## How to Compute Fibonacci Numbers?

Some inductive proofs and some light program derivation about Fibonacci numbers. If you think the fastest way to compute Fibonacci numbers is by a closed-form formula, you should read on!

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# Program Derivation

## How to Compute Fibonacci Numbers?

## Deriving Monadic Programs

## Calculating a backtracking algorithm: an exercise in monadic program derivation

## Equational reasoning for non-determinism monad: the case of Spark aggregation

## Formal derivation of greedy algorithms from relational specifications: a tutorial

## Calculating a linear-time solution to the densest segment problem

## Programming from Galois connections

## Calculating Programs from Galois Connections

## Programming from Galois connections — principles and applications

Some inductive proofs and some light program derivation about Fibonacci numbers. If you think the fastest way to compute Fibonacci numbers is by a closed-form formula, you should read on!

I started to take an interest in reasoning and derivation of monadic programs around 2016-17. Several years having passed, I collaborated with many nice people, managed to get some results published, failed to publish some stuffs I personally like, and am still working on some interesting tiny problems. This post summaries what was done, and what remains to be done.

Shin-Cheng Mu. Tech. Report TR-IIS-19-003, Institute of Information Science, Academia Sinica, June 2019.

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Shin-Cheng Mu. Tech. Report TR-IIS-19-002, Institute of Information Science, Academia Sinica, June 2019.

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Yu-Hsi Chiang, Shin-Cheng Mu. *Journal of Logical and Algebraic Methods in Programming*, 85(5), Part 2, pp 879-905, August 2016.

[Paper(doi:10.1016/j.jlamp.2015.12.003) | Code]

Sharon Curtis and Shin-Cheng Mu. Calculating a linear-time solution to the densest segment problem. Journal of Functional Programming, Vol. 25, 2015.

[Paper (doi: 10.1017/S095679681500026X)| Supplementary Proofs | Code]

The problem of finding a densest segment of a list is similar to the well-known maximum segment sum problem, but its solution is surprisingly challenging. We give a general specification of such problems, and formally develop a linear-time online solution, using a sliding window style algorithm. The development highlights some elegant properties of densities, involving partitions that are decreasing and all right-skew.

Shin-Cheng Mu and José Nuno Oliveira. Programming from Galois connections. In the *Journal of Logic and Algebraic Programming* , Vol 81(6), pages 680–704, August 2012.

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One is often amazed that, once two functions are identified as a Galois connection, a long list of nice and often useful properties follow from one concise, elegant defining equation. But how does one construct a program from a specification given as a Galois connection?

Shin-Cheng Mu and José Nuno Oliveira. Programming from Galois connections — principles and applications. Technical Report TR-IIS-10-009, Academia Sinica, December 2010.

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