Here we look at some software projects and theory that relate mathematics and computation. Such as:
Information about computation of mathematics on this site:
- Lambda Calculus
- Combinatory Logic
- Computation using Axiom/FriCAS language
- Computation using Scala language
|I tend to think of computation as some sort of machine (or person) that takes some input and uses it to produce some output.|
This is like a mathematical function in that it always generates the same output for a given input (no side effects in Haskel parlance) so an example would be a sine function which generates the ratio of the opposite and hypotenuse for a given angle.
For computer theory we need to make this very general, a Turing machine can be looked at as either:
- An acceptor - accepts a recursively enumerable set.
- A generator - computes a recursive function
- An algorithm - solves yes/no problems.
So the input/output could be anything including an infinite sequence.
More about mathematical proof on page here.
Some sites with introduction texts:
This is concerned with 'meaning' of programs as opposed to their syntax.
|The computations are modeled as a state with different types of command acting on that state.|
There are two flavors of this big-step and small-step semantics,
This tends to be a recursive process of combining the commands.
For instance, in a program we might put one command after another separated by a semicolon. So we can compose commands in a sequence defined like this:
An assignment (assign variable 'x' to value 'a') is more complicated because we need to do something inside the state, so this is like a second level:
The 'if' statement depends on the state:
|In addition to the state we hold the commands still to be executed. This allows us to model individual steps.|
Shows how to reduce program slightly
- Abstract machine with execution state
- Reduction rules
- iterativley do reduction steps until it cant be reduced further