Introduction

So, the "dot" product will simply corresponds to its mathematical definition:

let (<.>) xs ys = sum (xs <*> ys)    
printfn "%A" ([1;2] <.> [3;4])

This one can next be used to model a single neuron with weights ws:

let remind ws xs = ws <.> (1.0::xs)
let ws = [0.3;-0.5;0.1]
let xs = [1.0;1.0]
let y  = 1.0

let y1 = remind ws xs  
let e  = y-y1

printfn "%A (%A)" y1 e  (** -0.1 (1.1) **)

The learning algorithm has to reduce the error between the desired output y and the one returned by the network y1, by adapting the weights ws:

let k   = 0.01
let ws1 = ws <+> (List.map (fun v->v*k*e) (1.0::xs))
let y2  = remind ws1 xs  
let e1  = y-y2