Deviation theorems for pfaffian sigmoids

D. Yu. Grigoriev

Algebra i analiz, Tome 6 (1994) no. 1, pp. 127-131.

Résumé

By a Pfaffian sigmoid of depth $d$ we mean a circuit with $d$ layers in which rational operations are admitted at each layer, and to jump to the next layer one solves an ordinary differential equation of the type $v'=p(v)$ where $p$ is a polynomial whose coefficients are functions computed at the previous layers of the sigmoid. Thus, a Pfaffian sigmoid computes Pfaffian functions (in the sense of A. Khovanskii). A deviation theorem is proved which states that for a real function $f$, $f\not\equiv 0$, computed by a Pfaffian sigmoid of depth (or parallel complexity) $d$ there exists an integer $n$ such that for a certain $x_0$ the inequalities $(\exp(\dots(\exp(|x|^n))\dots))^{-1}\leq|f(x)|\leq\exp(\dots(\exp(|x|^n))\dots)$ hold for all $|x|\geq x_0$, where the iteration of the exponential function is taken $d$ times. One can treat the deviation theorem as an analogue of the Liouville theorem (on algebraic numbers) for Pfaffian functions.

Détail

Keywords: Pfaffian sigmoid, deviation theorems, parallel complexity.