K.J. Lange, P. McKenzie, and A. Tapp, Reversible space equals deterministic space, Proc. 28th ACM Symp. Theory of Computing (1996) 212-219.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... S is the space usage of the original machine and f is very close to linear (say, n log n or smaller) We generalize the previous results on this reversibility problem by proving a general theorem incorporating two simulations: one is space e#cient (O(S) and is due to Lange, McKenzie, and Tapp[5]; the other is time e#cient (O(T ) for any # 0, where T is the time usage of the original machine) and is due to Bennett[2] Corollaries of our general theorem give interesting new time space tradeo#s. One is that for any unbounded space constructible f(n) o(T (n) there is a reversible ....

....to perform our usual irreversible operations, erasing and writing data with abandon. To this end, simulations of deterministic (irreversible) computation suitable for reversible computers have been studied. The most celebrated work in this area includes Bennett [2] and Lange, McKenzie, and Tapp [5] (hitherto referred to as LMT) The former (along with an analysis of Levine and Sherman[6] showed that a reversible simulation of an arbitrary deterministic TM M using space S and time T could be performed in space #(c(#)S[1 log(T S) and time #(T ) for any # 0, where c(#) #2 . ....

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K.J. Lange, P. McKenzie, and A. Tapp, Reversible space equals deterministic space, Proc. 28th ACM Symp. Theory of Computing (1996) 212-219.

....inducing normpreserving operators on the associated Hilbert space) Quantum transformations will consist of a single step, so it will be trivial to argue that each quantum transformation can be performed as claimed. For reversible transformations, we rely on the result of Lange, McKenzie and Tapp [7], which implies that any logspace deterministic computation can be simulated reversibly in logspace. However, because the interference patterns produced by a given QTM depend greatly upon the precise lengths of the various computation paths comprising that machine s computation, we must take care ....

....paths comprising that machine s computation, we must take care to insure that these lengths are predictable in order to correctly analyze the computation. In the remainder of this section, we discuss reversible transformations somewhat more formally, and state a theorem based on the main result of [7] that will simplify this task. For a given space bound f and work tape alphabet #, define W f( x ) #) to be the set of all mappings of the form w : Z # # taking the value # (blank) outside the interval [ f( x ) f( x ) i.e. those mappings representing the possible contents of the work tape of ....

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K. Lange, P. McKenzie, and A. Tapp. Reversible space equals deterministic space (extended abstract). In Proceedings of the 12th Annual IEEE Conference on Computational Complexity, pages 45--50, 1997. To appear in Journal of Computer and System Sciences.

....T . If T is maximal, that is, exponential in S, then the space use is S 2 . This method can be modelled by a reversible pebble game. Reference [12] demonstrated that Bennett s method is optimal for reversible pebble games and that simulation space can be traded off against limited erasing. In [9] it was shown that using a method by Sipser [16] one can reversibly simulate using only O(S) extra space but at the cost of using exponential time. Results: Previous results seem to suggest that a reversible simulation is stuck with either quadratic space use or exponential time use. This ....

....on the order of the space S used by the simulated machine. In that case bridge(s; t) is easily implemented with the help of an additional history tape of size m which records the sequence of transitions. Instead, we allow an arbitrary choice of n and resort to the space efficient simulation of [9] to bridge the pebbled checkpoints. 4 Space Parsimoneous Simulation Lange, McKenzie and Tapp, 9] devised a reversible simulation, LMT simulation for short, that doesn t use extra space, at the cost of using exponential time. Their main idea of reversibly simulating a machine without using more ....

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K.J. Lange, P. McKenzie, and A. Tapp, Reversible space equals deterministic space, J. Comput. System Sci., 60:2(2000), 354--367.

....inducing unitary operators on the associated Hilbert space. Quantum transformations will consist of a single step, so it will be trivial to argue that each quantum transformation can be performed as claimed. For reversible transformations, we rely on the result of Lange, McKenzie, and Tapp [9], which implies that any logspace deterministic computation can be simulated reversibly in logspace. However, because the interference patterns produced by a given QTM depend greatly upon the precise lengths of the various computation paths comprising that machine s computation, we must take care ....

....paths comprising that machine s computation, we must take care to insure that these lengths are predictable in order to correctly analyze machines. In the remainder of this section, we discuss reversible transformations somewhat more formally, and state a theorem based on the main result of [9] that will simplify our analyses greatly. For a given space bound f and work tape alphabet Gamma, define W f(jxj) Gamma) to be the set of all mappings of the form w : Z Gamma taking the value # (blank) outside the interval [ Gammaf (jxj) f(jxj) i.e. those mappings representing the ....

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K. Lange, P. McKenzie, and A. Tapp. Reversible space equals deterministic space (extended abstract) . In Proceedings of the 12th Annual IEEE Conference on Computational Complexity, pages 45--50, 1997.

....for a 2 given space bound s, the class of languages recognizable in space O(s) by an RTM. It was later proved [8] that nondeterministic Turing machines can also be simulated reversibly with the same increase in space, i.e. NSPACE(s) RevSPACE(s 2 ) Recently, Lange, McKenzie and Tapp [18] proved that, at the cost of a possibly exponential increase in running time, DTMs can be simulated by RTMs with only a constant factor increase in space, i.e. DSPACE(s) RevSPACE(s) Various relationships regarding quantum simulations of probabilistic machines follow from derandomization, given ....

....Note that the restriction implies that U x is necessarily a norm preserving operator for every input x, following from the fact that each V oe is unitary. We define reversible Turing machines (RTMs) to be QTMs having transition functions that take only the values 0 and 1. The RTMs considered in [2, 3, 18] have transition functions that may be expressed as QTMs in this way, in accordance with the abovementioned restriction. In order for a QTM to reveal any information about its computation, it must be observed; the information revealed by a particular observation is described by an observable. For ....

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K. Lange, P. McKenzie, and A. Tapp. Reversible space equals deterministic space (extended abstract). In Proceedings of the 12th Annual IEEE Conference on Computational Complexity, pages 45--50, 1997.

....which have error probability approaching zero. We also note that the non contextfree language fa m b m c m j m 1g can be recognized by bounded error, linear time 2qfa s, based on the same technique. In order to prove 2, we apply a technique from a recent result of Lange, McKenzie and Tapp [14] regarding reversible simulation of deterministic Turing machines to the finite automaton case. A corollary of 2 is that reversible 2 way finite state automata are equivalent in power to (1 way or 2 way) deterministic finite state automata. This is in contrast to the fact that 1 way reversible ....

....in fb m c m j m 1g, and finally checks the initial part of the string to see that it is in fa m b m j m 1g. Details will appear in the final version of this paper. 5 Reversible simulation of 1dfa s In this section, we use a technique from a recent result due to Lange, McKenzie and Tapp [14] regarding space efficient reversible simulation of deterministic Turing machines to show that an arbitrary 1 way deterministic finite automaton (1dfa) can be simulated by a 2 way reversible finite automaton (2rfa) which we may simply define as a well formed 2qfa whose transition amplitudes may ....

K. Lange, P. McKenzie and A. Tapp. Reversible space equals deterministic space (extended abstract) . In Proceedings of the 12th IEEE Conference on Computational Complexity, 1997. To appear.

....for a given space bound s, the class of languages recognizable in space O(s) by a RTM. It was later proved [9] that nondeterministic Turing machines can also be simulated reversibly with the same increase in space, i.e. NSPACE(s) RevSPACE(s 2 ) Quite recently, Lange, McKenzie and Tapp [22] proved that, at the cost of a possibly exponential increase in running time, DTMs can be simulated by RTMs with only a constant factor increase in space, i.e. DSPACE(s) RevSPACE(s) Given that DSPACE(s) RevSPACE(s) we may deduce various relationships among probabilistic and quantum ....

....V oe i j q; oe w i d i = D i (q 0 ) dw = Dw (q 0 ) Z(q 0 ) 0 otherwise: This restriction is analogous to unidirectionality for the single tape QTM model, discussed in [6] therein it is shown that this restriction does not decrease the power of QTMs. Similarly, the RTMs considered in [3, 4, 22] obey this restriction (where each V oe is a permutation in this case) In the interest of simplicity, we prefer to include this restriction as part of the definition of multi tape QTMs. In short, the restriction requires that the output and movement of tape heads be determined by whatever ....

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K. Lange, P. McKenzie and A. Tapp. Reversible space equals deterministic space, In Proceedings of the 12th Annual IEEE Conference on Computational Complexity, pages 45--50, 1997.

....We conjectured that all reversible simulations of an irreversible computation can essentially be represented as the pebble game defined below, and that consequently the lower bound of Corollary 2 applies to all reversible simulations of irreversible computations. This conjecture was refuted in [11] using a technique due to [18] to show that there exists a general reversible simulation of an irreversible computation using only order S space at the cost of using a thoroughly unrealistic simulation time exponential in S. In retrospect the conjecture is phrased too general: it should be ....

....produces a reversible algorithm B such that B(x) A(x) for all input x and using T 0 = O(T ) time and S 0 = O(S) space. In the extreme cases of time and space use this is possible: If S = Theta(T ) then the simulation in [1] does the trick, and if T = Theta(2 S ) then the simulation of [11] works. For all other cases the pebble game analysis below has been used in [7] to show that any such simulation, if it exists, cannot relativize to oracles, or work in cases where the space bound is much less than the input length. This is a standard method of giving evidence that the aimed for ....

K.J. Lange, P. McKenzie, and A. Tapp, Reversible space equals deterministic space, Proc. 12th IEEE Computational Complexity Conference, IEEE Comp. Soc. Press, 1997.

....have error probability approaching zero. We also note that the non context free language fa m b m c m j m 1g can be recognized by bounded error, polynomial time 2qfa s, based on the same technique. In order to prove 2, we apply a technique from a recent result of Lange, McKenzie and Tapp [15] regarding reversible simulation of deterministic Turing machines to the finite automaton case. As a corollary of 2, we have that reversible 2 way finite state automata are equivalent in power to (1 way or 2 way) deterministic finite state automata. This is in contrast to the fact that 1 way ....

.... can be simulated in polynomial time by a two way reversible finite automaton (2rfa) which we may simply define as a well formed 2qfa whose transition amplitudes are elements of the set f0; 1g) In order to prove this fact, we use a technique from a recent result due to Lange, McKenzie and Tapp [15] regarding space efficient reversible simulation of deterministic Turing machines. Here, the construction is considerably simpler than in the Turing machine case, due to the fact that 1dfa s are much simpler than Turing machines. A 1dfa can be formally specified by a quintuple A = S; Sigma; s ....

K. Lange, P. McKenzie, and A. Tapp. Reversible space equals deterministic space (extended abstract). In Proceedings of the 12th IEEE Conference on Computational Complexity, 1997. To appear.

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