We started with the idea of code — data consisting of instructions in a special language. Code can express an algorithm, a process consisting of instruction steps. That implies an engine that understands the code language and executes the steps in the code.
Last time we started with Turing Machines, the abstract computers that describe algorithms, and ended with the concrete idea of modern digital computers using stored-programs and built on the Von Neumann architecture.
Today we look into that architecture a bit…
Is that you, HAL?
Last time, in Calculated Math, I described how information — data — can have special characteristics that allow it to be interpreted as code, as instructions in some special language known to some “engine” that executes — runs — the code.
In some cases the code language has characteristics that make it Turing Complete (TC). One cornerstone of computer science is the Church-Turing thesis, which says that all TC languages are equivalent. What one can do, so can all the others.
That is where we pick up this time…
The previous post, Halt! (or not), described the Turing Halting Problem, a fundamental limit on what computers can do, on what can be calculated by a program. Kurt Gödel showed that a very similar limit exists for any (sufficiently powerful) mathematical system.
This raises some obvious questions: What is calculation, exactly? What do we mean when we talk about a program or algorithm? (And how does all of this connect with the world of mathematics?)
Today we’re going to start exploring that.
I’m on a numeric kick with regard to Sidebands. Don’t worry, it can’t last past 13, because 14 and the numbers that follow are fairly uninspiring. Maybe #21 will be special (or not); Sidebands become adults or something. I’ll have to think about that.
I’ve also noticed that Sidebands have taken on more life than I intended. The plan was for them to be short and frequent, but some of them are pretty meaty. That’s okay, meaty words are the point here. And I’m still searching for my “voice” as I write. Many years of more formal writing, much of it technical, seem to push me away from the personal writing I wanted for this blog.
Well it’s a journey, isn’t it; a process.