On Nc-real Complexity Classes for Additive Circuits and Their Relations with Nc on Nc-real Complexity Classes for Additive Circuits and Their Relations with Nc on Nc-real Complexity Classes for Additive Circuits and Their Relations with Nc

Based on the results of Blum, Shub and Smale 1], Meer 5], Cucker and Matamala 3] and Koiran 7], we develop the study of real computation models restricted to additive operations. More speciically we introduce some complexity classes deened by algebraic circuits and we study their relationships with the real computation model. We show that the languages accepted by nonuniform additive circuits of polynomial size and polylogarith-mic depth are those accepted by uniform additive circuits of polynomial size and polylogarithmic depth with advice. Moreover, we prove that binary languages accepted by real uniform circuits of polynomial size and polylogarithmic depth, when the test nodes in the circuit are equality test, are the languages belonging to NC; when the test nodes are inequality test, the class obtained is NC/Poly. We also prove that the class deened by family of algebraic circuits with polynomial size and polylogarithmic depth is strictly contained in the class deened by real additive Turing machines working in polynomial time. A partir des r esultats de Blum, Shub et Smalee1], Meer 5], Cucker et Matamala 3] et Koirann7], nous pr esentons une etude des mod eles de calculs r eels restreints aux op erateurs additifs. Nous d eenissons des classes de complexit e a l'aide des circuits alg ebriques et nous etudions leurs relations avec le mod ele r eel. Nous mon-trons que les langages accept es par des circuits additifs non uni-formes de taille polynomiale et de profondeur polylogarithmique sont les m^ emes que ceux accept es par des circuits additifs uni-formes avec conseil de taille polynomiale et de profondeur poly-logarithmique. De plus, nous d emontrons que la classe des lan-gages binaires accept es par des circuits r eels uniformes de taille polynomiale et de profondeur polylogarithmique est la classe NC si les noeuds de test sont des tests d' egalit e et la classe NC/Poly si les tests d'in egalit e sont autoris es. Nous montrons ennn que la classe d eenie par les familles de circuits alg ebriques de taille polynomiale et de profondeur polylogarithmique est strictement incluse dans la classe des langages accept es par une machine de Turing additive re elle en temps polynomial. Mots-cl es: Complexit e des calculs parall eles, mod eles de calculs r eels, circuits alg ebriques. Abstract Based on the results of Blum, Shub and Smalee1], Meerr5], Cucker and Matamalaa3] and Koirann7], we develop the study of real …