1. Programming from Galois Connections
Anisa Allahdadi
Thematic Seminar
21 June 2012

2. Objectives Problem Statements:
one defining a broad class of solutions (the easy part)
requesting one particular optimal solution, (the hard part)
This article introduces a binary relational combinator which
mirrors this linguistics structure and
exploits its potential for calculating programs by optimization.

3. Some skills of a good programmer
Abstraction
Abstract Modeling, in the early stages of thinking about a software problem
Induction
Solving a complex program by imagining some smaller parts already solved
Divide and conquer programming strategy

4. The “Best” Solution
Whole number division: x ÷ y is the largest natural number when multiplied by y, is at most x
takeWhile p: longest prefix of the input list such that all elements satisfy predicate p
Scheduling: The best schedule for a collection of tasks, given their time spans and an acyclic graph

5. Easy / Hard E: the collection of solution candidates
H: criteria under which a best solution is chosen

6. Example #1
x ÷ y is the largest natural number that, when multiplied by y, is at most x.

7. Galois Connection
A Galois connection is a pair of functions f and g satisfying:
f and g are adjoints of each other (f is the lower adjoint and g is the upper adjoint)

8. Example #2 Indirect Equality
Galois connection blending with indirect equality

9. Cont.

10. Example #3
Take (n, x): yields the longest prefix of its input sequence up to some given length n.

11. Cont.

12. Shrink Operator Galois adjoints
By the easy part E is “shrunk” by the requirements of the hard part H

13. Cont.
Factoring Galois connection into two parts

14. Cont. Abbreviate to : the left hand operator: the right hand operand:
Given two relations of and ,
“Shrunk by R”:

15. Conclusion
Galois connection as a well-known mathematical device capable of scaling up
“programming from Galois connections” is proposed as a way of calculating programs from specifications in form of Galois connections
The main contribution of this work: using the algebra of programming expressed in closed formula which records what is “easy” and “hard” to implement
A new relational combinator (shrinking) expressing such formula at pointfree level
Cons: not every problem casts into a Galois connection

16. Questions?
Thanks for your time!