We present a mathematical model of the intuitive notions such as the knowledge or the information arising at different stages of computations on branching programs (b.p.). The model has two appropriate properties:

i) The "knowledge" arising at a stage of computation in question is derivable from the "knowledge" arising at the previous stage according to the rules of the model and according to the local arrangement of the b.p.

ii) The model confirms the intuitively well-known fact that the knowledge arising at a node of a computation depends not only on it but in some cases also on a "mystery" information. (I. e. different computations reaching the same node may have different knowledge(s) arisen at it.)

We prove that with respect to our model no such information exists in read-once b.p.`s but on the other hand in b. p.`s which are not read-once such information must be present. The read-once property forms a frontier. More concretely, we may see...

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