The recognition problem for the set of perfect squares

Cobham, Alan

doi:10.1109/swat.1966.30

Cited by 77 publications

(40 citation statements)

References 5 publications

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“…Such a result already appears in Cobham [4], where he shows that for recognizing the set of perfect squares (or for recognizing {w $ WR}) we must have T. S l)(n 2) for any computational device (including a multitape T. M.) having a separate one head-mad only input tape. Here T number of steps, $ "capacity"= log2 (number of configurations the machine enters when processing all strings of length n).…”

mentioning

confidence: 86%

“…Each node of the computation graph represents a distinct state of the computation. Like Cobham [4], it is profitable for us to ignore completely how (and So this will be our approach for establishing the analogous main lemma: We assert that, with sufficiently high probability, at a leaf of an R-tree program the elements that we have seen on this path will be "spread out" in such a way that there is only a small probability (i.e., for only a small fraction of all possible input sequences) that we will correctly output S ranks.…”

mentioning

confidence: 99%

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A time-space tradeoff for sorting on a general sequential model of computation

Borodin

Cook

1980

Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing - STOC '80

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mentioning

confidence: 86%

mentioning

confidence: 99%

A time-space tradeoff for sorting on a general sequential model of computation

Borodin

Cook

1980

Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing - STOC '80

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“…Let BP(f ) be the minimal size of any branching program computing f . A Boolean function f has a polynomialsize branching program iff f belongs to L/poly [9], the complexity class of logarithmic space machines with a polynomial amount of advice. Branching program for functions f : {0, 1} χ → {0, 1} with non-Boolean output can be constructed in a natural way.…”

Section: Branching Programsmentioning

confidence: 99%

Optimal Rate Private Information Retrieval from Homomorphic Encryption

Kiayias

Leonardos

Lipmaa

et al. 2015

Proceedings on Privacy Enhancing Technologies

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Abstract:We consider the problem of minimizing the communication in single-database private information retrieval protocols in the case where the length of the data to be transmitted is large. We present first rate-optimal protocols for 1-out-of-n computationallyprivate information retrieval (CPIR), oblivious transfer (OT), and strong conditional oblivious transfer (SCOT). These protocols are based on a new optimalrate leveled homomorphic encryption scheme for largeoutput polynomial-size branching programs, that might be of independent interest. The analysis of the new scheme is intricate: the optimal rate is achieved if a certain parameter s is set equal to the only positive root of a degree-(m + 1) polynomial, where m is the length of the branching program. We show, by using Galois theory, that even when m = 4, this polynomial cannot be solved in radicals. We employ the Newton-Puiseux algorithm to find a Puiseux series for s, and based on this, propose a Θ(log m)-time algorithm to find an integer approximation to s.

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“…As but four examples: Cobham [6] discovered a time-space trade-off for the problem of palindrome recognition; Borodin et al [2] found a similar trade-off for the problem of sorting; Thompson [24] discovered an area-time trade-off for the problem of implementing the DFT in VLSI; and Hong [8] uncovered a trade-off involving reversal and space complexity for Turing machines. It is not a priori obvious that cost tradeoffs exist in the simple world of graph embeddings; but in fact, such trade-offs can be found quite easily.…”

Section: Introductionmentioning

confidence: 99%

Cost tradeoffs in graph embeddings, with applications

Jiang¹,

Mehlhorn

Rosenberg³

1981

Automata, Languages and Programming

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Abstract. An embedding of the graph G in the graph H is a one-to-one association of the vertices of G with the vertices of H. There are two natural measures of the cost of a graph embedding, namely, the dilationcost of the embedding: the maximum distance in H between the images of vertices that are adjacent in G; and the expansion-cost of the embedding: the ratio of the size of H to the size of G. The main results of this paper illustrate three situaUons wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost. The first result establishes the following: There is an embedding of n-node complete ternary trees in complete binary trees with dilation-cost 2 and expansion cost O(n~), where ~ = 1og3(4/3); but any embedding of these ternary trees in binary trees that has expansion-cost c < 2 must have dilation-cost G(logloglogn). The second result provides a stronger but less easily stated example of the same type of trade-off. The third result concerns generic binary trees, that is, complete binary trees into which all n-node binary trees are "efficiently" embeddable. There is a generic binary tree into which all n-node binary trees are embeddable with dilauon-cost O(1) and expansion-cost O(n ~) for some fixed constant c; if one insists on embeddings whose dilation-cost is exactly 1, then these embeddings must have expansion-cost f~(n¢~°*~)/~); tf one insists on embeddmgs whose expansion-cost is less than 2, then these embeddings must have dilation cost ~(log log log n) An interesting application of the polynomial size genenc binary tree m the first part of this three-part result is to yield simplified proofs of several results concerning computational systems with an intrinsic nouon of "computation tree," such as alternating and nondeterministic Turing machines and context-free grammars.

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The recognition problem for the set of perfect squares

Cited by 77 publications

References 5 publications

A time-space tradeoff for sorting on a general sequential model of computation

A time-space tradeoff for sorting on a general sequential model of computation

Optimal Rate Private Information Retrieval from Homomorphic Encryption

Cost tradeoffs in graph embeddings, with applications

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