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?

?

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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

?

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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

?

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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??

?

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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??

??

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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

??

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