Notes on the Combinator Birds: 1. The combinatory birds were borrowed from To Mock A MockingBird, by Raymond Smullyan. 2. Some additional information about combinator birds can be found in To Dissect a Mockingbird by David C Keenan. 3. Some of the SK Combinatory terms were first reduced using the Combinatory Logic Tutorial by Chris Barker.
To Dissect a Mockingbird:A Graphical Notation for the Lambda Calculus with Animated Reduction David C Keenan, 27-Aug-1996 last updated 10-May-200 The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation. Introduction In the 1930s and 40s, around the birth of the "automatic computer", mathematicians wanted to formalise what we mean when we say some result or some function is "effectively computable", whether by machine or human. A "computer", originally, was a person who performed arithmetic calculations. The "effectively" part is included to indicate that we are not concerned with the time any particular computer might take to produce the result, so long as it would get there eventually. They wanted to find the simplest possible system that could be said to compute.
With all that scope for reasonable disagreement, is there anything we can all agree on? How much of the hierarchy in the medal table is indisputable, and how much depends on your point of view? So we want to say that one country has done strictly better than another if the medal score of the latter can be transformed into the former by a sequence of medal additions and medal upgrades. A bit of thought shows that this is exactly equivalent to defining a partial order on triples of medals, in which a triple (G,S,B) is considered at least as good as another triple (g,s,b) if and only if it satisfies the three conditions * G ≥ g * G + S ≥ g + s * G + S + B ≥ g + s + b
MA 1114 - Conceptual Mathematics - Fall Semester 2006, Polytechnic University. Jonathan Bain * Syllabus * Lectures o 01. The Branches of Mathematics o 02. Paradoxes o 03. Greeks & Aristotle o 04. Calculus o 05. Proofs o 06. Naive Set Theory o 07. Ordinals and Cardinals o 08. Formal Set Theory o 09. Problems: The Skolem Paradox o 10. Problems: Godel's Incompleteness Theorems o 11. Category Theory: Intro o 12. Isomorphisms, Sections, Retractions o 13. More Categories o 14. Generalized Elements o 15. Terminal Objects & Initial Objects o 16. Products o 17. Sums
In the fall of 1986, Eric Graham discovered that the Amiga 1000 personal computer, running at a meager 7.16 MHz, was powerful enough to ray trace simple 3D scenes. This is a recreation by coding from first principles.