1. Data Types http://cis.k.hosei.ac.jp/~yukita/

2. 2 Arithmetic funtions ans laws

3. 3 The Associative Law

4. 4 The Identity Law

5. 5 The Inverse Law

6. 6 The Commutative Law

7. 7 Data Types Algebras such as groups, rings, etc. are familiar data types in mathematics. CS handles other data types other than those algebras. –stacks –queues –arrays –binary trees –other useful data structures A data type consists of various sets of data together with operations between them definable in a distributive category and equations expressible in a distributive category.

8. 8 Stacks (Rough View)

9. 9 Pop revised to avoid stack underflow

10. 10 Push revised for symmetry

11. 11 Axioms

12. 12 Definition.

13. 13 Ex. 1. Infinite stacks

14. 14 Note Just from the axioms, we do not know whether the stacks are finite.

15. 15 Definition. Stacks of X

16. 16 Ex. 2.

17. 17 Notation

18. 18 Ex. 3. The usual operation of pushing an element on a stack

19. 19 Ex. 4. Imperative computing of the depth of a stack

20. 20 Ex. 5. Tests on stacks

21. 21 Arrays - the store function

22. 22 Arrays - the read function

23. 23 Binary Trees

24. 24 Two operations make and break

25. 25 Def. Binary Tree

26. 26 Queues

27. 27 Pointers

28. 28 Turing Machines