## On the power of real Turing machines over binary inputs (1997)

Venue: | SIAM Journal on Computing |

Citations: | 23 - 4 self |

### BibTeX

@ARTICLE{Cucker97onthe,

author = {Felipe Cucker and Dima Grigoriev},

title = {On the power of real Turing machines over binary inputs},

journal = {SIAM Journal on Computing},

year = {1997},

volume = {26},

pages = {243--254}

}

### OpenURL

### Abstract

this paper is to prove that BP(PAR IR ) = PSPACE/poly where PAR IR is the class of sets computed in parallel polynomial time by (ordinary) real Turing machines. As a consequence we obtain the existence of binary sets that do not belong to the Boolean part of PAR IR (an extension of the result in [20] since PH IR ` PAR IR ) and a separation of complexity classes in the real setting.

### Citations

377 |
On a theory of computation and complexity over the real numbers
- Blum, Shub, et al.
- 1989
(Show Context)
Citation Context ... PA 16802 USA e-mail: dima@cse.psu.edu In recent years the study of the complexity of computational problems involving real numbers has been an increasing research area. A foundational paper has been =-=[4]-=- where a computational model ---the real Turing machine--- for dealing with the above problems was developed. One research direction that has been studied intensively during the last two years is the ... |

300 |
Topology from the Differentiable Viewpoint
- Milnor
- 1965
(Show Context)
Citation Context ... 1 (a) = \Delta \Delta \Delta = @f @X k (a) = 0 In this case the value b = f(a) is said to be a critical value of f . In the case when f is a polynomial function Sard's lemma ([5] th'eor`eme 9.5.2 or =-=[22]-=-) implies that there are only a finite number of critical values of f . This last fact was used in [15] (and in several subsequent papers) to reduce the dimension of non-empty semialgebraic sets to ze... |

173 |
On the computational complexity and geometry of the first-order theory of the reals
- Renegar
- 1992
(Show Context)
Citation Context ...since the degrees and the coefficients of the polynomials appearing in this system are bounded by d 2 and O(L 2 d 2 ) respectively, if we apply the quantifier elimination algorithm 4 given in [16] or =-=[23]-=- along X 1 ; : : : ; X k onto Z we get a finite set of points in IR (just the critical values) such that each non zero one has absolute value greater than i L d 2 kfl 2 2 j \Gamma1 2 Remark 1 In the p... |

159 | On the computational power of neural nets
- Siegelmann, Sontag
- 1995
(Show Context)
Citation Context ... computational power of neural networks. The first results characterized the power of nets with rational weights working within polynomial time by showing that they compute exactly the sets in P (cf. =-=[26]-=-). The same problem was then considered for neural networks with real weights and it was shown that the power of these nets working within polynomial time is exactly P/poly (cf. [27],[28] and [21]). T... |

157 |
Géométrie algébrique réelle
- Bochnak, Coste, et al.
- 1987
(Show Context)
Citation Context ...f g and L = jcoeff(g)j. Then there exists a positive integer fl 1 such that any connected component of V has a non-empty intersection with the ball B(R) where R = L d 1 kfl 1 3 Let us recall now (see =-=[5]-=- section 9.5) that a point a 2 IR k is a critical point for a function f : IR k ! IR when it satisfies @f @X 1 (a) = \Delta \Delta \Delta = @f @X k (a) = 0 In this case the value b = f(a) is said to b... |

95 | On relating time and space to size and depth
- Borodin
- 1977
(Show Context)
Citation Context ...ign of g(Z 1 ; : : : ; Z k ) + Y on the point coded in (s5). The above considerations show that the described algorithm runs in parallel polynomial time. Since this is equivalent to polynomial space (=-=[6]-=-, [2] ch.4) we have shown that the set decided by the algorithm above belongs to PSPACE/poly. 2 3 Binary inputs for parallel real Turing machines Our next goal is to extend our previous result to the ... |

87 | Analog computation via neural networks
- Siegelmann, Sontag
- 1994
(Show Context)
Citation Context ...he sets in P (cf. [26]). The same problem was then considered for neural networks with real weights and it was shown that the power of these nets working within polynomial time is exactly P/poly (cf. =-=[27]-=-,[28] and [21]). This latter problem considers in a natural way a setting in which an algebraic model having real constants operates over binary inputs. A next step was then taken by P. Koiran who pas... |

60 | Bounds for the computational power and learning complexity of analog neural nets
- Maass
- 1992
(Show Context)
Citation Context ...cf. [26]). The same problem was then considered for neural networks with real weights and it was shown that the power of these nets working within polynomial time is exactly P/poly (cf. [27],[28] and =-=[21]-=-). This latter problem considers in a natural way a setting in which an algebraic model having real constants operates over binary inputs. A next step was then taken by P. Koiran who passed from a str... |

56 |
Solving systems of polynomials inequalities in subexponential time
- Grigov’ev, Vorobjov
- 1988
(Show Context)
Citation Context ... jcoeff(f)j the maximal absolute value of its coefficients. The aim of this section is to show how to find real algebraic points in the connected components of non-empty open sets. We closelly follow =-=[15]-=-. Thus, let g 1 ; : : : ; g N 2 ZZ[X 1 ; : : : ; X k ] and let V = fx 2 IR k : g 1 (x) ? 0& : : : &gN (x) ? 0g be an open non-empty semialgebraic set. For the rest of this section we consider d a boun... |

51 |
Turing machines, that take advice. L'Enseignement Mathematique, 28 2nd series:191{209
- Karp, Lipton
- 1982
(Show Context)
Citation Context ... real Turing machines. Because of clarity of exposition we will however first show the inclusion for the Boolean part of P IR . We begin by recalling the definition of non-uniform classes as given in =-=[17]-=-, that we extend to complexity classes over the reals. Definition 1 Let C ` \Sigma (resp. C ` IR 1 ) be a class of sets and F be any class of functions from IN to Sigma (resp. from IN to IR 1 ). The c... |

41 |
Sur la complexité du principe de Tarski–Seidenberg
- Heintz, Roy, et al.
- 1990
(Show Context)
Citation Context ...a. Now, since the degrees and the coefficients of the polynomials appearing in this system are bounded by d 2 and O(L 2 d 2 ) respectively, if we apply the quantifier elimination algorithm 4 given in =-=[16]-=- or [23] along X 1 ; : : : ; X k onto Z we get a finite set of points in IR (just the critical values) such that each non zero one has absolute value greater than i L d 2 kfl 2 2 j \Gamma1 2 Remark 1 ... |

36 |
Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets
- Coste, Roy
- 1988
(Show Context)
Citation Context ...ms given in [16] or [23]. For step (s5) one can apply the algorithms given in [16] or [24]. However, we remark here that a cylindrical algebraic decomposition together with the coding `a la Thom (see =-=[7]-=- for the algorithms, and [25], [11] for complexity analysis) suffices because the double exponential behaviour of this algorithm is only in the number of variables ---which is constant in our case--- ... |

33 |
Complexity of deciding Tarski algebra
- Grigoriev
- 1988
(Show Context)
Citation Context ...f the resulting 9 W is zero, and then we select the first ~v that gives a positive answer. Note that the determination of the dimension of each W can be done in PSPACE just combining the main idea of =-=[14]-=- section 6 with the parallel algorithms given in [16] or [23]. For step (s5) one can apply the algorithms given in [16] or [24]. However, we remark here that a cylindrical algebraic decomposition toge... |

30 | A weak version of the Blum, Shub & Smale model
- Koiran
- 1993
(Show Context)
Citation Context ...stants) must have degree and coefficient size bounded by the running time. For this weak model he considered the class PW of sets accepted in polynomial time and he proved that BP(P W ) = P/poly (see =-=[19]-=-). Subsequently, several papers exhibited new results on Boolean parts. In [12] it was shown that BP(PARW ) = PSPACE/poly where PARW is the class of subsets of IR 1 decided in weak parallel polynomial... |

23 | Computing over the reals with addition and order: Higher complexity classes
- Cucker, Koiran
- 1995
(Show Context)
Citation Context ... machines are order-free, i.e. they are required to branch only on equality tests, we now have that BP(P = add ) = P and that BP(NP = add ) = NP ([18]). These results were subsequently generalized in =-=[10]-=- to all the levels of the polynomial hierarchy constructed upon NP add (or NP = add ). None of the mentioned results was done for the (unrestricted) real Turing machine. In fact, for this case, it was... |

15 | Neural Networks with real weights : analog computational complexity
- Sontag, Siegelmann
- 1992
(Show Context)
Citation Context ...ts in P (cf. [26]). The same problem was then considered for neural networks with real weights and it was shown that the power of these nets working within polynomial time is exactly P/poly (cf. [27],=-=[28]-=- and [21]). This latter problem considers in a natural way a setting in which an algebraic model having real constants operates over binary inputs. A next step was then taken by P. Koiran who passed f... |

12 |
Accessible telephone directories
- Goode
- 1994
(Show Context)
Citation Context ... None of the mentioned results was done for the (unrestricted) real Turing machine. In fact, for this case, it was even asked whether it existed a subset of \Sigma not belonging to the BP(P IR ) (cf. =-=[13]-=-). First steps in this direction 2 were done in [20] where it is shown that if we consider order-free machines then we have the inclusion BP(P = IR ) ` BPP (the class of sets decided by randomized mac... |

11 |
Complexity of computation on real algebraic numbers
- Roy, Szpirglas
- 1990
(Show Context)
Citation Context ...r step (s5) one can apply the algorithms given in [16] or [24]. However, we remark here that a cylindrical algebraic decomposition together with the coding `a la Thom (see [7] for the algorithms, and =-=[25]-=-, [11] for complexity analysis) suffices because the double exponential behaviour of this algorithm is only in the number of variables ---which is constant in our case--- being NC in the rest of the p... |

7 |
On the complexity of quantifier elimination: the structural approach
- Cucker
- 1993
(Show Context)
Citation Context ...ongs to PSPACE/poly. 2 3 Binary inputs for parallel real Turing machines Our next goal is to extend our previous result to the class PAR IR of sets decided in parallel polynomial time. We recall from =-=[9]-=- the definition of a computational model for parallelism in the real Turing machine setting together with the complexity class it defines when restricted to polynomial time. Definition 2 An algebraic ... |

4 |
Simulations among classes of random access machines and equivalence among numbers succinctly represented
- Bertoni, Mauri, et al.
- 1985
(Show Context)
Citation Context ...or several sets of primitive operations. However, in all these cases, there is a primitive operation that can not be efficiently simulated by a real Turing machine. Thus, for instance, it is shown in =-=[3]-=- that integer RAM's with operations (+; \Gamma; ; \Xi) have the power of PSPACE. However, the simulation of the integer division by a real Turing machine over integers of exponential length take expon... |

1 |
NC algorithms for real algebraic numbers
- M-F
- 1992
(Show Context)
Citation Context ... (s5) one can apply the algorithms given in [16] or [24]. However, we remark here that a cylindrical algebraic decomposition together with the coding `a la Thom (see [7] for the algorithms, and [25], =-=[11]-=- for complexity analysis) suffices because the double exponential behaviour of this algorithm is only in the number of variables ---which is constant in our case--- being NC in the rest of the paramet... |

1 |
Complexity separations in Koiran's weak
- Cucker, Shub, et al.
(Show Context)
Citation Context ...this weak model he considered the class PW of sets accepted in polynomial time and he proved that BP(P W ) = P/poly (see [19]). Subsequently, several papers exhibited new results on Boolean parts. In =-=[12]-=- it was shown that BP(PARW ) = PSPACE/poly where PARW is the class of subsets of IR 1 decided in weak parallel polynomial time. Also, for additive machines (i.e. real Turing machines that do not perfo... |

1 |
Computing over the reals with addition and
- Koiran
(Show Context)
Citation Context ...here PARW is the class of subsets of IR 1 decided in weak parallel polynomial time. Also, for additive machines (i.e. real Turing machines that do not perform multiplications at all), it was shown in =-=[18]-=- that BP(P add ) = P/poly and that BP(NP add ) = NP/poly. Here P add and NP add denote the obvious classes but we recall that the nondeterministic guesses in this model are real numbers. Moreover, if ... |