Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

0.56 MB·

by  Christoph Haase

Subclasses of Presburger Arithmetic and the
Weak EXP Hierarchy
ChristophHaase∗
LaboratoireSpe´cificationetVe´rification(LSV),CNRS
E´coleNormaleSupe´rieure(ENS)deCachan,France
[email protected]
4
1
0 Abstract burgerarithmetic;forthatreasonthecomplexityofPresburgerarith-
2
metichasextensivelybeenstudiedintheliteraturefromthe1970’s
y Ibtuirsgesrhaorwitnhmtheatitcf,oir.ea.,niytsfirexsetdriciti>on0to,tih+eΣ1iq+u1a-nftriafigemreanltteronfatPiorenss- onwards.Inordertofullycapturethecomputationalcomplexityof
a beginningwithanexistentialquantifier,iscompleteforΣEXP,the Presburgerarithmetic,BermanevenintroducedtheSTAmeasureon
M i thecomplexityofadecisionproblem,sincePresburgerarithmetic
i-thleveloftheweakEXPhierarchy,ananaloguetothepolynomial-
“maynothaveprecisecomplexitycharacterisationsintermsofthe
timehierarchyresidingbetweenNEXPandEXPSPACE.Thisre-
usualtimeandtapemeasures”[3].TheclassSTA(s(n),t(n),a(n))
2 sultcompletesthecomputationalcomplexitylandscapeforPres-
istheclassofallproblemsoflengthnthatcanbedecidedbyan
1 burgerarithmetic,alineofresearchwhichdatesbacktotheseminal
alternatingTuringmachineinspaces(n)andtimet(n)usinga(n)
workbyFischer&Rabinin1974.Moreover,weapplysomeofthe
alternations,where“∗”actsasawildcardinordertoindicatean
] techniquesdevelopedintheproofofthelowerboundinorderto
O unboundedavailabilityofacertainresource.Basedonthework
establishboundsonsetsofnaturalsdefinableintheΣ -fragment
1 by Fischer & Rabin [11] and Ferrante & Rackoff [10], Berman
L ofPresburgerarithmetic:givenaΣ1-formulaΦ(x),itisshownthat establishedthefollowingresult.
. thesetofnon-negativesolutionsisanultimatelyperiodicsetwhose
s periodisatmostdoubly-exponentialandthatthisboundistight.
c Proposition1(Berman[3]). Presburgerarithmeticiscompletefor
[ CategoriesandSubjectDescriptors F.4.1[Mathematicallogic]: STA(∗,22nO(1),n).

Computationallogic
2
In terms of the usual time and space measures, this settles
v Keywords Presburgerarithmetic,boundedquantifieralternation, Presburgerarithmeticbetween2-NEXPand2-EXPSPACE.Despite
6 weakEXPhierarchy,ultimatelyperiodicsets,context-freecommu- thesehighcomputationalcosts,onthepositivesidewhenlooking
6 tativegrammars at fragments Presburger arithmetic becomes more manageable.
2
There are two dimensions in which we can constraint formulas
5
1. Introduction inordertoobtainfragmentsofPresburgerarithmetic:thenumber
.
1 of quantifier alternations and the number of variables in each
0 P(cid:104)Nre,s0b,u1r,g+er,<ar(cid:105)i.thTmheistictheisortyhewafisrssth-oorwdenrtothbeoerdyecoifdathbelesbtryucPtruerse- quantifier block. For i,j ∈ N∪{∗}, let PA(i,j) denote the set
4 of formulas of the Σ -fragment of Presburger arithmetic1 such
burger in his seminal paper in 1929 by providing a quantifier- i
1 that at most j different variables occur in each quantifier block,
elimination procedure [31]. Presburger arithmetic is central to a
: where“∗”isusedasawildcardforanunboundednumber.Hence,
v vast number of different areas in computer science and is often
Proposition1characterisesthecomputationalcomplexityofPA(∗,∗)
i employedasatoolforshowingdecidabilityandcomplexityresults.
X ThecentraldecisionproblemforPresburgerarithmeticisvalid- with n being the number of symbols required to write down
theformula.Subsequently,PAandPA(i)abbreviatePA(∗,∗)and
r ity,i.e.,todecidewhetheragivensentenceistruewithrespectto
a thestandardinterpretationinarithmetic.Thetwomostprominent PA(i,∗),respectively.
OneofthemostprominentfragmentsofPresburgerarithmeticis
waystodecidevalidityareeitherquantifier-eliminationbased[8]or
itsexistentialorquantifier-freefragment,whichiscomputationally
automatabased[9,23,40].AnydecisionprocedureforPresburger
notmoreexpensivethanstandardBooleansatisfiability.
arithmeticisinherentlytiedtothecomputationalcomplexityofPres-
Proposition2(Scarpellini[33],Borosh&Treybing[5]). Forany
∗TheauthorissupportedbytheFrenchAgenceNationaledelaRecherche fixedj ∈N,PA(1,j)isinP[33].PA(1)isNP-complete[5].
(ANR),REACHARD(grantANR-11-BS02-001).
Duetoitscomparablylowcomputationalcomplexity,quantifier-
freePresburgerarithmeticisthefragmentthatismostcommonly
found in application areas which aim at a practical impact. The
Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalor
classroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributed existentialfragmentofPresburgerarithmeticcanevenbeextended
forprofitorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitation withafulldivisibilitypredicatewhileretainingdecidability[25,26].
onthefirstpage.CopyrightsforcomponentsofthisworkownedbyothersthanACM AnothersubclassofPresburgerarithmeticwhichhasextensively
mustbehonored.Abstractingwithcreditispermitted.Tocopyotherwise,orrepublish,
been studied is obtained by allowing for an arbitrary but fixed
topostonserversortoredistributetolists,requirespriorspecificpermissionand/ora
[email protected]. numberofquantifieralternations.
CSL-LICS2014, July14–18,2014,Vienna,Austria.
Copyright(cid:13)c 2014ACM978-1-4503-2886-9...$15.00.
http://dx.doi.org/10.1145/2603088.2603092 1AllresultsobtainedaresymmetricwhenconsideringΠi-formulas.
Proposition 3 (Gra¨del [17], Scho¨ning [34], Reddy & Love- systems [15] or in formal language theory [20]. For instance,
land [32]). For any fixed i > 0 and j > 2, PA(i+1,j) is ΣP- analysingsuchboundshasbeencrucialinordertoobtainoptimal
i
complete2[17,34].PA(i)isinSTA(∗,2nO(i),i)[32]. complexityresultsformodel-checkingproblemsofaclassofone-
counterautomatain[15].Inmoredetail,in[15]ithasbeenshown
Thus, when fixing the number of quantifier alternations, the thatthesetofnon-negativeweightsofpathsbetweentwonodesina
complexity of Presburger arithmetic decreases roughly by one weightedgraphisultimatelyperiodicwithaperiodthatisatmost
exponent, and when additionally fixing the number of variables, singly-exponentialbounded.AresultbySeidletal.[35]onParikh
we obtain every level of the polynomial-time hierarchy. Notice imagesofnon-deterministicfinite-stateautomataimplicitlystates
thatthereisanobviousgap:acompletenessresultforPresburger thatthoseultimatelyperiodicsetsaredefinableintheΣ -fragment
1
arithmetic with a fixed number of quantifier alternations and an of Presburger arithmetic. It would thus be desirable to establish
arbitrarynumberofvariablesineachquantifierblockismissing. a generic upper bound for ultimately periodic sets definable in
The study of lower bounds for PA(i) goes back to the work PA(1)yieldingthesameoptimalbounds.Inthiscontext,Theorem2
ofFu¨rer[13],whoshowedaNEXPlowerboundforsomefixed providesanegativeresultinthatitshowsthatageneralboundon
i > 1. Later, Gra¨del showed NEXP-hardness and EXPSPACE ultimatelyperiodicsetsdefinableinPA(1)cannotyieldtheoptimal
membership of PA(2), but tight lower and upper bounds for the boundsrequiredfornaturalconcreteultimatelyperiodicsetslike
wholeclassofPA(i)formulashavenotyetbeenestablished.The thoseconsideredin[15].
purposeofthefirstpartofthispaperistoclosethisgapandestablish Thispaperisstructuredasfollows.InSection2weprovidemost
thefollowingtheorem. oftheformaldefinitionsrequiredinthispaper;howeverthereaderis
expectedtohavesomeleveloffamiliaritywithstandardnotionsand
Theorem1. Foranyfixedi>0,theΣ -fragmentofPresburger
i+1 conceptsfromlinearalgebra,integerprogramming,first-orderlogic
arithmeticisΣEXP-complete.
i andcomputationalcomplexity.Eventhoughweprovideaslightly
Here,ΣEXPdenotesthei-thleveloftheweakEXPhierarchy[19], moreelaboratedaccountonsuccinctencodingsviaBooleancircuits,
i
ananaloguetothepolynomial-timehierarchy[36]residingbetween it will be beneficial to the reader to be familiar with Chapters 8
NEXP and EXPSPACE; a formal definition will be provided in and20inPapadimitriou’sbookoncomputationalcomplexity[28].
Section 2.4. Equivalently, we obtain that PA(i+1) is complete Section 3is thengoing toestablish the lower andupper bounds
forSTA(∗,2nO(i),i).Determiningtheprecisecomplexityofthe ofTheorem1,andTheorem2isshowninSection4.Thepaper
concludesinSection5.Subsequenttothebibliography,aproofofa
Σ -fragmentofPresburgerarithmeticforafixedihasbeenlisted
i
technicalcharacterisationoftheweakEXPhierarchyisoutlinedin
as a problem that “deserves to be investigated” by Compton &
theappendixforthesakeofcompleteness.
Henson[7,Prob.10.14].However,aspointedoutin[7],theirgeneric
methodsforprovinglowerboundsdonotseemtobeapplicableto
thisfragment,andourhardnessresultisbasedonratherspecific 2. Preliminaries
propertiesof,forinstance,distributionsofprimenumbers.
2.1 Generalnotation
The second part of the paper diverts from the first part and
focusesontheΣ -fragmentofPresburgerarithmetic.Morespecif- By Z and N we denote the set of integers and natural numbers,
1
ically, we consider sets of naturals definable by formulas in the respectively.Wewillusuallyusea,b,cfornumbersinZandN.
Σ1-fragmentofPresburgerarithmeticopeninonevariable.Given Givena∈N,wedefine[a]d=ef{0,...a−1}.GivensetsM,N ⊆N,
aΣ1-formulaΦ(x),denoteby Φ(x) thesetofthosea∈Nsuch as is standard M + N d=ef {m + n : m ∈ M,n ∈ N} and
thatreplacingxwithainΦ(x)i(cid:74)svalid(cid:75).Itiswell-knownthat Φ(x)
isanultimatelyperiodicset,seee.g.[4].AsetN ⊆Nisult(cid:74)imately(cid:75) M·N d=ef{mn:m∈M,n∈N}.Moreover,wewillusestandard
periodicifthereexistsathresholdt∈N,abaseB ⊆{0,...t−1}, notationforintegerintervalsand,e.g.,fora ≤ b ∈ Ndenoteby
aperiodp ∈ N,andasetofresidueclassesR ⊆ {0,...p−1} [a,b)theset{a,...b−1}.Forvectorsa=(a1,...,an)∈Zn,we
suchthatN =U(t,p,B,R)with willdenoteby(cid:107)a(cid:107)thenormofa,whichisthemaximumabsolute
valueofallcomponentsofa,i.e.,(cid:107)a(cid:107) d=ef max{|a |} .For
U(t,p,B,R)d=efB∪{t+r+kp:r∈R,k≥0}. m×n integer matrices A, (cid:107)A(cid:107) denotes the maximium1≤ia≤bnsolute
valueofallcomponentsofA.Finally,givenasetM ⊆ Zn,we
GivenaΣ -formulaΦ(x),byapplyingsomeinsightsfromthe
1 denoteby(cid:107)M(cid:107)themaximumofthenormofallelementsofM.All
firstpart,wecanestablishadoubly-exponentialupperboundon
functionsinthispaperareassumedtomapnon-negativeintegers
theperiodoftheultimatelyperiodicsetequivalentto Φ(x) and
showthatthisboundistight,whichiscapturedbythes(cid:74)econd(cid:75)main tonon-negativeintegers.Unlessstatedotherwise,weassumeall
integersinthispapertobeencodedinbinary,i.e.,thesizeorlength
theoremofthispaper.
towritedowna∈ZisO(log|a|).
Theorem2. ThereexistsafamilyofΣ -formulasofPresburger
1
arithmetic (Φ (x)) such that each Φ (x) is a PA(1,O(n)) 2.2 PresburgerArithmetic
n n>0 n
formula with |Φn(x)| ∈ O(n2) and Φn(x) is an ultimately Usually, x,y,z will denote first-order variables, and x,y,z vec-
periodicsetwithperiodpn ∈22Ω(n).Mo(cid:74)reoverf(cid:75)oranyΣ1-formula tors or tuples of first-order variables. Let x = (x1,...,xn) be
Φ(x),wehave Φ(x) = U(t,p,B,R)suchthatt ∈ 2poly(|Φ(x)|) ann-tupleoffirst-ordervariables.Inthispaper,formulasofPres-
andp∈22poly(|Φ(cid:74)(x)|).(cid:75) burgerarithmeticarestandardfirst-orderformulasoverthestruc-
ture(cid:104)N,0,1,+,<(cid:105)obtainedfromatomicexpressionsoftheform
The most interesting part about this theorem is the doubly- p(x)<b,wherep(x)isalinearmultivariatepolynomialwithinte-
exponentiallowerboundoftheperiodofultimatelyperiodicsets gercoefficientsandabsolutetermzero,andb∈Z.Ifthedimension
definable by PA(1) formulas. Establishing bounds on constants ofxisclearfromthecontext,forbrevitywewilloftenomitstating
of ultimately periodic sets naturally occurs when analysing the itexplicitly.Leta=(a ,...,a )∈NnandΦ(x)beopeninthe
1 n
computationalcomplexityofdecisionproblemsforinfinite-state first-ordervariablesx,wedenotebyΦ(a/x)theclosedformula
obtainedfromreplacingeveryoccurrenceofx inΦ(x)witha .By
i i
2Inordertoestablishhardness,j > 2isonlyrequiredontheinnermost Φ(x) wedenotetheset{a∈Nn :Φ(a/x)isvalid}.Thesize|Φ|
quantifier. (cid:74)ofafo(cid:75)rmulaofPresburgerarithmeticisdefinedasthenumberof
symbolsrequiredtowriteitdown,andthenorm(cid:107)Φ(cid:107)isthelargest Theaforementionedtimehierarchiesarenowdefinedas
absolutevalueofallconstantsoccurringinΦ.
ΣP d=efΠP d=efP ΣEXP d=efΠEXP d=efEXP
0 0 0 0
Remark. Fornotationalconvenience,whenstatingconcreteformu- ΣP d=efNPΣPi ΣEXP d=efNEXPΣPi
laswewillpermitourselvestouseatomicformulasp(x) < q(x) i+1 i+1
forlinearpolynomialsp(x),q(x).Moreover,allresultsonthecom- ΠP d=efcoNPΣPi ΠEXP d=efcoNEXPΣPi
i+1 i+1
plexityofvalidityofformulasofPresburgerarithmeticcarryover
ifweassumeunaryencodingofnumbers,sincebinaryencoding PH d=ef (cid:91)ΣPi EXPHd=ef (cid:91)ΣEiXP.
ofnumberscanbe“simulated”bytheintroductionofadditional i≥0 i≥0
first-order variables and repeated multiplication by two, causing
For our lower bounds, we will rely on the following equivalent
onlyasub-quadraticblowupintheformulasize.Inaddition,an characterisationofΣEXP.
equalitypredicate“=”canbeexpressedintermsof<causinga i
linearblowup,sincex=y↔x<y+1∧y<x+1.Likewise, Lemma1. Foranyi>0,alanguageL⊆{0,1}∗isinΣEXPiff
i
x > y andx < y < z abbreviatey < xandx < y∧y < z, thereexistsapolynomialqandapredicateR⊆({0,1}∗)i+1such
respectively. thatforanyw∈{0,1}n,
2.3 Semi-LinearSetsandSystemsofLinearDiophantine w∈Liff∃w1 ∈{0,1}2q(n).∀w2 ∈{0,1}2q(n)···
Inequalities ···Q w ∈{0,1}2q(n).R(w,w ,...,w )
i i 1 i
AcentralresultduetoGinsburgandSpanierstatesthatthesetsof
andR(w,w ,...,w )canbedecidedindeterministicpolynomial
naturalnumbersdefinablebyaformulaofPresburgerarithmetic 1 i
time.
open in n variables are the n-dimensional semi-linear sets [14],
whichwejustcallsemi-linearsetsifthedimensionisclearfrom Despitebeinginthespiritofanelementaryresultoncomputa-
thecontext.Asemi-linearsetisafiniteunionoflinearsets.The tionalcomplexity,theauthorwasunabletofindaformalproofof
latteraredefinedintermsofabasevectorb∈Nnandafiniteset Lemma1inthestandardliterature.Itissomewhatstatedinformally
ofperiodvectorsP ={p1,...pk}⊆Nn,anddefinetheset withoutaproofin[19].Inordertokeepthispaperself-contained
andforthereader’sconvenience,aproofsketchofLemma1based
L(b;P)d=efb+λ1p1+···λkpk, λi ∈N,1≤i≤k. onaproofofananaloguecharacterisationofthepolynomial-time
hierarchygivenin[2]isprovidedintheappendix.
LetAbeanm×nintegermatrixandc ∈ Zm.Asystemof
linearDiophantineinequalitiesisgivenasS :Ax≥c.Thesize|S| 2.5 BooleanCircuits
ofSisthenumberofsymbolsrequiredtowritedownSassuming
Astandardapproachtoraisethecomplexityofaproblemknownto
binaryencodingofnumbers.ThesetofpositivesolutionsofS is
denotedby S ⊆ Nn andisthesetofalln-tuplessuchthatthe becompleteforacomplexityclassbyoneexponentistosuccinctly
inequalities(cid:74)ine(cid:75)veryrowofShold. representtheinput,seee.g.[16,29].Awell-knownconceptisto
representtheinputbyBooleancircuits.Inthispaper,fortechnical
The following proposition is due to Frank & Tardos and es-
convenienceweadaptthedefinitionprovidedin[16].
tablishesastronglypolynomial-timealgorithmforthefeasibility
problemofasystemoflinearDiophantineinequalitiesinafixed Definition 1. A Boolean circuit C of size r with n ≤ r inputs
dimension,i.e.,decidingwhether S (cid:54)=∅. is a function f : [r] → {&,(cid:107),∼,↑,↓ } × [r] × [r], where
1
(cid:74) (cid:75) f(i)=(t,j,k)iffthegatewithindexiisoftypet,i.e.,anand,or,
Proposition4(Frank&Tardos[12]). LetS :Ax≥cbeasystem
not,inputorconstantgate,respectively,andj,k<iareinputsof
oflinearDiophantineinequalitiessuchthatAisanm×nmatrix.
thegate,unlesst=∼inwhichcasewerequirej =k.
ThenfeasibilityofScanbedecidedusingn2.5n+o(n)|S|arithmetic
operationsandspacepolynomialin|S|. We identify each gate of C with an index from [r], and by
convention the first n ≤ r gates are input-gates, and the r-th
Whenweareinterestedinrepresentingthesetofallsolutionsof gate,i.e.,thegatewithindexr−1,istreatedastheoutputgate
S,wewillemploythefollowingproposition,whichprovidesbounds ofC.Moreoverfortechnicalconvenience,wesometimesidentify
onthesemi-linearrepresentationof S andisaconsequenceof the various types of the gates by natural numbers ordered as in
Corollary1in[30]. (cid:74) (cid:75) Definition1,i.e.,&isidentifiedas0,(cid:107)as1,etc.Byusingconstant
gatesasgateswithconstantvalue1,aninputw ∈ {0,1}n toC
Proposition5(Pottier[30]). LetS :Ax≥cbeasystemoflinear inducesauniqueevaluationmappinge : [r] → {0,1}defined
w
Diophantine inequalities such that A is an m×n matrix. Then in the obvious way, and C evaluates to true (false) on input w
(cid:83)
S = i∈IL(bi;Pi)suchthatforalli∈I, if ew(r − 1) = 1 (ew(r − 1) = 0). For brevity, we define
(cid:74) (cid:75)
(cid:107)b (cid:107),(cid:107)P (cid:107)≤(n(cid:107)A(cid:107)+(cid:107)c(cid:107)+2)m+n. C(w)d=efew(r−1),andifm1,...,mk ∈NthenC(m1,...mk)is
i i theoutputofC(w1···wk),whereeachwi ∈{0,1}(cid:100)logmi(cid:101)isthe
binary,ifnecessarypadded,representationofm .
i
2.4 TimeHierarchies For the remainder of this section, we will briefly recall and
Let us recall the definitions of the polynomial-time hierarchy elaborateonsomeresultsandconceptsaboutcircuitsandsuccinct
PH[36]andtheweakEXPhierarchyEXPH[19]intermsoforacle encodingsfromPapadimitriou’sbook[28]oncomputationalcom-
complexityclasses.Asusual, plexity.GivenacircuitCandaninputw∈{0,1}nforsomen≥0,
itiswell-knownthatdeterminingC(w)isP-complete[28,Thm.
P= (cid:91) DTIME(nk) EXP= (cid:91) DTIME(2nk) 8.1].In[28],theproofofP-hardnessisestablishedbyshowingthat
thecomputationtableofapolynomial-timeTuringmachinecanbe
k>0 k>0
NP= (cid:91) NTIME(nk) NEXP= (cid:91) NTIME(2nk). emnaccohdineedMas,aaBcoomolpeuatnatcioirnctuaibt.leFiosrananff(n(n)×)-tfim(ne)-bgoriudnodfecdeTllsurTing
i,j
k>0 k>0 fromanalphabetthatallowsforuniquelyencodingconfigurationsof
.. .. .. .. .. .. intothethreeframedgray-shadedboxesonthebottomofthedashed
. . . . . .
boxes.Here,weassumethatthreebitsaresufficienttorepresent
(cid:46) 1 1 0 (cid:3) (cid:3)
(cid:46) 1 1q1 1q0 (cid:3) (cid:3) tchoedeadlpahsa1b1et1.oCfothnesecqoumenptulyta,ttihoengtaatbelsewoifthMinwd,eaxn(d3,th4a)t,((cid:46)3q,05i)saennd-
(cid:46)(cid:46)q0 11q2 00 11 (cid:3)(cid:3) (cid:3)(cid:3) (a3re,6g)a,treespwreistehnctionngsttahnetcveallluTe1o,1neo,fathseincdoimcapteudtatiinonFitgaubrlee2o.fNMoww,
wewantthevaluesofthecellsT ,T ,etc.ofthecomputation
1,2 1,3
Figure1. Graphicalillustrationofacomputationtableofatime- tableofMw tobeequivalenttow1···wn,whicharerepresented
TbohuencdoendtTroulrisntagtemaanchdintheeMhe.aFdorpeoxsaitmiopnleo,fhMereiwseinhdaivceaTte2d,2b=yt0apq2e. bcoyrrtehsepognadteinsgwtiothT1in,2daicnedsT(10,,30h)a,v.e..in(d0i,cie2sq((n3),)1.5T),h(e3g,1at6e)s,(in3,D17w)
symbols with some q as subscript. The four gray-shaded cells and(3,26),(3,27),(3,28),respectively.Thosegateshavethegates
i
illustratethatsuccessivecellsonlydependonthreeprecedingcells. (0,0)and(0,1)astheirinputs,respectively.Supposethatinour
encoding1isrepresentedas101and0as010,thesequenceof(cid:107),
∼and(cid:107)gatesensuresthat1ismappedto101and0to010.Con-
M suchthattheconfigurationofM instepiwhilerunningonwis sequently,thegates(3,15),(3,16),(3,17)cancorrectlytransfer
encodedinthei-throw.Figure1graphicallyillustratestheconcept thealphabetsymbols{0,1}ofMwintotheinternalrepresentation
ofacomputationtable,where0and1arealphabetsymbolsofM, ofthecomputationtable,andinparticularcopythefirstsymbol
and(cid:46)and(cid:3)areleftdelimitersandblanksymbols,respectively.The oftheinputstringw1···wiintotheinternalrepresentationofthe
crucialfactforencodingcomputationtablesasBooleancircuitsis computationtable.IntheexampleinFigure2,thegateswithindex
thatfori,j >1,thesymbolatT onlydependsonafixednumber (3,15),(3,16),(3,17)wouldoutput1,0and1,respectively,since
i,j
ofcells,namelyT ,T andT ,illustratedbythe thegate(0,0)hasvalue1whichcorrespondstothefirstsymbol
i−1,j−1 i−1,j i−1,j+1
gray-shadedcellsinFigure1.Itisthenclearthatthealphabetofa oftheinputstringw1.Asstatedbefore,inourreductionthisvalue
computationtablecanbeencodedintoabinaryalphabetoftruth isprovidedon-the-fly.Therestofthereductionfollowsstandard
values,andthataconstantbasiccircuitcanbeconstructedfrom arguments.EachdashedboxcontainscircuitsT1,T2andT3which
M whichensuresthatthevaluesofthecellsarecorrectlypropa- computetheconsecutivecellofthesimulatedcomputationtableof
gatedalongthey-axis.ItthenfollowsthatM acceptswiffthere Mw,i.e.,thevaluesofthethreegatesrepresentingthiscell.The
existsacomputationtableendinginanacceptingstateiffthecircuit dashedboxesontheleftusedifferentcircuitsU1,U2andU3since
encodingthiscomputationtableevaluatestotrue. theydonothavealeftneighbor.Allunusedgatescanassumedto
Inthenextsection,forourlowerboundwewillapplyLemma1, bedummygates,i.e.gateswithconstantvalue1,asindicatedin
whichentailsdeciding(w,w1,...,wi)∈R,wherethewj areof Figure2.ItfollowsthatMwaccepts(w1,...,wi)iffCwevaluates
sizeexponentialinn=|w|.LetM beapolynomial-timeTuring totrueontheinputprovided,i.e.,thevalueofthegatewiththe
w
machinedecidingRforafixedw.Thew1,...wj willimplicitlybe highestindexofCwisequalto1.
codedintonaturalnumbers,soitwillnotbepossibletoconstruct InordertoencodeCwsuccinctly,itisclearthatduetotheregular
aBooleancircuitCw upfrontthatcanevaluateMw ontheinput structureofCw,thetypeandinputgatestoanygatecanbecomputed
w ,...w ofexponentialsize,sincewearerequiredtoestablish fromagivenindexofagatebyapolynomial-timealgorithm.The
1 j
a polynomial-time reduction. Instead, we will succinctly encode circuitDw cannowbetakenasthecircuitcorrespondingtothis
C viaanotherBooleancircuitD .Moreprecisely,C isencoded algorithm.
w w w
viaD asfollows:D has3r(n)+3inputgatesforsomefixed
w w
polynomialrdependingonMwsuchthatfori,j,k∈[2r(n)]and 3. CompletenessoftheΣi+1-Fragmentof
t ∈ [5],Dw(t,i,j,k) = 1iffthedefiningfunctionf ofCw gives PresburgerArithmeticforΣEXP
i
f(i)=(t,j,k),i.e.,thatthegatewithindexiofC isoftypetand
w
hasinputgateswithindexjandk.Inparticular,Dwandhenceforth Inthissection,weshowthatPA(i+1)isΣEiXP-completeforevery
C onlydependonwandM,andareindependentofw ,...w . fixed i > 0. We begin with the lower bound and first note that
w 1 i
Notethatwecanviewanassignmentoftruthvaluestothegatesof it is not possible to adapt Berman’s hardness proof [3] in order
C asastringoflength2r(n). to get the desired result, since it relies on a trick by Fischer &
w
More generally, it is known that if C with no input gates is Rabin[11]inordertoperformarithmeticoperationsonabounded
succinctlygivenbysomecircuitD,determiningwhetherCevaluates intervaloverlargenumberswhichlinearlyincreasesthenumberof
to true is EXP-complete [28, Thm. 20.2 & Cor. 2]. The idea quantifieralternations.Instead,wewillpartlyadaptconceptsand
underlyingthehardnessproofisastraight-forwardgeneralisation ideasintroducedbyGra¨delinhishardnessproofforPA(2)in[18]
of the approach outlined in the paragraph above. The circuit C and Gottlob, Leone & Veith in [16]. Roughly speaking, we aim
encodesthecomputationtableofanEXPTuringmachineM.Since for“implementing”Lemma1viaaPA(i+1)formula,whichwill
theindicesofthegatesofC canberepresentedinbinary,viaD entailencodingbitstringsofexponentialsizeintonaturalnumbers
we can encode C by implicitly encoding an exponential number and evaluating Boolean circuits in Presburger arithmetic on-the-
oftheconstantbasiccircuitensuringproperpropagationbetween fly. The upper bound does not follow immediately and requires
consecutivecells.Thisapproachcannowbeadaptedforourpurpose, combiningsolutionintervalsestablishedbyWeispfenningin[38]
i.e.,toevaluateapolynomial-timeTuringmachineonaninputof withProposition4.
exponential size. The major challenge is to transfer input to the
3.1 LowerBounds
succinctlyencodedcircuiton-the-fly.
ReferringtoLemma1andgivenM asabove,wecanconstruct Thegoalofthissectionistoprovethefollowingproposition.
w
inlogarithmicspaceaBooleancircuitD encodingC suchthat
w w Proposition6. LetL⊆{0,1}∗bealanguageinΣEXP,i>0and
theinput(w1,...,wi)toCwisobtainedfromthefirsti2q(n)gates, w∈{0,1}∗.Thereexistsapolynomial-timecomputiablePA(i+1)
andC encodesacomputationtableofM onthisinput.Figure2
w w formulaΦ suchthatw∈LiffΦ isvalid.
illustrateshowthiscanberealised.EachboxinFigure2isagate, L,w L,w
andacellofthecomputationtableofM whilebeingexecutedon Tothisend,weemploythecharacterisationofΣEXPinLemma1.
w i
(w ,...,w )isencodedintothedashedboxes,ormorespecifically, LetM bethedeterministicpolynomial-timeTuringmachinedecid-
1 n
Figure2. IllustrationoftheapproachofhowtosuccinctlyencodeaBooleancircuitencodingthecomputationtableofapolynomial-time
Turingmachineonaninputofexponentialsize.Eachsquarerepresentsagate,allgatesnotsurroundedbyboxesareassumedtobegateswith
constantvalue1.
ingRfromLemma1,andletM besuchaTuringmachinedecid- implicitlyreplacedby∃x.¬(ϕ¯(x,y)),and¬(ϕ¯(x,y))istreated
w
ingRforafixedinputw∈{0,1}n,whichcanbecomputedfrom inthesamewayifitisnotyetquantifier-free.Inthisway,wecan
M inlogarithmicspace.Thebitstringsw tow fromLemma1 alwaysmakesuretoresultinΣ -orΠ -formulas.
1 i 1 1
constitutingtheinputtoM arerepresentedinourreductionvia We now turn towards the details of our reduction and begin
w
naturalnumbersassignedtofirst-ordervariablesx=(x ,...,x ). with discussing the encoding of bit strings as natural numbers
1 i
Thepreciseencodingofaw viax isdiscussedbelow.Fornow,it weusesubsequently.TheencodingweuseisduetoGra¨del[18].
j j
isonlyimportanttomentionthatnoteverynaturalnumberencodes In his NEXP lower bound for PA(2) he exploits a result due to
abitstring.Letusfocusonthehigh-levelstructureofΦL,w: Ingham [6, 22] that for any sufficiently large3 i ∈ N there is at
least one prime in the interval [i3,(i+1)3). Given a bit string
Φ d=ef∃x .∀x ···Q x . (cid:94) Ψ (x )∧ w=b1···bn ∈{0,1}n,anaturalnumbera∈Nencodeswiffor
L,w 1 2 i i valid,r(n) j all1≤i≤nand
1≤j≤i,jodd
(cid:16) (cid:94) (cid:17) forallprimesp∈[i3,(i+1)3):a≡bi modp.
∧ Ψ (x ) →Ψ (x ,...,x ). (1)
valid,r(n) j Mw 1 i
The existence of such an a is then guaranteed by the Chinese
1≤j≤i,jeven
remainder theorem. Given a fixed n > 0, we call a ∈ N a
Unsurprisingly,thealternationofquantifiersinLemma1isreflected valid encoding if for every 1 ≤ i ≤ n, either a ≡ 0modp or
bythealternationofquantifiersin(1),soQ = ∃ifiisoddand a≡1modpforallprimenumbersp∈[i3,(i+1)3).
i
Q =∀ifiiseven.TheformulaΨ (x )isaΠ -formula, Inordertoenabletheextractionofbitsofbitstringsencodedas
i valid,r(n) i 1
andΨ (x ,...,x )isaformulaintheBooleanclosureofΣ if naturals,weshowhowtocheckfordivisibilitywithanaturalnumber
Mw 1 i 1
iisoddandaΣ -formulaifiiseven.Thefirstconjunctensures whosenumberofbitsisfixed.Next,weshowhowtoevaluatea
1
thattheexistentiallyquantifiedvariablesrepresentencodingsofbit BooleancircuitinPresburgerarithmetic.Thisservestwopurposes:
stringsandthesecondconjunctthat,undertheadditionalassumption first, it allows for deciding if a given number lies in an interval
thattheuniversallyquantifiedvariablesencodevalidbitstringsas [i3,(i+3)3)andfortestingwhetheragivennumberisaprimedue
well,M acceptsthebitstringsencodedinx ,...,x .Forthegiven totheAKSprimalitytest[1].Second,itallowsforsimulatingM
w 1 i w
w∈{0,1}n,thoseformulasareconcreteinstancesofafamilyof discussedaboveonaninputofexponentialsizeusingitssuccinct
formulas,andr(n)isanindexinthisfamilyforsomepolynomial encodingviaacircuitasdiscussedinSection2.5.Puttingeverything
r(n)whichdominatesq(n)inLemma1andismademoreprecise togethereventuallyyieldsthedesiredreduction.
atalaterstage.Consequently,forafixedi>0,wehavethatΦ Webeginwithafamilyofquantifier-freeformulasΦ (x,x)
L,w bin,n
isaPA(i+1)formula. suchthatgivenb∈{0,1}nandb∈N,Φ (b,b)holdsifbis
bin,n
In our reduction, we have to take extra care to prevent the thebinaryrepresentationofb.Consequently,thisformulaimplicitly
“accidental”introductionofquantifieralternations.Ingeneralwhen constraintsbsuchthatb∈[2n]:
pnreouvtridailnfgorfmorumlausl,asw,hwicehadaraeptoGperna¨dpeol’lsynapopmroiaalclyheinqu[1iv8a]leanntdΣpr1o-vaindde Φbin,n(x,x)d=ef (cid:94) (xi =0∨xi =1)∧x= (cid:88) 2ixi. (2)
Π1-formulas. This ensures that, for instance, we do not have to i∈[n] i∈[n]
careaboutwhetherwecouldpossiblyintroduceanewquantifier Next,weprovideafamilyofneutralformulasΦ (x,y)such
mod,n
alternationifaformulaisusedontheleft-handsideofanimplication. thatfora,bwithb ∈ [2n],Φ (a,b)holdsiffa ≡ 0modb.
mod,n
WdehneontepbryovΦ¯id(ixn)g=an∀eyut.rϕ¯al(xΣ,1y-f)oirtmsnuelautΦra(lxe)qu=iv∃alyen.ϕtΠ(x1,cyo)u,nwteerpwairltl. Etiossne.nItniaclolyn,tΦramstotdo,na(xfo,rym)urleaawlisiethstahfeosrammuelapfuorrpbooseungdiveednminul[t1ip8l]i,cait-
For the sake of consistent naming, whenever Φ(x) occurs as a
subformulainsomeotherformula,weimplicitlyassumethatitis 3Cheng[6]providesexplicitboundsonIngham’sresult[22]andshowsthat
appropriatelyreplacedsuchthattheresultingformulaiseitheraΣ1- thisstatementholdsforalli∈Nsuchthati>2215.Asin[18],forbrevity
oraΠ1-formula,dependingonthecontext.Likewise,ifforinstance wewilluseIngham’sresultasifitweretrueforalli>0.Itwillbeclear
Φ(x)occursasanegatedsubformulainaformulathatissupposed thatwecouldaddCheng’soffsettoallnumbersinvolved,causingaconstant
to be existentially quantified, we assume that this subformula is blowuponly.
isnotrecursivelydefinedandofsizeO(n)asopposedtoO(nlogn) ThefirstlineofΦ (x,y)convertsxandyintotheirbinary
prime,n
whenbinaryencodingofnumbersisassumed.Thelatterfactwillbe representation. Next, the second line first concatenates these bit
usefulinSection4.Theunderlyingideaofthesubsequentdefinitions representationsviatheadditionalvariablezbyappropriatelyshifting
isthatifthebinaryexpansionofbisb=(cid:80)i∈[n]2ibiandbk=a the value of y by m bits, and finally z is passed to Cprime,n.
forsomek ≥ 0thenacanbewrittenasa = (cid:80)i∈[n]2iai with Cao∈ns[ebq3,u(ebnt+ly,1w)3e).havethatΦprime,n(a,b)holdsiffaisprimeand
a =kb :
i i
Wearenowinapositioninwhichwecandefineafamilyof
Φ (x,y,k)d=ef (cid:94) (y =0→x =0∧y =1→x =k) Π -formulasΨ (x)usedin(1)thatallowfortestingwhether
dig,n i i i i 1 valid,n
some a ∈ N represents a valid respectively invalid encoding of
i∈[n]
a bit string of length 2n. For valid encodings, we wish to make
Φmod,n(x,y)d=ef∃x.∃y.∃k.Φbin,n(y,y)∧ surethatallprimesineveryrelevantinterval[b3,(b+1)3)have
∧Φ (x,y,k)∧x= (cid:88) 2ix (3) uniformresidueclassesinaforall1 ≤ b ≤ 2n,i.e.,foranytwo
dig,n i primesp ,p ∈ [b3,(b+1)3)weeitherhavea ≡ 0modp and
1 2 1
i∈[n]
a ≡ 0modp ,ora−1 ≡ 0modp anda−1 ≡ 0modp .Let
2 1 2
Φ¯ (x,y)d=ef∀x.∀y.∀k.(cid:0)Φ (y,y)∧ mbeasabove,
mod,n bin,n
∧Φ (x,y,k)(cid:1)→x= (cid:88) 2ix .
dig,n i Ψ (x)d=ef∀y.∀p .∀p .(cid:0)1≤y≤2n∧
valid,n 1 2
i∈[n]
(cid:1)
∧Φ (p ,y)∧Φ (p ,y) →
WenowturntowardsevaluatingBooleancircuitswithsuitable prime,n+1 1 prime,n+1 2
(cid:0)
formulas in Presburger arithmetic. The subsequent formulas for → (Φmod,m+3(x,p1)∧Φmod,m+3(x,p2))∨
evaluatingacircuitCwithninputandrgatesintotalareessentially (cid:1)
∨(Φ (x−1,p )∧Φ (x−1,p )) .
an adaption of a construction given by Gottlob, Leone & Veith mod,m+3 1 mod,m+3 2
in [16]. It is easily checked that for a ∈ [2n], ΦC(a) holds iff In order to complete our hardness proof for ΣEXP for a sub-
C(a)=1.InΦ ,theinputtoCisencodedviaadimensionnvector i
C sequently fixed i > 0 via its characterisation in Lemma 1 and
offirst-ordervariablesxandtheBooleanassignmenttothegatesvia
Φ in (1), we will now define the remaining Π -formula
L,w 1
adimensionrvectoroffirst-ordervariablesy,whichareimplicitly Ψ (x ,...,x )foragivenw∈{0,1}n.LetC betheBoolean
assumedtorangeover{0,1}.First,weprovideaformulaensuring Mw 1 i w
circuit succinctly encoded by a Boolean circuit D (t,y,z ,z )
w 1 2
thatthestructureofthegatesofCiscorrectlyencodediny: decidingM onaninputoflength2q(n)isuchthatC consistsof
w w
2r(n) gatesforsomepolynomialr : N → N.Recallthatwecan
Φ (x,y)d=ef
C,gates viewanassignmentoftruthvaluestothegatesofthesuccinctly
(cid:94)  yyii ==11↔↔((yyjj ==11∧∨yykk ==11)) iiffff((ii))==(((cid:107)&,,jj,,kk)) eapnpcorodperdiacteirlceunigttCh.wInasthaefboitllostwriinngg,leotrase=qu(ean1c,e..o.f,baiit)s∈triNngisb,eoaf
yi =1↔yj =0 iff(i)=(∼,j,k) valuation,forany1≤j <ieachaj willbeusedtoencodetheval-
i∈[r] yyii ==1xi iiffff((ii))==((↑↓,10,0,0,0)). uanesdoafitwheililnepnuctogdaetethsewviathluiensdoexft2hqe(ng)a(tejs−w1i)thuipndtoex2q2(qn()nj)−(i1−o1f)Cuwp,
(4) tothegatewithindex2r(n)−1ofC .Soinparticulartheinternal
w
Next,theformulaΦC(x)definedbelownowenablesustodetermine gatesofCwareencodedinai.
whetherCacceptsagiveninputencodedintothefirst-ordervariable Inordertoextractencodingsofbitstringsfromnaturalnum-
x: bΦ¯ers,a(sxa,yfi)rswthsitcehpawsseumpreovxidtoebneeautvraallidfoernmcoudlainsgΦ.TCwhe,0s(exf,oyrm)ualnads
ΦC(x)d=ef ∃x.∃y.∃y.Φbin,n(x,x)∧Φbin,r(y,y)∧ (5) enCawb,l0eustotestwhetherabitofabitstringwhoseindexisgiven
∧Φ (x,y)∧y =1 by y is encoded to be zero in x. Formally, for a valid encoding
C,gates r
a∈Nandforb∈[2r(n)],wehaveΦ (a,b)iffthereisaprime
Φ¯C(x)d=ef ∀x.∀y.∀y.(cid:0)Φbin,n(x,x)∧Φbin,r(y,y)∧ (6) p∈[(b+1)3,(b+2)3)anda≡0mCowd,0p,ora≡0modpforall
∧Φ (x,y)(cid:1)→y =1. primesp∈[(b+1)3,(b+2)3),respectively4.Letr(cid:48)(x)=r(x)+1,
C,gates r
wedefine:
We now show how a predicate determining whether a given
number a ∈ N is a prime number in the interval [b3,(b+1)3) ΦCw,0(x,y)d=ef∃p.Φprime,r(cid:48)(n)(p,y+1)∧Φmod,r(cid:48)(n)(x,p)
forsomeb>0representablebynbitscanberealised.Itiseasily
verifiedthatanynumberinthisintervalcanberepresentedbyatmost Φ¯Cw,0(x,y)d=ef∀p.Φprime,r(cid:48)(n)(p,y+1)→Φmod,r(cid:48)(n)(x,p).
m=3(n+1)bits.Moreoverasdiscussedabove,bothconditions
canbedecidedinpolynomialtime.Thereforewecanconstructin The formulas ΦCw,1(x,y) and Φ¯Cw,1(x,y) testing whether the
logarithmicspaceaBooleancircuitC withm+ninputgates bit with index y is set to 1 in the encoding x can be defined
prime,n
implementingthispredicate[28,Thm.8.1]anddefine analogouslybynegatingΦCw,0(x,y).Thepreviouslyconstructed
formulasnowenableustodefineformulasthatallowforevaluating
Φ (x,y)d=ef∃x.∃y.∃z.Φ (x,x)∧Φ (y,y)∧ the succinctly encoded Cw on an input that is provided on-the-
prime,n bin,m bin,n flyviaa.Givenanindexbimplicitlylessthan2r(n) ofagateof
∧z= (cid:88) 2ixi+2m (cid:88) 2iyi∧ΦCprime,n(z) (7) Cw,representedbythefirst-ordervariabley,andavectorofvalid
i∈[m] i∈[n] encodingsa=(a1,...,ai)representedbythefirst-ordervariables
x,thefollowingformulaΦ (x,y)checkswhetherthevalueof
Cw,(cid:62)
thegatewithindexbissettotrueunderthevaluationaaccording
Φ¯ (x,y)d=ef∀x.∀y.∀z.(cid:0)Φ (x,x)∧Φ (y,y)∧
prime,n bin,m bin,n
∧z= (cid:88) 2ix +2m (cid:88) 2iy (cid:1)→Φ (z). (8)
i i Cprime,n 4Inordertoproperlyhandlethecaseb=0,wehavetoshifttheintervalwe
i∈[m] i∈[n] usefortheencodingbyonefrom[b3,(b+1)3)to[(b+1)3,(b+2)3).
totheconventiondescribedbefore: Algorithm1Deciding∃x .ϕ(x ,...,x )foragiveninstanti-
i+1 1 i+1
Φ (x,y)d=ef (cid:94) (cid:16)(cid:0)2q(n)(j−1)≤y<2q(n)j → ationa1,...ai ∈Nj ofx1,...xi.
Cw,(cid:62) 1≤j<i 1: ϕ(x1,...,xi+1):=DNF(ϕ(x1,...,xi+1))
→ΦCw,1(xj,y)(cid:1)∧(cid:0)2q(n)(i−1)≤y→ΦCw,1(xi,y)(cid:1)(cid:17). 23:: forfoarllaclllaluisteersaψls(tx=1,.p.(.x,1x,i.+.1.),xoif+ϕ1)d<o bofψdo
4: replacetinψwith−p(x1,...,xi+1)≥−b+1
AformulaΦCw,⊥(x,y)testingwhetherthevalueofagateissetto 5: endfor
falsecanbedefinedanalogouslybynegatingΦCw,(cid:62)(x,y).Building 6: forallliteralst=¬(p(x1,...,xi+1)<b)do
uponthoseformulas,wecannowconstructBooleanconnectives 7: replacetinψ(x1,...,xi+1)withp(x1,...,xi+1)≥b
thatallowforcheckingthatthegatesofCwareconsistentlyencoded. 8: endfor
Given a ∈ Ni as above, ΦCw,&(a,b,c1,c2) holds if the logical 9: (S :Axi+1 ≥c):=ψ[a1/x1,...,ai/xi]
and-connectiveholdsforthetruthvaluesofthegateswithindex 10: if S (cid:54)=∅then
b,c1andc2encodedviaa: 11: (cid:74)re(cid:75)turn true
12: endif
Φ (x,y,z ,z )=
Cw,& 1 2 13: endfor
(cid:0) (cid:1)
ΦCw,(cid:62)(x,y)↔ΦCw,(cid:62)(x,z1)∧ΦCw,(cid:62)(x,z2) . 14: return false
TheremainingBooleanconnectivesfoundinDefinition1canbe
reflectedviatheadditionalformulas
Together with Lemma 1, this immediately gives that for any
ΦCw,(cid:107)(x,y,z1,z2), ΦCw,∼(x,y,z1,z2)andΦCw,↓1(x,y,z1,z2) fixedi>0,validityinPA(i+1)isinΣEi+XP1.Wenowshowhowto
whicharedefinedanalogouslytoΦ (x,y,z ,z ).Thesefor- decreasethenumberoforaclecallsbyone.
mulasnowenableustodefineaΠ -Cawna,&logueto1Φ 2 (x,y)in Tothisend,letΦbeaPA(i+1,j)formulainprenexnormal
1 C,gates
(4)inordertocheckiftheBooleanassignmentofthesuccinctly formforafixedi>0andsomej,i.e.,
encodedcircuitCwisconsistent: Φ=∃x .∀x ···Q x .ϕ(x ,...,x ).
1 2 i+1 i+1 1 i+1
ΨCw,gates(x)=∀t.∀y.∀z1.∀z2.ΦDw(t,y,z1,z2)→ InordertodecidevalidityofΦ,byapplicationofProposition7,a
→(cid:94) tt==01→→ΦΦCCww,,&(cid:107)((xx,,yy,,zz11,,zz22)) ΣfoEirXtPh-ealxg1o,r.it.h.mxicasuncahlttehrantat(cid:107)iangkl(cid:107)y≤gu2escs|Φv|a(3lju)ai+ti1onfsoar1a,ll.1..≤aik∈≤Nji
 tt==24→→ΦΦCCww,,(cid:62)(cid:62)((xx,).y)↔ΦCw,⊥(x,z1) wanidthsloemadeincgonzsetraonst,cw>em0,aaynadssbuymaeddthitaiotnaanlylynpuamdbdeinrginvaelvueartyioanks
Here,Φ (t,y,z ,z )isaninstantiationofΦ (x)definedin(5) is represented using 2poly(|Φ|) bits. Consequently, it remains to
acnledan(6a)sfDoΦrwt(hxe)ciirtcius1iotnD2lywo.pTehneiunsxe.oHfoΦwDewve(tr,,tyh,iszC1c,azn2e)aissilnyobtetofitaxleldy sbheodwectihdeadt vinalpidoilytynoomfiQalit+im1xe.i+T1h.iϕs(isa,1o/fxc1o,u.r.s.e,,anoi/txthie,xcai+se1u)ncdaenr
C
byconcatenatingt,y,z andz intoasinglefirst-ordervariableas standardassumptionsfromcomplexitytheory.However,thefinal
1 2
itwasdonein(7)and(8),anddetailshaveonlybeenomittedfor calltotheΣP0-oracleofaΣEiXPalgorithmgetsϕandallakasinput,
thesakeofreadability.Alsonotethatthevaluesoft,y,z andz thelatterbeingofexponentialsizein|Φ|.Informallyspeaking,this
1 2
arethenimplicitlyboundedthroughΦ (t,y,z ,z ). providesuswithsufficientadditionaltimeinordertodecidevalidity
Finally, we can define Ψ (x ,.D.w.,x ), th1e l2ast remaining of
Mw 1 i
formulafrom(1),asfollows Q x .ϕ(a /x ,...,a /x ,x ) (9)
i+1 i+1 1 1 i i i+1
Ψ (x ,...,x )d=ef inpolynomialtimewithrespecttothesizeoftheinput.
M(cid:26)w Ψ1Cw,gatesi(x)∧ΦCw,(cid:62)(x,2r(n)−1) ifiisodd algoArligthomritdhemci1d,inwghvicahlidtiatkyeosfϕaafnodrmthuelaaaksains(i9n)pufot,riesvaenpsie,ui.deo.,
ΨCw,gates(x)→ΦCw,(cid:62)(x,2r(n)−1) ifiiseven Qi+1 =∃.ThecaseQi+1 =∀canbederivedsymmetrically.Let
Inspecting the construction outlined in this section, it is not dif- usdiscussAlgorithm1andanalyseitsrunningtime.InLine1,the
ficult to see that for a given w ∈ {0,1}n the construction of algorithmconvertsϕintodisjunctivenormalform.Thisstepcan
Φ (x ,...,x )istedious,butcanbeperformedinpolynomial beperformedinexponentialtimeDTIME(2O(|Φ|))andthustakes
Lw 1 i
timewithrespectton.Weleaveitasanopenproblemwhetherthis polynomialtimewithrespecttotheinput.StartinginLine2,the
reductioncanactuallybeperformedinlogarithmicspace,though algorithmiteratesoverallclausesψofϕ,andsincethereareatmost
theredonotseemtobeanymajorobstacles.Followingtheargumen- 2O(|Φ|) clauses this iteration is performed at most a polynomial
tationofthissection,weconcludethatΦ istheformularequired number of times with respect to the size of the input. In each
L,w
inProposition6. iteration, in Lines 3–8 the algorithm transforms the disjuncts of
ψ into linear inequalities by eliminating negation. After Line 8,
3.2 UpperBounds ψ isaconjunctionoflinearinequalitiesandthusgivesrisetoan
We will now show that the previously obtained lower bounds equivalent system of linear Diophantine inequalities S in which
havecorrespondingupperbounds.Letusfirstrecallanimproved thefirst-ordervariablesx1,...xiareinstantiatedbythea1,...ai.
version of a result by Reddy & Loveland [32] established by Clearly,Lines3–9canbeexecutedinpolynomialtimewithrespect
Weispfenning[38,Thm.2.2],whichboundsthesolutionintervals tothesizeoftheinput.Finally,inLine10feasibilityofSischecked.
ofPresburgerformulas. To this end, we invoke Proposition 4 from which it follows that
feasibilityofeachScanbedecidedinDTIME(2p(|Φ|)|S|)forsome
Proposition7(Weispfenning[38]). Thereexistsaconstantc>0
polynomialp.Thisstepisagainpolynomialwithrespecttotheinput
suchthatforanyPA(i,j)formulaΦandN ={0,...2c|Φ|(3j)i},Φ totheoraclecall.IfSisfeasiblethealgorithmreturnstrueinLine11.
isvalidiffΦisvalidwhenrestrictingthefirst-ordervariablesofΦ Otherwise,ifnoSisfeasibleforalldisjunctsofϕ,thealgorithm
tobeinterpretedoverelementsfromN. returnsfalseinLine14.Consequently,wehaveshownthefollowing
proposition,whichtogetherwithProposition6completestheproof 4. Ultimately-PeriodicSetsDefinableinthe
ofTheorem1. Σ -fragmentofPresburgerArithmetic
1
Proposition8. Foranyfixedi>0,PA(i+1)isdecidableinΣEiXP. WewillnowapplysometechniquesdevelopedinSection3inorder
toproveTheorem2,i.e.,giveboundsontherepresentationofprojec-
3.3 Discussion tionsofPA(1)formulasopeninonevariableasultimately-periodic
Weconcludethispartofthepaperwithashortdiscussiononthe sets.Formally,givenaPA(1)formulaΦ(x),weareinterestedinthe
relationshipofourproofofthelowerboundofPA(i+1)totheproof representationoftheset
ofaNEXPlowerboundforPA(2)byGra¨del[18],andapplications Φ(x) ={a∈N:Φ(a/x)isvalid}.
ofandresultsderivablefromProposition8. (cid:74) (cid:75)
As it emerged in Section 3.1, at many places we can apply Subsequently,weshowthatthissetisanultimatelyperiodicset
andreuseideasofGra¨del’sNEXP-hardnessproofforPA(2)given whose period is at most doubly-exponential and that this bound
in[18]forourlowerbound.Onemaindifferenceisthatforhishard- is tight. Throughout this section we assume binary encoding of
nessproof,Gra¨delreducesfromaNEXP-completetilingproblem numbersinΦ(x)
thathespecificallyintroducesinordertoshowhardnessforPA(2). WebeginwiththefirstpartofTheorem2andprovethefollowing
Inourpaper,weareintheluckypositionofhavingaccesstotwenty- proposition.
fiveadditionalyearsofdevelopmentsincomputationalcomplexity,
Proposition9. ThereexistsafamilyofΣ -formulasofPresburger
inwhichitturnedoutthatsuccinctencodingsviaBooleancircuits 1
arithmetic (Φ (x)) such that each Φ (x) is a PA(1,O(n))
provideacanonicalwayinordertoshowhardnessresultsforcom- formula withn|Φ (xn)>|0∈ O(n2) and Φn(x) is an ultimately
pdliescxoitvyecrylaossfeaspthoalytninocmluidale-tEimXePa,lsgeoereit.hgm.[1fo6r,2d8e,ci2d9i]n.gMporirmeoavlietry,t[h1e] periodicsetwithpneriodpn ∈22Ω(n). (cid:74) n (cid:75)
alsoenablesustouseBooleancircuitsencodedintoΣ1-respectively Tothisend,wecombineΦmod,n(x,y)from(3)inSection3.1
Πi-formulasinordertodecideprimalityofapositiveintegerofa withthefollowingstatement.
boundedbitsize,whilein[18]thisisachievedbyanapplicationof
theLucasprimalitycriterion,cf. Lehmer’smoregeneralproof[24]. Proposition10(Nair[27]). Letn≥9,then2n ≤lcm{1,...n}≤
Inaddition,Gra¨del’sstrongerstatementthatPA(2)isNEXP-hard 22n.
alreadyforan∃∀∗-quantifierprefixcanberecoveredfromourlower
Wedefine
bound.Evenmoregenerallyfori>1,wecanderiveΣEXP-hardness
i
fromourconstructionfora(∃∀)((i−1)/2)∃∗∀∗quantifierprefixif Φn(x)d=ef∃y.Φmod,n(x,y)∧y>1.
iisodd,andfora∃(∀∃)(i/2−1)∀∗∃∗quantifierprefixifiiseven.
We have |Φ (x)| ∈ O(n2), and, since numbers are encoded in
EventhoughessentiallyalltechnicalresultsrequiredtoproveTheo- n
binary,thatΦ (x)isaPA(1,O(n))formula.Nowa∈ Φ(x) iff
rem1wereavailablewhen[18]waspublished,aswehaveseenin n
thereis1<m<2nsuchthata≡0modm,andconse(cid:74)quent(cid:75)ly
thissectiontheproofofthelowerboundrequiressomesubstantial
technicalefforts,whichisprobablyareasonwhythisresulthasnot Φ (x) = (cid:91) U(0,m,∅,{0})
n
beenobtainedearlier. (cid:74) (cid:75)
1<m<2n
WithregardstoapplicationsofProposition8,wegiveanexample
=U(0,p,∅,{a:a∈[p],m|a,1<m<2n}),
ofaresultwhichcanbeobtainedasacorollaryofthisproposition.
In[21],Huynhinvestigatesthecomplexityoftheinclusionprob- where p d=ef lcm{1,...2n −1}. By Proposition 10, p ∈ 22Ω(n),
lem for context-free commutative grammars. Given context-free
whichyieldsthelowerboundforTheorem2.
grammarsG ,G ,thisproblemistodeterminewhethertheParikh
1 2 Turning now towards the upper bound, the remainder of this
image5ofthelanguagedefinedbyG isincludedintheParikhim-
1 sectionisdevotedtoprovingthesecondpartofTheorem2,i.e.,the
ageofthelanguagedefinedbyG .Buildinguponacarefulanalysis
2 followingstatement.
ofthesemi-linearsetsobtainedfromParikhimagesofcontext-free
grammarsduetoGinsburg[14]andbyestablishingaCarathe´odory- Proposition 11. For any Σ -formula Φ(x), we have Φ(x) =
1
typetheoremforintegercones,Huynhshowsthatthecomplement U(t,p,B,R)suchthatt∈2poly(|Φ(x)|)andp∈22poly(|Φ(cid:74)(x)|). (cid:75)
ofthisproblemisinNEXP.Thisresultcanhowevernoweasilybe
obtainedasacorollaryofProposition8:Vermaetal.haveshown As a first step, we consider projections of sets of solutions
thattheParikhimageofacontext-freegrammarcanbedefinedin of systems of linear Diophantine inequalities. To this end, let
termsofaΣ -formulaofPresburgerarithmeticlinearinthesizeof S : Ax ≥ c be such a system. From Proposition 5, we have
1 (cid:83)
thegrammar[37].Non-inclusionthenreducestocheckingvalidity that S = i∈IL(bi;Pi)forsomeindexsetI.LetMi bethe
ofaΣ2-sentence,whichyieldsthefollowingcorollary. proje(cid:74)cti(cid:75)onofL(bi;Pi)onthefirstcomponent.WegetthatMican
beobtainedas
Corollary1. Non-inclusionbetweenParikhimagesofcontext-free
grammarsisinNEXP. Mi ={bi+λi,1p1+···λrpi,ri :λk ∈N}
=b +g ·{λ p /g +···λ p /g :λ ∈N}
Of course, the “hard work” of the upper bound is done in i i 1 i,1 i r i,ri i k
Proposition4,butneverthelessweareabletoobtainasuccinctproof forsomeb ,p <···<p andg =gcd{p ,...p }.Since
of Huynh’s result. In general, the NEXP upper bound for PA(2) gcd{p /ig ,i.,.1.p /g }=i,ri1,itisifolklorethai,t1 i,ri
providesagenericupperboundfornon-inclusionproblemsthatcan i,1 i i,ri i
bereducedtocheckinginclusionbetweensemi-linearsetsdefinable Mi =bi+gi·U(t(cid:48)i+1,1,Bi(cid:48),{0})
via PA(1) formulas. For context-free commutative grammars, it forsomeB(cid:48) ⊆ [t(cid:48)]andt(cid:48) ∈ NknownastheFrobeniusnumber
shouldhoweverbenotedthatitisnotknownwhetherthisupper i i i
ofp /g ,...p /g .Givenco-primepositiveintegers1<a <
boundistight,thebestknownlowerboundbeingΣP2 [21]. a <i,1···i< a i,∈riN,itheFrobeniusnumberf ∈ Nisthelarg1est
2 k
positiveintegernotexpressibleasapositivelinearcombinationof
5TheParikhimageofawordw∈Σ∗isavectorofnaturalsofdimension a ,...a andcanbeboundedasfollows.
1 k
|Σ|countingthenumberoftimeseachalphabetsymboloccursinw.
Proposition12(Wilf[39]). Let1 < a1 < a2 < ··· < ak ∈ N [3] LeonardBerman. Thecomplexityoflogicaltheories. Theoretical
bepairwiseco-prime.ThentheFrobeniusnumberf isboundedby ComputerScience,11(1):71–77,1980.
f ≤a2k. [4] AlexisBe`s. Asurveyofarithmeticaldefinability. InAtributeto
Hence, for some t ≤ b + g (p /g )2 ≤ b + p2 we MauriceBoffa,pages1–54.Socie´te´Mathe´matiquedeBelgique,2002.
consequentlyhave i i i i,ri i i i,ri [5] ItshakBoroshandLeonB.Treybing. Boundsonpositiveintegral
solutionsoflinearDiophantineequations.ProceedingsofttheAmerican
Mi =U(ti,pi,Bi,{0}) (10) MathematicalSociety,55:299–304,1976.
forsomep ≤p . [6] Yuan-YouFu-RuiCheng.Explicitestimateonprimesbetweenconsec-
i i,ri utivecubes.RockyMountainJournalofMathematics,40(1):117–153,
LetΦ(x)=∃x.ϕ(x,x)beaPA(1)formula.FromAlgorithm1
2010.
wecanderivethat
(cid:91) (cid:91) [7] KevinJ.ComptonandC.WardHenson.Auniformmethodforproving
ϕ(x,x) = Sj = L(bi;Pi), lower bounds on the computational complexity of logical theories.
(cid:74) (cid:75) j∈J(cid:74) (cid:75) i∈I AnnalsofPureandAppliedLogic,48(1):1–79,1990.
where each S : A (x,x) ≥ c is a system of linear Diophan- [8] D.C.Cooper. Theoremprovinginarithmeticwithoutmultiplication.
j j j
MachineIntelligence,7:91–99,1972.
tine inequalities obtained from one disjunct of the disjunctive
normal form of ϕ, similar as in Line 9 of Algorithm 1. Clearly, [9] AntoineDurand-GasselinandPeterHabermehl.Ehrenfeucht-Fra¨ısse´
(cid:107)A (cid:107),(cid:107)c (cid:107)≤(cid:107)Φ(x)(cid:107)+1foralli∈I.Moreover,fromProposi- goes elementarily automatic for structures of bounded degree. In
j j
tion5wederivethat S = (cid:83) L(b ;P )forsomeindexset ChristophDu¨rrandThomasWilke,editors,29thInternationalSym-
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i i∈I
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