So your argument is, because it is unclear how to "physically implement the arbitrary-function minimization operator", Turing is the better "theoretical backbone" and has founded computer science?
Church had piles of stuff that he and his students produced that were computable with the lambda calculus. Basically, all of the natural number functions that a person thinks are intuitively mechanically computable, those folks had showed how to lambda compute. With this evidence, he proposed to Godel (they were working together at Princeton at the time), who was considered the world's expert, taking "lambda-calculable" as a mathematically precise version of "mechanically computable." But Godel was notoriously careful, and he did not accept the thought as perfectly clear.
That is, they had a subset of the things that could be mechanically computed. But was it the entire set? Or was there something that some discrete and deterministic mechanism could be made to do that would lead to more than Church's set?
Imagine you are Dedekind and you are looking at the primitive recursive functions and (1) any such function is intuitively mechanically computable, and (2) you are able to work out how to define a pile of things like prime factorization of an integer using this system. You might well conjucture that this is it. But we know (and Godel and Church and Turing knew) that this is not it, that you need to add unbounded search of some kind (this is what minimization does) to get more things that are intuitively mechanically computable.
I agree that the minimization operator is not as easy to picture with gears and levers as some of the other operations. But the issue in 36 was that a person could worry that there was even more. Just as minimization is not as easy to picture and the need for it didn't hit Dedekind with great force, could there be something else out there that we have all missed?
That worry disappeared when Godel read Turing's masterful analysis. It convinced him that this is what a machine can do. He wrote, "That this really is the correct definition of mechanical computability was established beyond any doubt by Turing.'' Church felt the same way, writing that Turing machines have "the advantage of making the identification with effectiveness ... evident immediately.''
The Rekursiv advantage is its ability to do recursion on the bare-metal. It's described as an object-oriented architecture. Its memory was a persistent store of objects, with each object having its size, type, and position known in memory.
If I may ask, from your perspective, what's more elegant about Rekursiv's design? Is it the philosophy or implementation? I'd also love any links to orthogonal persistence!
I came into programming/computer science from a mathematics degree; I read some old treatises on recursion theory[1] and fell in love. I couldn't ever quite wrap my head around the Turing Machine formalism, but kept at it for a while. Finding Barendregt's paper [2] was a huge shock! I grasped it much quicker. So, yes, lambda calculus and the Turing Machine formalism are equivalent in explanatory power, but there are also reasons someone might prefer one to the other. So, yes, for me, the value _is_ the formalism.
As to why I think the Rekursiv would provide a good platform for implementing lambda calculus on the bare-metal, that's entirely due to Rekursiv's memory model advantage and the fact that it has a user-writable ISA. Why would someone choose to implement the lambda calculus on bare-metal? You call it "fetishism," I call it fun!
More generally, I just really like the idea of having a machine with a user-writable ISA.
[1] Theory of Recursive Functions and Effective Computability: https://openlibrary.org/books/OL2738948M/Theory_of_recursive...
[2] Introduction to Lambda Calculus: https://www.cse.chalmers.se/research/group/logic/TypesSS05/E...
Thanks for clearing up that it's the formalism you find interesting. Also, to offer a counterpoint, I'm also from a math background, but I was more of an analysis person (as much as one can be in mathematics where it's all related) than an algebra person, and when I did some FP research, it often felt like where all the algebraists go to play computer science. I feel like analysts are underrepresented in PLT (and overrepresented in complexity theory!) but this is already going off-topic, so cheers.
Out of curiosity, can you identify any areas in PLT that could be made more analyst-friendly?
Intuitively, it feels that PLT is almost necessarily of the algebraist; to me, one of the big divides is the discreteness of algebra vs the continuity of analysis. Would it help if there was a PLT that exhibited a greater degree of continuousness? If so, what do you think that might look like?
That's also a very interesting perspective on analysis. To get a better feel: is your joy of analysis in getting into the "internals" of an algebra, to directly derive properties of elements within the algebra as opposed to relying solely on global properties endemic to the construction of the algebra?
I'd also love to see your talk! Do you think a RISC Rekursiv could be achieved? What value, if any, do you think such might have in our current world?
http://www.wolczko.com/mushroom/
The slides for my 2019 talk about Smalltalk computers (in LibreOffice and PDF formats):
http://www.merlintec.com/download/2019_slides_jecel_fast1v2....
http://www.merlintec.com/download/2019_slides_jecel_fast1v2....
and the video (1 hour and 13 minutes):
https://www.youtube.com/watch?v=tATpzsyC6OA
If you replace the Rekursiv's special microcode memory with a cache to allow microcode to live in main memory you will essentially have a RISC version of the machine. I have adopted this solution in several of my own designs.
I'm also deeply in love with the merlintec website. It's refreshing to see a website which has persisted since 1999(!), and without a lick of JavaScript it would seem!
Could I purchase a Merlin 6 or a Pegasus 2000?
Speaking of which, the way that the Pegasus 2000's eGUI documentation[1] is written is incredibly dear. The "world" and "heaven" analogies are great!
Yes, the set of functions computable with mu recursion, or with lambda calculus, is the same as the set of functions computable with a Turing machine. (Turing in his original paper showed that the set is the same for his system and Church's, and the proofs for mu recursion and many other systems are well-known.)
> I'll totally agree that from a teacher's perspective, the Turing machine is the better explanation.
When I learned this, the instructor did not use Turing machines. They used mu recursion. That has the advantage that you don't first define a machine and then derive the set of functions, instead you go straight to the functions. But I agree that Sipser (which I understand to be the most popular text today) defines a computable function as one that is computed by Turing machine, and to my mind that has the advantage of honestly suggesting to students what they are about.
It does seem weird to me though that we're letting our engineering limitations determine how we think about these things mathematically.
Turing machines do have mathematical advantages in some areas. See this siblings comment: https://news.ycombinator.com/item?id=28541800
That theory continues to bear fruit as the history of programming languages is almost entirely about bringing new and "better" abstractions to problems and then translating them to "dumber, more practical" machines. We have programming languages today modeled directly off the elegance of (though now sometimes still a few steps removed from being direct implementations of) the lambda calculus and mu-recursive functions, and the amazing thing is that they work great even given how "dumb" and "inelegant" our machines can be in practice.
See also https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis
This is already proven to be the case. Mu-recursion (which IIRC is not Godel's general recursive functions, despite what Wikipedia says; Kleene was the originator of the mu operator. Godel's general recursive functions are defined in a separate, but yet again equivalent way that directly extends primitive recursion), Turing machines, and the lambda calculus are all proven to be exactly equivalent to each other. The fact that these three independent approaches to computability are all equivalent is why we have strong informal justification that the Church-Turing Thesis (a non-mathematical statement) holds.
Separately, there's a sentiment that I've seen come up several times on HN that somehow the lambda calculus, mu-recursion, or general recursion is more "mathematical" and Turing machines are less "mathematical."
I want to push back on that. The mathematical field of computability is based almost entirely off of Turing machines because there are many classes of mathematical and logical problems that are easy to state with Turing machines and extremely awkward/almost impossible to state with the lambda calculus and mu-recursion (this is consistent with my previous statement that the three are all equivalent in power because computability theory often deals with non-computable functions, in particular trying to specify exactly how non-computable something is). The notion of oracles, which then leads to a rich theory of things like Turing jumps and the arithmetical hierarchy, is trivial to state with Turing machines and very unwieldy to state in these other formalisms.
Likewise the lambda calculus and mu-recursion (but not general recursion) provide a very poor foundation to do complexity theory in CS. Unlike Turing machines, where it is fairly easy to discern what is a constant time operator, the story is much more complicated for the lambda calculus, where to the best of my knowledge, analyzing complexity in the formalism of the lambda calculus, instead of translating it to Turing machines, is still an open problem.
There is indeed a mathematical elegance to Turing machines that makes it so that most of the mathematics of computability is studied with Turing machines rather than the lambda calculus.
The lambda calculus on the other hand is invaluable when studying programming language theory, but we should not mistake PLT to be representative of the wider field of mathematics or theoretical CS.
EDIT: I should perhaps make clear that if I put on my mathematical hat, mu-recursive functions seem like the most familiar formalism, because they align with a common way families of things are defined in mathematics (specify individual members and then generate the rest through some relationship). However, I would contend that for the majority of mathematicians outside of computability theory, the lambda calculus and Turing machines seem equally "strange."
However, yes, I do think that 'mechanization' or physical implementation is a crucial piece of Turing's contribution that is wrongly ignored in this article. And I think without mechanization, there is no CS as we understand it.
"Likewise, Konrad Zuse never got a Turing award despite having created the world's first working programmable general computer 1935-41. [...] It was pointed out that none of the computers built during the 1940s were influenced in any way by Turing's 1936 theoretical paper, [...]"
In my view, Turing's contribution is providing a plausible definition of computation along with a deep and comprehensive theoretical characterization of the properties of this model of computation. This is why Turing machines form the basis of theoretical computer science, and not other models such as lambda calculus. I think saying that Turing machines were adopted since they were merely more convenient is highly misleading.
I think this pattern repeats a lot: There are many cases where you can point to multiple people who invented similar ideas around the same time, but it is typically the person who provided the most deep and comprehensive treatment of the subject that ultimately gets most of the credit. This depth is not conveyed in pop science attributions such as "Turing invented computation", but this doesn't mean Turing doesn't deserve the credit.
His Turing award citation: "Professor Wilkes is best known as the builder and designer of the EDSAC, the first computer with an internally stored program."
The 2009 award to Chuck Thacker, on the other hand, was clearly based on his contributions to hardware. I have the impression that ACM had a change in policy around that time.
But officially I seem to be wrong:
https://amturing.acm.org/bysubject.cfm?cat=16 https://amturing.acm.org/bysubject.cfm?cat=15
In the "hardware" category Wilkes is listed as the only one, while Thacker, with Brooks, Cocke and Wilkes are in the "computer architecture" category.
The word computer means multiple things. In one sense it the abstraction of universal computation. Imagine a world where actual physical computers didn't progress to universal computation, but were stuck being purpose built to the present day. The field of computer science would be utterly different because they couldn't actually compute anything with their science. They could just discuss computability in an abstract sense. It'd be like physics without the particle colliders or telescopes or lasers.
I think of the founders of computer science more like the founding fathers of America, rather than a single guy named Turing, but some are more memorable than others.
"Likewise, Konrad Zuse never got a Turing award despite having created the world's first working programmable general computer 1935-41. [...] It was pointed out that none of the computers built during the 1940s were influenced in any way by Turing's 1936 theoretical paper, [...]"
It’s not clear that you need a theory of computation to build a stored process computer. You might not clearly understand the theoretical capability of such a machine , but that wouldn’t prevent you from doing many useful things with it.