Distributive Categories http cis k hosei ac jpyukita

Distributive Categories http: //cis. k. hosei. ac. jp/~yukita/

Products and Sums • Products are concerned with operations on the data. • Sums are concerned with control. – Decisions on which operation to perform • How do they interact when both products and sums are involved? 2

The Distributive Law 3

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Axiom 1 [abstracted from Rule (i)] • The arrow d is an isomorphism. • Rule (i) is not automatic in categories other than Sets. 5

Axiom 2 [abstracted from Rule (ii)] 6

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Def. Distributive Categories 8

Ex. 3 Category 2 X is distributive. 9

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In Sets 12

The & function 13

Sets as a Distributive Category • Using – sums – products – composition – distributive law – and some additional primitive functions, • we can construct various functions. 14

Some Primitives in the category of Sets 15

Divide 16

Ex. 5. Realization of testx>0　 17

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Imperative Programming 19

Construction of factorial as an imperative program 20

Implementation of factorial in a distributive category 21

Construction of f 0 22

Construction of h and f 0 23

Def. Imperative program 24

Ex. 11. while t(y) is true do … 25

Implementation of Ex. 11 26

Ex. 12. A Circuit with Feedback x nor u v nor y 27

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Ex. 13. Flow chart 29

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Term Paper • Implement the function in example 13. • Use only – add – multiply – constants – constructions in a distributive category. 31