Alternating Hierarchies for Time-Space Tradeoffs

Alternating Hierarc hies for Time-Space T radeoffs Chris P ollett Departmen t of Computer Science San Jose State Univ ersit y 1 W ashington Square San Jose, CA 95192 p ollett@cs.sjsu.edu Eric Miles Departmen t o f Compu ter Science San Jose State Univ ersit y 1 W ashington Square San Jose, CA 95192 enmiles@gmail.com Octob er 25, 2018 Abstract Nepo mnja ˇ sˇ ci ˇ ı’s Theorem states that for all 0 ≤ ǫ < 1 and k > 0 the class of languages recognized in nondeterministic time n k and space n ǫ , NTISP [ n k , n ǫ ], is contained in the linear time hierar chy . B y consider ing r estrictions on the size of the universal qua ntifiers in the linear time hierarch y , this paper refines Nep omnja ˇ sˇ ci ˇ ı’s re s ult to giv e a sub- hierarch y , Eu - LinH , of the linear time hierarch y that is con tained in NP and whic h contains NTISP [ n k , n ǫ ]. Hence, Eu - LinH cont a ins NL and SC . This pap er investigates basic s tructural prop erties of Eu - LinH . Then the re la tionships b et ween Eu - LinH and the classes NL , SC , a nd NP are considered to see if they can shed light on the NL = NP or SC = N P questions. Finally , a new hie r arch y , ξ - L inH , is defined to reduce the space requirements needed f o r the upper b ound on Eu - LinH . Mathematics Subje ct Classific ation: 03F30, 68 Q15 Keywor ds: str uctural complexity , linear time hierarch y , Nep omnja ˇ sˇ ci ˇ ı’s Theor em 1 In tro duction F ortnow [3] b egan the stud y of time-space trade-offs as an app roac h to sho wing L 6 = NP . His pap er giv es the first non-trivial time-space trade-offs for satisfiabilit y . These results w ere sub s equen tly impro v ed u p on and the in terested r eader can consult F ortno w et al. [4] and Williams [16] f or an in tro duction to th e literature as w ell as curren t results. One tool used in these time-space results is Nep omnja ˇ s ˇ ci ˇ ı’s Theorem. The form of this theorem w e are int erested in s h o ws that NTISP ( n k , n ǫ ), the class of languages decidable simultaneously in nondeterministic time n k and space n ǫ for 0 < ǫ < 1 and k > 0, is con tained in the linear time hierarch y , Lin H . If one examines the p r o of of th is result, one is struc k that although a language L in N TIS P ( n k , n ǫ ) is seemingly m app ed into the Σ lin 2 k + 1 lev el o f the linear-time hierarc h y , the univ ersal quan tifiers that app ear are all of log arithmic size. It is thus n atural to ask what is the complexit y of the sub-class of Σ lin k where w e restrict the unive rs al quanti fi ers to b e logarithmic size. I t is this hierarch y that we in vestig ate in the present pap er. Giv en a class of languages C , let E C (resp ., U C ) denote the class of languages of the f orm { x | ∃ y ∈ { 0 , 1 } k ·| x | , h x, y i ∈ L } (resp ., { x | ∀ y ∈ { 0 , 1 } k ·| x | , h x, y i ∈ L } ) for some L ∈ C and fixed in teger k . (Note that we say { 0 , 1 } n to refer to the set of all b inary strings with length no greater th an n .) W e use lo w er case, e or u to denote the case where the b ounding term k · | x | is r eplaced with log( k · | x | ) for some fixed in teger k . W e will call these kind of quantifiers sharply b ounde d . Using this nota tion, we write ( Eu ) lin k for the cla ss languag es giv en b y k quan tifier blocks whose v alues are fi nally fed in to a deterministic linear time languag es, where eac h quan tifier blo c k is of th e form an E qu an tifier follo w ed b y an u quant ifier. By examining the pro of of Nep omn ja ˇ s ˇ ci ˇ ı’s theorem, w e sh o w ev ery language in NTISP [ n k , n ǫ ] can b e repr esen ted as a language in Eu - LinH := ∪ k ( Eu ) lin k . W e in fact sho w a lev el-wise result with an almost matc hing upp er b ound: NTISP ( n k (1 − ǫ ) , n ǫ ) ⊆ ( Eu ) lin k ⊆ NTIS P [ n k +1 , n ]. So our hierarc h y is closely asso ciated w ith the class of languages that can b e solve d using nondeterministic p olynomial time and linear sp ace. As it is simultaneously con tained in the linear time hierarc hy and in NP , it also contai n s NL an d S C := ∪ k ∪ m DTISP [ n k , log m n ]. As such our h ierarc h y might b e a usefu l to ol in trying to separate th ese classes fr om NP . F ur ther as this h ierarch y has a v ery synta ctic defin ition, it lends itself to b e used in areas suc h as b ounded arithm etic. Bounded arithm etic are w eak formal systems that are often used to m od el th e reasoning needed to carry out complexity argumen ts. So ev en if ( Eu ) lin k basic p r op erties don’t immediately help one separate complexit y classes they migh t b e us efu l in b oun ded arithmetic to qu an tifier wan t kin d of reasoning abilit y in terms of strengths of theories are needed to sep arate complexit y classes. Quant ifier s su c h as e and u hav e b een considered b efore in the context of linear time and quasi-linear in Blo c h, B u ss, Goldsmith [1]. In particular, they lo ok at a so-called sharply b ounded h ierarc h y o ve r qu asi-linear time, using the notation ˜ ∃ and ˜ ∀ for sharply b ounded quan tifiers. Their class is con tained in P and has interesti n g closure pr op erties, bu t it is unclear if it conta ins a w ell-known class lik e NL . F or this pap er, we are in terested in the structural and closure pr op erties o f Eu - LinH and the connections b etw een Eu - LinH and the classes NL , S C , and NP . As ind icated ab o v e, one has the con tainmen ts NL , SC ⊆ Eu - LinH ⊆ NP . Since NL and S C are closed u nder complemen t one also has NL , SC ⊆ ∪ k ( Ue ) lin k ⊆ co - NP . W e first show s ome b asic str u ctural results suc h as if ( Eu ) lin k = ( Eu ) lin k +1 then NL 6 = NP and SC 6 = NP . W e further sho w if ( Eu ) lin 1 is con tained in ( Ue ) lin k for some fi x ed k , then NP = co - NP . W e then consider p ossible inclusions b et we en our class and the classes NL , SC , and NP . W e sho w if NL or SC = Eu - LinH then NP = co - NP , the p olynomial time hierarc hy is equal to the linear time hierarc hy , and the linear time hierarc hy is infin ite. W e then turn ou r atten tion to whether it is p ossible for Eu - Li nH to equal NP . W e sho w that if Eu - LinH = NP , then the Eu - LinH hierarc h y is infinite, LinH con tains the b o olean h ierarc h y o v er NP , and either the linear time hierarc h y is infi n ite or N P 6 = co - NP . Th en w e sho w th at if the tot al sum of the length boun ds on the univ ersal quan tifiers in Eu - Li nH hierarch y is b ounded to k log n , then the restricted hierarc hy , Eu - LinH k , is strictly con tained in NP . So in particular, if Eu - LinH k con tains NL , whic h wo u ld happ en if Eu - LinH k = Eu - LinH , then NL 6 = NP . It migh t b e p ossible to further restrict the sizes of the unive rs al quanti fi ers if the space r equiremen ts in the con tainment ( Eu ) lin k ⊆ NTISP [ n k +1 , n ] could b e reduced. T o this e n d we d efi ne a new quantifier ξ and obtain partial results in th is dir ectio n . This pap er is organized as follo ws: The next section con tains the notations and main definitions used in th is pap er. This is f ollo wed b y a section with some b asic conta inments and closure p rop erties of Eu - LinH . Then Nep omnja ˇ s ˇ ci ˇ ı’s T heorem is prov en so as to connect 2 it with Eu - LinH . Th is is follo wed by a short section reminding the reader o f some useful padding and diagonalizat ion tec hniques. The next t w o sec tions co ntain our main r esults, the first of these give s conditional results concerning what w ould happ en if th e Eu - LinH collapses, the second of these pro ves our results concerning Eu - LinH and the classes NL , S C , and NP . The section whic h follo ws defines a new quan tifier ξ and a new hierarc h y ξ - LinH , and uses these to show a con tainment for Eu - LinH with a slightly lo wer space b ound. Finally , there is a conclusion with some suggestions for f u rther inv estigatio n. 2 Preliminaries A go o d survey of b asic mac hine m o d els and complexit y classes can b e foun d in Johnson [10]. W e briefly summarize here the mo dels, classes and notation whic h will b e used in this pap er. W e w ill use d eterministic, nondeterministic, or alternating m ulti-tap e T uring Mac hines as our basic mac hin e mo dels. These mac hines are assumed to h a v e a r ead only input tap e, a write only output tap e, and some fin ite num b er of wo rk tap es. S p ace is m easured in terms of the total num b er of tap e squares used on the w ork tap es. F or the sak e of simplici ty , w e assu me without loss of generalit y that all languages are ov er the binary alphab et. W e will often n eed at our d isp osal the abilit y to enco de and deco de pairs or fi nite sequences of strings. T o do this, the sequen ce of v alues h x 1 , . . . , x n i — when n = 2, one has a pairing op eration — is defin ed to b e the strin g obtained by replacing 0s and 1s in th e x i s by 00 and 10 resp ectiv ely and by inserting a 01 in b etw een n u m b ers. W e write DTIME [ t ( n )] and NTIME [ t ( n )] for the languages decided b y mac hin es whic h are r esp ectiv ely deterministic or nond etermin istic, and which r un for at most t ( n ) steps on inputs of length n . W e wr ite Σ k - TIME [ t ( n )] and Π k - TIME [ t ( n )] for the languages decided in t ( n ) steps b y mac hines whic h alternate at most k − 1 times, in the former case the outermost alternation b eing existen tial, and in the latter case univ ersal. W e define DSP ACE [ s ( n )] and NSP A CE [ s ( n )] similarly except n o w the b ound is on space rather th an time. The classes DTISP [ t ( n ) , s ( n )] and NTISP [ t ( n ) , s ( n )] are the languages decided by mac hines in t ( n ) time steps using at most s ( n ) work tap e squares in resp ectiv ely th e d eterministic or nondeterministic m ac hine mo del. F or a language L , ¯ L is the complemen t of L , consisting of those strings x o ver the alphab et that are n ot in L . Given a class of languages C , w e wr ite co - C for the languages { ¯ L | L ∈ C } . Giv en these basic definitions, the follo wing w ell-studied complexit y classes can now b e defined: DLIN := DTIME [ n ], P := ∪ k DTIME [ n k ], NLIN := NTIME [ n ], NP := ∪ k NTIME [ n k ], L := DS P ACE [lo g n ], NL := NSP ACE [log n ], and SC := ∪ k ∪ m DTISP [ n k , log m n ]. It is kno wn that L ⊆ NL = co - NL ⊆ P ⊆ NP and that L ⊆ SC ⊆ P . As SC , w hic h stands for Stev e’s Class in honor of Stev e C o ok, is less-kno wn w e men tion that SC , in addition to the con tainmen ts ab o v e, conta ins all the deterministic con text fr ee languages. Finally , we will b e in intereste d in the linear and p olynomial time hierarchies: LinH := ∪ j Σ j - TIME [ n ] and PH := ∪ j Σ p j , w here Σ p j := ∪ k Σ j - TIME [ n k ] and Π p j := ∪ k Π j - TIME [ n k ]. A t the b ottom lev els of th ese h ierarc hies w e ha v e NLIN = Σ 1 - TIME [ n ], co - NLIN = Π 1 - TIME [ n ] a nd Σ p 1 = NP , Π p 1 = co - NP . W e also ha ve trivially that Σ j - TIME [ n ] ⊆ Π j +1 - TIME [ n ] ⊆ Σ j +2 - TIME [ n ] and Σ p j ⊆ Π p j +1 ⊆ Σ p j +2 , s o giv en our defi n ition for eac h j , Π j - TIME [ n ] ⊆ Li nH an d Π p j ⊆ PH . W e c onsid er strings w built out of the qu an tifier alphab et E , U , e , and u . Let τ b e a collect ion of nondecreasing functions on the nonnegativ e in tegers. W e w ill use τ to interpret 3 quan tifiers. The quan tifier E (resp., U ) represent s a quantifier of the form ∃ y ∈ { 0 , 1 } t ( | x | ) (resp., ∀ y ∈ { 0 , 1 } t ( | x | ) ) for some t ∈ τ ; the quantifier e (resp., u ) is sup p osed to repr esen t a quantifier of the form ∃ y ∈ { 0 , 1 } log t ( | x | ) (resp., ∀ y ∈ { 0 , 1 } log t ( | x | ) ). Give n a class C of languages, a s tring w = w k w k − 1 · · · w 0 o v er our quantifier alph ab et, and a set of f u nctions τ as ab o ve , w e defin e ( w C ) τ b y induction on k . If k = 0 then ( w C ) τ = C . F or k > 0, let w ′ = w k − 1 · · · w 0 . Then one h as fou r cases d ep ending on the quantifier w k : F or instance, if w k is E , then ( w C ) τ is the class of languages of the form L = { x | ∃ y ∈ { 0 , 1 } t ( | x | ) h x, y i ∈ L ′ } for some t ∈ τ and L ′ ∈ ( w ′ C ) τ . T he other three cases are similarly d efined usin g our wa y of inte rp reting U , e , and u explained ab ov e. The main choic es for τ w e will b e intereste d in are lin whic h consists of functions of th e form ℓ ( n ) := k · n for s ome k > 0, and p oly wh ic h consists of the p olynomials p ( n ) := n k for some k > 0. The p r incipal starting choic es for the complexit y cl ass C we w ill b e in terested in a re DLIN and P . W e will us e tw o furth er w a ys t o abb r eviate our notation: (1) we will write ( w ) lin for ( w DLIN ) lin and wr ite ( w ) poly for ( w P ) poly ; (2) we will sometimes u se subscr ip ted n otatio ns suc h as ( w ) m as abb reviation for the string m concaten tations z }| { w · · · · · · · · · · · · w . As an example of our notation sc heme, the class NP = Σ p 1 could b e written as ( E ) poly and the class c o - NP = Π p 1 could b e written as ( U ) poly . The lev els of th e p olynomial hierarch y for k > 1 could b e defined by sa ying Σ p k = ( E Π p k − 1 ) poly and Π p k = ( U Σ p k − 1 ) poly . Finally , the new classes we will b e in terested in for this pap er are ( Eu ) lin k := (( Eu ) k ) lin and ( Ue ) lin k := (( Ue k )) lin for some int eger k ≥ 0. W e will wr ite Eu - LinH for the whole hierarch y ∪ k ( Eu ) lin k and Ue - LinH for the wh ole h ierarch y ∪ k ( Ue ) lin k . 3 Basic Closure Prop erties and Cont ainmen ts This sectio n explores so m e of the basic relationships b et w een, and clo su re prop erties o f, our newly defin ed complexit y classes ( Eu ) lin k , ( Ue ) lin k , Eu - LinH , Ue - LinH . T o b egin w e observe ( Eu ) lin k and ( Ue ) lin k are complements of eac h other. W e also connect these classes to the levels of the linear time h ierarc h y . Prop osition 1 (a) F or k ≥ 0 , ( Eu ) lin k = co - ( Ue ) lin k . (b) F or k ≥ 0 , ( Eu ) lin k ⊆ Σ 2 k - TIME [ n ] and ( Ue ) lin k ⊆ Π 2 k - TIME [ n ] . So b oth Eu - LinH and Ue - LinH ar e c ontaine d in LinH . Pr o of. F or (a), in th e k = 0 case, b oth classes are DLIN . F urther DLIN is closed un der complemen t since giv en a mac hine M for L ∈ DLIN we can swa p its accept and r eject states to o b tain a m ac hine for ¯ L . No w assume the pr op osition is true up to some k ≥ 0, and let t ∈ lin . Remem b er that x ∈ ( Eu ) lin k +1 iff ∃ y 1 ∈ { 0 , 1 } t ( | x | ) suc h that ∀ y 2 ∈ { 0 , 1 } log t ( | x | ) , h x, y 1 , y 2 i ∈ ( Eu ) lin k . T h en it is clear that x ∈ co -( Eu ) lin k +1 iff ¬ ∃ y 1 ∈ { 0 , 1 } t ( | x | ) suc h that ∀ y 2 ∈ { 0 , 1 } log t ( | x | ) , h x, y 1 , y 2 i ∈ ( Eu ) lin k . By the indu ction hyp othesis we h a v e h x, y 1 , y 2 i 6∈ ( Eu ) lin k iff h x, y 1 , y 2 i ∈ ( Ue ) lin k ; using that, and the fact that in fi rst order logic ¬∃∀ is equiv alen t to ∀∃ ¬ , we can sa y x ∈ co -( Eu ) lin k +1 iff ∀ y 1 ∈ { 0 , 1 } t ( | x | ) , ∃ y 2 ∈ { 0 , 1 } log t ( | x | ) suc h that h x, y 1 , y 2 i ∈ ( Ue ) lin k . T his sh o ws that x ∈ co -( Eu ) lin k +1 iff x ∈ ( Ue ) lin k +1 , so the induction step is prov ed. T o pro ve (b ), w e first note by increasing the size of the tap e alph ab et we can increase the n um b er of compu tations th at b e done in n steps b y a constan t factor (linear sp eed- up), so Σ 2 k - TIME [ O ( n )] = Σ 2 k - TIME [ n ]. F urther f or a language L in ( Eu ) lin k an alternating 4 mac hine op erating in time O ( n ) can guess fi rst the outermost quant ifier, then univ ersally guess the small un iv ersal quantifier, and so on, with at most 2 k alternations. Finally , it could s im ulate the innermost d eterministic linear time mac hine us ed in deciding L . Thus, ( Eu ) lin k ⊆ Σ 2 k - TIME [ n ].  Prop osition 2 (a) Lin H is close d under union, interse c tion, and c omplement. (b) ( Eu ) lin k , ( Ue ) lin k , Eu - LinH , and Ue - Li nH ar e close d under union and interse ction. Pr o of. LinH is closed und er complemen t since if L is in LinH then it is in Σ k - TIME [ n ] for some k , so ¯ L w ill b e in Π k - TIME [ n ]. As we already observ ed in the pr eliminaries Π k - TIME [ n ] ⊆ Σ k +1 - TIME [ n ] ⊆ Li nH . F or b oth parts (a) and (b) of the prop osition, the argumen ts for closure un der u nion and intersectio n are essentia lly the same, so w e ind icate the pro of f or ( Eu ) lin k . Let L 1 and L 2 b e t w o ( Eu ) lin k languages. On in p ut x we can u se pairin g to guess p airs of strings for eac h E or u quantifier, one of the t wo v alues to b e used to chec k if x is in L 1 and the other to c h ec k if x is in L 2 . T h is at most in creases the size of t h e b ound on eac h quantifier b y a constan t multiplic ativ e factor, so this sequence of existen tial and u niv ersal guesses can still b e done in ( Eu ) lin k . The deterministic mac hine for the un ion, first u npac ks eac h pair to p ro duce a string to start sim u lating the DLIN mac hin e f or L 1 . If this simulat ion accepts then the u n ion machine accepts; otherwise, the other comp onen ts of eac h pair are extracted and the mac h in e simulate s the DLIN m ac hine f or L 2 . If it accepts, then the union mac hine ac cepts; otherwise it r eject. The total runtime will b e at most the su m of the t w o r u n times so still linear. F or intersect ions the s ame sort of algorithm is done except that the mac hine alwa ys p erforms the simulatio n of the DLIN mac hine for L 1 follo w ed b y the DLIN mac hine f or L 2 and only accepts if b oth accepted.  Prop osition 3 (a) NLIN ⊆ ( Eu ) lin 1 and co - NLIN ⊆ ( Ue ) lin 1 . (b) F or k ≥ 0 , ( Eu ) lin k ⊆ NTISP [ n k +1 , n ] ⊆ NTIME [ n k +1 ] , so Eu - LinH is c ontaine d in NP and Ue - LinH is c ontaine d in co - NP . (c) F or k ≥ 1 , ( Eu ) poly k = NP . Pr o of. F or (a) n otice that us ing an E qu an tifier to guess a string w of nondeterministic mo v es of an NLIN mac hine M on inpu t x , a deterministic lin ear time mac hine could then u se these c hoices together with x to decide M ’s language. This s h o w NLIN ⊆ ( E ) lin ⊆ ( Eu ) lin 1 . By taking complemen ts w e hav e the co - NLIN r esu lt. T o prov e (b) we sho w by induction on k that ( Eu ) lin k is con tained in NTISP [ n k +1 , n ]. As Eu - LinH = ∪ k ( Eu ) lin k and NP = ∪ k NTIME [ n k +1 ], th is implies Eu - LinH is con tained in NP . The fact that Ue - LinH ⊆ co - NP then follo ws from Prop osition 1 ( a ). F or th e k = 0 case, we ob viously ha ve DLIN ⊆ NLIN = NTISP [ n , n ]. So assume the statemen t is true u p to some k ≥ 0. Let L b e a language in ( u ( Eu ) k ) lin . W e will argue L is conta ined in NTISP [ n k +2 , n ]. By the definition of L there must b e some ( Eu ) lin k languages L ′ suc h that x ∈ L if and only if ∀ y ∈ { 0 , 1 } log t ( | x | ) , h x, y i ∈ L ′ . Here t ( n ) is some gro wth r ate from lin so is of the form m · | x | . By our ind uction h yp othesis there is a NTISP [ n k +1 , n ] m ac hine M ′ whic h decides L ′ . Let M b e the mac h ine whic h cycles through the m · | x | strings y of length log ( m · | x | ), reusing the sp ace. F or eac h y , M then sim ulates M ′ on h x, y i . If M ′ ev er rejects, then M rejects. Otherwise, if M ′ accepts all such strings, then M ac cepts. This inv olv es less than a linear amoun t of additional space. F ur ther, M ’s ru ntime is O ( m · | x | ) times longer than the runtime of M ′ , so usin g linear sp eed-up , M will op erate in time n k +2 . F urther , if x ∈ L then 5 for ev ery string y with | y | ≤ log ( m · | x | ), h x, y i ∈ L ′ , so there must b e some p ath making M ′ accept. These accepting p aths of M ′ for eac h y , tak en together, can b e used to mak e an accepting path for M . On the other hand , if x 6∈ L , then for some y , there is no p ath which mak es M ′ accepts h x, y i , and so when this y is chec k ed by M , it to o will reject. Thus, w e ha ve sh o wn u ( Eu ) lin k is conta in ed in NTISP [ n k +2 , n ]. But then ( E u ( Eu ) k ) lin = ( Eu ) lin k +1 will also b e in N TISP [ n k +2 , n ] since giv en a language in ( Eu ) lin k +1 , an NTISP [ n k +2 , n ] mac hine could in nondeterministic linear time and sp ace guess the outermost existen tial quant ifier and then run the NTISP [ n k +2 , n ] mac hine for the corresp onding language in ( u ( Eu ) k ) lin . Hence, the indu ction step is p ro v ed and the r esult follo ws. F or (c), we first note b y the same argumen t as in (a), we hav e NP ⊆ ( E ) poly ⊆ ( Eu ) poly k for k ≥ 1. T o see ( Eu ) poly k ⊆ NP , w e can use the same algorithm as in (b). Guessing an E quan tifier will no w inv olv e guessing a str ing of p olynomial length, b ut this can b e handle in NP ; sim u lating a u qu an tifier will in volv e cycling ov er the at | x | m string of length log t ( | x | ) where t is a p olynomial of the form | x | m for some m , so can b e d one in p olynomial time.  Giv en the r esult (b) ab o ve it is reasonable to ask if ( Eu ) lin k = NTIS P [ n k +1 , n ] or if ( Eu ) lin k = NTIME [ n k +1 ] f or k > 0? W e conjecture the latter is n ot th e case and w e will giv e some evidence for this in Corollary 4. Nev ertheless, the results of the n ext section can b e view ed as telling us that ( Eu ) lin k migh t b e close to NTISP [ n k +1 , n ]. 4 Nep omnja ˇ s ˇ ci ˇ ı’s Theorem Theorem 1 F or k ≥ 1 and 0 ≤ ǫ < 1 , NTISP ( n k (1 − ǫ ) , n ǫ ) is c ontaine d in ( Eu ) lin k . Pr o of. Our pro of of Nep omnja ˇ s ˇ ci ˇ ı’s Theorem is essen tially th e same as found in sa y H´ ajek and Pud l´ ak [8] or F ortno w et al. [4 ]. Here, though, w e emphasize the details that imp ly our con tainment . Cons ider a k -tape T u ring Mac hine M for a language L in NSP AC E ( n ǫ ). T o make our argument easier, w e will assume that when M halts, it erases its tap es, mo v es to the left-most square on eac h tap e, and then either halts accepting or r ejecting. A configuration C of M is k + 3 tuple h q , i, c, t 1 , . . . , t k i represent ing the state o f M , th e index of the square b eing read on the in p ut tap e, the c haracter b eing wr itten to the output tap e, and the strings representing the visited con ten ts of eac h work tap e. T he p osition of w ork tap e heads are indicated in eac h t i b y us in g an alphab et sym b ol with an unders core b eneath it for that tap e s q u are. Consider the language L k consisting of 4-tuples of the form ( x, C s , C t , 1 | x | k (1 − ǫ ) ) suc h that C s and C t represent configurations of M on input x , and there is some | x | k (1 − ǫ ) -step computation of M b eginning in configur ation C s and end ing in configuration C t . W e pr oceed b y ind uction on k and argue that L k is in ( Eu ) lin k . When k = 1, we n eed to argue that L 1 is con tained in ( Eu ) lin 1 . V erifying that C s and C t are ind eed configurations of M can b e d one in linea r time in C s and C t . Using th e E quantifier w e can guess a sequence C 1 , . . . , C v of configurations o f M where C s = C 1 , C t = C v , v = | x | 1 − ǫ . This is possible since eac h configuration has size | x | ǫ and v = | x | 1 − ǫ , so the string w e are guessing has length O ( | x | ). Using a u quantifier to guess ev ery i ∈ { 1 , . . . , v − 1 } , one can then v erify in deterministic linear time that (for eac h i ) C i +1 follo ws from C i according to a transition in M , or, if C i is a halted configuration, th at C i = C i +1 . This shows the resu lt holds for k = 1. 6 Assume L k ∈ ( Eu ) lin k holds for k ≥ 1 and consider an instance ( x, C s , C t , 1 | x | ( k +1)(1 − ǫ ) ) of L k +1 . Again, with an E quan tifier we can guess a sequence C 1 , . . . , C v of configu r ations of M where C s = C 1 , C t = C v , v = | x | 1 − ǫ . T hen using a u quantifier w e can c hec k if for eac h i = 1 , . . . v − 1, that we can guess the 4-tuple ( x, C i , C i +1 , 1 | x | k (1 − ǫ ) ) and v erify that it is in L k . T his sh o ws L k +1 is in ( EuE ( Eu ) k ) lin , b ut ( E ( Eu ) k ) lin = ( Eu ) lin k b ecause we can alwa ys com bine outer tw o existent ial qu an tifiers of linear size into one existen tial quantifier still of linear size. Hence, we h a v e L k +1 is in ( Eu ) lin k +1 and the induction step is pr ov ed. No w if M w as for a language L that was not only in NSP ACE [ n ǫ ] but in NTISP [ n k (1 − ǫ ) , n ǫ ], then on inpu t x we could guess the 4-tuple ( x, C s , C h , | x | k (1 − ǫ ) ), w h ere C s is the starting configuration of M on x and C h is the u nique p ossible accepting, halting configuration; and c hec k if this 4-tuple is in L k . This wo u ld show L ∈ E ( Eu ) k ) lin , b ut as w e ha ve j ust argued, ( E ( Eu ) k ) lin = ( Eu ) lin k . Hence, the theorem follo ws.  Com binin g Prop osition 3 and Theorem 1 ab o ve , we ha v e an almost optimal contai n men t for the lev els of our hierarc h y: Corollary 1 F or k ≥ 1 and 0 ≤ ǫ < 1 , NTISP ( n k (1 − ǫ ) , n ǫ ) ⊆ ( Eu ) lin k ⊆ NTISP [ n k +1 , n ] . The next corollary also follo ws fr om Theorem 1 since NL = ∪ k NTISP ( n k , log n ) and S C := ∪ k ∪ m DTISP [ n k , log m n ]. Corollary 2 (a) NL is c ontaine d in Eu - LinH . (b) SC is c ontaine d in Eu - Li nH . 5 Common Argumen ts In this section, w e br iefly presen t w ell-kno wn results concerning padding and time h ier- arc hies whic h we will n eed for our late r results. T o b egin w e sa y a function t ( n ) is time c onstructible if ther e is a t ( n ) time b ounded T uring Mac hine M suc h that for eac h n , M runs for exactly t ( n ) steps on an input of length n . Giv en a complexit y class C , w e write R pad ( C ) for the class of languages L for whic h ther e is a logspace c omp utable, time con- structible function t ( n ) ≥ n , suc h that x ∈ L if and only if x 0 i ∈ L ′ where L ′ ∈ C and | x 0 i | = t ( n ). The string x 0 i here is the string x padd ed by a string of 0’s of length i . The next lemma col lects together the main results w e n eed ab out th e padded versions of th e complexit y classes of th is pap er. Lemma 1 (a) L et C 1 ⊆ C 2 b e classes of languages. Then R pad ( C 1 ) ⊆ R pad ( C 2 ) . (b) R pad ( NLIN ) = R pad ( Eu - LinH ) = R pad ( NP ) = NP . (c) R pad ( co - NLIN ) = R pad ( Ue - LinH ) = R pad ( co - NP ) = co - NP . Pr o of. (a) If L is a language in R pad ( C 1 ). Then ther e must b e some language L ′ in C 1 and a time constructible t ( n ) suc h that x ∈ L if and only if x 0 i ∈ L ′ where | x 0 i | = t ( n ). Since C 1 ⊆ C 2 , th is same L ′ and t show L is in R pad ( C 2 ). (b) As NLIN ⊆ Eu - LinH ⊆ NP b y (a) it suffices to sho w R pad ( NP ) ⊆ NP and NP ⊆ R pad ( NLIN ). The first inclusion follo ws since given a language L ∈ R pad ( NP ) and an inp ut x a nond eterministic T u ring mac hine can compute the logspace computable, time con- structible fun ction t ( n ) for L to get x 0 i . It can then simulat e the NP mac hine for this language on x 0 i . Th e total run time will b e b ound ed b y a comp osition of p olynomials, 7 sho wing L is in N P . F or the second inclusion, let L ∈ NP b e decidable in time b ounded b y a polynomial p ( n ). Then th e language { x 0 p ( | x | ) −| x | | x ∈ L } will b e in NLIN and sh o w L ∈ R pad ( NLIN ). The argument for (c) is essen tially the same as for (b), except that the inclusions sho wn are R pad ( co - NP ) ⊆ co - NP and co - NP ⊆ R pad ( co - NLIN ).  W e next turn to t w o diagonaliza tion results for time complexit y classes whic h we will use but not prov e. The first of these is due to Ch an d ra and Sto c kmeyer [2], whic h Williams [16] giv es the catc hy name: T he No C omplemen tary Sp eedup Theorem. The second of th ese is the Nondeterministic Time Hierarch y Th eorem [14, 17]. Theorem 2 L et k b e a p ositive inte ger and t a time c onstructible function. Then Π k - TIME [ t ] 6⊆ Σ k - TIME [ o ( t )] . Theorem 3 L et t 1 ( n ) and t 2 ( n ) b e functions with t 2 ( n ) time c onstr uc tible. If t 1 ( n + 1) ∈ o ( t 2 ( n )) then NTIME [ t 1 ] ( NTIME [ t 2 ] . 6 Lev els of Eu - LinH In this section w e present consequences of Eu - LinH b eing con tained in Ue - LinH and we consider the question of w hether or n ot th e Eu - LinH h ierarch y is in finite. Theorem 4 If ( Eu ) lin 1 is c ontaine d in ( Ue ) lin k then NP = co - NP . Pr o of. As NLIN ⊆ ( Eu ) lin 1 , this implies NLIN ⊆ ( Ue ) lin k , and th u s NLIN ⊆ Ue - LinH . So by Lemma 1 w e hav e NP = R pad ( NLIN ) ⊆ R pad ( Ue - LinH ) = co - NP , and h ence NP = co - NP  Theorem 5 L et k > 0 . If ( Eu ) lin k = ( Eu ) lin k +1 then NL 6 = NP and SC 6 = NP . Pr o of. First, if ( Eu ) lin k = ( Eu ) lin k +1 , th en ( Eu ) lin k +2 = ( Eu ( Eu ) k +1 ) lin = ( Eu ( Eu ) k ) lin = ( Eu ) lin k . Con tinuing up the hierarc hy in this f ashion one th us has ( Eu ) lin k = Eu - LinH . B y Prop osition 3 (b), this im p lies Eu - LinH ⊆ NTIME [ n k +1 ] ( NP . T he resu lt then follo ws, as b y Theorem 1, w e kn o w SC ⊆ Eu - LinH and NL ⊆ Eu - LinH .  7 Other complexit y classes and Eu - LinH In this section we in v estigate v arious consequences of certain in clusions holding among Eu - LinH , LinH , SC , NL , and NP . Prop osition 4 If Li nH = PH then for al l k , LinH 6 = Σ k - TIME [ n ] .That is, th e line ar time hier ar chy is infinite. Pr o of. By T h eorem 2 we kno w that Π k - TIME [ n 2 ] 6⊆ Σ k - TIME [ n ]. W e also kno w Π k - TIME [ n 2 ] ⊆ Π p k ⊆ Σ p k +1 . Therefore, if LinH = Σ k - TIME [ n ] fo r some k , then it is strictly con tained in PH .  8 Corollary 3 If co - NP ⊆ Eu - LinH then for al l k , LinH 6 = Σ k - TIME [ n ] . Pr o of. Prop osition 3 (b) and co - NP ⊆ Eu - LinH w ould imply co - NP ⊆ Eu - LinH ⊆ NP . Hence, NP = co - NP = Eu - LinH . F ur ther as NP = co - NP implies NP = PH and Eu - LinH ⊆ LinH ⊆ PH , we w ould ha v e Li nH = PH and so th e result follo ws from Prop osition 4.  F or the next result recall that the b o olean h ierarch y ab o v e NP is defined as BH := ∪ i BH i , where BH 1 = NP , where BH 2 i is the class of languages consisting of the inte rs ectio n of a language in BH 2 i − 1 with a language in co - NP , and where BH 2 i +1 is the class of languages consisting of th e in tersection of a language in BH 2 i with a language in NP . Kadin [6] h as sho wn if this hierarc h y colla p ses so d o es the p olynomial hierarc hy . W e next consider some consequences of Eu - LinH b eing equal to NP . Theorem 6 If Eu - LinH = NP then: (a) F or al l k ≥ 0 , Eu - LinH 6 = ( Eu ) lin k . (b) LinH c ontains BH . (c) Either for al l k , LinH 6 = Σ k - TIME [ n ] or NP 6 = co - NP . Pr o of. T o see (a) n otice Prop osition 3 (b) implies ( Eu ) lin k ⊆ NTIME [ n k +1 ] ( NP , where NTIME [ n k +1 ] ( NP follo ws by the nondeterministic time hierarc h y theorem (Theorem 3). F or (b) observ e that if Eu - LinH = NP then by Prop osition 3 (b) we h av e NP = Eu - LinH ⊆ LinH . T he result then f ollo ws as Li nH is closed un d er unions, in tersections, and complemen ts. Lastly , for (c) supp ose Eu - LinH = NP , NP = co - NP , and LinH = Σ k - TIME [ n ] for s ome k . Then Eu - LinH = NP = co - NP = PH , as Eu - Lin H ⊆ LinH ⊆ PH we h a v e LinH = PH . Th u s, LinH = Σ k - TIME [ n ] con tradicts Prop osition 4.  A t the end of S ectio n 3, w e said it w as unlik ely that ( Eu ) lin m = NTIME [ n m +1 ]. Th e next corollary giv es s ome consequen ces of this equ ality . Corollary 4 If for al l m ′ > 0 , ther e is a m > m ′ such that ( Eu ) lin m = NTIME [ n m +1 ] , then: (a) F or al l k ≥ 0 , Eu - LinH 6 = ( Eu ) lin k . (b) Li nH c ontains BH . (c) Either fo r al l k , LinH 6 = Σ k - TIME [ n ] or NP 6 = co - NP . Pr o of. The hyp othesis ab ov e implies Eu - LinH = NP as there are un b oun dedly large m satisfying ( Eu ) lin m = NTIME [ n m +1 ], so the un ion ∪ m ( Eu ) lin m for these m ’s is Eu - LinH and the union ∪ m NTIME [ n m +1 ] is NP . The result then follo ws from Theorem 6.  Theorem 7 (a) If Eu - LinH ⊆ co - NP then NP = PH . (b) If Eu - LinH = Lin H then NP = PH . (c) If S C = Eu - Lin H or NL = Eu - LinH then Eu - LinH = LinH = NP = PH . Pr o of. F or (a), if Eu - LinH ⊆ co - N P , then in p articular NLIN ⊆ co - NP and so by padd ing NP ⊆ co - NP . Thus, NP = co - NP = PH . T o see part (b ) notice that LinH con tains co - NLIN , so if Eu - LinH = LinH then co - NLIN ⊆ Eu - LinH . Thus, by Lemma 1, we h a v e co - NP ⊆ NP . Hence NP = co - NP = PH . F or (c) let C b e either SC or NL . In either case, C is closed under complemen t and closed under log sp ace (and hence, p olynomial length padding) redu ctions [9, 15]. S o if C = Eu - LinH then in particular NLIN ⊆ C and by Lemma 1 we get NP ⊆ C . Since in b oth cases, we kno w C ⊆ NP , we get C = NP . So by closure under complemen t of C = NP = co - NP = PH . As w e are assuming C = Eu - Li nH ⊆ LinH ⊆ PH , we also h a v e C = Eu - LinH = Li nH = PH .  The argument for part (c) th us shows: Corollary 5 If SC = NP or NL = NP then Eu - LinH = LinH = NP = PH . 9 If the conclusion were to happ en th en by Prop osition 4 and Theorem 6, b oth LinH and Eu - LinH m u s t not collapse. Corollary 6 F or k ≥ 1 , if Eu - LinH ⊆ Σ k - TIME [ n ] then NL 6 = NP and SC 6 = NP . Pr o of. By th e p revious corollary , SC = NP or NL = NP implies Eu - LinH = LinH = PH . By Prop osition 4 w e th en ha ve LinH 6 = Σ k - TIME [ n ] ∀ k ≥ 1. Since Σ k - TIME [ n ] ⊆ LinH by definition, w e ha v e LinH 6⊆ Σ k - TIME [ n ], and th us Eu - LinH 6⊆ Σ k - TIME [ n ].  In tuitivel y , Eu - LinH = LinH comes very close to sho wing that LinH collapses. W e already kno w by Theorem 7 (b) that in itself this condition implies NP = PH . So if Eu - LinH = LinH do es imply LinH collapses, th en together with the consequence NP = PH , this wo u ld force u s to conclude the p remise of Corollary 5 is false. i.e., SC 6 = NP and NL 6 = NP . In an attempt to giv e at least some justifi cation for this intuition that Eu - LinH is a small sub class of LinH , on e migh t consider a sligh tly weak er v ersion of Eu - Li nH with smaller sharp ly b ounded unive r s al quan tifiers and see if it can equal NP or not . Let s ∈ O (log n ). Extendin g our n otatio n sligh tly we will w rite u s in a quanti fi er strin g to mean the same as u except where the b ound on the quan tifier is s ( n ) rather k log n . C onsider the class C m,k := ( Eu s m · · · Eu s 1 ) lin where ( P m j =1 s j ( n )) ≤ k log n . Let Eu - LinH k := ∪ m C m,k . So Eu - LinH k is a nalogous to Eu - LinH , except that we ha ve forced the sharply b oun ded quan tifiers to b e slightly smaller. As an example, if eac h s 1 , . . . s m w ere log log n , then for eac h m ≥ 1, ( Eu s m · · · Eu s 1 ) lin w ould b e con tained in Eu - LinH 1 . T he n ext result sho w s if w e could tigh ten Nep omnja ˇ sˇ ci ˇ ı’s result to sho w SC and NL are in Eu - LinH k for an y k ≥ 1 then SC 6 = NP and NL 6 = NP . Theorem 8 F or k ≥ 1 , Eu - LinH k ⊆ NTIME [ n k +1 ] ( NP . Pr o of. Supp ose L is in Eu - LinH k and hence in C m,k for some m . L et s 1 ( n ) , . . . s m ( n ) b e the size b ound s on the sh arply b ounded qu antifiers. By the d efinition of Eu - LinH k , w e h a v e ( P m j =1 s j ( n )) ≤ k log n . If one examines the pr o of of Pr op osition 1, the run time of the NP mac hine given there to simulate L w ould b e O ( n + 2 s m ( n + 2 s m − 1 ( · · · ( n + 2 s 1 · n )))) , the n ’s coming from guessing the existent ial qu an tifiers, th e 2 s i ’s coming fr om lo oping o ver all the c h oices of strin gs of length s i . This run time can b e rewritten as O ( n (1 + 2 s m + 2 s m + s m − 1 + · · · + 2 ( P m j =1 s j ( n )) )) . whic h can b e b ound ed by O ( n · m 2 k log n ) = O ( n k +1 ), the latter equalit y follo wing as m is constan t. Th us, the result follo ws.  8 Reducing the S pace Requiremen ts for Eu - LinH With th e goal of r educing the space requirements needed for our upp er b ound on Eu - LinH , w e define a new quan tifier, and th us a new hierarc hy . Giv en a class C , define ξ C to b e the class of languages of the form { x | ∃ w ∈ { 0 , 1 } | x | d 1 ∀ i < | x | d 2 , h x, i, ( w ) f 1 ( i ) , ( w ) f 2 ( i ) i ∈ L } for 10 some L ∈ C , some constan ts d 1 and d 2 (where d 1 ≥ d 2 ), and some f 1 , f 2 : [0 , | x | d 2 − 1] → [0 , | x | d 2 − 1]. Note that ( w ) i denotes th e i th blo c k of w when w is split into | x | d 2 blo c ks, eac h of size | x | d 1 − d 2 , and the functions f 1 and f 2 are used to select blo c ks of w . With this new quan tifier ξ , we can d efine ( ξ ) lin k and ξ - LinH in exactly the same wa y that we defined ( Eu ) lin k and Eu - LinH . Notice that, lik e in ( Eu ) lin k , the universal quantifiers of ( ξ ) lin k are of logarithmic size, though the existent ial quanti fi ers are of p olynomial r ather than linear size. W e first sh o w the relationship b et w een the tw o hierarc h ies. Theorem 9 F or k ≥ 0 , ( Eu ) lin k ⊆ ( ξ ) lin k . Pr o of. The pr oof is by ind uction. When k = 0, we ha ve ( Eu ) lin k = DLIN = ( ξ ) lin k , so the theorem holds. No w assume the theorem holds up to some k − 1. Let L b e a language in ( Eu ) lin k . Then th er e is an L ′ ∈ ( Eu ) lin k − 1 ⊆ ( ξ ) lin k − 1 suc h that L = { x | ∃ y 1 ∈ { 0 , 1 } c 1 | x | ∀ y 2 ∈ { 0 , 1 } log ( c 2 | x | ) , h x, y 1 , y 2 i ∈ L ′ } for some constan ts c 1 , c 2 . W e shall d efine a ( ξ ) lin k -mac hine M that decides L . On input x , for M ’s outermost ξ quan tifier, t h e guessed w has length c 1 c 2 | x | 2 (whic h is within the p olynomial b ound), i ranges from 0 to c 2 | x | − 1 (also w ithin the p olynomial b ound), and we h a v e f 1 ( i ) = i , f 2 ( i ) = min( i + 1 , c 2 | x | − 1). ( f 1 and f 2 w ere c hose in this w a y to pick consecutiv e blo c ks from w , but to pic k the same blo c k when i is at its maxim um v alue.) Note that the set of all i s, when w ritten in binary , is exactly { 0 , 1 } log ( c 2 | x | ) . Each blo c k of w is of length c 1 | x | , and in fact w is mean t to r epresen t c 2 | x | concatenations of th e string guessed for y 1 in the defin ition of L . Giv en the ab ov e definitions, M op erates on strings of the form h x, i, ( w ) i , ( w ) i +1 i , wher e w e un derstand ( w ) i +1 to mean ( w ) i when i is at its maximum v alue. M ’s fir st step is to v erify that ( w ) i = ( w ) i +1 ; this is done to ensu r e that w is in d eed c 2 | x | concatenati ons of the same string. Th is chec k can b e done in linear time. Assu ming that that c hec k passes, M th en “renames” its inp uts by sa yin g y 1 = ( w ) i and y 2 = i (in binary), and c hec ks if h x, y 1 , y 2 i ∈ L ′ b y us ing the ( ξ ) lin k − 1 -mac hine for L ′ . T o ve r if y M ’s correctness, we need only consider the branc hes wh ere the guessed w is c 2 | x | concatenations of the same string, since w e know M w ill reject on all existen tial branc h es where this is not the case. If x ∈ L , th en there is some fixed y 1 suc h th at h x, y 1 , y 2 i ∈ L ′ for eac h y 2 ∈ { 0 , 1 } log ( c 2 | x | ) , and so M w ill accept x when the outer ξ c ho oses the w wh ic h consists of rep eated y 1 s. Conv ersely , if x 6∈ L , then for all w whic h consist of concatenati ons o f the same y 1 , there is some y 2 ∈ { 0 , 1 } log ( c 2 | x | ) suc h that h x, y 1 , y 2 i 6∈ L ′ , and so M will r eject x .  Corollary 7 Eu - Li nH ⊆ ξ - LinH . Theorem 9 also giv es us an analogous resu lt to Nep omnja ˇ s ˇ ci ˇ ı’s Th eorem for ξ - LinH . Corollary 8 F or k ≥ 1 and 0 ≤ ǫ < 1 , NTISP ( n k (1 − ǫ ) , n ǫ ) is c ontaine d in ( ξ ) lin k . In order to pro ve an upp er b ound on this new hierarch y , we no w sh o w that the ξ quant ifi er s are collapsable. Theorem 10 F or any class C , ξ ξ C ⊆ ξ C . 11 Pr o of. Firs t, let us b e precise ab out the defin ition of ξ ξ C . A mac hine M which decides a language L ∈ ξ ξ C computes the follo wing on input x : ∃ w ∈ { 0 , 1 } | x | d 1 ∀ i < | x | d 2 ∃ w ′ ∈ { 0 , 1 } | x | d ′ 1 ∀ i ′ < | x | d ′ 2 , R ( x, i, ( w ) f 1 ( i ) , ( w ) f 2 ( i ) , i ′ , ( w ′ ) f ′ 1 ( i ′ ) , ( w ′ ) f ′ 2 ( i ′ ) ) where R is a predicate computable w ithin the b ounds of the class C . In order to compress the t wo ξ s in to one, w e define the string v to b e h w ′ 0 , w ′ 1 , . . . , w ′ | x | d 2 − 1 i , wh ere w ′ n is the string guessed by the inn er ξ when the i (from the outer ξ ) is equal to n . Then , we can rewrite th e quan tifier string ∃ w ∀ i ∃ w ′ ∀ i ′ as ∃ w ∃ v ∀ i ∀ i ′ . No w, w e use the follo wing notation for our new “compressed” ξ : ∃ y ∈ { 0 , 1 } | x | e 1 ∀ j < | x | e 2 , S ( x, j, ( y ) g 1 ( j ) , ( y ) g 2 ( j ) ) T o sho w that this is the same, w e will define the new existen tial string y to contai n all of w and v , though y will not b e a sim p le concatenation of th ese t wo strings. As w e will see shortly , the size of y is 2( | x | d 2 + d ′ 2 )( | x | d 1 − d 2 + | x | d ′ 1 − d ′ 2 ), and so we choose e 1 so that the b ound on y , | x | e 1 , is bigger than this quant ity . Eac h blo c k of y con tains one blo c k of w and one blo c k of one of the w ′ k s, so eac h blo c k has size | x | d 1 − d 2 + | x | d ′ 1 − d ′ 2 . And th us, we c h oose e 2 suc h that | x | e 2 ≥ 2( | x | d 2 + d ′ 2 ). F or the string y , w e d efine it as follo ws. The first 2 | x | d ′ 2 blo c ks con tain the necessary information w hen i = 0 and i ′ ranges fr om 0 to | x | d ′ 2 − 1. T h e next 2 | x | d ′ 2 blo c ks con tain the necessary in formation when i = 1 and i ′ again ranges f rom 0 to | x | d ′ 2 − 1. (These groups con tain 2 | x | d ′ 2 blo c ks r ather than ju st | x | d ′ 2 blo c ks b ecause w e need t wo b lo cks fr om w and t wo from one of the w ′ k s, but eac h blo c k of y con tains only half this information.) More generally , if w e let α r an ge from 0 to | x | d 2 − 1 (the r ange of i ) and w e let β b e the ev en n umb ers in the range 0 to | x | d ′ 2 − 1 (the range of i ′ ), then the (2 α | x | d ′ 2 + β )th blo c k of y is h ( w ) f 1 ( α ) , ( w ′ α ) f ′ 1 ( β ) i , and the (2 α | x | d ′ 2 + β + 1)th b lock of y is h ( w ) f 2 ( α ) , ( w ′ α ) f ′ 2 ( β ) i . Due to our selection of e 2 ab o v e, we can treat eac h v alue of the in dex j as b eing of the f orm h i, i ′ i , ignoring an y extra bits that result from the fact that log(max( j )) ≥ log(max( ii ′ )). Then we defin e g 1 ( j ) = g 1 ( h i, i ′ i ) = 2 i | x | d ′ 2 + 2 i ′ , and we define g 2 ( j ) = g 1 ( j ) + 1. And fin ally , by viewin g the ind ex j and the blo c ks of y as w e ha v e defined them ab o v e, the predicate S ( x, j, ( y ) g 1 ( j ) , ( y ) g 2 ( j ) ) is e qu al to R ( x, i, ( w ) f 1 ( i ) , ( w ) f 2 ( i ) , i ′ , ( w ′ i ) f ′ 1 ( i ′ ) , ( w ′ i ) f ′ 2 ( i ′ ) ).  Corollary 9 F or al l k ≥ 0 , ( ξ ) lin k = ( ξ ) lin 1 = ξ DLIN ⊆ NP . Pr o of. The fir s t equalit y follo ws fr om Th eorem 10, and the second equalit y from the defi- nition of ( ξ ) lin 1 . The inclusion ξ DLIN ⊆ NP follo ws from the fact that an NP mac h ine could existen tially guess the string w , and then d eterministicall y try eac h of the p olynomially- man y v alues of i to d ecide a language in ξ DLIN .  W e no w u se the follo win g result from [5 ], wh ic h was largely based on w ork o riginally published in [13]. Theorem 11 D LIN ⊆ Σ 2 - TISP [ n, n log ∗ n ] . 12 The resu lt actually prov ed was that D TIME [ t ( n )] ⊆ Σ 2 [ t ( n ) log ∗ n ] for all time-constructible t ( n ) ≥ n log ∗ n , bu t as the lo wer b ound on t ( n ) is not relev an t to the space b oun d on th e sim ulating machine in the p ro of, the same pro of can b e used to s h o w Theorem 11. Corollary 10 Eu - LinH ⊆ ξ Σ 2 - TISP [ n, n log ∗ n ] . Pr o of. F rom Theorem 11, w e ha v e that ( ξ ) lin 1 ⊆ ξ Σ 2 - TISP [ n, n log ∗ n ]. F r om Corollary 9, w e then ha ve that ( ξ ) lin k ⊆ ξ Σ 2 - TISP [ n, n log ∗ n ] for a ll k ≥ 0. Finally , since ev ery languag e L ∈ Eu - LinH m u st b e in some pr ecise lev el of Eu - LinH , and th us in some precise lev el o f ξ - LinH , the result f ollo ws.  9 Conclusion W e b eliev e w e ha ve sho wn that the hierarc hy Eu - LinH is an in teresting hierarc hy closely connected with the NL = NP question. This is evid en ced b oth by the con tainments NTISP [ n k (1 − ǫ ) , n ǫ )] ⊆ ( Eu ) lin k ⊆ NTISP [ n k +1 , n ] and b y the fact that if our hierarch y coll apses th en NL 6 = NP . W e would lik e b riefly t o conclude w ith some f uture directions f or research. One immediate question is whether the con tainmen ts NTISP [ n k (1 − ǫ ) , n ǫ ] ⊆ ( Eu ) lin k ⊆ NTISP [ n k +1 , n ] can b e m ade tigh ter? W e ha v e giv en some evidence for this in the preceding sectio n , b ut is an exact relatio n s hip b etw een our hierarch y and a nondeterministic time space class p ossible? It w ould also b e in teresting to study what happ ens when one considers the hierarc h ies lik e Eu - LinH where one has nonconstan tly m any alternations. Giv en that the total size of the string that would b e fed in to the mac hine in this case could b e nonlinear in the input size, it w ould probably be w orthwhile to consid er qu asi-linear v arian ts of our class as w ell. References [1] S. Blo c h, J. Bu s s and J. Goldsmith. Sharply b oun ded alternation and quasilinear time. Theory of Comp u ting Sys tems, 31(2):187 -214, Marc h 1998. [2] A. K . Chandra and L. J. Sto c kmeyer. Alternation. IEEE Symp osium on F oun dations of Computer Science, p p. 98–106, 197 6. [3] L. F ortno w. Time-space tradeoffs for sa tisfiab ility . Journal of Computer and System Sciences, 60(2):337- 353, April 2000. [4] L. F ortno w and R. Lipton and D. v an Melk eb eek and A. Vig las. Time-space lo w er b ounds for satisfiabilit y . Journal of the ACM, 52(6):835-8 65, No ve mb er 2005. [5] S. Gupta. Alternating Time V ersus Deterministic Time: A Separation. Mathematic al Systems The ory , pp. 661-672 , 1996. [6] Jim Kadin. The Pol yn omial Time Hierarch y Collapses if the Boolean Hie r arch y Col- lapses. SIAM Journ al on Computing, 17(6): 1263–1282 (198 8). 13 [7] J. Hartmanis and R. Stearns. O n th e computational complexit y of algo rithm s. T rans- actions of the American Mathemat ical So ciet y , V ol. 117, pp . 285–306, 1965. [8] P . H´ ajek and P . Pu dl´ ak. Metamathematics of First-O r der Arithmetics . Springer-V erlag, 1993. [9] N. Immerman. Nondeterministic Space is Closed Un d er Complemen t. SIAM Journal on Compu ting, V ol. 17, pp. 935–938, 1988. [10] D. S. Johnson. A Catal og of Complexit y Classes. In The Handb o ok of The or e tic al Computer Scienc e , V olume A. J . V an Leeuw en, Ed., pp. 68–161, MIT P ress, 1990. [11] R. Lipton and A. Viglas O n the C omplexit y of SA T . IEEE Symp osiu m on F oundations of Computer Science, p p. 459–464. 1999. [12] V.A. Nep omnja ˇ s ˇ ci ˇ ı. Rudiment ary p redicates and T urin g compu tatio n s. Dokl. A c ad. Nauk , V ol. 195, pp. 282–284, 197 0. transl. V ol. 11, pp. 1462–146 5, 1970. [13] W. J. P aul, N. Pipp enger, E. S zemeredi and W. T. T rotter. On d eterminism v ersu s non- determinism and related problems. In Pr o c e e dings of the 24th Annual IEEE Symp osium on F oundations of Computer Scienc e , p p. 429–438 , 1983 . [14] J . Seiferas and M. Fischer and A. Meye r . Separating nondeterministic time complexity classes. Journ al of the ACM, 25:146167 , 1978. [15] R. S zelepcsenyi. Th e metho d of forcing for nondeterministic automata. Bulletin EA TCS 33, pp. 96–100, 1987. [16] Ryan Williams: Better T im e-Sp ace Low er Boun ds for SA T and Related Problems. IEEE Conference on C omputational Comp lexit y (CCC 2005), p p. 40–49, 2005. [17] S . ˇ Z` ak. A T u ring mac hine time h ierarc h y . Theoretical Compu ter Science, 26:327 333, 1983. 14