REVISED: Thursday, February 8, 2024
Haskell And Bottom.
I. IS ⊥ BOTTOM THE HASKELL RED-HEADED STEPCHILD?
Undefined or bottom written with an up tack or falsum ⊥ refers to a computation which never completes successfully. Think of bottom as representing a computation which will not halt, it does not terminate. The glyph of the up tack appears as an upside down tee symbol, and is sometimes called eet, the word tee in reverse. Bottom or whatever you prefer to call it represents a computation which causes our program to fail. However, the ⊥ bottom type is a sub-type of all types! For example, the type Bool does not just include True, False it really includes True, ⊥, False. The type () with only one element actually has two: ⊥, (). The type Integer really includes [⊥, 0, 1, 2, 3..]. The same inclusion can be said of all types. Adding ⊥ to a set of values is called lifting. All normal Haskell types are lifted.
Why do we rarely see discussions devoted to bottom? Maybe it is related to our technological evolution. In the past we appeared to need information on large objects we could see with the unaided eye, we did not seem as interested in the very small, the bottom. Our technology has evolved, now we need information on both the very large and the very small. We need to rethink the importance of bottom. Granted Haskell does not have sub-types; however, bottom should be thought of as more than just not terminating or an error message. Seeing a bottom should spark our creative thought processes. There is more to a bottom than meets our unaided eye.
Why do we rarely see discussions devoted to bottom? Maybe it is related to our technological evolution. In the past we appeared to need information on large objects we could see with the unaided eye, we did not seem as interested in the very small, the bottom. Our technology has evolved, now we need information on both the very large and the very small. We need to rethink the importance of bottom. Granted Haskell does not have sub-types; however, bottom should be thought of as more than just not terminating or an error message. Seeing a bottom should spark our creative thought processes. There is more to a bottom than meets our unaided eye.
One area of creative thought processes is the math field of calculus which calculates limits and derivatives. How often have you recently seen Haskell used to solve non lambda calculus, calculus problems? These of course are problems calculating numbers approaching infinity or zero. What is infinity, is infinity bottom? What is zero? We need Haskell to help us with the creative what is ... problem types. Haskell is based on math; however, I have seldom seen easy to understand, Haskell non lambda calculus, calculus limits or derivative functions outside of Haskell Cabal. We have a lot of work ahead of us.
I love Haskell, it is my favorite programming language; however, is the ⊥ bottom type the Haskell red-headed stepchild?
II. CONCLUSION
In this tutorial, you have been introduced to Haskell ⊥ bottom.
III. REFERENCES
Bird, R. (2015). Thinking Functionally with Haskell. Cambridge, England: Cambridge University Press.
Davie, A. (1992). Introduction to Functional Programming Systems Using Haskell. Cambridge, England: Cambridge University Press.
Goerzen, J. & O'Sullivan, B. & Stewart, D. (2008). Real World Haskell. Sebastopol, CA: O'Reilly Media, Inc.
Hutton, G. (2007). Programming in Haskell. New York: Cambridge University Press.
Lipovača, M. (2011). Learn You a Haskell for Great Good!: A Beginner's Guide. San Francisco, CA: No Starch Press, Inc.
Thompson, S. (2011). The Craft of Functional Programming. Edinburgh Gate, Harlow, England: Pearson Education Limited.
