Alan Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in in the context of sorting [DNS91] selectivity estimation [PIHS96] query optimization [SALP79] and in providing online user feedback [Hel] The survey by Yannakakis [Yan90] is a comprehensive account of graph theoretic methods in database theory. Classical work on time space tradeoffs [Cob66, Tom80] may be interpreted as lower bounds on workspace for problems such as verifying palindromes, perfect squares and undirected st connectivity. Paterson and Munro [MP80] studied the space required in selecting the kth largest out of n elements using at most P passes over the data. They showed an ....

A. Cobham. The recognition problem for the set of perfect squares. IBM Research Report RC 1704, T.J. Watson Research Center, 1966.

....bits to be computed in any order. The model is strong enough to eciently simulate any other reasonable model of computation, such as multitape Turing machines or RAMs. In Section 3, we recall the de nition of the model, and the notions of time and space in this model. With the exception of [10], rst time space tradeo s were obtained on structured models of compuation. Tompa [23] contains a discussion of these results. In the case of non structured models, Cobham [10] shows a timespace tradeo for any computational device having one head read only input tape. The boolean branching ....

....Section 3, we recall the de nition of the model, and the notions of time and space in this model. With the exception of [10] rst time space tradeo s were obtained on structured models of compuation. Tompa [23] contains a discussion of these results. In the case of non structured models, Cobham [10] shows a timespace tradeo for any computational device having one head read only input tape. The boolean branching program removes this restriction to accessing the input bits. The rst non trivial time space tradeo s in this model were given by Borodin and Cook [6] who proved that sorting ....

A. Cobham. The recognition problem for the set of perfect squares, conference record. In IEEE 7th Annual Symposium on Switching and Automata Theory, pages 78-87, 1966.

.... algorithms and lower bounds (see Knuth [1] and Andersson [2] for an overview of classical as well as more recent work in the area) One fruitful line of research has been the investigation of the trade off between the two most fundamental complexity measures; time and space pioneered by Cobham [3]. Accordingly, time space trade offs for sorting is a much studied problem [4, 5, 6, 7, 8, 9] Despite the successes of this work, a discrepancy of at least a logarithmic factor (referred to in [4, 6, 7, 10, 11] between the best known upper bound O(n log n) 5] and the best known lower ....

A. Cobham, "The Recognition Problem for the Set of Perfect Squares," in Conference Record of

....of previously known time space tradeo s for speci c languages proved using combinatorial methods. Such methods have been used extensively to prove lower bounds or tradeo s on restricted models for example, in [3] 6] 8] and [11] An early result of this kind was the tradeo proved by Cobham [1] for the acceptance of languages on Turing machines. Cobham s examples worked only for Turing machines with one read write 1 head per tape; tradeo s for Turing machines with multiple (but constant) read write heads per tape were proved by Duris and Galil [2] We use the results of [1] and [2] ....

....by Cobham [1] for the acceptance of languages on Turing machines. Cobham s examples worked only for Turing machines with one read write 1 head per tape; tradeo s for Turing machines with multiple (but constant) read write heads per tape were proved by Duris and Galil [2] We use the results of [1] and [2] and construct ecient reductions to SAT from the languages considered by them, to prove new time space tradeo s for the recognition of SAT by Turing machines. Speci cally, we prove that if SAT is accepted by a non deterministic Turing machine using time T (n) and space S(n) then T ....

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A.Cobham. The recognition problem for the set of perfect squares. In Conference Record of the Seventh Annual Symposium on Switching and Automata Theory, pages 78-87. IEEE, New York, 1966.

....It is further argued that such programs are generally no more powerful than LOGCFL. 1 Introduction An important open problem in complexity theory concerns the relationship between logarithmic space (class L) and polynomial time (class P) Is L indeed a proper subset of P A well known strategy [Co66] which would answer this question affirmatively requires proving that a given language Y 2 P cannot be accepted by a polynomial size branching program. Although unsuccessful to this date because of the notorious difficulty of obtaining significant branching program size lower bounds, this strategy ....

A. Cobham, The recognition problem for the set of perfect squares, Research Paper RC-1704, IBM Watson Research Center, Yorktown Heights, New York (April 1966).

.... lower bounds (see Knuth [10] and Andersson [1] for an overview of classical as well as more recent work in the area) One fruitful line of research has been the investigation of the trade off between the two most fundamental complexity measures, time and space, an investigation pioneered by Cobham [7]. Accordingly, time space trade offs for sorting is a much studied problem [2, 4, 6, 8, 13, 15] Despite the successes Supported by the ESPRIT Long Term Research Programme of the EU under project number 20244 (ALCOM IT) E mail: fpagter,theisg brics.dk. y Part of this work was done while the ....

A. Cobham. The Recognition Problem for the Set of Perfect Squares. In Conference Record of 1966 Seventh Annual Symposium on Switching and Automata Theory ("FOCS'7"), pages 78--87. IEEE, 1966.

....or determined by flipping a fair coin, resp. More precise definitions are given later on. It is a well known fact that sequences of functions which can be computed by BPs of polynomial size can also be computed within logarithmic space on the nonuniform variant of Turing machines and vice versa [18, 42]. Hence, it is an important problem to prove superpolynomial lower bounds on the size of BPs for explicitly defined sequences of functions. So far, not even superlinear lower bounds for deterministic BPs are known. Nevertheless, superpolynomial and exponential lower bounds could be proven for ....

....be made deterministic by setting the probabilistic variables to constants in an appropriate way. # 22 Randomized BPs are defined in a such a way that they may be simulated by probabilistic nonuniform Turing machines and vice versa analogously to the well known result for the deterministic case [18, 42]. Especially, this gives us the following results (where PL Poly is the class of all sequences of functions computable by a probabilistic nonuniform Turing machine with unbounded two sided error using at most logarithmic space) Proposition 5.6: NP BP = NL Poly, PP BP = PL Poly. This is ....

A. Cobham. The recognition problem for the set of perfect squares. In Proc. of the 7th Symposium on Switching an Automata Theory (SWAT), 78--87, 1966.

....a single resource (such as time or space) does not always correctly capture the entire picture regarding the complexity of the problems at hand. Consequently, some work was directed to considering two resources simultaneously, most commonly time and space product, denoted by TS. In 1966 Cobham [Co66] showed that any computational device with one read only input tape must satisfy TS = Omega (n 2 ) in order to be able to recognize the set of perfect squares (n is the length of the input number) Although working within the severe limitation of tape input (and deriving his result from the ....

....issue when discussing lower bounds. As described above, much of the earlier results were derived for models of a very restrictive type, such as tape input, or oblivious programs. The models adopted in this paper are variants of the quite general branching program model, as introduced by Cobham [Co66], described by Borodin et al. BFKLT81] and generalized by Borodin and Cook [BC82] This model is formally defined in section 2. Our results are based on techniques that were used previously to provide bounds for 2 two problems, namely Element Distinctness (ED) and Unique Elements (UE) ....

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A. Cobham, The Recognition Problem for the Set of Perfect Squares, 7th IEEE Symp. on Switching and Automata Theory, 1966, pp. 78-87.

....by a sequence of branching programs of polynomial size. The depth of a branching program obviously measures the time of computation required in the worst case. On the other hand, nodes of branching programs closely correspond to configurations of other sequential models of computation. Cobham [35] and Pudlak and Zak [91] have independently used this correspondence to prove that the size of branching programs is essentially the logarithm of the space complexity on nonuniform Turing machines. Especially, we have the following important result: Theorem 1.6 (Cobham Pudl ak and Z ak) P BP = ....

A. Cobham. The recognition problem for the set of perfect squares. In Proc. of the 7th Symposium on Switching an Automata Theory (SWAT), 78--87, 1966.

....[22] under the assumption of the Extended Riemann Hypothesis) and over finite fields [17] unconditionally) to obtain a polynomial time test deciding whether a given integer M of the form (1) or a polynomial F of the form (3) are perfect squares. Our algorithm is somewhat related to that of [8] 4 which deals with a similar problem for integers in their standard representation. However, in our situation that approach produces an exponential time algorithm even if computing Jacobi symbols is for free . Roughly speaking, in this paper, instead of using bounds on the smallest quadratic ....

....with a similar problem for integers in their standard representation. However, in our situation that approach produces an exponential time algorithm even if computing Jacobi symbols is for free . Roughly speaking, in this paper, instead of using bounds on the smallest quadratic nonresidue (as in [8]) we use some results about the density of quadratic nonresidues in small intervals. It is known that many natural problems with sparse integers and polynomials are NP hard [14, 18, 26, 27, 28] and it is of ultimate interest to study problems which admit polynomial time algorithms. We also ....

A. Cobham, `The recognition problem for the set of perfect squares', Proc. of the 7th ACM Symp. on Switching and Automata Theory , 1966, 78--87.

....on the type of the general machine. Acceptance is defined as it is for the corresponding types of Turing machines. Time is defined as the number of moves, and space as log 2 Q, where Q is the number of states. General machines are almost identical to the recognition machines defined by Cobham [23], except that recognition machines require the input to be accessed sequentially, whereas general machines allow completely random access to the input. It is also easy to see that Turing machines are a special case, by including the worktape contents and head positions as part of the state of the ....

A. Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.

....is a quadratic non residue. Quadratic residuosity can be tested by computing the Legendre symbol. The expected number of Legendre symbol computations for proving that a given integer is not a square is on the order of 2. The problem of proving a number is not a perfect square was considered in [19]. For related results and more references we refer to [9] Results that are similar to this lemma are described in [17] and [22] but the proofs appear to be more complicated. Note that Lemmas 1 and 2 can easily be generalized to allow a to be different for each q j [16] but we will only make use ....

A. Cobham, The recognition problem for the set of perfect squares, Proc. 7th Annual Syp. on Switching and Automata Theory, pp 78-87, 1966.

....crucially on the sequential access feature may not say anything meaningful about resource requirements on real machines. We will give two examples. As our first example, consider the problem of checking whether two halves of a given input string of length 2n are equal. Cobham s classical result [Cob66] states that for any Turing machine solving this problem, the product of its time and space requirement is at least Omega0 n 2 ) However, the proof crucially relies on the fact that in order to read input symbols located far apart on the tape, the Turing machine has to spend a lot of time. In ....

....of Turing machines, we also need to define nonuniform versions of Turing machines. An s(n) space bounded nonuniform Turing machine is allowed to hardwire 2 O(s(n) bits of advice. See page 279 of Wegner [Weg87] for a definition of nonuniform Turing machines. 14 Proposition 2. 2 (Cobham [Cob66] Pudl ak and Z ak [PZ83] For any s = Omega0 log n) branching programs using space O(log s) compute exactly the same class of functions as nonuniform Turing machines using space O(log s) Proof sketch: We will state the basic intuition of the proof; a complete proof appears on page 415 of ....

Alan Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.

....to separating complexity classes like NL and NP. This paper shows that separating these classes might be not nearly as difficult as previously believed, perhaps considerably easier than separating P from NP. 1. 1 Related Work The question of time space tradeoffs goes back to 1966 when Cobham [Cob66] showed that palindromes required quadratic time space tradeoffs on a Turing machine with a single head on a readonly input tape. Much of the work on time space tradeoffs deals with restricted machine models such as comparison models (see [Yao94] and JAG models (see [EP95] For Turing machines, ....

A. Cobham. The recognition problem for the set of perfect squares. In Conference Record of the Seventh Annual Symposium on Switching and Automata Theory, pages 78--87. IEEE, New York, 1966.

....have infinite worst case time. Hence, with the random bits provided at the beginning of the computation, the expected time would be infinite. However, a probabilistic WAG can easily traverse such a graph in expected n 2 time. bounds say something about time and space simultaneously. Cobham [Co66] established the first time space tradeoff. The model used was a one tape Turing Machine. Tompa [Tm80] established a number of time space tradeoffs for both the Boolean and the arithmetic circuit models. Borodin, Fischer, Kirkpatrick, Lynch, and Tompa [BFKLT81] introduced a framework that has been ....

Alan Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.

....on the type of the general machine. Acceptance is defined as it is for the corresponding types of Turing machines. Time is defined as the number of moves, and space as log 2 Q, where Q is the number of states. General machines are almost identical to the recognition machines defined by Cobham [18], except that recognition machines require the input to be accessed sequentially, whereas general machines allow completely random access to the input. It is also easy to see that Turing machines are a special case, by including the worktape contents and head positions as part of the state of the ....

A. Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.

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Alan Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.

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A. Cobham. The Recognition Problem for the Set of Perfect Squares. Conference Record of the Seventh Annual Symposium on Switching and Automata Theory ("FOCS"), pages 78--87, 1966.

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Cobham, A. (1966). The recognition problem for the set of perfect squares. In Proc. of 7th Symposium on Switching and Automata Theory, 78--87.

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Cobham, A. (1966). The recognition problem for the set of perfect squares. 7th IEEE Symp. on Switching and Automata Theory, 78-87.

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Alan Cobham. The recognition problem for the set of perfect squares. Research Paper RC-1704, IBM Watson Research Center, 1966.

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