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Programming from Galois Connections: A New Approach to Calculating Programs

Explore the innovative concept of programming from Galois connections, utilizing a binary relational combinator to optimize solutions. Learn about abstract modeling, divide and conquer strategies, and the algebra of programming for efficient problem-solving. Discover how Galois connections can scale up programming tasks and categorize solutions effectively.

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Programming from Galois Connections: A New Approach to Calculating Programs

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

  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!

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