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Comment by imode | Hacker News Reader

Because the Lambda calculus is, in large part, rather messy to implement when it comes to physical computing.

When we look at what computation is, physically, we find it's a lot like a Turing machine underneath (physical rewriting systems), albeit with different kinds of structures that are being rewritten.

Anything involving trees/terms in general seems to be rather messy in terms of physical computation, because there's no straightforward way to represent a manipulatable tree regardless of the medium.

Part of the reason TM-like computing devices sprung up is because strings, i.e linear sequences of symbols, are surprisingly easy to represent and manipulate physically. Turing himself appealed to physical intuition in his original paper.

Just my thoughts as a person who's tried to bootstrap his own LC-based computing environment. I did succeed, but I'm not entirely happy with the results.

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I'm under the impression that LC is closer to how humans think of computation, and it seems that even though it's easier to implement TM as a bare-metal machine, isn't it a trade-off? If humans mostly produce LC-like high-level code, then compile them to TM-like machine code, wouldn't it be better at some point to produce LC-like machines? Because even though they're slower at face value, wouldn't it be faster on "human software".

This comment assumes 'humans mostly produce LC-like high-level code', but I think this is valid; other than C, I cannot think of a language in which programmers don't use lambdas, higher-order functions ubiquitously. It seems like our CPUs are optimized for C, but I'm curious how would it change if we optimize them for javascript, python or haskell...

Here's a test: do you think in terms of step-by-step instructions to solve a process?

There are ways of composing Turing Machines much like composing functions to yield higher abstraction levels in the Lambda Calculus. It comes in the form of building up larger forms of instructions out of simpler machines.

I don't think humans think in a LC-like manner, I think we think compositionally: small parts make up big parts, use the parts you make to make bigger parts. Whether those come in the form of a physical machine or an applicative system doesn't matter, what matters is how you string primitives together.

Check this[1] out. This walks you through building up a more usable language out of the rough-and-ready Turing Machine. There are many ways to go about this idea of composition, but this is a good introduction.

https://pdfs.semanticscholar.org/presentation/98e5/6df9c1c30...

I disagree with what you're suggesting. It's tempting to think humans think procedurally, because that's how we study algorithms. I think this is very misleading. Algorithm is an abstraction of a step-by-step process; studying algorithms in an applicative fashion would be harder simply because they're different sort of things (algorithm is abstracted out of something else). Computation, on the other hand, is a larger category where you're not interested in a particular algorithm in isolation; instead, you have N algorithms and M data.

Your hypothesis doesn't pass any test, if you look at most of the code written today, this very minute, you'll see that it is mostly written in LC-like manner. I'm also not suggesting it is purely applicative, but most PLs seem to be a cross between two models, LC getting more traction recently (look at programming languages I listed above). Compare how popular C was a few decades ago vs now. Our web browsers' native language is javascript, which is essentially lisp in a C-suit, and people started using the same language on desktop and even in embedded devices. Where is C, Fortran, Algol?

If you can't define an algorithm as a step-by-step process, what can you define it as? That's the whole point: it is a process, manifested as sets of discrete steps, towards an end result by following sets of rules.

Lambda Calculus, Turing Machines, etc. are no different in this regard. One has its steps realized in reduction and substitution rules, the other has its steps realized in movements of the head and manipulations of the tape.

You'll need more to disprove my hypothesis. All software is built compositionally, up from smaller pieces. It's why we commonly refer to everything as a "software stack": we can reason about smaller making up larger. If you examine most of the code written today, this very minute, you'll see it's compositional, in the way that larger codebases are built from components.

Neither LC or TMs are "natural ways of thought", they're mechanisms. Turns out that TMs won out not only because they admit an easy physical implementation, but also because they're compositional. I can wire up two TMs and get them to perform operations in sequence, perform conditional branching, etc.

JavaScript is hardly Lisp in a C-suit, but that's beside the point, because C is pretty terrible (despite it being my favorite) regardless of what paradigm you choose. I'm not going to devote much of this comment towards that.

All software is compositional, regardless of the medium, and all computation is purely mechanical, regardless of the model of computation, because unless you want to stare at a lambda expression and infer the value, you need to perform some kind of step-wise reduction. That, after all, was Church's point.

> Neither LC or TMs are "natural ways of thought"

I agree. I think it's like Moravec's paradox: we tend to forget how hard it was to first grasp the concepts of programming, and go on to treat whatever's most familar as being "obvious", even if it's not.

Still, there are a few specific points I take issue with:

> Lambda Calculus, Turing Machines, etc. are no different in this regard. One has its steps realized in reduction and substitution rules, the other has its steps realized in movements of the head and manipulations of the tape.

There is a very big difference between TMs and LC in this regard: each step of a TM takes a constant amount of resources (time, energy, whatever https://en.wikipedia.org/wiki/Blum_axioms ); whilst a single beta-reduction "step" in LC may require an arbitrary amount of work (depending on how many times a variable occurs, whether we need to rename to avoid name capture, etc.). My (perhaps naive) assumption is that the work required for beta-reduction is only bounded by the busy beaver function.

This is probably why most computational complexity research sticks with machine models (operational semantics) like TMs. Still, there are alternative approaches which are very natural for LC-style programming, like "cost semantics" (essentially an alternative set of reduction rules for LC, which evaluate a program into its the algorithmic complexity, rather than into its return value).

> Turns out that TMs won out not only because they admit an easy physical implementation, but also because they're compositional.

I'd hesitate to call TMs compositional. As I wrote at https://cstheory.stackexchange.com/questions/21705/what-is-t... TMs can't really be re-used. Consider a TM (or program for a universal TM) for adding two numbers: it's hard to actually use that as part of any other TM/program, since it may clobber the rest of the tape, we need to setup the contents of the tape, state and read-head just right before invoking it, and clean up afterwards, those may require that we shuttle the rest of the tape contents along to make room, and so on. In contrast, LC lives in an abstract world where subexpressions can expand and contract without bumping into each other, terms can be duplicated and rearranged without having to drag them across the intermediate symbols, etc.

I think you're right w.r.t re-usability, though I've found semi-Thue systems more useful for formulating Turing Machines in order to avoid shifting things around. The string can just expand as needed, though the complexity probably changes.

Humans think procedurally because that's how the real world works. Assembling something, cooking, developing land, treating a patient, you name it.

A doctor doesn't get to destructure you, and then construct a new you that is exactly like you except without the bacterial infection.

Cons-based tree data structures have little to do with data processing in Lisp. Specifically, it is a compile time feature (which is available at runtime, too) that source code is contained in these structures in a sort of AST gives it its metaprogramming capabilities.

The runtime data crunching, GC, and optimization techniques aren't substantially different from other modern dynamic (or semi-dynamic) languages, though Lisp tends to have more features (dynamic bindings, complex numbers, in-language macros, etc). Cons cells at runtime are usually used as plain linked lists, as any dynamic list/array would be in other languages.

On a related note, the SPEC CINT92 benchmark suite included running a tree walking algoritm in interpreted XLISP. I have no idea what the benchmark author thought it demonstrated.

The problem is all of this relies on a representation of the data structure you're manipulating. What exactly is the structure that a Lisp machine is manipulating?

The basis of Lisp is the Lambda Calculus, and a generalization of that gets you term rewriting. Both the basis and the generalization of Lisp admit to requiring a tree (or a graph, in the case of Cons-cells) to get on with the computations, because that's the "state" that you're working with.

The problem is that while these things in the minds of humans are pretty easy to work with, implementing them physically has you rely on things like the RAM model, which, while pretty far away from the Turing Machine, still has machine-like qualities.

That's my argument. We can't really build "Lisp Machines", we can build "Random Access Machines with Lisp Interfaces", because there's no physically implementable version of a tree or graph memory outside of linking randomly-accessible cells.

> implementing them physically has you rely on things like the RAM model

Not at all. In fact, electronic circuitry is quite similar to "LC style" programming, for example we can think of both as a sort of "pipeline for data to flow through".

On the other hand, Turing machines, RAM machines, etc. aren't particularly well suited to being implemented electronically. We end up introducing clock signals, addressing schemes, etc. and it all ends up very complicated, slow and power hungry. I think the only reason we put up with it at all is because electronics, especially ICs, are so incredibly small and fast to begin with, that these inefficiencies aren't too noticable at the human scale.

The basis of Lisp isn't lambda calculus. Lambda calculus doesn't have a concept of its own expressions being data manipulated by lambda calculus. There is no quote operator in lambda calculus.

All the terms in lambda calculus are functions; there are no conses, symbols, numbers, strings, nothing. (Immutable) cons cells can be simulated in lambda calculus using functions. This being in the sense that functions resembling cons, car and cdr can be expressed such that car(cons(x, y)) = x, and cdr(cons(x, y)) = y.

Lisp arises from the realization that expressions built out of trees that consist of pairs can be executed using an eval function, in a way that incorporates some concepts from lambda calculus, such as anonymous functions.

> Lambda calculus doesn't have a concept of its own expressions being data manipulated by lambda calculus. There is no quote operator in lambda calculus.

McCarthy himself noted this about Lambda Calculus and came up with EVAL. In fact several people have also noticed this. They simply shrugged and delta-ruled all the things.

I have no idea how any of that is unique to Lisp or lambda calculus. If you want a tree or graph structure in another language, you'll have random access of nodes in memory, too.

If you want to manually optimize such a linked data structure on top of an array of node structs using offsets, or hash tables, or whatever, you can do that in Lisp or any other language as well.

Any form of non-serialized memory access will penalize you in today's memory heirarchies, no matter the language. Care can be taken in any language to have cache-aligned data structures and preprocessing to attempt to linearize the most common access patterns, usually with preallocated arrays. But generally that level of fine-grained microoptimization isn't needed, and basic data structure best practices are language-agnostic.

> When we look at what computation is, physically, we find it's a lot like a Turing machine underneath (physical rewriting systems), albeit with different kinds of structures that are being rewritten.

I don't think that's the case at all. We see sequential TM-like processes when we look at mainstream electronic computers, but that's kind of a circular argument. When we look at physical processes in general, the computations they're performing tend to be massively parallel.

Pure computation (e.g. lambda calculus, and especially combinatory logic) is well suited to running on massively parallel circuitry (e.g. https://en.wikipedia.org/wiki/Graph_reduction_machine ).

It's perhaps also worth mentioning that such languages can be interpreted as (descriptions of) circuitry! E.g. see http://conal.net/papers/compiling-to-categories

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