On Bounded Nondeterminism and Alternation

Transcription

1 On Bounded Nondeterminism and Aternation Mathias Hauptmann May 4, 2016 Abstract We continue our work on the combination of variants of McCreight and Meyer s Union Theorem with separation resuts aong the ine of Pau, Pippenger, Szemeredi, Trotter and S.Gupta. Under assumption NP=PSPACE, we prove a union theorem for the cass PSPACE=AP with respect to a particuar subfamiy S i) ) of aternating machines. This yieds a union function F which is nondeterministicay computabe in time F n) C for some constant C. We show that for each probem L AP, there exists a poynomia time bounded machine S i) such that L = L S i) ). Moreover, we prove a variant of Gupta s resut who showed that DTIMEt) Σ 2 t) for time-constructibe functions tn). Our variant of this resut hods with respect to the subfamiy S i) ) of aternating machines. We show that these two resuts contradict each other, whence our starting assumption cannot hod. Keywords: Aternating Turing Machines, Nondeterministic versus Aternating Time Compexity, Union Theorem, Bounded Nondeterminism. 1 Introduction Let us first reca the two starting points for our previous paper [H16]. The Union Theorem of McCreight and Meyer [McCM69] states that whenever f i ) is a famiy of functions f i : N N such that λi, n.f i n) is recursive, then there exists a singe recursive function F such that DTIMEF ) = i DTIMEf i). Indeed this does not ony hod for deterministiime compexity casses but for any Bum compexity measure. The second starting point is the resut of Pau, Pippenger, Szemeredi and Trotter [PPST83] who proved that DLIN NLIN. This proof is based on their remarkabe resut that deterministic computations can be simuated by Σ 4 -computations with a sma speedup. Namey, for every time-constructibe function tn), DTIMEt og t)) Σ 4 t). Gupta [G96] improved on this resut and showed that for each such t, DTIMEt og t)) Σ 2 t). Moreover he proved that for every time-constructibe t, DTIMEt) Σ 2 t). See aso the discussion at the end of [H16]. Our approach in [H16] was now the foowing. We assumed P = Σ p 2. The idea was to construct then a union function F such that P = DTIMEF ) = Σ 2 F ) = Σ p 2 such as to obtain a contradiction to the resut of Gupta. However, a direct construction of such a union function F aong the ines of McCreight and Meyer yieds a function F which is recursive but not timeconstructibe. The second idea was then that it might be possibe to construct F such that F n) is computabe in time poynomia in F n), namey by making use of the assumption P = Σ p 2. If the function F is, say, deterministicay computabe in DTIMEF C ), then one might try to use Padding in order to show that DTIMEF ) = Σ 2 F ) aso impies DTIMEF C ) = Σ 2 F C ), which then yieds the desired contradiction to Gupta s resut. However, for such a padding construction to work, it is necessary that the function F satisfies an inequaity of the kind F n) C F n h ). In Dept. of Computer Science, University of Bonn. e-mai: hauptman@cs.uni-bonn.de 1

2 [H16] we caed this a Padding Inequaity. Now again, if we construct a union function directy aong the ines of McCreight and Meyer, then the resuting function F woud not satisfy such a Padding Inequaity. The third idea is then that a padding inequaity might be reachabe provided the underying machines satisfy some additiona property. The construction of a union function for poynomia time compexity casses is essentiay a diagonaization against a the machines whose running time is not poynomiay bounded. Now suppose that a the machines under consideration have the foowing additiona property: Whenever a machine vioates a given poynomia time bound at some input ength and say both the poynomia and the input ength are sufficienty arge), then the machine has aready vioated a simiar but somewhat smaer poynomia time bound at a number of smaer input engths. It turns out in [H16] that such a property can be formaized and the construction of the union function can be adjusted that it wi satisfy a Padding Inequaity. But this means now that we have switched from a standard enumeration of, say, a aternating machines to a subfamiy, namey of machines which actuay satisfy this additiona property. Then it is of course necessary to show that this does not change the compexity casses under consideration. We give now an outine how to appy the method from [H16] to the casses NP and AP = PSPACE. We wi assume NP = AP. Then we start from a standard enumeration S i ) of aternating machines and construct a subfamiy of machines, containing for each origina machine S i and every integer d a machine S i,d. These machines wi satisfy the additiona property, which is the same as in [H16] and which we ca Property [ ]. We wi show that going from a genera famiy S i ) of aternating machines to a subfamiy S i,d ) wi not change the compexity casses under consideration. Then we construct a union function F such that NP = NTIMẼF ) = ATIMẼF ) = AP. Here by NTIMẼF ) and ATIMẼF ) we denote the casses with respect to our restricted famiy of machines. We show that this function F satisfies a Padding Inequaity. This yieds that we aso have NTIMẼF C ) = ATIMẼF C ). The function F C is not time-constructibe, but it can be computed nondeterministicay and co-nondeterministicay in time F n) C. Thus we wi show an anaogon of Gupta s resut, namey that NTIMẼt) ATIMẼt) for every function t which is nondeterministicay and co-nondeterministicay computabe in time tn) 1 ɛ for some ɛ > 0. For such a resut to work, we aso have to require that the nondeterministic machines in our subcass have bounded nondeterminism. It suffices to restrict the number of nondeterministic steps to the squareroot of the running time of the machine, and again this wi not change the compexity cass NP. 2 Impications from the Assumption NP=AP We et S i ) be a standard enumeration of aternating machines. We et Check be the foowing decision probem: Check = { i, n, a, b) time Si n) > an b} Since AP=PSPACE is cosed under compement, NP=AP impies NP=coNP. Via some standard padding argument, this yieds that the decision probem Check can be soved by some NTIME contime-agorithm which has running time c 0 i an b ) c 0 on inputs i, n, a, b), for some constant c 0. This means that this agorithm is a guessing agorithm which has three possibe outputs 0, 1,?. For each input x, either there exists no computation path of the agorithm on input x with output 0, or there exists no path with output 1, and furthermore, for each input x there exists at east one path with output either 0 or 1. Since we are not aware of any existing short notation for casses NTIMEt) contimet) from the iterature, we wi denote them as ZNTIMEt) =NTIMEt) contimet), where ZNTIME stands for zero-error nondeterministic 2

3 time. Moreover, abusing notation, we write Check ZNT IMEc 0 i an b ) c 0 ). In this case, ZNT IMEc 0 i an b ) c 0 ) does not denote a compexity cass but just the fact that the probem Check can be soved by such a zero-error nondeterministic agorithm within the time bound c 0 i an b ) c 0 ) for inputs of the form i, n, a, b). 3 A New Programming System for AP We start from a standard programming system aso caed Göde numbering) S i ) of aternating machines. Without oss of generaity we assume that this numbering has the foowing additiona property: Whenever S i is a nondeterministic machine, meaning that it does not have any universa states, then the number of guesses made by machine S i is aready bounded by the squareroot of its running time. As we have described in [H16], in order to ensure that the union function F satisfies the Padding Inequaity F n) c F n h ), we want to work with a new famiy of aternating machines such that if tn) is a running time function of some machine from this famiy, then tn) satisfies the foowing condition which we caed Property [ ]. Definition 3.1. [H16] We say function tn) satisfies Property [ ] with parameters c, d, p if for every n 2 c2 and a pairs of integers a, b with c a b ogn) c, if tn) > an b, then there exist og ogn) c pairwise distinct integers m 1,..., m og og n in the interva c I n,d = n 1/h, n 1/h 1 + ogn) ) ) d n 1/h d) such that m is not an h-power and tm ) > a p)m b p og og n, = 1,..., c. Starting from the famiy S i ) of aternating machine, we wi construct a new famiy S i,d ) of aternating machines which contains for each machine S i and every integer d a machine S i,d. This new famiy of machines wi have the foowing properties: For every machine index i and every integer d, the machine S i,d satisfies Property [ ] with some parameters, d, p i. For every i and d, the running time of the machine S i,d wi be bounded by twice the running time of machine S i. If S i is a nondeterministic machine, then for each integer d, the machine S i,d is aso a nondeterministic machine for which the number guess Si,d n) of guesses made by S i,d satisfies guess Si,d n) time Si,d n). If the running time function time Si n) aready satisfies Property [ ] with parameters 2, d, p i 2, then L S i,d ) = LS i ). We sha work with the associated nondeterministic and aternating time compexity casses NTIMẼ t) and ATIMẼt), which are defined as foows: L is in ATIMẼt) if there exists some aternating machine S i,d such that L = L S i,d ) and time Si,d n) = Otn)). L is in NTIMẼ t) if there exists some nondeterministic machine S i,d such that L = L S i,d ) and time Si,d n) = Otn)). Especiay, since S i,d is a nondeterministic machine, we have guess i,d n) time i,d n) = O tn)). 3

4 3.1 Construction of Machines S i,d Before we describe the construction of the new famiy S i,d ) of aternating machines, et us give a characterization of Property [ ] which we have aready used in [H16]. This wi be usefu in the construction of the machines S i,d. Intuitivey, the machine S i,d wi just perform a step-by-step simuation of the machine S i, but at the same time it wi check if it is aowed to continue this simuation, such as to satisfy Property [ ]. In the foowing we just give a name to the property which machine S i,d has to check. Here we give first the genera formuation for functions t: N N. Definition 3.2. The predicate P N N) N 6 is defined as foows. P t, c, d, p, n, a, b) hods if n < 2 c2 or a < c or b > ogn) c or n 2 c2 and there exist pairwise distinct integers m 1,..., m og ogn) c I n,d which are not h- powers and such that tm ) > a p)m b p and P t, c, d, m, a p, b p), = 1,..., og ogn) c. We observe that t: N N satisfies Property [ ] with parameters c, d, p iff for a n 2 c2 and c a b ogn) c, tn) > an b impies P t, c, d, p, n, a, b) Now we can describe the construction of the machines S i,d. In the case of P versus Σ p 2 in [H16], the machines S i,d just performed a simuation of the machines S i and in parae checked deterministicay) if the property P ) was satisfied such as to continue the simuation. Here the situation is sighty different. Since we work under the assumption NP=AP, this especiay means that we do not have a deterministic agorithm at hand to check the predicate P ). Instead, we are given now an NTIME contime-agorithm for the predicate P ), and the machine S i,d is supposed to simuate machine S i, which may now be any aternating machine. This means that checking the predicate P ) wi cause the machine S i,d to perform some additiona aternations. We construct the machine S i,d as foows: Given an input x of ength n, consider two consecutive pairs of integers a, b ), a, b) with a b ogn) and a b ogn) consecutive with respect to the order b < b or b = b and a < a)). Within the time interva a n b, an b ] the machine S i,d uses the first haf of the time in this interva to check if the predicate P ) hods, i.e. if it is aowed to continue the computation for more than an b steps. The remaining haf of the time within this interva is used to simuate computation steps of the machine S i on input x. Thus, as ong as the machine S i,d is aowed to continue its computation, it wi make twice as many computation steps as S i on the same input x. We are now ready to give the definition of the predicate P for machines S i,d. Definition 3.3. Predicate P for machines S i,d ) P i, d, n, a, b) hods if n 2 c2 i,d and a b og n ) impies that there exist pairwise distinct non-h-power integers m 1,..., m og ogn) I n,d such that for = 1,..., og ogn), time Si m ) > 1 2 a p i)m b p i and for a pairs of integers α, β) ex a p i, b p i ), P i, d, m, α, β) hods. On input x of ength n, if n 2 c2 i,d, then S i,d just simuates S i on input x and makes in tota twice as many computation steps as S i. On the other hand, if n > 2 c2 i,d, then S i,d does the foowing: Let L = L i,d n) denote the number of pairs of integers a, b) with a b ogn), 4

5 and et a, b ), = 1,..., L be these pairs in the exicographic order described above. The associated intervas are T 0 = 0, a 1 n b 1], T = a n b, a +1 n b +1], < L and TL = a L n b L, ). Now S i,d uses the first haf of the computation steps within each interva T, < L to simuate computation steps of S i on input x. The second haf of the time within interva T is used to check if the predicate P i, d, n, a +1, b +1 ) hods. If within some interva T the computation of S i on input x terminates, then S i,d terminates as we, with the same output accept/reject). If within some interva T, the computation of S i does not yet terminate but the predicate P i, d, n, a +1, b +1 ) does not hod, Si,d aso competes this interva and then terminates and rejects. Otherwise, it continues within the next inerva T +1. If the computation reaches the interva T L, then it just continues to simuate the computation of S i and does not check the predicate P ) anymore. The computation of S i,d is organized in such a way that whie being in an interva T, < L, it aways makes precisey twice as many computation steps as S i. This wi be particuary important when we show in Lemma 3.2 that for those machines S i which aready satisfy Property [ ], the machine S i,d wi compute the same as machine S i, i.e. L S i,d ) = LS i ). We give a pseudo-code description of the machine S i,d. Machine S i,d Input: x of ength n If n < 2 c2 i,d, simuate the computation of machine S i on input x and make in tota twice as many computation steps as S i. If n 2 c2 i,d For = 0,... L 1 where L = L i,d n)) Use the first haf of the interva T to continue the simuation of computation of S i on input x If the computation of S i terminates after t steps in T, then S i,d performs t additiona dummy steps and then aso terminates. Otherwise, use the second haf of the interva T to check if P i, d, n, a +1, b +1 ) hods and fi the rest of T with dummy steps. If P i, d, n, a +1, b +1 ) does not hod, stop and reject. / Now we are in the interva T L = a L n b L, ) / Continue the simuation of computation of S i on input x. We wi now show that for each < L, one haf of the interva T suffices to check if the predicate P i, d, n, a +1, b +1 ) hods. Lemma 3.1. Predicate P i, d, n, a, b) can be checked in NT IMEn b p i ), where p i = i 2. Proof. In order to check if P i, d, n, a, b) hods, it suffices to sove the instances i, m, a, b ) of the probem Check for a b b p i and m R i,d n), where the set R i,d n) - which we ca the set of reevant input engths for i, d, n - is defined as foows: R i,d n) = R i,d n) with ) ) d Ri,d 1 n) = I n,d = n 1/h, n 1/h 1 + ogn) n 1/dh) R +1 i,d n) = m R i,d n),m 2c2 i,d I m,d We give a bound on the cardinaity of the set R i,d n) as foows. We have R i,d n) [L, R ], 5

6 where L = n 1/h and R = h h 1 n 1/h = 2 1 1/h 1 1/h = h ) h h 1 n 1/h 2 h h 1 n 1/h n 1/h Moreover, R < 2 c2 i,d impies that the eve set R +1 i,d n) is empty. Now the condition R < 2 c2 i,d foows - by taking ogarithms and combining it with the previous inequaity - from 1 h ogn) + h h 1 < c2 i,d 1 h > > c 2 i,d h h 1 ogn) 1 ogh) og ogn) γ i,d) where γ i,d is a constant that ony depends on i, d and the goba constant h. Thus we obtain R i,d n) n 1/h og ogn) Thus P i, d, n, a, b) can be decided nondeterministicay by soving at most n 1/h og ogn) n 2/h instances of the probem Check of the form i, m, a, b ) with a a p i, b b p i, m 2 n 1/h, which can be done nondeterministicay in time n 2/h c ia p i )2n 1/h) b p i) ) c n b p i. Remark. This emma aso yieds P i, d, n, a, b) ZNT IMEn b p i ). Moreover, if S i is a nondeterministic machine, then for every d, the machine S i,d is nondeterministic as we. Lemma 3.2. Properties of the machines S i,d ) Machine S i,d satisfies Property [ ] with parameters, d, p i. Furthermore, time Si,d n) 2 time Si n). If the machine S i aready satisfies Property [ ] with parameters 2, d, p i 2, then L S i,d ) = LS i ). Coroary 1. Under assumption NP = AP we have NP = N P = A P = AP. Proof. of Lemma 3.2) In order to show that S i,d has Property [ ] with parameters, d, p i, we first prove the foowing Auxiiary Caim: For every n, for a pairs a, b) of integers with a b, time Si n) > 1 2 anb and α, β) ex a, b) P i, d, n, α, β)) impies that time Si,d n) > an b. Proof of the Auxiiary Caim. For n < 2 c2 i,d, we have time Si,d n) = 2time Si n), and thus the caim hods. On the other hand, for n 2 c2 i,d, it foows directy from the construction of the machine S i,d that if time Si n) > 1 2 anb and the property P i, d, n, α, β) hods for a α, β) ex a, b), then this impies time Si,d > an b. Now we use the Auxiiary Caim in order to show that S i,d satisfies Property [ ] with parameters, d, p i. Thus suppose that time Si,d n) > an b and a b ogn). Since time Si,d n) 2 time Si n), this impies that time Si n) > 1 2 anb, and moreover, P i, d, n, a, b) hods. This means that there exist pairwise distinct non-h-power integers m 1,..., m og og n I n,d such that time Si m ) > 1 2 a p i)m b p i and α, β) ex a p i, b p i ) P i, d, m, α, β) hods for = 1,..., = 1,..., og og n. Now the Auxiiary Caim yieds that time Si,d m ) > a p i )m b p i og og n, which means that S i,d satisfies Property [ ] with parameters, d, p i. 6 for

7 Now suppose that S i satisfies Property [ ] with parameters 2, d, p i 2. We want to show that L S i,d ) = LS i ), i.e. that for every input x, S i,d x) competey simuates S i x). This hods in case when x < 2 c2 i,d. Now suppose n = x 2 c2 i,d. If S i,d x) S i x), then there exists some {0,..., L i,d n)} such that time Si x) > 1 2 a +1n b +1, P i, d, n, a +1, b +1 ) does not hod and P i, d, n, a, b ) hods for a 0 <. Since n 2 c2 i,d > ) 2 and 2 a +1 2 b +1 ogn) /2, we obtain that there exist pairwise distinct integers m 1,..., m og og n in I n,d which are not /2 h-powers and such that time Si m ) > a +1 2 p i 2 )mb i p i /2 this property, then we have time Si,d m ) > a +1 p i )m b i p i /2. If we assume n to be minima with a +1 p i )m b i p i. Now the construction of S i,d impies that P i, d, m, α, β) hods for a α, β) ex a p i, b p i ), and, using the Auxiiary Caim, this yieds that S i,d x) wi continue and go to the + 1)st iteration of the for oop, a contradiction. Atogether we have shown that L S i,d ) = LS i ). 4 The Union Function F n) Now we wi briefy describe the construction of the union function. The construction is the same as in [H16]. The function F is constructed in stages. In stage n, the function vaue F n) is defined. The whoe construction is a diagonaization against a those machines S i,d which vioate a pre-defined poynomia upper bound for the running time of the machine, namey a variant of the origina construction from [McCM69]. The construction is based on the notion of guesses. A guess is just a pair S i,d, b), where S i,d is a machine from our famiy of aternating machines and b is a positive integer. The guess hods in stage n if time Si,d n) b n b, otherwise the guess is vioated in stage n. Our union function F is supposed to have the foowing property. N P = NTIMẼ F ) = ATIMẼF ) = AP 1) Moreover, the function F n) wi be computabe in ZNT IMEF n) C ) for some constant C. Using Padding, we want to show then that NTIMẼ F ) = ATIMẼF ) aso impies NTIMẼ F C2 ) = ATIMẼF C2 ). As we sha see beow, for such a padding construction to work, we wi aso have to satisfy the foowing additiona requirement: N P = NTIMẼ F n + 1)) = ATIMẼF n + 1)) = AP 2) Whie 1) is the s property of a standard union function in this context, property 2) wi be used when we assign to each given probem L ATIMẼF C2 ) an associated poynomiay padded version L. Whie it turns out to be easy to guarantee that L ATIMEF ), showing that we aso have L ATIMẼF ) wi require making use of property 2). We arrange the machines S i,d in a ist and et S j) denote the j th machine in this order. This is done in such a way that from j we can efficienty compute i, d such that S i,d = S j) and furthermore, j. During the construction of the union function F, we maintain a ist L of guesses. In stage n of the construction, the function vaue F n) wi be determined. In the stage n, guesses for the first og n) machines from the ist wi be taken into account. The overa strategy is to seect a exicographicay smaest vioated guess, to diagonaize against the associated machine and then to repace this guess by one with a higher vaue b j. In order to satisfy both 1) and 2), we wi maintain for each machine S j) two guesses. In stage n, one is used for diagonaizing against vioations at input ength n and one for input ength n 1. We wi denote these guesses as S j), b j,1, 1) and S j), b j,2, 2). In stage n, the first of the two guesses, S j), b j,1, 1), wi be checked for vioation at ength n i.e. if time Sj) n) b j,1 n b j,1 is 7

8 vioated), and the second guess S j), b j,2, 2) wi be checked for vioation at ength n 1, i.e. it wi be checked if time Sj) n 1) b j,2 n b j,2 hods. We et L n denote this ist at the beginning of stage n of the construction. Thus the invariant wi be that L n wi contain guesses S j), b j,1, 1) and S j), b j,2, 2) for j = 1,..., og n) with b j og n). The foowing notations turn out to be usefu, cf. [H16]. We et og n) = min{t ] t n}, the minimum height of a tower of 2 s which is n. For t N we et I t be the set of integers n N for which og n) = t. This means that I t = [δ t 1 + 1, δ t ) with δ 0 = 0, δ 1 = 2, δ t+1 = 2 δt, t 1. In the construction of the union function, we consider two cases. If n is an h-power, i.e. n = n h for some integer n, then from the ist L n the exicographicay smaest vioated guess S j), b j,, ) is seected first ordered by the vaue b j, then by and then by j). If a guess S j), b j,1, 1) is seected, then we define F n) := n b j,1, repace the guess S j), b j,1, 1) by S j), og n, 1) and continue. If a guess S j), b j,2, 2) is seected, then we define F n) := n 1) b j,2, repace the guess S j), b j,2, 2) by S j), og n, 2) and continue. If n is not an h-power, we proceed sighty differenty. We consider the foowing set of guesses: a the guesses from L n and additionay guesses of the form S j), b j, p i), ), where S j), b j,, ) is a guess in L n and within the interva I t with og n = t the guess S j), b j, p i), ) has not yet been seected. In order to keep track of this, we et σ = σ 1,1,..., σ t,2 ) with σ j, = 0 iff the machine S j) has not yet been seected within a guess of the form S j), b j, p j), ) within the interva I t, and σ j, = 1 otherwise. Now in such a stage n we seect the smaest vioated guess from this set with respect to the foowing order: first ordered by the vaue b j,, then by the second entry of the guess b j, or b j, p j) ), then by {1, 2} and then by j. If a guess S j), b j,1 p j), 1) is seected, we set F n) := n b j p j) and σ j,1 := 1. If a guess S j), b j,2 p j), 2) is seected, we set F n) := n 1) b j p j) and σ j,2 := 1. If a guess S j), b j,1, 1) is seected, we set F n) := n b j,1. If a guess S j), b j,2, 2) is seected, we set F n) := n 1) b j,2. In both cases, the guess S j), b j,, ) being seected is repaced by S j), og n, ) in the ist L. Finay, if n = δ t is the ast stage within the interva I t, then two new guesses S t+1), t + 1, ), = 1, 2 are added to the ist. Initiay, in stage 1, we set L = { S 1), 1, 1), S 1), 1, 2)}, σ 1,1 := 1 and F 1) := 1. Moreover, at the beginning of every stage n = δ t 1 +1, i.e. the first stage within the interva I t, we initiaize the vector σ = σ 1,1,..., σ t,2 ) as 0,..., 0) and then proceed with stage n as described above. For more information about the construction as we as for proofs of the foowing resuts which are identica to those in the case of P versus Σ p 2 ) we refer to [H16]. Lemma 4.1. We have NP = N P = NTIMẼ F ) = AT IMẼF ) = A P = AP and NTIMẼ F ) = NTIMẼ F n + 1)) = AT IMẼF n + 1)) = A P = AP. Proof. Since in the construction of the union function, guesses are added and repaced with increasing b-vaue, F majorizes every poynomia. On the other hand, if the running time of a machine S i,d vioates every poynomia bound infinitey often, then in the construction of the union function, machine S i,d wi be seected infinitey often within a guess S i,d, b, 1) and infinitey often within a guess S i,d, b, 2), both with increasing vaues of b. Thus in that case, neither time Si,d n) = OF n)) nor time Si,d n) = OF n + 1)). For a more detaied version of the proof we refer to [H16]. Lemma 4.2. There exists a constant C such that F n) can be computed in ZNT IMEF n) C ). Proof. We define the functions F b n) in the same way as in [H16]. Thus, F b n) is the function which is computed when we repace a the guesses b j, b j p j) which occur in the computation 8

9 of F n) by guesses min{b j, b + 1} and min{b j p j), b + 1} respectivey. Functions F n) and F b n) are reated as foows: If F n) = n a for some integer a, then F b n) = F n) for a b 3a. The ony difference is that in the case of P versus Σ p 2 in [H16], the functions F bn) coud be computed deterministicay in time F b n) O1). Since here we work under the assumption NP = AP, the functions F b n) are now computabe in time F b n) O1) by what we caed zeroerror nondeterministic machines, i.e. in ZNT IMEF b n) O1) ). This means that there exists a nondeterministic agorithm which has running time F b n) O1) on every computation path and either returns? or some output, which is then the function vaue F b n). For each n, there is at east one computation path of this agorithm which returns the vaue F b n). Now we can construct a nondeterministic agorithm for computing F n) as foows: For b = 1, 2,... this agorithm runs the agorithm for computing the function vaue F b n), unti it finds some b such that F b n) = n a, b 3a. Then it returns n a, which is then the correct function vaue F n). Whenever some of the computations of function vaues F b n) returns?, the whoe agorithm stops and aso returns?. Lemma 4.3. Padding Inequaity) For every h-power n, F n 1/h ) C F n), where C is the constant from Lemma 4.2. Sketch of Proof. Suppose that n is an h-power such that F n 1/h ) C > F n). Suppose in stage n 1/h of the construction of the union function F, some guess of the form S i), b i, i ) or S i), b i p i), i ) is vioated and seected, and in stage n a guess of the form S j), b j, j ) is vioated and seected. Note that since n is an h-power, in stage n none of the guesses from the extended ist of the form S j), b j p j) ) is taken into account. Now it is straight forward to see that in each of the possibe cases, this impies that b i > b j. But then the guess S j), b j, j ) was aready contained in the ist L n 1/h in stage n 1/h of the construction. Since it is vioated at input ength n, Property [ ] yieds that the guess S j), b j p j, j ) is vioated sufficienty often within the stages in the interva I n,d, i.e. in stages n 1/h + 1,... such that eventuay this guess woud have been seected earier, a contradiction. For further detais we refer to [H16]. Lemma 4.4. Padding Lemma) NTIMẼ F ) = AT IMẼF ) impies NTIMẼ F C2 ) = AT IMẼF C2 ). In order to prove the Padding Lemma, we wi first show that the Property [ ] satisfies certain cosure properties. Lemma 4.5. Cosure Properties of Property [ ]) Let t: N N be a function that satisfies Property [ ] with parameters c, d, p. a) For every k N the function t k satisfies Property [ ] with some parameters c k, d k, p k. b) Let the function T n) be defined as T n h2 1) = tn) for a n, T m) = m γ for some constant γ otherwise Then the function T n) satisfies Property [ ] with some parameters c T, d T, p T Proof. Both the proof of a) and b) are technica yet straight forward. In this extended abstract we ony give the proof of b). Suppose that we have aready chosen the parameters c T, d T, p T. Suppose that T N) > an b, where c T a b ogn) c T. If we choose c T sufficienty arge compared to the constant γ, we can concude that this impies that N is of the form N = n h2 1 9

10 and T N) = T n h2 1) = tn) > an h2 1) b an h2 b 1). We want to assure that this impies a h 2 b 1) ogn). Since it is sufficient to require that h 2 b 1) h 2 b h 2 ogn) c T = h 2 ognh2 1) c T h 2 h2 ogn) c T, c T, h4 ogn) c T ogn) h 4 c T 3) Moreover, in order to appy Property [ ] for the function t, we have to assure that N 2 c2 T impies n 2 c2 t. We have 2 c 2 T N = n h2 1 n h2, thus it suffices to require that c T h It is cear that 3) impies 4). We assume now that 3) hods. Then Property [ ] for the function t yieds that there exist non h-power integers which means m 1,... m og og n T m h2 Thus we want to assure that a p t )m h2 b 1) p t is sufficient to require that I n,d with tm ) > a p t )m h2 b 1) p t og og n, = 1,..., 1) > a p t )m h2 b 1) p t og og n, = 1,... a p T )m h2 4) 1) b p T. For this purpose, it h 2 b 1) p t h 2 b p T ), which is equivaent to h 2 + p t h 2 p T. 5) We can satisfy 5) by choosing p T p t. Moreover we have to assure that sufficienty many integers M = m h2 1 are within the interva I N,dT. Since the impication m I n,dt M I N,dT does not hod, we take the foowing approach. We require the number of integers m I n,dt be sufficienty arge compared to the required number of integers M I N,dT. By definition, we have I N,dT = N 1/h, N 1/h 1 + ogn) ) ) dt N 1/d =: L T h) N,dT, R N,dT ) and We require now that I n,dt = n 1/h, n 1/h 1 + ogn) ) ) dt n 1/dt h) og ogn) og ogn) 3 c T =: L n,dt, R n,dt ) Then it is sufficient to repace the requirement {m h2 1 m I n,dt } I N,dT which we cannot satisfy) by the foowing weaker version: L n,dt + 1 < m < R n,dt 6) og ogn) M = m h2 1 I N,dT 7) We observe that the first part of the impication in 7) is satisfied. Namey, if m > n 1/h + 1, then M = m h2 1 > n 1/h + 1) h2 1 > N 1/h = n h2 1) 1/h n 1/h + 1) h2 1) h > n h2 1 10

11 and this ast inequaity hods. For the second part, suppose that We want to show that this impies m < n 1/h 1 + ogn) ) dt og ogn) n 1/dt h) 8) ) dt m h2 1 < n h2 1) 1/h 1 + ognh2 1) n h2 1) 1/d 9) T h) For such m satisfying 8) we have m h2 1) h < n 1/h 1 + ogn) ) dt n 1/dt h) ) h 2 og ogn) 1 h 10) We require c T to be sufficienty arge such that N 2 c2 og ogn) T impies that > 1 + ogn) ) dt n 1/dt h) note that since N = n h2 1, we have that n is monotone increasing in N). Then the right hand side in 10) is upper bounded by n 1/h 1) 1 + ogn) ) ) dt h 2 n 1/dt h) 1 h n h2 1) 1 + ogn) ) dt h 3 n 1/dt h) 1 Thus in order to show 9), it suffices to show that n h2 1) 1 + ogn) ) ) dt h 3 dt h n 1/dt h) 1 n h2 1) 1 + ognh2 1) n h2 1) 1/d T h) 11) Finay, in order to satisfy 11), we can just choose d T sufficienty arge such that the inequaities d t h 3 d T h, ogn) ogn h2 1) and n 1/dt h) n h2 1) 1/d T h) hod separatey. This concudes the proof of the emma. Lemma 4.6. Powers of F satisfy Property [ ]) For each q N, there exist integers c q, d q, p q such that the function F q n) satisfies Property [ ] with parameters c q, d q, p q. Proof. First we prove that F satisfies Property [ ]. The rest of the Lemma then foows from Lemma 4.5. Suppose we have aready fixed integers c 1, d 1, p 1. Now suppose that n 2 c2 1 and c 1 a b ogn) c 1 are such that F n) > an b. Say in stage n of the construction of the union function F, a guess S i), b i, i ) or S i), b i p i), i ) has been seected. By definition, in any of the resuting cases we have F n) n og n). Now it is easy to check that the associated interva I n,d1 is contained in I og n) I og n) 1, so one of the two intervas I og n), I og n) 1 contains at east haf of the integers in I n,d1. Suppose that I og n) 1 contains at east haf of the integers in I n,d1, the other case is treated anaogousy. The size of the interva I n,d1 is at east og d 1 n). Thus in I n,d1 I og n) 1 there are at east ogd 1 n) 2 Oog n)) integers m for which F m) > m 1) og n) 1 > m og n) 2. Thus for a given d 1, it suffices to choose p 1 2. Hence F satisfies Property [ ]. For further detais we refer to [H16]. 11

12 Proof of the Padding Lemma 4.4. Suppose L AT IMẼF C2 ). Consider L = {x10 x h2 x 2 x L} Then we have L AT IMEF C2 n + 1) 1/h2 )). Now since F satisfies Property [ ], it foows from the Cosure Lemma 4.5 a) that the function F C2 satisfies Property [ ]. Then it foows from the Cosure Lemma 4.5 b) that aso the function tn) := F C n + 1) 1/h ) satisfies Property [ ]. Moreover, the function tn) is computabe in ZNT IMEt 1 ɛ ) for ɛ = 1 C 1. Therefore we aso have L AT IMẼF C2 n + 1) 1/h )) AT IMẼF n + 1)), where the ast incusion foows from the inequaity F C n 1/h ) F n). Thus we have L NTIMẼ F n + 1)) = N P. Since L is a padded version of L, this impies L NP = N P = NTIMẼ F ) NTIMẼ F C2 ) 12) This concudes the proof of the Lemma. 5 A Separation Resut This section gives a proof of the foowing theorem, which is a variant of the separation resut from [G96] for casses NTIMẼ t) and AT IMẼt). Theorem 1. Separation Theorem) For every function t: N N such that tn) is computabe in ZNT IMEtn) 1 ɛ ) for some ɛ > 0, If additionay tn) satisfies Property [ ], NTIME t) AT IMEt) NTIME t) AT IMẼt) The proof consists of adapting the methods from [PPST83] and [G96] to the case of bounded nondeterministic versus aternating time compexity casses. In particuar we wi first provide the Simuation Lemma 5.1 and the Hierarchy Lemma 5.2. Lemma 5.1. Simuation Lemma) For every tn) being computabe in ZNT IMEt 1 ɛ ) for some ɛ > 0, If additionay tn) satisfies Property [ ], then NTIME t og t) AT IMEt) NTIME t og t) AT IMẼt) Proof. Let L NTIME t og t). We et bn) = tn) 1/3 and an) = tn) 2/3 og tn). According to [PPST83], a tn) og tn)-time bounded machine S is caed a, b)-bock preserving if the foowing hods: For every input x of ength n, we partition both the time interva [1, c tn) og tn)] and the tapes of the machine S into an) bocks of ength bn). Then for every bock in the time interva, the read/write-heads of the machine stay within a singe bock on their working tapes for different tapes, this bock might be different). As it was shown in [PPST83], for every c tn) og tn)-time bounded deterministic machine there exists a 3 c tn) og tn)-time bounded a, b)-bock preserving machine S such that LS ) = LS). The 12

13 same hods for aternating machines, and in that case the number of guesses and the number of aternations made by S are the same as for S. Thus we may assume that L = LS), where S is a c tn) og tn)-time bounded a, b)-bock preserving nondeterministic machine with guess S n) c time S n). Foowing the ines of [G96], we construct an Otn))-time bounded aternating machine S for L as foows: Input: x of ength n 1) Use the ZNT IMẼt1 ɛ )-agorithm to compute tn). If this agorithm returns?, reject. Otherwise, compute the bock size bn) = tn) 2/3 and the number of bocks an) = tn) 1/3 og tn). 2) Existentiay guess a binary string g of ength c tn) og tn) and a partition of g into bocks g 1,... g an). Note that some of the strings g j wi be empty. The string g contains the guesses made in the computation of machine S on input x. For 1 j an), g j contains the guesses made by S in the bock j of the time interva [1, c tn) og tn)]. 3) Perform phases 1 and 2 from the agorithm in [G96], where in step 3 of phase 1 and step 2 of phase 2, the guess strings g j are used to simuate the bocks of the computation of machine S. Whenever for some bock to be simuated, the number of bits in the associated guess string g j is too sma, then the simuation stops and rejects. We obtain that S is a Otn))-time bounded aternating machine with LS ) = LS). Moreover, we may assume that time S n) = c tn) for a n, for some constant c. Thus the function time S n) = c tn) satisfies Property [ ]. Therefore, there exists a machine S j,d with L S j,d ) = L and time Sj,d n) = Otn)), whence L ATIMẼtn)). This concudes the proof of the Simuation Lemma. Lemma 5.2. Hierarchy Lemma) If g = of) and f is computabe in ZNT IMEf), then AT IMEg) AT IMEf). Moreover, if f aso satisfies Property [ ], then AT IMEg) AT IMẼf). Proof. An Of)-time bounded aternating machine can be constructed in the standard way which diagonaizes against a AT IM Eg)-machines. Moreover, if f aso satisfies Property [ ], then the diagonaizing machine can be constructed in such a way that it makes precisey fn) steps on a inputs of ength n, which impies that the running time function of this machine satisfies Property [ ]. Proof. of the Theorem 1) First we note that if tn) satisfies Property [ ] and is computabe in ZNT IMEtn) 1 ɛ ), then AT IMEt) = AT IMẼt). Suppose now that tn) is computabe in ZNT IMEtn) 1 ɛ ) and furthermore, NTIME t) = AT IMEt). We have NTIME t) NTIME t og t) NTIME t og t) AT IMEt) Thus by assumption, we have equaity in this chain of incusions. However, due to the Simuation Lemma 5.1, NTIME t ) og t t) AT IME og, t which now yieds ) t AT IME og t = AT IMEt), 13

14 a contradiction to the Hierarchy Lemma 5.2. Coroary 2. NP AP. Proof. Suppose that NP = AP = P SP ACE. Then this impies NP = NTIMẼ F ) = AT IMẼF ) = AP. Making use of padding we obtain NTIMẼ F C2 ) = AT IMẼF C2 ). Since F and therefore aso F C2 is computabe in ZNT IMEF C ), this is a contradiction to the Separation Theorem Thm. 1). Discussion. We may ask if the methods presented here can be used to separate P H from AP = P SP ACE, or even to prove strictness of P H. We are currenty not abe to answer this question. If we start from any assumption which does not directy impy NP = conp, then it is not cear how to guarantee that the union function F n) is computabe in time F n) c for some constant c. Reca that NP = conp impies that the probem Check can be soved by a ZTIME-agorithm. This means whenever we have to sove an instance of the Check probem within the computation of F n), we can run both the nondeterministic agorithm for the instance of Check and for the instance of the compement. Thus there aways exists a computation path that yieds an output, and every path which does not reject yieds the correct answer. If we cannot make use of the impication NP = conp, this does not seem to work anymore. Nevertheess, we can modify the construction presented here such as to obtain the foowing refinement. We assume NP = Σ p 2. Again this impies NP = NP = Σ p 2. Now we start from a standard famiy S i ) of Σ 2 -machines and construct an associated subfamiy S i,d ) which contains again for each machine S i and every integer d a machine S i,d whose running time function satisfies Property [ ]. The construction of the machines S i,d is structuray the same as before. Namey, the computation on an input of ength n is spit into time intervas T 0 = 0, a 1 n b 1 ], T 1 = a 1 n b 1, a 2 n b 2 ],..., T L 1 = a L 1 n b L 1, a L n b L ], T L = ) a L n b L, For < L, the first haf of the time interva T is used to simuate the computation of the origina machine S i on the given input. The second haf of the time interva is used to check if the predicate P i, d, n, a +1, b +1 ) hods, i.e. if the machine is aowed to continue its computation in the next time interva T +1. Now the difference to the previous construction is the foowing. Suppose that the machine is in some interva T of its computation and has to use the second haf of this interva to decide if it is aowed to continue the computation in the next interva T +1. For this purpose, the machine has to check if the associated instance of the predicate P i, d, n, a +1, b +1 ) hods. Before, this was done nondeterministicay, which might increase the number of aternations made during the computation. Now we proceed differenty such as to avoid this. Namey, if at the beginning of the second hat of the interva T, the machine is in an existentia state, then it tests existentiay i.e. nondeterministicay) if the predicate P ) hods. In a the branches which do not yied a decisive answer, the computation stops and the machine rejects. We know that there is at east one computation path which yieds an answer, and this answer is then correct. However, if the machine is in a universa state at the beginning of the interva T, it aso runs the nondeterministic agorithm for the predicate P ), but now in every computation path where this does not yied a decisive answer, the machine stops its computation and accepts. Again, there is at east one path which yieds an answer to the question if the predicate P ) hods, and this answer is correct. In any such computation path which yieds the correct answer, the machine stops and rejects or continues its computation, depending on whether the answer is no or yes. Atogether we obtain that if S i is a nondeterministic machine, then S i,d is aso nondeterministic, and if S i is a Σ 2 -machine, then S i,d is aso a Σ 2 -machine. Thus we obtain in this way that the assumption NP = Σ p 2 impies that NP = N P = Σ p 2. Now we construct 14

15 the union function F as before. We obtain N P = NTIMẼ F ) = Σ 2 F ) = Σ p 2. F is now sti computabe in ZNTIMEF C ), and a padding construction yieds NTIMẼ F C2 ) = Σ 2 F C2 ). This wi then be a contradiction to the foowing version of Gupta s resut: For every function tn) such that t satisfies Property [ ] and t is computabe in ZNTIMEt 1 ɛ ) for some constant ɛ > 0, NTIMẼ t) Σ 2 t). Thus the assumption NP = Σ p 2 cannot hod. References [BDG89] J.L. Bacazar, J. Diaz, J. Gabarro, Structura Compexity II, Springer, [B67] M. Bum, A Machine-Independent Theory of the Compexity of Recursive Functions, J. ACM, XIV, No. 2, 1967, pp [BGW70] R.V. Book, S.A. Greibach, B. Wegbreit, Time and Tape Bounded Turing Acceptors and AFL s, J. Com. and Sys. Sci., ), pp [CKS81] A.K. Chandra, D.C. Kozen, L.J. Stockmeyer, Aternation, Journa of the ACM, ), pp [CS76] A.K. Chandra, L.J. Stockmeyer, Aternation, Proc. 17th Symp. on Foundations of Computer Science, 1976, pp [G96] [H16] [K81] [K83] [K85] S. Gupta, Aternating Time Versus Deterministic Time: A Separation, Math. Systems Theory 29, pp ). M. Hauptmann, On Aternation and the Union Theorem, Preprint, submitted to Computationa Compexity 11/2015, aso avaiabe at Computing Research Repository CoRR), arxiv: , R. Kannan, Towards Separating Non-Deterministic Time from Deterministic Time, FOCS, ), pp R. Kannan, Aternation and the Power of Non-Determinism, STOC, ), pp K. Kobayashi, On proving time constructibiity of functions, Theoretica Computer Science 35, pp , [K80] D. Kozen, Indexing of subrecursive casses, Theoretica Computer Science 11, pp , [McCM69] E.M. McCreight, A.R. Meyer, Casses of Computabe Functions Defined by Bounds on Computation: Preiminary Report, Proc. 1st Annua ACM Symposium on Theory of Computing, pp , [PPST83] W.J. Pau, N. Pippenger, E. Szemeredi, W.T. Trotter, On Determinism versus Non-Determinism and Reated Probems, Proc. IEEE FOCS, pp , [PPR80] W.J. Pau, E.J. Prauss and R. Reischuk, On Aternation, Acta Informatica 14, pp , 1980 [PR81a] W.J. Pau and R. Reischuk, On Aternation, II, Acta Informatica 14, pp ,

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