A similar technique shows that deterministic space S(n) is in EREW(c^S(n), S(n)). In this case the computation of the space-bounded Turing machine can be modeled as a directed tree, and an input is accepted if and only if the final configuration is reachable from the initial one. Since reachability in a directed tree is a special case of expression evaluation, the tree contraction algorithm of Section 2.2 gives the required result.

The paragraph confused me, because both expression evaluation and tree contraction can be done in NC^1. However, NC^1 = Logspace would be a major result. And I didn’t see how those arguments would suffice to show that Logspace is in EREW^1, if they should be insufficient to show that NC^1 = Logspace. Now I finally took the time to investigate this question in detail. My conclusion is that Logspace is really in EREW^1, but that the proof is missing a sort step at the beginning, which makes crucial use of the fact that EREW^1 can compare two numbers in constant time, which explains why that proof doesn’t go through for NC^1.

A smaller issue with that paragraph was that the tree contraction algorithm of Section 2.2 only worked for binary trees, even so there was a remark that it is easy to generalize for arbitrary trees. OK, but would that generalized algorithm still be in EREW^1? A newer paper made the generalization precise and also highlighted the points where things can go wrong:

A Practical Tree Contraction Algorithm for Parallel Skeletons on Trees of Unbounded Degree

Akimasa Morihata, Kiminori Matsuzaki – 2011

The conclusion is that it can indeed be done in EREW^1. Can this algorithm also be implemented in NC^1? I guess so. However, I will first work through the three papers by Sam Buss (et al.) on that problem again, before I form my own opinion and conclusion. Basically, my guess is that EREW^1 like pointer jumping/manipulation can be emulated in NC^1 while CREW^1 like pointer jumping cannot.

Unrelated, already in 2. Jan. 2018, the following paragraph from section “3.4 Relation between circuits and PRAMS” caught my interest:

It is shown in [HoKlPi] that any bounded fan-in circuit of size S(n) and depth D(n) can be converted into an equivalent circuit of size O(S(n)) and depth O(D(n)) having both bounded fan-in and bounded fan-out. Using this result, it is easy to see that NC^k ⊆ EREW^k.

Then I read the references paper. At the moment, that paper is freely available:

Bounding Fan-out in Logical Networks

Howard James Hoover, Maria M Klawe, Nicholas J Pippenger – 1984

It is funny. It uses a sort of Huffman compression to determine the optimal ordering that minimizes the depth.

In his corresponding slides, his remark can be found at the bottom of slide 5:

[Why not use continued fractions? Even worse:

can form an object , but is complemented in it

(in fact it’s the union of and its Heyting negation).

And we can’t even define addition on .]

I find this interesting, since the article Exact Real Computer Arithmetic with Continued Fractions by Jean Vuillemin, 1987 mentioned in the main post explains how to overcome those issues. Even that public link above was already available since 2006, so Peter Johnstone could have found it back then.

This also shows that even so I keep repeating in the main post that using signed digits is not a new idea, it may not be as widely known as it deserves.

]]>However, our models already come with a natural order relation, but it does not satisfy Q8. The other standard axioms of Robinson arithmetic

are satisfied by our models. That natural order relation does satisfy 1.6 Theorem (4) “” and (5) “” in addition to the typical equations for an ordered semiring including asymmetry ().

But the question remains whether open induction holds for our models. Finding a counterexample by further study of easily accessible existing literature is unlikely, since nearly everybody takes Q8 for granted. And without Q8, open induction might even hold. Why? Because it holds in every model of nsEFA, and our models define a subset in each of those models. Equality and the order relation are still dangerous, because their definition depends on the entire collection of models of nsEFA. For example, holds in each model of nsEFA, but it doesn’t hold in our models.

]]>Seriously, looks like a nice piece of philosophy of math but a bit orthogonal to my concerns. I’d always taken it as a given that, if you don’t have a pre-existing understanding of what’s meant by a “finite positive integer,” then you can’t even get started in doing any kind of math without getting trapped in an infinite regress.

Thinking about my goals, this was indeed one point which was very important to me, and that I tried to make explicit: The intuition of encoding a natural number in binary (or any other positional base b numeral system) gets one from one preexisting natural number to an exponentially bigger natural number (or even a finitely iterated exponentially bigger natural number), but it cannot replace a pre-existing understanding of what’s meant by a natural number.

—

However, my previous attempt to explain that circularity was better suited to illuminate that point: You have a string of digits, and you don’t know whether it is finite or infinite. If it is finite, then you can eventually become aware of that fact. This post highlights that a proof in a given axiom system is one way to learn that it is finite. But that was not the motivation for this post. The goal was to do something non-trivial. And there was the curiosity whether different representations would lead to different properties of the resulting numbers (in the sense of different first order sentences satisfied by them).

At some point during working on it, another interesting question emerged: There are different ways to implement addition and multiplication, even for unary encoding. One could interlace the execution of the n-machines in a way such that the commutativity of the operation becomes more obvious, or one could use the more straightforward implementation. This is similar to the different types of addition and multiplication for ordinal numbers (natural/surreal vs continuous). Turned out that it didn’t make a difference for addition and multiplication (basically because EFA is sufficiently powerful). But MIN clearly needs the natural/interlaced/coroutine type of implementation. Related to this, there was also the hope that a connection to ordinal numbers would somehow magically appear. This didn’t happen, but just when finishing the post, it turned out that open induction should have been investigated too. This is related, because open induction should be sufficient to prove commutativity, associativity and distributivity of addition and multiplication over Robinson arithmetic, and some related laws for exponentiation (if POW is available in the system).

]]>Thanks for your friendly words. Especially since finishing the “other post” was hard for me. First I had to fight with myself to finish it, and accept that the result would be less complete and less readable than I had initially hoped. And then I read the sometimes acrimonious discussions (mostly without my participation) which had motivated that post, after which I had to mutilate that post even more to clarify my own position (and also to clarify where I failed to achieve my goals).

Therefore, I like the underlying light hearted realistic instrumentalism of this post even more. It starts from the construction of integers as difference (a-b) of natural numbers, interprets this difference on the level of the digits of the number, and then dives into how this plays out on the level of bit-twiddling, with reference to further material explaining why it gets even more complicated in practice. And at the end, it connects it to foundational issues, and uses the construction of rational numbers as quotient (a-b)/(c-d) of integers as a motivation to investigate further representations like (relaxed) continued fractions (or the factorial/LCM like representations of the other post).

]]>Best

Dick

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