Page 1

ReportontheDagstuhl?Seminar

?Structure and Complexity?

Organizers?

Eric Allender ?RutgersUniv ersity?

Uwe Sch? oning?Universit?at Ulm?

KlausW?W agner?Universit?atW? urzburg?

Septemb er???October ??????

The seminar ?StructureandComplexit y?w as thethird DagstuhlSeminar

dev oted tothe structural asp ects ofComputational Complexity Theory? It

w as atten tedby ??scientists whoin ?? talkspresen ted newresults inthis

?eld? Thefollo wingtopicsw ere amongthemain sub jectscoveredby thetalks?

Kolmogorov complexity? isomorphismtheory? resource?bounded measures?

relativizations? randomness?leaflanguagecharacterizations?circuit theory?

logicalc haracterizationsofcomplexityclasses?interactiveproofsystems?one?

wayfunctions?andcomputationalmodels?

?

Page 2

PROGRAM

F aridAblay ev

OnthePow erof RandomizedBranc hing Programs

EricAllender

Recent Progress ontheIsomorphism Conjecture

Jos?e L?Balc?azar

Kolmogorov?Easy Circuit Expressions

BerndBorchert

Lookingfor anAnalogue ofRice?sTheorem inComplexity Theory

HarryBuhrman

SixHypotheses

StephenF enner

AV ariant ofImm unity? Btt?Reductions? andMinimal Programs

LanceF ortnow

Two Queries

JudyGoldsmith

TheComplexity ofDeterministically Observ ableFinite?Horizon

Markov DecisionProcesses

Mon tserratHermo

Compressibility ofIn?niteBinary Sequences

UlrichHertrampf

The Shap esofT rees

Klaus?J?orn Lange

Mangrove Deforestation?Algorithms forUnambiguousLogspace

Classes

Jack H? Lutz

Equivalence ofMeasures ofComplexityClasses

PierreMcKenzie

NondeterministicNC

?

Computation

?

Page 3

R? udigerReisch uk

Interactive Proofs with PublicCoins andSmallSpace Bounds

J?org Rothe

Characterizations oftheExistence ofP artial andT otalOne?Way

P ermutations

JamesRoy er

Complexity atHigherT yp es

Miklos Santha

A DecisionPro cedurefor Unitary Linear QuantumCellular Au?

tomata

Rainer Sch uler

IsT estingMore ComplexThanQuerying?

ThomasSchw entick

AlgebraicandLogicalCharacterizations ofDeterministicLinear

Time Classes

Martin Strauss

AnInformation?TheoreticT reatment ofRandom?Self?Reducibility

DenisTh? erien

TheCrane Beach Conjecture

Thomas Thierauf

TheIsomorphism Problem forOne?Time?Only Branc hingPrograms

andArithmetic Circuits

NicolaiV ereshchagin

Boolean DecisionT rees andStructure ofRelativizedComplexity

Classes

Herib ertVollmer

Lindstr? omQuan ti?ers in Complexity Theory

JieWang

RandomIsomorphisms

?

Page 4

OsamuW atanabe

SomeCombinatorialProblem fromtheStudy ofLo callyRandom

Reducibility

GerdW echsung

Query Order

?

Page 5

ABSTRACTS

OnthePow erofRandomized BranchingPrograms

F aridAblay ev

Kazan University

?jointw orkwith Marek Karpinski?Univ ersit? atBonn?

We de?nea notionof randomized branching programsina naturalway

similartothe notionofrandomized circuits?We presenttwo explicitbo olean

functionsf

n

?f???g

?n

?f???g andg

n

?f???g

n

?f???g such that?

??f

n

canbe computedbya randomizedordered read?oncebranching pro?

gram ofsizep olynomialinn and witha small?constan t?error?

??anynondeterministicorderedread?k?timesbranchingprogramthatcom?

putesf

n

needsexponentialsize?thesizeis??exp?n???k???????

??g

n

canbecomputedbyanondeterministicread?oncebranchingpro?

gramofsizepolynomialinn?and

??anyrandomizedorderedread?oncebranchingprogramthatcomputes

g

n

withaconstanterror?hassizenolessthanexp?c???n?logn??

RecentProgress on the Isomorphism Conjecture

EricAllender

RutgersUniversity

http???www?cs?rutgers?edu? ?all ender

?jointwork withManindra Agraw aland Stev enRudic h?

In thistalkI willdiscuss recentprogressby Manindra Agrawal? Steven

Rudich? andm yself?sho wing unexpected similarities inthe structureof com?

plete sets?We show that forany complexity classC closed underman y?one

?

Page 6

reductions computablein uniformTC

?

? thefollo wing aretrue?

Gap?The setsthat arecompleteforC underAC

?

andNC

?

reducibility co?

incide?

Isomorphism?Thesets complete forC underAC

?

reductionsare alliso?

morphicunder isomorphismscomputable andinv ertiblebyAC

?

circuitsof

depth three?

OurGapTheorem do esnotholdforstronglyuniformreductions?we

show thatthere areDlogtime?uniformAC

?

?completesets for NC

?

thatare

notDlogtime?uniform NC

?

?complete?

An important op en problemisthe question of whetherthere isany natural

complexityclass suchthat thesetscompleteunderpolynomial?time ?ormore

powerful?reductionsare notalreadycompleteunderNC

?

reductions?

?Thiswork extends thepaperby Agraw alandmyselfinthe???? Com?

putationalComplexityconference?Afull pap er isavailable at

http???www?cs?rutgers?edu??al lender?publications ??

Kolmogorov?EasyCircuitExpressions

Jos?eL?Balc?azar

UniversitatPolit? ecnica de Cataluny a? Barcelona

http???www?lsi?upc?es??balqui

?jointw orkwithHarry Buhrmanand Mon tserratHermo?

Circuitexpressionsw erein tro duced to providea naturallinkb etw een

ComputationalLearningand certainaspects ofStructuralComplexity? Up?

p er and low erb ounds on thelearnability of circuitexpressions arekno wn?

We studyherethe casein which thecircuit expressionsare oflow ?time?

bounded? Kolmogorov complexity?We show that these arep olynomial?time

learnableifand only ifacceptingcomputations of acceptingnondeterminis?

tic exp onen tial?timemac hinescanbe found deterministically inexp onential

time?Wealsoexactlycharacterize? in termsofadvice classes and of doubly

tallypolynomial?timedegrees? thesetsthat have sucheasycircuit expres?

sions? andobtain consequencesregardingtheir lowness?

?Afull paper isav ailableat

http???www?lsi?upc?es??balqui ?pos tscript ?q?chara ct?ps??

?

Page 7

Lo oking foranAnalogue ofRice?s Theorem in

Complexity Theory

BerndBorchert

Universit? atHeidelberg

http???math?uni?heidelberg ?de?log ic?b b?bb ?html

?jointw orkwithFrank Stephan? Univ ersit? atHeidelberg?

Rice?sTheoremsa ys that every non trivialseman ticalprop erty of pro?

grams isundecidable? In thisspiritweshow thefollo wing?Every non trivial

absolute ?gap?relativ e? coun tingprop erty ofcircuitsis UP?hardwith resp ect

top olynomial?timeT uring reductions?

?A fullpap erisavailable at

http???math?uni?heidelber g?de ?logic?bb?p aper s?Ri ce? ps??

SixHypotheses

HarryBuhrman

CWIAmsterdam

?jointwork withLanceF ortno w?University ofChicago?

We considerthefollowing sixh ypotheses?

???P? NP?

??? SAT?

p

tt

PSEL?

??? SAT?

p

tt

K?APPRO X?

??? SAT isO?logn??APPRO X?

??? FP

NP?log?

? FP

NP

jj

? and

?????SAT? SAT?? P?

Itiskno wn that ????? ????? ????? ??????????? ????

We show the follo wing?

?

Page 8

??If??NP??? ?? then???????are false?

??If??NP???? andsymmetry of information w?r?t?p olynomial?time

CD

p oly

holds ?i?e????x?y? ?CD

p

?xy?? CD

p

?x?? CD

p

?yjx??O ?logn????

then??????? arefalse?

?? Symmetryofinformation alone implies????? ????

??Consider theconjecture thatthere existsahierarchy theorem for com?

pressibilit y?F oreachpolynomialp

?

thereexistsapolynomialp

?

such

that foreveryn thereexistsastringx of lengthnsatisfying

CD

p

?

?x??O?logn??CD

p

?

?x??n?O?logn??

Theconjectureimpliesthat

?a?R?P and

?b? ?????????

Thepro ofs use essentially anupp erb ound lemmafor CDcomplexity anda

construction of Zuck erman?

AV ariant of Immunity? Btt?Reductions? andMinimal

Programs

StephenF enner

University of Southern Maine

Wew ork witha generalizationofimmunity calledk ?imm unity?k? ???

Itis sho wn thatifa setA isk ?imm une?thenK ??

k?tt

A? whereK is the

haltingproblem?It is also shownthat MIN? theset ofminimal indicesfor

partial computablefunctions?isk ?imm uneforallk? andhenceK ??

btt

MIN

?MIN

df

?fej??i?e???

i

???

e

?g??Ashort historyof MINis given? andit

ismentioned that MINis notregressiv e? allregressive setsareeithercom?

putably enumerable ork ?immune for allk? andevery computablyen umerable

non?computable T?degreecontainsak ?immune?non??k? ???immune set?for

allk?

?Current sources aretwo tec hnicalreports found ontheWWW at

http???www?cs?usm?maine?edu??

??

Page 9

Two Queries

LanceF ortnow

CWI? Amsterdam? and TheUniversity ofChicago

http???www?cs?uchicago?edu??fo rtnow

?jointw orkwith Harry BuhrmanatCWI?

Hemaspaandra?Hemaspaandra and Hempelshow ed that fork? ?? if

P

?

p

k

???

tt

?P

?

p

k

???

then?

p

k

??

p

k

?We extendtheir techniquesto show that if

P

?

p

?

???

tt

?P

?

p

?

???

then?

p

?

??

p

?

?

Howev er? thetechniques cannotbe pushed down tok? ??We showa

relativizedw orldwhereP

NP ???

tt

?P

NP???

but NP?? coNP?

What do eshappen whenP

NP???

tt

?P

NP???

? Kadin showsthat

NP? coNP ?poly and thus PH??

p

?

?Y ap??Beigel? Chang and Ogihara

buildingonChang andKadinimprove thisto show that every language in

thep olynomial?timehierarchy canbe solv edby an NPqueryand an?

p

?

query?

Building on the techniques of the above pap erswe show several new

collapsesifP

NP???

tt

?P

NP???

including?

? Lo cally eitherNP? coNP orNP hasp olynomial?sizecircuits?

?P

NP

?P

NP???

?

??

p

?

? UP

NP???

? RP

NP???

?

? PH? BPP

NP???

?

?The paper canbefound at

http???www?cs?uchicago?ed u??f ortnow? papers??

??

Page 10

TheComplexity ofDeterministically Observable

Finite?Horizon Markov Decision Processes

Judy Goldsmith

Univ ersity of Kentuc ky

http???www?cs?engr?uky?edu??gold smit

?jointw orkwithMartinMundhenk?Univ ersit? atTrier?and Chris Lusena?

University ofKentuc ky?

Weconsider the complexity of the decisionproblemfor di?erentt yp es of

partially?observ able Markov decision processes ?MDPs?? given anMDP? does

there existapolicy withperformance? ??Low erandupp erb oundson the

complexity ofthedecisionproblems aresho wn interms ofcompleteness for

NL? P? NP?PSPA CE? EXP? NEXPor EXPSPA CE? dependent onthet ype of

theMarkovdecision process?F orseveral NP?completet ypes?we show that

theyare not ev enp olynomial?time??approximablefor any ?xed?? unless

P? NP? Theseresults also reveal interesting trade?o?sb etw een thepow er

ofp olicies?observ ations?andrew ards?

Compressibility of In?niteBinary Sequences

MontserratHermo

Universit?atT rier

?jointw orkwith RicardGav ald?a andJos?e L? Balc?azar? Barcelona?

It is known thatin?nite binarysequences ofconstant Kolmogorov com?

plexity are exactlythe recursive ones? Suchakind ofstatement no longer

holdsin thepresence of resourceb ounds?Con traryto whatin tuition might

suggest? there are sequencesofconstan t?polynomial?timebounded Kol?

mogorovcomplexity that are notpolynomial?time computable? Thismo?

tiv atesthestudy ofseveral resource?b oundedvarian tsin search forac har?

acterization?similar inspirit? ofthep olynomial?time computablesequences?

We proposea de?nition?basedon Kobay ashi?s notionof compressibility? and

compare ittob oth thestandardresource?b oundedKolmogorov complex?

ity ofin?nite strings? andtheuniform complexity intro ducedby Lov eland?

Some nontrivial coincidences and disagreementsareprov ed?The resource?

unb ounded case isalso considered?

??

Page 11

The Shapes ofT rees

Ulrich Hertrampf

Universit? at Stuttgart

We investigate thein?uence ofthet ypesofcomputation trees? allow ed

inaleaflanguage description ofa certain complexity class?on thecom?

putationalpow er of thisclass?To this end?we in tro ducethreep otentially

di?erentclasses Leaf

p

?A??BalLeaf

p

?A??and FBTLeaf

p

?A?? theclassesof sets

recognizedusingpolynomial?time machineswithacceptance de?nedby leaf

languageA? andarbitrary?balanced? orfull binarytrees astheircomputation

trees?F orlanguageclassesC?wefurther de?neLeaf

p

?C?

df

?

S

A?C

Leaf

p

?A? and

analogously BalLeaf

p

?C ?? andFBTLeaf

p

?C ??

F orA?N ?theclass of languagesha vinga so?calledneutr al letter?? it

is easy tosee thatLeaf

p

?A?? BalLeaf

p

?A?? FBTLeaf

p

?A?? On the other

hand? in ?U?Hertrampf? H?V ollmer? K?W?W agner? OnBalancedVersus

UnbalancedComputationT rees? Mathematical Systems The ory? ????? itw as

sho wnthat BalLeaf

p

?DLOGTIME?? FBTLeaf

p

?DLOGTIME?? P?but

Leaf

p

?DLOGTIME??P

PP

?

F or thesetREG ofregularlanguages? itw assho wnthatLeaf

p

?REG??

BalLeaf

p

?REG??FBTLeaf

p

?REG?? PSPA CE ?cf??U? Hertrampf etal?? On

thePow er ofP olynomialTime BitReductions? Structures???????

Ourmainresultsare?

???L? REG???L

?

? REG??Leaf

p

?L?? Leaf

p

?L

?

?? BalLeaf

p

?L

?

??

FBTLeaf

p

?L

?

???

???L? REG???L

?

?REG? ?BalLeaf

p

?L?? BalLeaf

p

?L

?

?? FBTLeaf

p

?L

?

???

Moreov er?de?ning LEAF

p

?C?

df

?fLeaf

p

?A?jA?Cg? andBALLEAF

p

?C?

andFBTLEAF

p

?C?analogously?we obtain the existenceof sets

C

?

?C

?

?C

?

? REGsuchthat

? FBTLEAF

p

?C

?

??BALLEAF

p

?C

?

?? LEAF

p

?C

?

??

? FBTLEAF

p

?C

?

?? LEAF

p

?C

?

?? BALLEAF

p

?C

?

??and

? LEAF

p

?C

?

?? BALLEAF

p

?C

?

?? FBTLEAF

p

?C

?

??

If?P? NP? then allthese inclusionsare strict?

?Theseresultsare from?U? Hertrampf?RegularLeaf Languages and

?Non?? RegularT ree Shap es? TR??????Universit?atL? ub eck???????

??

Page 12

Mangrove Deforestation?Algorithms forUnam biguous

Logspace Classes

Klaus?J?orn Lange

Univ ersit? atT?ubingen

http???www?fs?informati k?uni?tuebi ngen?de??lan ge

?jointw orkwithEricAllender?

Therearesev eralv ersionsof unambigouoslog?spaceclasses?We givea

completeproblem for the classR USPA CE?logn??making it the ?rst non?

syntactical classwithacomplete set?In thetime?b oundedcasethereare

relativizationsexcluding theexistence ofcomplete sets?

Further on? inajointw ork withERICALLENDER?deterministic al?

gorithms for thememb ershipproblems of theelemen ts inR USPA CE?logn?

usingO?log

?

n? logn??o?log

?

n?space areconstructed? Asa consequence

we get parallelalgorithms running onrather simple machinemodels?

?Thesecond part hasapp earedasTR ??????at

http???www?eccc?uni?trier ?de? the ?rstpart willapp earinthe Chicago

JournalofThe oretical Computer Science??

Equiv alenceofMeasures ofComplexity Classes

Jack H? Lutz

Iowa State Univ ersity

?jointw ork with JosefBreutzmann?

The resource?b ounded measures ofcomplexity classesaresho wn tobe ro?

bustwithrespect tocertainc hanges inthe underlyingprobabibility measure?

Sp eci?cally? forany realn umb er?? ??anyuniformlypolynomial?time com?

putable sequence

??

????

?

??

?

??

?

????? of realn umb ers?biases??

i

????????

and any complexity classC ?such asP? NP? BPP? P?poly? PH?PSPA CE?

etc?? thatis closed underp ositivep olynomial?timetruth?table reductions

withqueries of atmostlinear length?it is sho wn thatthe follo wingtwo

conditions areequiv alen t?

??

Page 13

??C hasp?measure? ?resp ectively? measure? in E? measure? inE

?

?

relative tothe coin?toss probability measuregiv enby the sequence

??

??

??C hasp?measure? ?respectiv ely? measure? in E?measure? inE

?

?

relative to theuniformprobabilitymeasure?

Theproofintro duces three techniques thatmaybeuseful in other con?

texts? namely? ?i? thetransformation of ane?cient martingalefor oneproba?

bilitymeasure in toane?cient martingalefora?nearb y?probabilitymeasure?

?ii? theconstruction ofap ositivebiasre duction?a truth?tablereduction that

enco desap ositiv e? e?cient?appro ximatesim ulationof onebias sequenceby

another?and ?iii? theuse of sucha reductionto dilateane?cient martin?

gale forthesimulated probability measure in toane?cient martingale forthe

simulatingprobabilitymeasure?

Nondeterministic NC

?

Computation

Pierre McKenzie

Univ ersit?e de Mon treal

http???www?iro?umontrea l?ca ??mckenz ie

?jointw ork withHerv?eCaussin us? Montreal?Denis Th? erien? McGill?and

Herib ertV ollmer?W? urzburg?

We present and extendresults fromour ????ComputationalComplex?

ityConference pap erwiththesametitle?We de?nethe coun tingclasses

?NC

?

?GapNC

?

? PNC

?

? andC? NC

?

?We prove thatbo oleancircuits? alge?

braiccircuits? programsov ernondeterministic ?nite automata?and programs

ov er constant integer matricesyieldequiv alent de?nitions of thelatterthree

classes? Alternativede?nitions ofnondeterministic NC

?

computationlead to

the op enquestionof whetherevaluating log?depthf?? ?g form ulasov erIN

reduces tomultiplying constant size matricesov erIN?

Thenwe adapt theleaf languageconcept tothelev el ofNC

?

?We show

how knowncharacterizations implyA CC

?

? MODPHand TC

?

?CH?We

thenextend thesetec hniquestoprove thatMODPH ?resp??CH? contains

languages which have noA CC

?

?t ype circuits?resp?? TC

?

?t ypecircuits? of

??

Page 14

size?

g?n?

? providedg?n?satis?es?

??????p olynomialsp??p?n?

g?p?n??g ??

p?n?

??

?o??

n

?

???

This ine?ectmatchesEricAllender?sCOCOON??? low erbounds? but

withan ev ensimpler proof which do esrequireEric?sstronger form of the

time hierarchy theorem? Eric?sconstructiveCOCOON??? low erb ounds can

alsobededuced?

Interactive Proofs withPublic Coinsand SmallSpace

Bounds

R? udigerReisch uk

Medizinische Univ ersit? atzuL? ubeck

?jointw orkwithMaciej Liskiewicz?

Weconsider in teractive pro of systemswithab ound on thespace used

by thev eri?er?While such systemswitha logarithmic spacebound seem

tobe extremelypow erful if the random coin ?ipsarek ept secret?restricting

topublic coinsone sta yswithinP? An alternativec haracterizationcanbe

giv enby Arthur?Merlin?Games? that meansstoc hasticT uring machines that

alternateb etw eenprobabilistic?A? andexisten tial ?M?con?gurations?

Let AM

k

Space?S? ?resp?MA

k

Space?S ?? denotethecorresp ondingcom?

plexity classes? wherethemac hines startina probabilistic?resp?existen?

tial?con?guration and useat most

?

spaceS?W

?

e prove forthe language

P ATTERN

df

?fw

?

?w

?

????w

m

??u??BIN ??

l

?jl?IN? u?w

i

?f???g

?

?

juj?l??iu?w

i

g thefollowingresults?

P ATTERN? MA

?

Space ?loglog??

P ATTERN ??AM

?

Space?o?log???

P ATTERN ??AM

?

Space?o?log???

Thisyields thehierarchy AMSpace?S?? AMSpace?S?? AM

?

Space?S?

for any sublogarithmicspaceboundS?At theendwe discusshow this

hierarchy mightbe extended?

??

Page 15

Characterizations of theExistence ofPartialandT otal

One?WayP ermutations

J?org Rothe

F riedric h?Schiller?Universit? atJena

http???www?minet?uni?jena? de??rothe

?jointwork withLane A?Hemaspaandra?Univ ersity of Roc hester?

We study theeasycerti?cateclasses introducedbyHemaspaandra? Rothe?

andW echsung ?cf??EasySets and HardCerti?cate Sc hemes??to app ear in

A ctaInformatica ?? withregardto the questionofwhether or notsurjective

one?wayfunctions exist?This isan imp ortant op enquestion incryptology?

We show that theexistence of partial one?wayp ermutations canbec harac?

terizedby separatingP from theclass ofUP sets that? forall unambiguous

p olynomial?timeT uring machines accepting them?alwa ys have easy ?i?e??

p olynomial?timecomputable?certi?cates? Similarresultsc haracterizing cer?

taint ypes ofp oly?one one?way functions are given?

This extendsthew orkofGrollmann andSelman ??Complexity Measures

for Public?KeyCryptosystems?? SIAMJournalof Computing? ?????and Al?

lender??The Complexity ofSparse SetsinP ??Structur es??????By Gr? adel?s

recent results ab outone?way functions??De?nability onFinite Structures

and theExistence of One?WayF unctions??Metho ds ofLo gicinComputer

Science? ?????? thisis also linked tostatemen ts in?nite mo deltheory?

Finally?we establisha conditionnecessary and su?cient for theexistence

of total one?wayp ermutations?

Complexity at HigherT ypes

James Roy er

Syracuse University

http???top?cis?syr?edu?peo ple?royer?r oyer ?html

Constable? in?????p osedthe problemofw orkingouta computational

complexity theoryfor functionalsand operatorsoft yp e?? and higher?In

particular?hew anteda goodt yp e??analogue ofpolynomial?time?Progress

??

Page 16

onthesequestions hasb eenslow incoming? In largepartthis isb ecause

understanding thedynamic complexity oft ype?? computationsissurprisingly

tric ky? In???? Kapronand Cook prov eda machinecharacterization ofBFF

?

?

a particulart yp e??analogueofpolynomial?time and inthe process intro duced

several lov elyideasthathaveb een central torecent progressonConstable?s

problems?I will surv eythisw ork? focusingon the recent resultsofmyself

and others?I willalso illustratewhy sortingoutcomplexity att yp e??and

above willbe anev entric kier en terprise?

A Decision Pro cedurefor UnitaryLinear Quan tum

Cellular Automata

Miklos Santha

Universit?eP aris?Sud

Linear quantum cellular automataw ere intro ducedrecen tlyas oneof the

models ofquan tum computing?A basicp ostulateof quan tum mechanics

imp osesa strongconstraint on any quantummachine? ithas tobe unitary?

that is itstime evolutionop eratorhas tobea unitary transformation?In

this paperwe give an e?cient algorithm todecideifa linearquan tum cellular

automaton is unitary? The complexity of the algorithmisO?n

?r??

r ??

??O?n

?

?

iftheautomaton hasa con tin uousneighb orhood of sizer?

IsT estingMoreComplex ThanQuerying?

Rainer Sch uler

Univ ersit? atUlm

In thistalkwe consider thecomplexity of generatinginstances of NP

search problemswithone ofits solutionsunder some pre?c hosendistribution?

In particular?ifthe distribution is su?cientlynatural? i?e??polynomial?time

samplable? thengenerating onlyinstances?askingquestions? is ?easy?for

ev eryproblem?We observe that forevery probleminR ?randomp olynomial

??

Page 17

time?? generatinginstances withsolutions iseasy forpolynomial?time sam?

plabledistributions? It turnsout thatevery problemthat allo wsgenerating

instances withsolutions ?certi?ed instances? iscon tainedin co?AM?where

AM isthe classofArthur?Merlin gamesin tro ducedby Babai? Thisshows

thatitis unlik elythatev eryNPsearch problemallows generatingcerti?ed

instances? Nevertheless? it isstillp ossiblethat natural problems notkno wn

tobe inP? like graphisomophism? fallinto this class?

AlgebraicandLogicalCharacterizations of

Deterministic Linear TimeClasses

Thomas Schw entick

Univ ersit?atMainz

http???www?informatik?uni ?mainz?d e?P ERSONEN? Schwent ick?eng? html

Analgebraicc haracterizationoftheclass DLIN of functionsthatcan

be computed inlinear timebya deterministic RAMusing onlyn umb ers of

linearsize isgiven? Thisclassw asin troducedby Grandjean?whoshow ed

that it isrobust andcon tainsmost computationalproblems that are usually

consideredtobe solv able indeterministiclineartime?

Thecharacterization is interms ofa recursion schemefor unary functions?

Av ariationof this recursionschemecharacterizes theclassDLINEAR?which

allo wsp olynomially largen umbers?A secondv ariationde?nesa classthat

stillcon tains DTIME?n?? theclassoffunctions thatare computable inlinear

timeonaT uring machine?

F rom thesealgebraicc haracterizations logicalc haracterizationsof DLIN

andDLINEAR asw ellas completeproblemsfor theseclasses?under

DTIME?n? reductions? arederiv ed?

??

Page 18

AnInformation?TheoreticTreatment of

Random?Self?Reducibility

MartinStrauss

Iowa StateUniv ersity

http???www?cs?iastate?edu? ?ms trauss?h omepage ?html

?jointw orkwith JoanF eigenbaum?A T?TLabs?

Informally?afunctionf isr andom?self?re ducibleif the ev aluationoff at

any giv eninstancex canbe reducedinp olynomialtimeto the evaluation

off atone or morer andom instancesy

i

? each one of which isuncorrelated

withx?

Random?self?reduciblefunctions have manyapplications?including

av erage?casecomplexity? low erb ounds?cryptography? interactive pro ofsys?

tems? andprogramc heck ers?self?testers? andself?correctors?

Inthis paper?we initiatethe studyofrandom?self?reducibility from an

information?theoreticp oint ofview?Sp eci?cally?we formallyde?nethe no?

tionofa random?self?reduction that? withresp ect toa given ensemble of

distributions? leaksa limitedn umb er ofbits?i?e?? producesy

i

?s in sucha man?

ner thateach hasa limitedamount ofm utualinformationwithx?We argue

thatthis notion is usefulinstudying the relationshipsb etw een random?self?

reducibility andother properties of in terest?including self?correctability and

NP?hardness? Inthe case of self?correctability?we show thattheinformation?

theoreticde?nition ofrandom?self?reducibilityleads to somewhatdi?erent

conclusions from thosedra wnbyF eigenbaum?F ortno w? Laplan te? and

Naik ???? whoused thestandardde?nition? In the caseofNP?hardness?we

use theinformation?theoretic de?nitiontostrengthen the resultofF eigen?

baumandF ortnow ????who prov ed? using thestandardde?nition?that the

polynomialhierarchycollapses ifan NP?hardset israndom?self?reducible?

?The full paper canbe foundat

http???www?research?att?c om?libra ry?trs?TRs????? ??????? ??????body? ps??

References

???J?Feigenbaum and L?Fortno w?Random?self?reducibilityof complete

sets?SIAM Journalon Computing? ?????????????????

??

Page 19

???J?Feigen baum? L?F ortnow?S?Laplan te? andA?Naik? Oncoherence?

random?self?reducibility? and self?correction? InPro c???thConfer ence

onComputational Complexity?pages ?????? IEEEComputer Society

Press? LosAlamitos? ?????

TheCrane Beach Conjecture

DenisTh? erien

McGill Univ ersity? Mon treal

LetL??

?

assumption

?e? ??e is neutral forL if and only if?forall u?v??

?

?

uv isinL if and onlyif uevis inL? The CraneBeach Conjecture says

thefollo wing?IfL isa language witha neutralletter? thenL is inAC

?

if andonly ifL isa regular??free set?Inlogicalterms? thismeans that?

assumingthe presence ofa neutralletter? any language de?nablebya ?rst?

order sen tenceusing arbitraryn umericalpredicates canalsobe represen ted

bya ?rst?order form ulausing? only? UsingtheF urst?Saxe?Sipser?Ajtai

circuitlow erb ound forP ARITY?we know the conjecturetobetrue under

theadditional thatthelanguage is regular?On theotherhand?

an independent proof of theCraneBeach Conjecturew ould implythe circuit

low erb ound?

The IsomorphismProblemfor One?Time?Only

Branching Programsand ArithmeticCircuits

Thomas Thierauf

Universit? at Ulm

We inv estigate thecomputational complexity oftheisomorphism prob?

lem for one?time?onlybranchingprograms ???BPI??On inputoftwo such

programs?B

?

andB

?

? decide whetherthere existsap erm utationof thev ari?

ablesofB

?

such that itb ecomes equivalent toB

?

?

??

Page 20

We showthat??BPI cannotbe NP?hard?unless thepolynomial hierarchy

collapses tothe second lev el?The resultis extended to theisomorphism

problemfor arithmeticcircuitsov er largeenough ?elds?

Bo olean DecisionT rees andStructure ofRelativized

Complexity Classes

NicolaiV ereshc hagin

Moscow StateUniv ersity

We de?neanalogs of thecomplexity classesP? NP? AM? and IPforthe

Booleandecision treesmo del? Theanalogs aredenoted respectiv elybyP

dt

?

NP

dt

? AM

dt

? andIP

dt

? Itis known thatP

dt

?? NP

dt

? but NP

dt

? coNP

dt

?

P

dt

? BPP

dt

andev enIP

dt

? coIP

dt

?P

dt

? Theop en problem iswhether

IP

dt

? NP

dt

ornot?T ryingto prove that IP

dt

?? NP

dt

?we de?netwo classes

C

?

andC

?

b othcon tainingall ofIP

dt

forwhichwe managed to prove that

NP

dt

??C

?

andNP

dt

??C

?

andev en MA

dt

??C

?

andMA

dt

??C

?

?

Lindstr? om Quan ti?ersin Complexity Theory

Herib ertV ollmer

Universit? atW? urzburg

http???haegar?informatik?uni? wuer zbu rg?de?pe rson ?mitarb eiter?vo llmer

?jointw ork withHans?J? org Burtschic k?T ec hnische Univ ersit?at Berlin?

We show that examinationsof theexpressivepow er oflogicalform ulae

enrichedby Lindstr? om quanti?ersov er ordered?nite structures haveawell?

studiedcomplexity?theoretic coun terpart? theleaflanguageapproach tode?

?necomplexityclasses? Model classesofformulae withLindstr? omquanti?ers

arenothingelse thanleaf languagede?nable sets?Alongthewaywetighten

thebest up to now kno wnleaflanguagec haracterizationof the classesofthe

p olynomialtime hierarchy and givea new model?theoreticc haracterization

ofPSPA CE?

??

Page 21

RandomIsomorphisms

JieW ang

University of NorthCarolina atGreensboro

We extendtheNP?isomorphism theoryofBerman and Hartmanisin the

settingof randomizedreductions for distributionalNPproblems? Earlyre?

sultson isomorphisms ofaverage?caseNP?complete problemsareobtained

w?r?t?deterministicreductions?We de?ne?ina standardway? whatit means

fortwodistributional NPproblems tobeisomorphicunderrandomized reduc?

tions?We then show that allthe kno wnav erage?caseNP?completeproblems?

whether they arecompleteunder deterministic or randomizedreductions? are

all?randomly?isomorphic? and so theseproblems? regardlesstheir origins?

are?random enco dings? ofeach other?

Some Com binatorial Problemfrom theStudyof

Lo cally Random Reducibility

OsamuW atanabe

T okyo Institute ofT echnology

http???watanabe?www?cs?ti tech?ac?jp??wat anab e?my home?intro?e?ht ml

Abadi?F eigenbaum? andKilian ?inM? Abadi?J?F eigen baum?and J?Kil?

ian? On HidingInformation from anOracle? Journalof Computer andSys?

tems Scienc es?V ol????????????????? formallyde?ned thenotionof ?lo cally

random reducibility? and studiedit froma complexitytheoreticp oint of

view?Since then?ithasb een op en whetherev eryfunction isk ?locally re?

ducibleto somek functions?even forsome constantk? ?? Basedon theidea

ofY ao?F ortnow andSzegedy show edthat there isa Boolean functionthatis

??lo cally reducibleto nopair of Bo olean functions?buttheir argument does

notseem tow orkfor thequestion whetherthere isa Bo oleanfunction that

is ??lo callyreducible to nopairof functionswhoserange isf?????g? HereI

asked onecombinatorialquestion?which seems tobeak eyforsolving this

problem?

??

Page 22

QueryOrder

GerdW ec hsung

F riedrich?Schiller?Univ ersit?atJena

http???www?minet?uni?jena?de??wec hsung

?jointw ork withLaneA?Hemaspaandra and HaraldHempel?

LetP

C ????D ???

be theclass of all setsacceptedbyap olynomial?timeoracle

T uring mac hineasking onequery to someoracleC?C follow edby one query

to some oracleD?D?We ask? Do esqueryorder matter? i?e? does

P

C ????D ???

?P

D ????C ???

hold?

We show forC?D fromthe Boolean hierarchyov er NP

P

BH

i

????BH

j

???

?

?

?

?

?

?

R

p

?i??j????tt

?NP? ifi? ?????j? ????

R

p

?i??j??tt

?NP? otherwise?

F romthis itfollo wsthat theonlynon trivialcase for

P

BH

i

????BH

j

???

?P

BH

j

????BH

i

???

isi even andj?i? ?? provided thati?j? In allremaining caseswe have

nonequality? unless theBH? andth us alsothePH? collapse?

We similarlyc haracterize oracleclasses witha tree likequerystructure

andoracles fromthe BH?

??

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