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## On relating time and space to size and depth (1977)

Venue: | SIAM JOURNAL ON COMPUTING |

Citations: | 115 - 1 self |

### Citations

387 |
Relationship between nondeterministic and deterministic tape classes.
- Savitch
- 1970
(Show Context)
Citation Context ... model which uses a separate tape for inputs to an oracle, nonuniformity for time takes form in a {0}* oracle. We can use Theorems 2 and 4 to make explicit the role of transitive closure in Savitch’s =-=[24]-=- construction. Throughout the remainder of this paper we let a (1 _<-a _-<2) be such that the NN transitive closure problem can be realized (uniformly) with depth -<_ c log N. COROLLARY 2. Let a be as... |

348 |
Formal Languages and their Relation to Automata.
- Hopcroft
- 1969
(Show Context)
Citation Context ...ude with some observations concerning arithmetic complexity. 2. Turing machines, circuits and the basic simulation. We assume that the reader is familiar with Chapters 6 and 10 of Hopcroft and Ullman =-=[11]-=-. Our TM model is an "off-line" machine with a two-way read only input tape. For time bounded computations we allow an arbitrary but finite number of work tapes. For tape bounded computations, it is s... |

117 |
The Computational Complexity of Algebraic and Numeric Problems," Amer
- BORODIN, MUNRO
- 1975
(Show Context)
Citation Context ...e inputs are indeterminates x 1,’’ ", x (and possibly constants c F, F a field), the internal gates are +, -, , :-, and the , x) (see Borodin and Munro outputs are considered to be elements of F(x 1, =-=[1]-=-). For definiteness, let’s take F-Q, the rationals. Strictly speaking, the size-depth question for arithmetic circuits is not a problem. Kung [14] has shown that x 2 requires k depth (depth is called ... |

105 |
Fast parallel matrix inversion algorithms.
- Csanky
- 1975
(Show Context)
Citation Context ...gue the case that arithmetic complexity and the more traditional studies of computational complexity are related, let us consider a current problem concerning parallel arithmetic computations. Csanky =-=[5]-=- has shown that if PWR (A) {A 2, A 3,..., A ]A an n n matrix} is computable in L (n) depth (parallel steps) then A 1, det A, coefficients of char (A) would all be computableON RELATING TIME AND SPACE... |

81 |
Complete Problems for Deterministic Polynomial Time.
- Jones, Laaser
- 1977
(Show Context)
Citation Context ... bounded machine. Can we realize A by circuits with SIZEA (n) ciT(n) k’ (simultaneously) DEPTHa (n)2T(n) k2 for some constants cl, c2, kl, k2. 9 represent log space reducibility (see Jones and Laaser =-=[12]-=-). We Let -<-log could allow nondeterministic transduction here but we might as well follow the standard meaning of deterministic log space many-one reducibility. By converting every log space transdu... |

64 |
An observation on time-storage trade off.
- Cook
- 1973
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Citation Context ...Following standard notation, let P be the class of languages recognizable in deterministic polynomial time. B is called log space complete for P if (i) B is in P, (ii) A in P implies A -< log B. Cook =-=[3]-=-, Jones and Laaser [12] and Ladner [29] exhibit a variety of natural sets which are log space complete for P. We can define an analogous concept for circuits. Namely, let us say that B is depth comple... |

45 |
On time-hardware complexity tradeoffs for boolean functions.
- Spira
- 1971
(Show Context)
Citation Context ...roblem for Turing machines is "roughly" equivalent to an "efficiently constructive" version of the SIZE-DEPTH problem for circuits. Circuits with a fan-out 1 restriction correspond to formulas. Spira =-=[26]-=- has shown that a formula of size T(n) >-_ n can be transformed to an equivalent formula of depth =< c log T(n). (Consider also Brent’s [30] analogous result for arithmetic expressions). Moreover, it ... |

40 |
Boolean matrix multiplication and transitive closure.
- Fischer, Meyer
- 1971
(Show Context)
Citation Context ...arithmetic matrix multiplication), we then have a 0* min (0, 1). Starting with a 0-1 matrix A, we know ij <-- n". We would like to simulate (integer) arithmetic as in Munro [18] and Fischer and Meyer =-=[6]-=- but "mod n n’’ arithmetic is n. log n bit arithmetic and costs depth log n. Instead following another suggestion by S. Cook, we can simulate the arithmetic modpi (1 <-i <= m) where {pi,..., p,,} are ... |

36 | Storage requirements for deterministic polynomial time recognizable languages.
- Cook, Sethi
- 1976
(Show Context)
Citation Context ...e simulations of time bounded computations. In particular, there is a conjecture that DTIME(T(n))c_.DSPACE(log’T(n)) for some constant k. (Indeed, k 1 is still possible.) Cook [3], and Cook and Sethi =-=[4]-=- present important evidence that the conjecture is false, and this represents the concensus of opinion _ _ at this time. On the positive side, Hopcroft, Paul and Valiant [10] have shown that DTIME (T(... |

27 | Computational work and time on finite machines - Savage - 1972 |

27 |
The circuit value problem is logspace complete for P
- Ladner
- 1977
(Show Context)
Citation Context ...the class of languages recognizable in deterministic polynomial time. B is called log space complete for P if (i) B is in P, (ii) A in P implies A -< log B. Cook [3], Jones and Laaser [12] and Ladner =-=[29]-=- exhibit a variety of natural sets which are log space complete for P. We can define an analogous concept for circuits. Namely, let us say that B is depth completeforpolynomial size circuits if (i) SI... |

22 |
On the power of multiplication in random access machines
- Hartmanis, Simon
- 1974
(Show Context)
Citation Context ... time.) In Pratt and Stockmeyer [22], we are introduced to vector machines, a general parallel machine model; general in the sense that inputs can be of arbitrary length (see also Hartmanis and Simon =-=[9]-=-). It is then shown that polynomial time (that is, parallel time) for these vector machine corresponds to TM polynomial space. Motivated by the simulations of Pratt and Stockmeyer [22] and Hartmanis a... |

22 |
The network complexity and Turing machine complexity of nite functions
- Schnorr
- 1976
(Show Context)
Citation Context ...duction. Fischer and Pippenger [7] have shown that a T(n) time bounded Turing machine (TM) can be simulated on n bits by a combinational (Boolean) circuit with O(T(n)log T(n)) gates (see also Schnorr =-=[25]-=-). In this paper, we observe that nondeterministic S (n) tape bounded Turing machines can be simulated by circuits of depth O(S(n)2). In doing so, we relate the power of nondeterminism for space bound... |

13 |
On parallelism in Turing machines
- Kozen
- 1976
(Show Context)
Citation Context ...on between TM space and circuit depth. This can be interpreted as another piece of evidence (see Pratt and Stockmeyer [22], Hartmanis and Simon [9] and more recently Chandra and Stockmeyer [2], Kozen =-=[13]-=-, and Tourlakis [28]), that parallel time and space are roughly equivalent within a polynomial factor. The simplicity of the circuit model focuses our attention on the importance of the transitive clo... |

8 |
Circuit size is nonlinear in depth
- Paterson, Valiant
- 1976
(Show Context)
Citation Context ...ime. On the positive side, Hopcroft, Paul and Valiant [10] have shown that DTIME (T(n))_c DSPACE T(n)/log T(n)). Independent of this result (and independent of our observations), Paterson and Valiant =-=[19]-=- proved that SIZE (T(n))c_ DEPTH (T(n)/log T(n)), noting that their result only had significance when T(n) n. The known relationships between TIME-SIZE, and SPACE-DEPTH are not refined enough to show ... |

4 |
Vermeidung yon Divisionen
- Strassen
- 1973
(Show Context)
Citation Context ...hat during the computation we might be dividing by a very large y for which y---0 mod (pg) for all small primes pi. For the computation of a polynomial of degree n, one can use the method of Strassen =-=[27]-=- to eliminate /, but this results in an O(log n) factor in the depth bound. It is not known whether this factor can be improved. Yet in spite of all our restrictions, one has "the feeling" that an A7... |

3 |
Short formulae for symmetric functions
- Pippenger
- 1974
(Show Context)
Citation Context ...unctions f(xl,’’’ ,x,)of degree- < n.) Is depth logT(n) possible? In general, one cannot expect that positive results for Boolean computations always have arithmetic analogues. For example, Pippenger =-=[20]-=- shows that every Boolean symmetric function on n variables has formula size _-< n :’6 (and hence depth -< c. log n) whereas the elementary symmetric function Y’.<__<...<=,xx...x,/ appears to need O(l... |

2 |
The LBA problem and its importance
- Hartmanis, Hunt
- 1974
(Show Context)
Citation Context ...//www.siam.org/journals/ojsa.php O (log n) depth method for any of the problems det A, A-l, A would lead to a positive solution to the LBA problem (i.e. NSPACE (n)= DSPACE (n)? See Hartmanis and Hunt =-=[8]-=-). The present consensus is that this is very unlikely; that is, NSPACE (S (n )) DSPACE (S (n )) for constructible S(n), and indeed any improvement to Savitch’s NSPACE (S(n))_ DSPACE (S(n)2) would be ... |

1 |
Log Space Recognitton and Translation of Parenthesis Languages, preprint
- LYNCH
(Show Context)
Citation Context ...t formula of depth c log n. (This transformation should be compared with the hypothesis of Corollary 4.) We do not see how to construct a depth c. log n formula in space log n. However, using Lynch’s =-=[15]-=- log n formula evaluation in conjunction with Theorem 2, we can construct a depth c. logZn formula in space log n. 4. Some comments on arithmetic circuits. An arithmetic circuit is like a Boolean circ... |

1 |
Efficient determination of the transitive closure of a graph
- MUNRO
- 1971
(Show Context)
Citation Context ...ng (0) A" (with respect to arithmetic matrix multiplication), we then have a 0* min (0, 1). Starting with a 0-1 matrix A, we know ij <-- n". We would like to simulate (integer) arithmetic as in Munro =-=[18]-=- and Fischer and Meyer [6] but "mod n n’’ arithmetic is n. log n bit arithmetic and costs depth log n. Instead following another suggestion by S. Cook, we can simulate the arithmetic modpi (1 <-i <= m... |

1 |
A characterization ofthe power ofvector machines
- PRATT, STOCKMEYER
- 1976
(Show Context)
Citation Context ...y means parallel time, since several gates in a combinational circuit can operate in parallel. (In arithmetic complexity, depth is almost always referred to as parallel time.) In Pratt and Stockmeyer =-=[22]-=-, we are introduced to vector machines, a general parallel machine model; general in the sense that inputs can be of arbitrary length (see also Hartmanis and Simon [9]). It is then shown that polynomi... |

1 |
A universal parallel machine and the efficient simulation of storage bounded sequential computations, Dept
- TOURLAKIS
- 1976
(Show Context)
Citation Context ...and circuit depth. This can be interpreted as another piece of evidence (see Pratt and Stockmeyer [22], Hartmanis and Simon [9] and more recently Chandra and Stockmeyer [2], Kozen [13], and Tourlakis =-=[28]-=-), that parallel time and space are roughly equivalent within a polynomial factor. The simplicity of the circuit model focuses our attention on the importance of the transitive closure problem. As a r... |

1 |
The parallel evaluation ofseveral arithmetic expressions
- BRENT
- 1974
(Show Context)
Citation Context ... with a fan-out 1 restriction correspond to formulas. Spira [26] has shown that a formula of size T(n) >-_ n can be transformed to an equivalent formula of depth =< c log T(n). (Consider also Brent’s =-=[30]-=- analogous result for arithmetic expressions). Moreover, it should be clear that formula size -<_ 2depth. Hence, if we are looking for an example where formula size is exponentially larger than (arbit... |