April 2012
Intermediate to advanced
416 pages
10h 40m
English
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5.3 ADDITIONAL EXERCISES
1. Fix an alphabet Σ with m > 1 elements. Show that the following functions are in Cobham’s class 
Σ (cf. 5.1.1.7):
(i) λx.x
(ii) λx.0
(iii) The left successors: λx.d ∗ x, for all d 
Σ
(iv) λx.xR, where xR is the number whose m-adic notation is the reverse of that of x.
2. Fix an alphabet Σ with m > 1 elements. Show that Cobham’s class Σ is closed under bounded left recursion on notation, that is, under the schema below, where the h, (gd)d Σ, and B are in Σ.
f (0, 
n) = h(n)f (d ∗ x,n)= gd(x,n, f(x, n)), for all d Σ| f (x,n)| ≤ |B(x,n)|
Hint. (iv) of the preceding exercise ...
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