By Marshall C. Yovits

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**Example text**

3 is a rooted tree, because it can be generated in three steps as shown in Fig. 6 a) to c). Rooted trees have a single source which is called the root of the tree. The sinks of trees are called leaves of the tree. If (u, v) is an edge of a tree one calls u the father of v and one calls v a son of u. 5 Final Remarks We recommend to remember as key technical points: • about modulo computation – the definition of equivalence modulo k – that we consider two systems R of representatives, namely R = [0 : k − 1] and R = [−n/2 : n/2 − 1] and that the representatives rep(n) for n in these systems are named (n mod k) and (n tmod k) – the trivial fact ‘if x and y are equivalent mod k and x lies in a system R of representatives, then x is the representative of y in R’ is applicable to any system of representatives.

A subset B ⊂ A is called a system of representatives if and only if for every a ∈ A there is exactly one r ∈ B with aRr. The unique r ∈ B satisfying aRr is called the representative of a in B. Lemma 11. For i ∈ Z and k ∈ N, the interval of integers [i : i + k − 1] is a system of representatives for equivalence mod k. Proof. Let a ∈ Z. We define the representative r(a) by f (a) = max{ j ∈ Z | a − k · j ≥ i}, r(a) = a − k · f (a). Then r(a) ≡ a mod k and r(a) ∈ [i : i + k − 1]. Uniqueness follows from Lemma 10.

We define the cost of a switching function f as the cost of the cheapest pure Boolean expression which computes f: L( f ) = min{L(e) | e ∈ BE ∧ e is pure ∧ f ≡ e}. Show for f : Bn → B a) L( f ) ≤ n · 2n+1 . b) f (x1 , . . , xn ) = (xn ∧ f (x1 , . . , xn−1 , 1)) ∨ (xn ∧ f (x1 , . . , xn−1 , 0)). c) L( f ) ≤ (5/2) · 2n − 4. Hint: to show (c), apply the formula from (b) recursively and conclude by induction on n. 5 Hardware In Sect. 1 we introduce the classical model of digital circuits and show — by induction on the depth of gates — that the computation of signals in this model is well defined.