Category Archives: collatz

collatz, tightening screws on general invariant

this tightens the screws some )( on the prior findings and show they generalize nicely. prior analysis seems very solid but there is always the shadow of question of bias in the generation algorithms which start from small seed sizes and build larger ones out of smaller ones. an entirely different strategy for generating sample seeds is used here based on a genetic algorithm. the idea is to start with a fixed bit width and alter the bits in it just like “genes”. fitness is based on glide length (or equivalently ‘h2’ for a constant seed width). it starts with 20 random parents of given length. there is 1 mutation operator and 2 crossover operators. 1 crossover operator selects adjacent/ contiguous bits from parents at a random cutoff/ crossover point (left to right) and the other just selects bits randomly from parents not wrt adjacent/ contiguous position. fit24 is again used for the linear regression fit. these runs are for 50, 80, 100 bit sizes and ~200 points generated for each. 50K iterations.

because of declining # of points for higher widths, this is circumstantial evidence that as widths increase long glides (relative to width, ie ‘h2’) are (1) increasingly hard to find and/ or (2) rare. these two aspects interrelate but are not nec exactly equivalent. hardness of finding seeds with certain qualities ie computational expense does not nec mean theyre rare. an example might be RSA algorithm. finding twin primes is somewhat computationally expensive (although still in P) but theyre not so rare. technically/ theoretically rareness and computational expense are related through probabilistic search aka “randomized algorithms”.

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collatz radial basis function back to basics

hi all. some extended wallowing in self appraisal/ reflection to begin with in this installment. last months installment of collatz had a big highlight, at least wrt this blogs history.  as has been stated in various places, & trying not to state the obvious here, part of the idea of this blog was to try to build up an audience… aka communication which (to nearly state the almost-canard) is well known to be a “two way street!” there are all kinds of audiences on the spectrum of passive to active, and in cyberspace those in the former camp are also long known semi-(un?)-affectionately as lurkers.

must admit do have some “blog envy” of some other bloggers and how active their audiences are wrt commenting. one that comes to mind is scott aaronson. wow! thought something like a fraction of that level would be achievable for this blog but now in its 5th year, and candidly/ honestly, it just aint really happening. have lots of very good rationalizations/ justifications/ excuses for that too. ofc it would help to have some breakthrough to post on the blog and drive traffic here through a viral media frenzy… as the beautiful women sometimes say, dream on… ah, so that just aint really happening either. 😐

however, there was a highlight from last month, for this blog something like a breakthrough, but also, as you might realize the subtext on reading further, with some major leeway on where the bar is set (cyber lambada anyone?). got an anonymous, openminded, even almost/ verging )( on encouraging comment from someone who wrote perceptively and clearly had a pretty good rough idea of what was going on in that significantly complicated collatz analysis blog post as if reading substantial part of it and comprehending it, and getting to some of the crux/ gist of ideas/ approach here. nice! 😎

(alas, full “open kimono”/ self-esteem challenging disclosure… admittedly that is a very rare event on this blog, and despite immediate encouragement and my marginal/ long gradually increasing desperation now verging on resignation acceptance, anonymous has so far not returned. this overall predicament is something of a nagging failure gap/ regret/ ongoing challenge wrt the original idealism/ enthusiasm/ conception of this blog. which reminds me, also, long ago there was an incisive/ discouraging/ naysaying/ cutting/ near-hostile/ unforgettable comment, and may get to “highlighting” that one too eventually as part of the overall yin/ yang balance etc after changing circumstances and/ or building up enough courage wrt my cyber-ego, long keeping in mind that other quirky aphorism, success is the best revenge…) 😈

anyway here is the comment again, suitably highlighted/ framed/ treasured forever at the top of this blog:

What is a “glide” and how is it related to the trajectory length? Have you defined it somewhere earlier? What are your input variables for the model? What’s the reason to believe that even if you have a good predictor for your “glide” it helps to prove the conjecture?

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collatz 2017, long thought idea realized

this is a hazy idea thats been pulling at me for several years, finally came up with a way to solidify it and then decided to try it out. results are unfortunately lackluster so far but it was a relief to finally see how it works (otherwise would just keep wondering if its the “one that got away”!). and anyway think the overall conceptual idea is at least somewhat sound and still has promise maybe with some other revision/ angle.

the prior runs showed that theres a roughly predictable linear relationship between decidable metrics for each iterate and the glide length (“horizontal ratio”).  so, one of my rules of thumb discovered the hard way over the years, once its proven at least a simple machine learning algorithm works, then one can look at more sophisticated ones that should at least outperform linear regression. (and conversely, if there is not even a weak linear relationship present, it seems unlikely that machine learning will “pull a rabbit out of a hat”. this is surely not an absolute phenomenon, but it seems to hold in practice, and think it would be very interesting to isolate exceptions, suspect they are rare.)

the idea explored here on specific collatz data is a general one for machine learning. suppose that one does not have very many coordinates in ones data. each coordinate is something like a “feature”. and one would like to increase the number of features using only the supplied ones. this is similar to what a neural network does but typically each neuron has many connections. one wonders, can it be done with few connections? the smallest case is 2 connections. is it possible that only 2-way analysis of input data, built on recursively, could have predictive capability? the answer here for this data is mostly no but maybe theres more to the story with further tweaking. also it might work for some other type of data.

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collatz linear multiple regression attack contd

this took quite a bit of effort and is an idea that builds on extend23.rb, some of the effort is related to more need for analyzing results. it works with the long trajectory database. it uses 5 of the glide algorithms that typically lead to longer glides of at least 10 steps. then it does a linear regression on the trajectory iterates metrics to fit the estimated iterations left in the glide, but scaled by the initial seed bit width. this was called the “horizontal scale” in some earlier experiments and seems to be stationary in the statistical sense. then the optimization algorithm attempts to find trajectories that maximally “thwart” the estimate ie maximize error.

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collatz, more trajectory limits

went a little while with no idea what to try next, then came up with some simple idea. earlier it was found that nonmonotone run lengths on left/ right side of the glide are unbounded. but then wondered a bit, in a sort of long-shot guess, that maybe this is a bit of trick going on (this is also hopeful in the sense that the transducer framework is very powerful working against “constant-count-or-less” computations, ie TM ID sequences or other basic arithmetic operations such as addition/ multiplication by constant). a later finding was that the div2 operations count compared to initial seed bit width may be quite significant. so maybe it makes sense to look at “left” and “right” of this key point, namely the trajectory index where div2 operations equals initial bit width.

since it seems to be something significant call it the “div2 trajectory index” or just “div2 index”. my 1st question was to look at nonmono run length in the right side of the div2 index wondering if it was bounded. (my general suspicion aka “hypothesis” is that maybe the behavior of the semirandom walk is significantly smoothed out after the div2 index and a lot of the randomness is “left” of it and attributable/ correlated to the initial seed. but also another meshing pov is that this randomness can allow greater flexibility for arbitrary structures eg unbounded nonmono run lengths on either left or right of the trajectory max.)

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