)y⇔def
∃z. xRz ∧ zR′y .
(cid:0)TheunionR+(cid:1)R′,alsodenotedR∪R′,isjusttheset-theoreticunion.Rn isthen-th
powerR.R...RofRandR−1istheinverseofR:xR−1y⇔defyRx.Thetransitiveclosure
S RnofRassumesY =X andisdenotedR+,whileitsreflexive-transitiveclosure
n=1,2,...
isR+∪Id ,denotedR∗.
X
Below we often use notations from relation algebra to state simple equivalences.
E.g.,wewrite“R=R′”and“R⊆S”ratherthan“xRyiffxR′y”and“xRyimpliesxSy”.
Ourproofsoftenrelyonwell-knownbasiclawsfromrelationalgebra,like(R.R′)−1=
R′−1.R−1,or(R+R′).R′′=R.R′′+R′.R′′,withoutexplicitlystatingthem.
3 Leftist Grammars
A leftist grammar (an LGr) is a triple G=(S ,P,g) where S ∪{g}={a,b,...} is a
finitealphabet,g6∈S isafinalsymbol(alsocalled“axiom”),andP={r,...}isasetof
productionrulesthatmaybeinsertionrulesoftheforma→ba,anddeletionrulesof
theformcd→d.Forsimplicity,weforbidrulesthatinsertordeletetheaxiomg(this
isnolossofgenerality[JL07,Prop.3]).
Leftist grammars are not context-free (deletions are contextual), or even context-
sensitive (deletions are not length-preserving). For our purposes, we consider them
as string rewritesystems, morepreciselysemi-Thuesystems. Writing S for S ∪{g},
g
the rules of P define a 1-step rewrite relation in the standard way: for u,u′ ∈S ∗, we
g
write u⇒r,p u′ wheneverr is some rule a →b , u is some u a u with |u a |= p and
1 2 1
u′ =u b u . We often write shortly u⇒r u′, or evenu⇒u′, when the positionor the
1 2
rule involvedin the step can be left implicit. On the other hand, we sometimes use a
subscript,e.g.,writingu⇒ v,whentheunderlyinggrammarhastobemadeexplicit.
G
A derivation is a sequence p of consecutive rewrite steps, i.e., is some u0 ⇒r1,p1
u1⇒r2,p2 u2···⇒rn,pn un,oftenabbreviatedasu0⇒n un, orevenu0⇒∗ un. A subse-
quence(ui−1⇒ri,pi ui)i=m,m+1,...,l of p is a subderivation.As with allsemi-Thuesys-
tems,steps(andderivations)areclosedunderadjunction:ifu⇒u′thenvuw⇒vu′w.
Two derivations p =(u ⇒∗ u′) and p =(v ⇒∗ v′) can be concatenated in the
1 2
obviousway (denoted p .p ) if u′ =v. They are equivalent,denoted p ≡p , if they
1 2 1 2
havesameextremities,i.e.,ifu=vandu′=v′.
Wesaythatu∈S ∗isacceptedbyGifthereisaderivationoftheformug⇒∗gand
wewriteL(G)forthesetofacceptedwords,i.e.,thelanguagerecognizedbyG.
WesaythatI⊆S ∗isaninvariantforanLGrG=(S ,P,g)ifu∈Iandug⇒vgentail
v∈I.KnowingthatIisaninvariantforGisusedintwosymmetricways:(1)fromu∈I
andug⇒∗vgonededucesv∈I,and(2)fromug⇒∗vgandv6∈Ionededucesu6∈I.
4 P.ChambartandPh.Schnoebelen
3.1 GraphsandTypesforLeftistGrammars
When dealing with LGr’s, it is convenient to write insertion rules under the simpler
form“a b”,anddeletionrulesas“d c”,emphasizingthefactthata(resp.d)isnot
modifiedduringtheinsertionofb(resp.thedeletionofc)onitsleft.Fora∈S ,welet
g
def def
ins(a)={b|P∋(a b)}anddel(a)={b|P∋(a b)}denotethesetofsymbols
thatcanbeinserted(respectively,deleted)bya.We writeins+(a)forthesmallestset
that containsb and ins+(b) for all b∈ins(a), while del+(b) is defined similarly. We
saythataisinactiveinaLGrifdel(a)∪ins(a)=∅.
ItisoftenconvenienttoviewLGr’sinagraph-theoreticalway.Formally,thegraph
ofG=(S ,P,g)isthedirectedgrapht havingthesymbolsfromS asverticesandthe
G g
rulesfromPasedges(comingintwokinds,insertionsanddeletions).Furthermore,we
oftendecoratesuchgraphswithextrabookkeepingannotations.
We say that G “has type t ” when t is a sub-graph of t . Thus a “type” is just a
G
restriction on what are the allowed symbols and rules between them. Types are often
givenschematically,groupingsymbolsthatplayasimilarroleintoasinglevertex.For
insertion:
S g
deletion:
Fig.1.Universaltype(schematically).
example,Fig.1displaysschematicallythetype(parametrizedbythealphabet)observed
byallLGr’s.
3.2 Leftmost,PureandEagerDerivations
We speak informallyof a “letter”, say a, when we really mean “an occurrenceof the
symbola”(insomeword).Furthermore,wefollowlettersalongstepsu⇒v,identifying
thelettersinuandthecorrespondinglettersinv.Hencea“letter”isalsoasequenceof
occurrencesinconsecutivewordsalongaderivation.
Aletteraisan-thdescendantofanotherletterb(inthecontextofaderivation)ifa
hasbeeninsertedbyb(whenn=1),orbya(n−1)-thdescendantofb.
Given a step u⇒r,p v, we say that the p-th letter in u, written u[p], is the active
letter: the one that inserts, or deletes, a letter to its left. This is often emphasized by
writingthestepundertheform(u=)u au ⇒u′au (=v)(assumingu[p]=a).
1 2 1 2
Aletterisinertinaderivationifitisnotactiveinanystepofthederivation.Aset
of letters is inert if it only contains inertletters. A derivationis leftmost if every step
u au ⇒u′au inthederivationissuchthatu isinertintherestofthederivation.
1 2 1 2 1
Aletterisusefulinaderivationp =(u⇒∗v)ifitbelongstouorv,orifitinserts
or deletes a useful letter along p . This recursive definition is well-founded:since let-
tersonlyinsertordeletetotheirleft,the“inserts-or-deletes”relationbetweenlettersis
acyclic.Aderivationp ispureifalllettersinp areuseful.Observethatifp isnotpure,
TowardaCompositionalTheoryofLeftistGrammarsandTransformations 5
itnecessarilyinsertsatsomestepsomelettera(calledauselessletter)thatstaysinert
andwilleventuallybedeleted.
Aderivationiseager if,informally,deletionsoccurassoonaspossible.Formally,
p =(u0⇒r1,p1 u1···⇒rn,pn un)isnoteagerifthereissomeui−1 oftheformw1baw2
where b is inert in the rest of p and is eventually deleted, where P contains the rule
a b,andwherer isnotadeletionrule.1
i
A derivation is greedy if it is leftmost, pure and eager. Our definition general-
izes[Jur07,Def.4],mostnotablybecauseitalsoappliestoderivationsug⇒∗vgwith
nonemptyv.Henceasubderivationp ′ ofp isleftmost,eager,pure,orgreedy,whenp
is.
Thefollowingpropositiongeneralizes[Jur07,Lemma7].
Proposition3.1 (Greedyderivationsaresufficient).Everyderivationp hasanequiv-
alentgreedyderivationp ′.
Proof. Withaderivationp oftheformu0⇒r1,p1u1⇒r2,p2u2···⇒rn,pnun,weassociate
its measure µ(p )d=efhn,p ,...,p i, a (n+1)-tuple of numbers. Measures are linearly
1 n
ordered with the lexicographic ordering, giving rise to a quasi-ordering,denoted ≤ ,
µ
betweenderivations.A derivationiscalled µ-minimalif anyequivalentderivationhas
greaterorequalmeasure.
We can now prove Prop. 3.1 along the following lines (see Appendix A for full
details):firstprovethateveryderivationhasaµ-minimalequivalent(LemmaA.1),then
showthatµ-minimalderivationsaregreedy(LemmaA.2). ⊓⊔
Observethat≤ iscompatiblewith concatenationofderivations:if p ≤ p then
µ 1 µ 2
p .p .p ′≤ p .p .p ′ whentheseconcatenationsaredefined.Thusanysubderivationofa
1 µ 2
µ-minimalderivationisµ-minimal,hencealsogreedy.
µ-minimalityisstrongerthangreediness,andisapowerfulandconvenienttoolfor
provingProp. 3.1.However,greedinessis easier to reason with since it only involves
localpropertiesofderivations,whileµ-minimalityis“global”.Theseintuitionsarere-
flectedby,andexplain,thefollowingcomplexityresults.
Theorem3.2. 1.Greediness(decidingwhetheragivenderivationp inthecontextofa
givenLGrGisgreedy)isinL.
2.µ-Minimality(decidingwhetheritisµ-minimal)iscoNP-complete,evenifwerestrict
toacyclicLGr’s.
Proof. 1.Beingleftmostoreageriseasilycheckedinlogspace(i.e.,isinL).Checking
non-puritycanbedonebylookingforalastinserteduselessletter,henceisinLtoo.
2. µ-minimality is obviously in coNP. Hardness is proved as Coro. 6.9 below, as a
byproductofthereductionweusefortheNP-hardnessofBoundedReachability. ⊓⊔
1Eagernessdoes not requirethatr deletesb: other deletionsareallowed,only insertionsare
i
forbidden.
6 P.ChambartandPh.Schnoebelen
4 Leftist Grammarsas Transformers
Someleftistgrammarsareusedascomputingdevicesratherthanrecognizersofwords.
Forthispurpose,werequireastrictseparationbetweeninputandoutputsymbolsand
speakofleftisttransformers,orshortlyLTr’s.
4.1 LeftistTransformers
Formally,anLTrisaLGrG=(S ,P,g)whereS ispartitionedasA⊎B⊎C,andwhere
symbolsfromAareinactiveinPandarenotinsertedbyP(seeFig.2).Thisisdenoted
G :A ⊢C. Here A contains the input symbols, B the temporary symbols, and C the
outputsymbols,andG ismoreconvenientlywrittenasG=(A,B,C,P,g).Whenthere
isnoneedtodistinguishbetweentemporaryandoutputsymbols,wewriteGunderthe
def
formG=(A,D,P,g),whereD=B∪Ccontainsthe“working”symbols,
A D g
Fig.2.Typeofleftisttransformers.
AconsequenceoftherestrictionsimposedonLTr’sisthefollowing:
Fact4.1 A∗D∗ isaninvariantinanyLTrG=(A,D,P,g).
WithG=(A,B,C,P,g), weassociate a transformation(arelationbetweenwords)
R ⊆A∗×C∗definedby
G
uR v ⇔def ug⇒∗ vg∧u∈A∗ ∧v∈C∗
G G
andwesaythatGrealizesR .Finally,aleftisttransformationisanyrelationonwords
G
realizedbysomeLTr.Bynecessity,aleftisttransformationcanonlyrelatewordswritten
usingdisjointalphabets(thisisnotcontradictedbye R e ).
G
Leftist transformations respect some structural constraints. In this paper we shall
usethefollowingproperties:
Proposition4.2 (Closure for leftist transformations,see App. B). If G:A⊢C is a
leftisttransformer,thenR = (⊒ .≈.R .≈).
G A G
4.2 Composition
We say thattwoleftisttransformationsR ⊆A∗×C∗ andR ⊆A∗×C∗ arechainable
1 1 1 2 2 2
ifC =A andA ∩C =∅.TwoLTr’sarechainableiftheyrealizechainabletransfor-
1 2 1 2
mations.
Theorem4.3. ThecompositionR .R oftwochainableleftisttransformationsisaleft-
1 2
ist transformation.Furthermore,onecan buildeffectivelya linear-sized LTr realizing
R .R fromLTr’srealizingR andR .
1 2 1 2
TowardaCompositionalTheoryofLeftistGrammarsandTransformations 7
Foraproof,assumeG =(A ,B ,C ,P ,g)andG =(A ,B ,C ,P ,g)realizeR and
1 1 1 1 1 2 2 2 2 2 1
R .Beyondchainability,weassumethatA ∪B andB ∪C aredisjoint,whichcanbe
2 1 1 2 2
ensuredbyrenamingtheintermediarysymbolsinB andB .ThecomposedLTrG .G
1 2 1 2
isgivenby
def
G .G =(A ,B ∪C ∪B ,C ,P ∪P ,g).
1 2 1 1 1 2 2 1 2
ThisisindeedaLTrfromA toC .SeeFig.3foraschematicsofitstype.SinceG .G
1 2 1 2
g
P P
P 1 2
1
P
2
A1 D1(⊇A2) D2
P P
1 2
P P
1 2
Fig.3.ThetypeofG .G .
1 2
hasallrulesfromG andG itisclearthat(⇒ +⇒ )⊆⇒ ,fromwhichwededuce
1 2 G1 G2 G
RG1.RG2 ⊆RG1.G2.Furthermore,theinclusionintheotherdirectionalsoholds:
Lemma4.4 (CompositionLemma,seeAppendixC).RG1.G2 =RG1.RG2.
Remark4.5 (Associativity).Thecomposition(G .G ).G iswell-definedifandonlyif
1 2 3
G .(G .G )is.Furthermore,thetwoexpressionsdenoteexactlythesameresult. ⊓⊔
1 2 3
5 Simple LeftistTransformations
AsatoolforSections6and7,wenowintroduceandstudyrestrictedfamiliesofleftist
grammars(andtransformers)wheredeletionrulesareforbidden(resp.,onlyallowedon
A).
AninsertiongrammarisaLGrG=(S ,P,g)wherePonlycontaininsertionrules.
SeeFig.4foragraphicdefinition.ForanarbitraryleftistgrammarG,wedenotewith
Gins theinsertiongrammarobtainedfromGbykeepingonlytheinsertionrules.
The insertion relation I ⊆ S ∗×S ∗ associated with an insertion grammar G =
G
(S ,P,g)isdefinedbyuIGv⇔defug⇒∗Gvg.Obviously,IG⊆⊑S .ObservethatIG isnot
necessarilyaleftisttransformationsinceitdoesnotrequireanyseparationbetweenin-
putandoutputsymbols.
AsimpleleftisttransformerisanLTrG=(A,B,C,P,g)whereB=∅andwhereno
ruleinPerasessymbolsfromC.SeeFig.4foragraphicdefinition.We give,without
proof,animmediateconsequenceofthedefinition:
Lemma5.1. Let G=(A,∅,C,P,g) be a simple LTr and assume ug⇒k vg for some
G
u∈A∗andv∈C∗.Thenk=|u|+|v|.
8 P.ChambartandPh.Schnoebelen
S g A C g
Fig.4.Typesofinsertiongrammars(left)andsimpleleftisttransformers(right).
Given a simple LTr G=(A,∅,C,P,g) and two words u=a ···a ∈A∗ and v=
1 n
c ···c ∈C∗, we say that a non-decreasing map h:{1,...,n}→{1,...,m} is a G-
1 m
witnessforuandvifPcontainstherulesc a andc c (foralli=1,...,n
h(i) i j+1 j
and j=1,...,m, with the convention that c =g). Finally, we write u(cid:209) v when
m+1 G
suchaG-witnessexists.Clearly,(cid:209) ⊆R .Indeed,whenGisasimpletransformer,(cid:209)
G G G
canbeusedasarestrictedversionofR thatiseasiertocontrolandreasonabout.
G
Lemma5.2 (SeeApp.D).LetG=(A,∅,C,P,g)beasimpleLTr.ThenR =(cid:209) .I .
G G Gins
Combining Lemma 5.2 with IdC∗ ⊆IGins ⊆⊑C, we obtain the following weaker but
simplerstatement.
Corollary5.3. LetG=(A,∅,C,P,g)beasimpleLTr.Then(cid:209) ⊆R ⊆(cid:209) .⊑ .
G G G C
5.1 UnionofSimpleLeftistTransformers
WenowconsiderthecombinationoftwosimpleLTr’sG =(A,∅,C ,P ,g)andG =
1 1 1 2
(A,∅,C ,P ,g) that transform from a same A to disjoint output alphabets, i.e., with
2 2
C ∩C =∅.WedefinetheirunionwithG +G d=ef (A,∅,C ∪C ,P ∪P ,g).Thisis
1 2 1 2 1 2 1 2
clearlyasimpleLTrwith(R +R )⊆R .Itfurthersatisfies:
G1 G2 G1+G2
Lemma5.4. IfuR vthenu(R +R )v′forsomev′⊑v.
G1+G2 G1 G2
Proof. AssumeuR v.WithCor.5.3,weobtainu(cid:209) v′forsomev′=c ···c ⊑
G1+G2 G1+G2 1 m
v.HenceG +G hasinsertionrulesc c forall j=1,...,m,anddeletionrules
1 2 j+1 j
oftheformc u[i].SinceC andC aredisjoint,eitheralltheserulesareinG (and
h(i) 1 2 1
u(cid:209) v′),ortheyareallinG (andu(cid:209) v′).Henceu(R +R )v′. ⊓⊔
G1 2 G2 G1 G2
6 Encoding 3SATwithAcyclicLeftist Transformers
Thissectionprovesthefollowingresult.
Theorem6.1. BoundedReachabilityandExactBoundedReachabilityinleftistgram-
marsareNP-complete,evenwhenrestrictingtoacyclicgrammars.
(Exact)BoundedReachabilityisthequestionwhetherthereexistsan-stepderiva-
tionu⇒nv(respectively,aderivationu⇒≤nvofnon-exactlengthatmostn)between
givenuandv.Thesequestionsareamongthesimplestreachabilityquestionsand,since
we consider that the input n is given in unary,2 they are obviously in NP for leftist
grammars(andallsemi-Thuesystems).
2Itisnaturaltobeginwiththisassumptionwhenconsideringfundamentalaspectsofreachabil-
itysincewritingnmoresuccinctlywouldblurthecomplexity-theoreticalpicture.
TowardaCompositionalTheoryofLeftistGrammarsandTransformations 9
Consequently,ourcontributioninthispaperistheNP-hardnesspart.Thisisproved
byencoding3SATinstancesinleftistgrammarswherereachingagivenfinalvamounts
toguessingavaluationthatsatisfiestheformula.Whiletheideaofthereductioniseasy
tograsp,thetechnicalitiesinvolvedareheavyanditwouldbedifficulttoreallyprove
thecorrectnessofthereductionwithoutrelyingonacompositionalframeworklikethe
onewedevelopinthispaper.Itisindeedverytemptingto“prove”itbyjustrunningan
example.
Ratherthanadoptingthiseasyway,weshalldescribethereductionasacomposition
of simple leftist transformers and use our composition theorems to break down the
correctnessproofinsmaller,manageableparts.Oncetheideasunderlyingthereduction
are grasped, a good deal of the reasoning is of the type-checkingkind: verifyingthat
theconditionsrequiredforcomposingtransformersaremet.
Throughoutthis section we assume a generic 3SAT instance F =Vmi=1Ci with m
3-clauses on n Boolean variables in X ={x ,...,x }. Each clause has the formC =
1 n i
W3k=1e i,kxi,k for some polarity e i,k ∈{+,−} and xi,k ∈X. (There are two additional
assumptions on F that we postpone until the proof of Coro. 6.5 for clarity.) We use
standardmodel-theoreticalnotationlike|=F (validity),ors |=F (entailment)whens
isaBooleanformulaoraBooleanvaluationofsomevariables.
We write s [x 7→b] for the extension of a valuation s with (x,b), assuming x 6∈
Dom(s ).Finally,foravaluationq :X →{⊤,⊥}andsome j=0,...,n,wewriteq to
j
denotetherestrictionq ofq onthefirst jvariables.
|{x1,...,xj}
6.1 AssociatinganLTrGF withF
Fortheencoding,weuseanalphabetS ={Tj,Uj,T′j,U′j|i=1,...,m∧ j=0,...,n},
i i i i
i.e., 4(n+1) symbols for each clause. The choice of the symbols is that a U means
“Undetermined”andaT means“True”,ordeterminedtobevalid.
For j=0,...,n,letV d=ef{Uj,...,Uj,Tj,...,Tj},V′d=ef{U′j,...,U′j,T′j,...,T′j},
j 1 m 1 m j 1 m 1 m
andWjd=efVj∪Vj′,sothatS ispartitionedinlevelswithS =Snj=0Wj.Witheachxj∈X
weassociatetwointermediaryLTr’s:
G⊤d=ef(W ,∅,V ,P,g), G⊥d=ef(W ,∅,V′,P′,g)
j j−1 j j j j−1 j j
withsetsofrulesP andP′.TherulesforG⊤aregiveninFig.5:somedeletionrulesare
j j j
conditional,dependingon whether x appearsin the clausesC ,...,C . The rulesfor
j 1 m
G⊥areobtainedbyswitchingprimedandunprimedsymbols,andbyhavingconditional
j
rules based on whether ¬x appears in the C’s. One easily checks that G⊤ and G⊥
j i j j
are indeed simple transformers. They have same inputs and disjoint outputs so that
theunion(G⊤+G⊥):W ⊢W iswell-defined.Hencethefollowingcompositionis
j j j−1 j
well-formed:
GF d=ef(G⊤1 +G⊥1).(G⊤2 +G⊥2)···(G⊤n +G⊥n).
WeconcludethedefinitionofGF withanintuitiveexplanationoftheideabehindthere-
duction.GF operatesonthewordu0=U10···Um0 whereeachUi0standsfor“thevalidity
ofclauseC isundeterminedatstep0(i.e.,atthebeginning)”.Atstep j,G⊤+G⊥picks
i j j
10 P.ChambartandPh.Schnoebelen
T1j−1 T′1j−1 T2j−1 T′2j−1 ··· Tmj−1 T′mj−1 T1j (ifxj|=C1) UU1′j−j−11
1
UT11jj UT22jj ······ UTmmjj g T2j ... (ifxj|=C2...) UU2′j2−j−11
U1j−1 U′1j−1 U2j−1 U′2j−1 ··· Umj−1 U′mj−1 Tmj (ifxj|=Cm) UUm′jm−j−11
Fig.5.P,therulesforG⊤:Fixedpartonleft,conditionalpartonright.
j J
avaluationforx :G⊤ picks“x =⊤”whileG⊥ picks“x =⊥”.ThistransformsUj−1
j j j j j i
intoUj,andTj−1intoTj,movingthemtothenextlevel.Furthermore,anundetermined
i i i
Uj−1 can be transformedinto Tj ifC is satisfied by x . In addition,and because G⊤
i i i j j
andG⊥musthavedisjointoutputalphabets,thesymbolsintheV ’scomeintwocopies
j j
(hencetheV′’s) thatbehaveidenticallywhentheyareinputinthetransformerforthe
j
nextstep.
The reductionis concludedwith the followingclaim that we proveby combining
Corollaries6.5and6.8below.
F issatisfiableiffU0U0···U0g⇒2mnTnTn···Tng
1 2 m GF 1 2 m
iffU0U0···U0g⇒≤2mnTnTn···Tng (Correctness)
1 2 m GF 1 2 m
iffU0U0···U0g⇒∗ TnTn···Tng.
1 2 m GF 1 2 m
ObservefinallythatGF isanacyclicgrammarinthesenseof[JL07],thatistosay,
its rules define an acyclic “may-act-upon”relation between symbols. Such grammars
are much weaker than generalLGr’s since, e.g., languagesrecognized by LGr’s with
acyclicdeletionrules(andarbitraryinsertionrules)areregular[JL07].
Remark6.2. TheconstructionofGF fromF ,mostlyamountingtocopyingoperations
fortheG⊤’sandG⊥’s,totype-checkingandsets-joiningoperationsforthecomposition
j j
oftheLTr’s,canbecarriedoutinlogarithmicspace. ⊓⊔
6.2 CorrectnessoftheReduction
Wesaythataworduis j-cleanifithasexactlymsymbolsandifu[i]∈{Tj,T′j,Uj,U′j}
i i i i
foralli=1,...,m.Itis⊤-homogeneous(resp.⊥-homogeneous)ifitdoesnotcontain
any(resp.,onlycontains)primedsymbols.
Let0≤ j≤nandq be a Booleanvaluationofx ,...,x :we say thata j-cleanu
j 1 j
respects(F under)q when,foralli=1,...,m,q |=C whenu[i]isdetermined(i.e.,
j j i
∈Tj+T′j).Finallyucodes(F under)q ifadditionallyeachu[i]isdeterminedwhen
i i j
q |=C.Thus,aworduthatcodessomeq exactlylists(viadeterminedsymbols)the
j i j
clausesofF madevalidbyq ,andtheonlyflexibilityinuisinusingtheprimedorthe
j
unprimedcopyofthesymbols.Hencethereisonlyone j-cleanucodingq thatis⊤-
j
homogeneous,andonlyonethatis⊥-homogeneous.Ifurespectsq insteadofcoding
j
it, morelatitudeexistssince symbolsmaybeundeterminedevenifthe corresponding