Program Development by Stepwise Refinement

1. Program Development byStepwise Refinement Niklaus Wirth

2. Motivation • Programming is usually taught via examples • Examples normally demonstrate what a computer can do • Examples are not normally chosen by their suitability for a technique • Programming is not just mastering of a language – more important to master the methods of design and construction.

3. Aim • The paper aims to demonstrate the development of an example program through a sequence of refinement steps • Today, we refer to this approach as top-down design (or top-down decomposition) • The program and data specification are refined in parallel

4. Refinement • Each refinement implies design decisions • It is important to make these explicit • Be aware of the critieria for the refinement • Be aware there are alternatives (after all, this might not work out) • Decompose decisions as much as possible • Defer decisions about data as long as possible

5. The 8-Queens Problem Given an 8X8 chessboard and 8 queens which are hostile to each other, find a position for each queen such that no queen may be taken by any other queen. NOTE: Eight-Queens on wikipedia has an animated example of the Recursive solution solving this problem

6. Brute Force Approach A = Set of candidates for the solution One exists that satisfies condition p Repeat Generate the next element of A and call it x until p(x) OR (no more elements in A) if p(x) then x = solution

7. Issues • Problem space is too large • Computers at the time would take ~ 7 hrs • We need a way to preselect solutions • Criteria • Smaller than A • Elements easily generated • Easier to test than p(x)

8. 8-Queens: Unique Columns • Every column must have exactly 1 queen • Now we are just testing rows and diagonals • We also need the ability to “try” a row for a queen and backtrack if the test for the row/diagonal fails.

9. Program At This Point repeat trycolumn; if safe then setqueen; considernextcolumn; else regress until lastcoldone OR regressoutoffirstcol

10. Define the Instructions • Considerfirstcolumn • Considernextcolumn • Reconsiderpriorcolumn • Advancepointer • Testsquare • Setqueen • removeQueen

11. Refining the Instructions Upon realizing that we can store the indexes of occupied rows, and thru carefully choosing our array structures, we can reduce the instruction set to: testsquare setqueen regress movequeen

12. A recursive alternative • View the problem as starting with one column • Extend the board by one column at each step

13. Generalizing the Problem • We often need to change our program based on new requirements… • What if we want ALL possible solutions?

14. Conclusions • Programs are constructed thru a sequence of refinement tasks • Each task is decomposed into subtasks • The degree of decomposition (modularity) will determine the ease with which we can modify the program

15. Conclusions • Use a notation that is natural for the problem • Each refinement implies a design decision, criteria include efficiency, storage, clarity • The development of this simple problem formed a long story, programming is not trivial – simplifying the problem is important.