K. Wagner and G. Wechsung. Computational Complexity. Reidel Publishing, 1986. 16  Home/Search   Document Not in Database   Summary   Related Articles   Check

....PSPACE, and EXPTIME. We also develop a logic, based on a finitary variant of the classical game quantifier a, that captures PSPACE. We will presume familiarity with the computational complexity classes NLOGSPACE, PTIME, PH, PSPACE, and EXPTIME: for an introduction to these, see [GaJ79] or even [WW86]. I would like to thank the referee for several useful suggestions and comments. 1 Computational Complexity Classes and the Languages that Capture Them This section consists of background material in Finite Model Theory: for more on this, see [EF95] or [I99] One begins with a schema (or ....

K. Wagner & G. Wechsung, Computational Complexity (D. Reidel, 1986). 13

....with (d Gamma1) dimensional tapes. Graedel [Gra90] studied the class of languages L such that for all ffl 0, L is acceptable in time O(n 1 ffl ) by one of the respective kinds of machines, observing a slightly better robustness picture. For background on these machines and simulations, see [WW86, vEB90] Our answer to this problem of non robustness is to arrange that all of our quasilinear time upper bounds be attainable by Turing machines, and that our lower bounds hold even for RAMs. A second difficulty compared to polynomial time is that different ways of formalizing and encoding ....

....predicate if R 2 P and there is a polynomial p such that L = fx : 9 p y) R(x; y)g, and a quasilinear witness predicate if R 2 DQL and there is a quasilinear function q such that L = fx : 9 q y) R(x; y)g. Now we note the following provision about oracle Turing machines made standard in [WW86] and [BDG88] see also [LL76, Wra77, Wra78] Convention 2 Whenever an OTM M enters its query state q with some query string z on its query tape, z is erased when the oracle gives its answer. If A and B are languages such that L(M B ) A and M B runs in quasilinear time under this ....

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K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986. 30

....first glance the property O(f(n) f(O(n) seems to be restrictive, but it is not. It is easily verified that almost all of the commonly considered time complexities have this property. As usual here we remark that even the family T (DTM 1 ) is very rich. More details can be found for example in [1, 15]. Due to the small time bounds the devices under investigation are too weak for diagonalization. In order to separate complexity classes counting arguments are used. The following equivalence relation is well known. At least implicitly it has been used several times in connection with real time ....

Wagner, K. and Wechsung, G. Computational Complexity . Reidel, Dordrecht, 1986. 13

....j S 2 L Gamma Mg (and vice versa) and for any rule C, M j= C iff vM j= C. The reason for the use of models (rather than valuations) in deductive database theory stems from the fact that only positive information is explicity stored in the database. 1. 5 Definition (The polynomial time hierarchy) [St77, Wr77, Wa85]. A P k formula (in L) is one of the form 9X 1 8X 2 9X 3 : Q k X k Phi where each X i is a string of predicates from L, and Phi is propositional in the predicates in X 1 ; X 2 ; X k . Q k = 8 if k is even, else Q k = 9. A problem is: a) in the class P P k iff it is ....

K. Wagner and G. Wechsung, Computational Complexity (Reidel, 1985).

....that nondeterministic qlin 1 time for these machines, namely NNLT, equals NQL. However, currently it appears that DNLT is larger than DQL, and that for all d 1, Turing machines with d dimensional tapes accept more languages in time qlin than do TMs with (d Gamma 1) dimensional tapes (cf. WW86] Our constructions all work for DQL as well as DNLT. Our main motivation is to ask: How much of the known theory of complexity classes based on polynomial time carries over to the case of quasilinear time Section 2 observes that the basic results for the polynomial hierarchy and PSPACE hold ....

....A 2 NP, B 2 P, and p is a polynomial such that for all x, x 2 A ( 9 p y) B(x; y) then we call B a witness predicate for A, with the length bound p understood. We use the same terms in the context of NQL and DQL. We note the following provision about oracle Turing machines M made standard in [WW86] see also [LL76, Wra77, Wra78] Whenever M enters its query state q with the 2 query string z on its query tape, z is erased when the oracle gives its answer. If the oracle is a function g, we suppose that g(z) replaces z on the query tape in the next step. If A and B are languages such that ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986. 13

....on the number of tapes or on the number of heads have been shown. Moreover, for multi or k tape TMs (k 2) lineartime is known to yield strictly more powerful acceptors than real time. For any time complexity n r(n) at most a speed up to n ( Delta r(n) 0) is 2 possible (e.g. see [19] for TM results) The remaining gap between real time and linear time was filled with an infinite dense hierarchy in [5] The basic notions and the model in question are defined in the next section. Section 3 is devoted to the hierarchy theorem and some examples. In Section 4 the proof of the ....

Wagner, K. and Wechsung, G. Computational Complexity . Reidel Publishing, Dordrecht, 1986. 10

....and output only serve data transport without any computational capabilities. But through communication, the computational capabilities of other machines can be utilized. Interaction can then be interpreted as a (subroutine) call. Assuming this view, interaction machines resemble oracle machines [Wagner and Wechsung 1986, p. 48 f] Oracle machines compute the solution of a problem with respect to an oracle. Only if the oracle is undecidable, oracle machines can solve non computable problems. A decidable oracle, however, coincides with a subroutine call to a Turing machine, so that there is a Turing machine for ....

Wagner, K. and Wechsung, G. (1986) Computational Complexity. VEB Deutscher Verlag der Wissenschaften, Berlin.

....the property O(f(n) f(O(n) seems to be restrictive, but it is not. It is easily verified that almost all of the commonly considered time complexities have this property. As usual here we remark that at least for k 2 the family T (DTM k ) is very rich. More details can be found for example in [1, 12]. 3 Hierarchies Between Real Time and Linear Time In this section we will present our main results, time hierarchies between real time and linear time. Due to the small time bounds the devices under investigation are too weak for diagonalization. In order to separate complexity classes counting ....

Wagner, K. and Wechsung, G. Computational Complexity . Reidel, Dordrecht, 1986. 29

.... the one of algebraic (or context free) linear algebraic (or linear context free) languages and CS the one of context sensitive languages (see [12] Moreover, d] means that the result is a consequence of two facts: PCA = DSPACE(n) that can be deduce from simulations in [26] and CS = NSPACE(n) [34] for example. We can add that the unary languages in RPOCA are the unary rational languages [27] Rat RPOCA Rat RSCA AlgL RPOCA [27] Alg ae SOCA [15] Alg 6ae RSCA [3] RSCA 6ae Alg [3] RPOCA 6ae Alg [3] Alg 6ae RPOCA [32] PCA = SCA CS So, connections between the Chomsky s hierarchy and ....

Wagner K. and Wechsung G. Computational Complexity. Reidel, 1986. 30

....realize, under 6 Recall that from the computability point of view Turing machines and cellular automata the latter ones here considered on finite configurations are equivalent, but from the complexity point of view, cellular automata are much more efficient. See Part 3 in this volume and [81] for example. 30 conditions, the connection between two devices (or two classes of devices) are ad hoc or, often, underhanded. In particular, different notions of universality are attached to different sorts of simulations. 4.3.1 Simulations A Turing machine which computes a partial function ....

....of) a Turing machine. Note that in dimension 2, the proof of computation universality of the Game of Life by Conway is in a similar vein. Carrying on with work and reflections initiated by Wolfram [85] some people try to find out connections between computational universality and complexity [81]. The above results lean on simulations of sequential models expressing computational universality, which masks or neglects the parallel nature of 33 cellular automata. Moreover it is interesting to compare the expressive power of cellular automata on their own. This leads to intrinsic ....

Wagner K. and Wechsung G. Computational Complexity. Reidel, 1986.

....random access machines, models, simulation, finite automata. 1. Introduction It is widely believed that random access machines (RAMs) are more efficient and powerful than multitape Turing machines (TMs) However, no conclusive separation has been proven. The standard RAM model ( ordinary RAM in [21]) has addition and subtraction as basic operations. Inputs x 2 f 0; 1 g n are initially placed in the first n registers, one bit per register. The best known result is that for time bounds t(n) which are Omega Gamma n log n] and time constructible by RAMs, a TM running in time t(n) can be ....

....simulation [6] can be carried out efficiently by a BM. Lemma 3.1. For any time function t, DTIME[t(n) D 1 TIME[t(n) log t(n) Proof Sketch. Let T be a multitape TM with alphabet Gamma which runs in time t(n) and consider the two tape TM T 0 in the Hennie Stearns theorem (see [6, 8, 21]) The k many tapes of T are simulated on 2k many tracks of the first tape so that all tape heads of T are maintained on cell 0 of each track. T 0 uses its second tape only to transport blocks of the form [2 j Gamma1 : 2 j Gamma 1] from one part of the first tape to another. The ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

.... that the IDs themselves do not become bigger than the initial bound t 0 (n) on the time for each stage of M , as might happen if we merely specify that A 2 DTIME[t(n) 2 ] This raises the question of whether any interesting collapses happen if, say, DTIME[n 2 ] DTISP[n 2 ; n] The text [WW86] devotes an entire chapter (26. to time space problems like this one, but has no mention of whether this implies P = DTISP[n O(1) n] or even P ae DSPACE[n] This is plausible on padding and translation grounds, but we have been unable to prove it. The obstacle to applying the above proof to ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

....showed that nondeterministic qlin time for these machines, namely NNLT, equals NQL. However, currently it appears that DNLT is larger than DQL, and that for all d 1, Turing machines with d dimensional tapes accept more languages in time qlin than do TMs with (d Gamma 1) dimensional tapes (cf. WW86] Our constructions all work for DQL as well as DNLT. Our main motivation is to ask: How much of the known theory of complexity classes based on polynomial time carries over to the case of quasilinear time Section 2 observes that the basic results for the polynomial hierarchy hold also for the ....

....R 2 P, and p is a polynomial such that for all x, x 2 A ( 9 p y) R(x; y) then we call R a witness predicate for A, with the length bound p understood. We use the same terms in the context of NQL and DQL. We note the following provision about oracle Turing machines M made standard in both [WW86] and [BDG88] see also [LL76, Wra77, Wra78] Whenever M enters its query state q with the query string z on its query tape, z is erased when the oracle gives its answer. If the oracle is a function g, we suppose that g(z) replaces z on the query tape in the next step. If A and B are languages ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

....and lower bounds are shown for various data processing primitives, and some interesting open problems are given. 1 Introduction Recent years have seen marked dissatisfaction with the computational realism of the classic machine models, such as Turing machines or the standard integer RAM (see [13, 12]) Many algorithms that theoretically run in linear time on the RAM scale non linearly when it comes time to implement them. Cook [5, 6] proposed replacing the usual unit cost RAM measure by the log cost criterion, by which an operation that reads an integer i stored at address a is charged log ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

....Following (Gottlob, 1995b) we also write Q B for Q 0 B FO. For a class of languages C we use the notation Q C with the obvious meaning, e.g. FO(Q 0 C ) denotes all first order sentences with arbitrary quantifiers Q 0 B for B 2 C. Succinct Representations A succinct representation (Wagner, 1986; Balcazar et al. 1992; Veith, 1996) of a binary word x is a boolean circuit given on input i the ith bit of x. The succinct version sA of a language A is the following: Given a boolean circuit describing a word x, is x = x 1 0x 2 0 Delta Delta Delta x n 1 0x n 1w for arbitrary w 2 f0; 1g ....

....P[1] etc. On the other hand, we can use Hertrampf s result to obtain logical characterizations of these classes, e.g. P NP[1] Phi P[1] can be characterized by a certain firstorder Lindstrom quantifier. Similarly we can use completeness results for certain succinct problems, e.g. those in (Wagner, 1986; Balcazar et al. 1992) to obtain immediately logical characterizations of the corresponding classes, either with first order or with second order Lindstrom quantifier. A naturally arising question now is of course: what about nested quantifiers We come back to this issue later. 4 Constant ....

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Wagner, K. W. and Wechsung, G. (1986). Computational Complexity. VEB Verlag der Wissenschaften, Berlin.

....time unit per bit of address and operand values in an instruction. Sources differ on how to represent an input x 2 f 0; 1 g n in a RAM based model: i) as an integer in register R 0 , ii) bitwise in the first n registers, or (iii) on a separate read only Turing tape; iii) is standardized in [WW86] and used in [KvLP88, TLR92, LL92b] 1 The author s own work reported here was partly supported by NSF Grant CCR 9011248 2 Having unit cost multiplication yields the MRAM, which can accept QBF and other PSPACE complete languages in polynomial time [HS74] and is regarded as a parallel model ....

.... D 1 TIME[O(t log t) b) D log TIME[t] TC TIME[O(t) D log TIME[O(t log t) c) For any d 1, D d TIME[t] DTIME d [t] Thus letting vary from log to 1 spans the whole range of models in Section 1. The second inclusion in (a) follows because the Hennie Stearns simulation [HS66, HU79, WW86] is memory efficient under 1 . We suspect that for d 1 the converse simulation in (c) requires notably more than the O(log t) overhead of the d = 1 case (a) see [PSS81] for related matters. The intuitive reason is that a d TM may often change its head direction, but in going to a BM this is a ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

....with (d Gamma1) dimensional tapes. Graedel [Gra90] studied the class of languages L such that for all ffl 0, L is acceptable in time O(n 1 ffl ) by one of the respective kinds of machines, observing a slightly better robustness picture. For background on these machines and simulations, see [WW86, vEB90] For related work on limited nondeterminism classes, see [KF80, DT90, BG93, BG94] Our answer to this problem of non robustness is to arrange that all of our quasilinear time upper bounds be attainable by Turing machines, and that our lower bounds hold even for RAMs. A second ....

....= f x : 9 p y) R(x; y) g, then we call R a polynomial witness predicate; while if R 2 DQL and there is a quasilinear function q such that L = f x : 9 q y) R(x; y) g, then R is a quasilinear witness predicate. Now we note the following provision about oracle Turing machines made standard in [WW86] and [BDG88] see also [LL76, Wra77, Wra78] Convention 2.2. Whenever an OTM M enters its query state q with some query string z on its query tape, z is erased when the oracle gives its answer. If A and B are languages such that L(M B ) A and M B runs in quasilinear time under this ....

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K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

.... for logarithmic bounds the corresponding question is still open, i.e. for any k it is unknown whether AAlterSpace(log k ; log ) ae AAlterSpace(log k 1 ; log ) It is well known that for any function S the complexity class Sigma 1 Space(S) is closed under union and intersection (see e.g. [25]) However, it is still an open problem whether for S 2 SUBLOG the class Sigma 1 Space(S) is closed under complementation. More general, for arbitrary k the classes Sigma k Space(S) are closed under union, and symmetrically the Pi k Space(S) are closed under intersection. In [14] we have ....

....2 Pi k 1Space(llog ) n Sigma k 1Space(o(log ) A Pik [ B Pik 2 Sigma k 1Space(llog ) n Pi k 1Space(o(log ) Proof. It is well known that for any function S the classes Sigma k Space(S) are closed under union, and symmetrically the Pi k Space(S) are closed under intersection (see e.g. [25]) Hence by (i) A Sigmak B Sigmak 2 Pi k 1Space(llog ) and A Pik [ B Pik 2 Sigma k 1Space(llog ) To prove that A Sigmak B Sigmak 62 Sigma k 1Space(o(log ) and A Pik [ B Pik 62 Pi k 1 Space(o(log ) first we modify Proposition 3.9 in the following way: Proposition 4.2. Let k ....

K. Wagner, and G. Wechsung, Computational complexity, Reidel, Dordrech, 1986.

....: j E k . We do show, however, that the containment problem for expressions of this form is exponential space complete. 2 Encoding Turing Machine Computations by Regular Expressions with Interleaving We assume familiarity with regular expressions and time and space complexity; see, e.g. 9] or [22] if needed. The interleaving of words x and y, denoted xjy, is the set of all words of the form x 1 y 1 x 2 y 2 : x k y k where x = x 1 x 2 : x k and y = y 1 y 2 : y k and where the words x i and y i , 1 i k, can be of arbitrary length (including the empty word) If X and Y are sets ....

K. Wagner and G. Wechsung, Computational Complexity (D. Reidel, Dordrecht, 1986).

....p. 10] Theorem 3.2. Let p be a prime number. Then for all n 0 we have p (n ) n Gamma s p (n) p Gamma 1 : One annoyance is that the canonical representation in base k suffers from the leading zeros problem that is, the map w [w] k is not one one if w 2 Sigma k . For example, [101] 2 = 0101] 2 = 00101] 2 = 5. One way around this difficulty is the following simple folk theorem , whose precise origins are unknown to me (but see [87, Note 9.1, pp. 90 91] 101, p. 24] and [40] Theorem 3.3. Let k be an integer 2. Then every non negative integer can be represented ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

....will be used. We begin with the results concerning the simulations of higher dimensional CA on lower dimensional CA and vice versa. Observe that in both cases the size of the computation cube remains invariant. 5.1 Proposition. Change of dimensionality) If d is space constructible in time t [25]: 1. Z r GammaCA GammaExt GammaTime(d; t) Z r Gamma1 GammaCA GammaExt GammaTime(d r= r Gamma1) d 1= r(r Gamma1) t) 2. Z r Gamma1 GammaCA GammaExt GammaTime(d; t) Z r GammaCA GammaExt GammaTime(d (r Gamma1) r ; t) Proof. 1. The basic idea for the proof of this result ....

K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, Dordrecht, 1986.

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K. Wagner and G. Wechsung. Computational Complexity. Reidel Publishing, 1986. 16

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Wagner, K. and Wechsung, G. 1986. Computational Complexity. D. Reidel, Dordrecht, The Netherlands.

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Wagner, K. and Wechsung, G. 1986: Computational Complexity, Reidel, Dordrecht.

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Klaus Wagner and Gert Wechsung, Computational complexity, Reidel, New York, 1986.

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Wagner, K. and Wechsung, G. 1986: Computational Complexity, Reidel, Dordrecht.

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K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.

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K. Wagner and G. Wechsung, Computational Complexity, Dordrecht: Reidel (1986).

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