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What Is The Centinel’s View Of The Three Party System
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The Sentinel, 1944
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Combinators: A Centennial View—stephen Wolfram Writings
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We’ve made several improvements by designing a new Explore page with larger images, easy-to-find information, and more. 1 of 3 Next Before Turing machines, before lambda calculus—even before Gödel’s theorem—were combinators. They were the first concrete examples of what is now known as universal computing — and were first demonstrated on December 7, 1920. In another version of the story, the entire computing infrastructure may have been built on top of them. But because of them, for a century, they remained a kind of interest – and the height of abstraction, and ambiguity.
, which follows a simple substitution rule (now represented cleanly in terms of patterns in the Wolfram Language):
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! And the important thing is that whatever symbolic construction we want, we can always arrange their combination.
‘ which will eventually do it for us – and eventually become a global reckoning. They are equivalent to Turing machines, lambda calculus, and all other systems we know to be universal. But they were created before this system.
Basically, this is how the Wolfram Language achieves the above result (//. the rule is repeated over and over again until something else changes):
And, yes, it’s no coincidence that working with combinators in the Wolfram Language is very easy and natural—because combinators are actually some of the ancestors of the core language of the Wolfram Language.
A Centennial View
For me, the combinator also has a deep personal resonance. They are examples of very simple computer systems that seem (as we will see at length here) to demonstrate the extraordinary behavioral complexity that I have studied over the years in the computer world.
A century ago—especially without a real computer to do my experiments—the conceptual framework I developed for thinking about the computational world didn’t exist. But I’ve always thought that of all systems, the combinator is probably the first “close miss” I’ve ever seen in the world of computing.
Let’s say you want to use a combinator to do a calculation on something. The first question is: how should we represent “things”? Well, the obvious answer is: just use the system built from the combinator!
For example, let’s say we want to represent a number. Here’s a (seemingly odd) way to do it. Take it
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) and the form e[s][k] This is what happens for example in the third case above when you apply
. . . . OK, so given two integers represented like this, how do you sum them? Well, that’s a combo! And here it is:
One might think that this is a crazy way of calculating things. But the important thing was to work, and along the way, the basic idea was created in 1920.
So far we have used the blank form of a number. But we can set the combinator to handle binary numbers. Or, for example, you can build a combinator to perform logical operations.
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And by combining these (or just a copy of them for Nand), we can create a combinator that calculates any Boolean function. And in general, one can – at least in principle – express all calculations of “creation” as a combinator.
. . . . The following represents a step in the evolution of Rule 110 mobile automation:
And, here from the book, is an illustration of the repeated application of this combinator to calculate – with great difficulty – three stages of the development of the 110 rules:
There are other things to do, including fixed point combinations, etc. But in general, since we know that the principle of 110 is a universal calculation, it shows that the combinator is no different.
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Now that a century has passed, what should we think about combinators? On the one hand, they are probably the cleanest way to represent calculations that we know of. However, it is difficult for us humans to understand them.
However, as computing and computing paradigms advance, and become more mainstream, it seems that in many areas we are getting closer to the basic idea of combinatorics. And the underlying symbolic structure of the Wolfram Language — and much of what I’ve built myself over the last 40 years — can finally be seen as a compelling information source from the ideas that first appear in it. in the combinator.
Computing is perhaps the single most powerful unifying concept in existence. But the actual technical development of computers and computing tends to separate the different aspects. Data is available. There are different types of data. There is a code. There is work. There are variables. There are controls. And, yes, it’s easy to separate these things from the traditional methods of computer systems engineering. But it doesn’t have to be that way. And the combinator shows us that it is not necessary to have these differences: they can all be combined, and they can be done with the same dynamic “computing object”.
This is a very powerful idea. But in its raw form it also confuses us humans. Because in order to understand reality, we tend to rely on “fixed anchors” that we can attach meaning to. And in the clear and ever-changing ocean
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However, there were some compromises – and to some extent they are what I can build a complete computer language with today’s Wolfram Language. The point is that if we want to represent everything in the world through calculations, we need the kind of unity and flexibility that builds like combinators. But we are not just looking for a raw and simple combination. In general, we need to define many constructions such as combinators that have a special meaning related to what we represent in the world.
At the practical level, the important idea is to represent things as symbolic expressions, and then to say that the evaluation of these expressions is to repeatedly apply changes to them. And, yes, symbolic expressions in Wolfram Language are just like the expressions we create from the combinator — except they don’t just include.
‘s, they include thousands of different symbolic constructions that we
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