Computational category theory by D. E. Rydeheard
By D. E. Rydeheard
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Val (x,y as ( ,p)) = ((2,3),(4,(5,6))) Exercise 2. Defining functions Define the following functions on integers: 1. The function sign which tests whether an integer is positive. 2. The function absvalue which returns the absolute value of an integer. 3. The function finding the maximum of two integers. 4. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, . . in which each number is the sum of its two immediate predecessors. Write a recursive definition of the n-th entry in the sequence. Exercise 3.
6. The function calculating the sum of a list of integers can be generalized. Suppose there is a binary function f: A*B -> B and an initial value v:B, then we may run through an A list accumulating a result by successively applying the binary function to the current element of the list and the value accumulated so far, starting with the initial value. Define this function – the definition is shorter than its explanation! Exercise 6. Binary trees For this exercise, a binary tree is a structure like: • ✁ ❆ ✁ ❆ ✁ ❆ • 1 ✁✁ ❆❆ ❆ ✁ ❆ ✁ 2 ❆ ❆ ❆ ❆ ❆ 3 It consists of binary branching nodes and values stored at the tips.
Transitivity) If x ≤ y and y ≤ z, then x ≤ z. e. order preserving functions. This forms a category Pos. 5 Partial orders as categories As well as the category Pos of partial orders, each partial order (a, ≤) may itself be considered to be a category as follows. The objects of the category are the elements of a. There is precisely one arrow from x ∈ a to y ∈ a iff x ≤ y. Transitivity ensures that composition is defined and reflexivity ensures that identities exist. In fact, we do not need the anti-symmetry axiom, so this construction works for pre-orders (reflexive, transitive relations).