ComputabilityandComplexity 18-1 ComputabilityandComplexity 18-2 MeasuringSpace Sofarwehavedefinedcomplexitymeasuresbasedonthetimeused bythealgorithm SpaceComplexity Nowweshalldoasimilaranalysisbasedonthespaceused IntheTuringMachinemodelofcomputationthisiseasytomeasure —justcountthenumberoftapecellsreadduringthecomputation Definition1 ThespacecomplexityofaTuringMachine Tisthe functionsuchthatSpaceT SpaceisthenumberofT (x ) distincttapecellsvisitedduringthecomputation T(x) ComputabilityandComplexity (Notethatif T(x)doesnothalt,thenisundefSpace T (x) ined.) AndreiBulatov ComputabilityandComplexity 18-3 ComputabilityandComplexity 18-4 Palindromes AnotherDefinition Considerthefollowing3-tapemachinedecidingifaninputstringisapalindrome •Thefirsttapecontainstheinputstringandisneveroverwritten ByDef.1,thespacecomplexityofthismachineisn •Themachineworksinstages.Onstage iitdoesthefollowing However,itseemstobefairtodefineitsspacecomplexityas log n -thesecondtapecontainsthebinaryrepresentationof i -themachinefindsandremembersthe i-thsymbolofthestringby (1)initializingthe3 rd string j =1 ;(2)if j< ithenincrement j; Definition2 Let TbeaTuringMachinewith2tapessuchthat (3)if j= ithenrememberthecurrentsymbolbyswitchingto thefirsttapecontainstheinputandneveroverwritten.The correspondentstate spacecomplexityof TisthefunctionsuchthatSpace T -inasimilarwaythemachinefindsthe i-thsymbolfromtheend isthenumberofSpace T (x) distinctcellsonthesecondtape ofthestring visitedduringthecomputation T(x) -ifthesymbolsareunequal,haltwith“no” •Ifthe i-thsymbolistheblanksymbol,haltwith“yes” ComputabilityandComplexity 18-5 ComputabilityandComplexity 18-6 Proof Blum’sAxioms 1.isobvious. Blumproposedthatanyusefuldynamiccomplexitymeasureshould 2. satisfythefollowingproperties: Notethatifthemachinehas kstatesand lsymbolsinthealphabet, ⋅ ⋅ r •isdefinedexactlywhenϕ (x) M(x)isdefined thentherearepossibleconfiguratk r l ioninvolvingatmost r M visitedcellsonthesecondtape ϕ = •Theproblem:forgiven M,x,r ,doesisdecidableM (x) r Ifthesameconfigurationoccurstwiceduringthecomputation M(x), thenthemachinegoesintoaninfiniteloopand M(x)(aswellas TheoremSpace T isapropercomplexitymeasure )isnotdefinedSpace M (x) Thereforeafterperformingstepsk ⋅ r ⋅l r M(x)either halts,or visitsmorethan rcells,or goesintoaninfiniteloop 1 ComputabilityandComplexity 18-7 ComputabilityandComplexity 18-8 SpaceComplexityofProblems SpaceComplexityClasses Aswithtimecomplexity,wecannotdefineanexactspacecomplexity foralanguage,butwecangiveanasymptoticform… Nowweareinapositiontodivideupthedecidablelanguagesinto classes,accordingtotheirspacecomplexity Definition Foranyfunction f,wesaythatthespacecomplexityofa Definition decidablelanguage Lisin O(f)ifthereexistsaTuring Thespacecomplexityclass SPACE[ f]isdefinedtobetheclass Machine Twhichdecides L,andconstantsandn0 c ofalllanguageswithtimecomplexityin O(f) > suchthatforallinputs xwith | x | n0 ≤ Space T (x) cf (| x |) (Note,itissometimescalled DSPACE[ f]—forDeterministicSpace) ComputabilityandComplexity 18-9 ComputabilityandComplexity 18-10 PolynomialSpace Timevs.Space Aswithtime,wecanobtainarobustspacecomplexityclassby •Visitingacelltakesatleastonetimestep,sowehave consideringallpolynomialspacecomplexityclasses Theorem TIME[ f ] ⊆ SPACE[ f ] Definition PSPACE = SPACE[ nk ] •Amoresubtleanalysis,basedonthefactthatthereareonly U f (n) k≥0 differentp|Q |⋅ f (n ⋅ |) Σ | ossibleconfigurationswith f(n)non- blanktapecells,gives Theorem SPACE[ f ] ⊆ TIME[ f ⋅k f ] ComputabilityandComplexity 18-11 ComputabilityandComplexity 18-12 ReusingSpace All Thepreviousresultsuggeststhatagivenamountofspaceismore Languages usefulthanagivenamountoftime DecidableLanguages Thisisbecausespacecanbereused Forexample, Satisfiability canbesolvedinlinearspace,bytrying P NP PSPACE eachpossibleassignment,oneatatime,reusingthesamespace Theorem NP ⊆ PSPACE 2 ComputabilityandComplexity 18-13 ComputabilityandComplexity 18-14 ProblemsinPSPACE Quantified Boolean Formula Generalized Geography Instance:AquantifiedBooleanformula Instance:Agamefor2playersconsistingofadirectedgraphG, Φ = andastartvertexs.Playerstaketurnstomovealong (Q1 X1 )( Q2 X 2 )K(Qn X n ) B(X 1, X 2 ,K, X n ) theedgesfromthecurrentvertex.Aplayerwhocannot whereeachisaBooleanvarX iable,B(X , X ,K, X ) l 1 2 n movewithoutreturningtoavertexalreadyvisitedloses isaBooleanexpressioninvolvingandX1, X 2 ,K, X n ∃ ∀ Question:Isthereawinningstrategyforthefirstplayer? eachisaquantifier(Ql or ). Question:Is Φlogicallyvalid? Complexity 13-15 Complexity 13-16 PSPACE-CompleteProblem QuantifiedBooleanFormula PSPACE-Completeness Instance:AquantifiedBooleanformula Φ = (Q1 X1 )( Q2 X 2 )K(Qn X n ) B(X 1, X 2 ,K, X n ) whereeachisaBooleanvarX l iable,B(X 1, X 2 ,K, X n ) Definition isaBooleanexpressioninvolvingandX1, X 2 ,K, X n Alanguage Lissaidtobe PSPACE -completeif,forany ∃ ∀ eachisaquantifier(Ql or ). ∈ ≤ A PSPACE , A L Question:Is Φlogicallyvalid? Theorem QBF isPSPACE-complete Complexity 13-17 Complexity 13-18 Proof Firstweshowthat QBF isin PSPACE, usingthefollowingrecursive Thisalgorithmispolynomialspace,because algorithm •oneveryroundofrecursionitneedstostoreonlyatransformed FortheinputBooleansentence Φ copyoftheinputformula •If Φcontainsnoquantifiers,thenitcontainsnovariables,only •thedepthofrecursionisthenumberofvariables, m constants.Evaluatethisexpression;acceptifitis true ; otherwisereject •Therefore,ituses m⋅|Φ|space Φ = ∃ Ψ Φ Φ ∨Φ •If X then is true ifandonlyifis|X =0 |X =1 true . Φ Φ Recursivelycallthealgorithmon|X =0 and,andaccept|X =1 ifoneofthemisaccepted;otherwisereject Φ = ∀ Ψ Φ Φ ∧Φ •If X then is true ifandonlyifis|X =0 |X =1 true . Φ Φ Recursivelycallthealgorithmon|X =0 and,andaccept|X =1 ifbothofthemisaccepted;otherwisereject 3 Complexity 13-19 Complexity 13-20 WhenprovingCook’stheoremwemanagedtodefineaBooleanCNF Φ Φ formulasuchthatT (X ) T (X ) istrueifandonlyifthevalues ofvariablesdescribeinacertainwayaleX galconfigurationofthe •Let Abealanguagein PSPACE Turingmachine T •Let TbeadeterministicTuringMachinethatdecides A ThenweconstructedaformulawΨ (X ,Y ) hichistrueifandonlyif withspacecomplexity f(n), where fisapolynomial T andrepresentlegalconfigurations,X Y sayand,andC1 C2 TcangofromtoinatmostonestC C ep •Chooseanencodingforthecomputation T(x)thatuses 1 2 k·f(| x|) symbolsforeachconfiguration Thelengthofbothformulasispolynomialintheamountoftapecells Ψ •Letbetheinitialconfiguration,andC0 betheacceptingCa involvedinthecomputation.DenotethelengthofbyT g(| x|) configuration WeconstructaformulawhiΩ( X ,Y ,t) chistrueifandonlyif andrepresentlegalconfigurations,X Y sayand,andC1 C2 t TcangofromtoinatmostC1 C2 2 steps Ω Finally, Taccepts xifandonlyif(C0 ,Ca , f (| x |)) istrue Complexity 13-21 Theformulaisbuiltrecursively Straightforwardapproach: Ω = Ψ (X ,Y )0, T (X ,Y ) Ω(X ,Y ,t) = ∃Z ( Ω(X ,Z ,t − )1 ∧ Ω(Z ,Y ,t − ))1 ItdoesnotworkbecauseΩ(X ,Y , f (| x isbuildof|)) 2 f (| x|) Ψ formulas T (X ,Y ) Thereareuniversalquantifierstodothejob! Ω(X ,Y ,t) = ∃Z ∀(U ,V ) (((( U ,V ) = ( X ,Z )) ∨ (( U ,V ) = (Z ,Y ))) ∧ Ω(U ,V ,t − ))1 ThelengthofthisformulaisO( f (| x |) g(| x |)) 4.