This was one of my first forays into modifying c code, trying to figure out why 350mb seemed to be the biggest brain size (32 bit memory limits and requiring a contiguous block for the entire brain).
I miss the innocence of those days. Just being a teen, tinkering with things i didn't understand.
I remember reading the source of the original MegaHAL program when I was younger - one of the tricks that made it stand out (particularly in the Loebner competitions [1]) was that it used both a backwards and forwards Markov chain to generate responses.
I once, probably 4-6 years ago, used exports from Slack conversations to train a Markov Chain to recreate a user that was around a lot and then left for a while. I wrote the whole thing in python and wasn't overly well-versed in the statistics and math side but understood the principle. I made a bot and had it join the Slack instance that I administrate and it would interact if you tagged it or if you said things that person always responded to (hardcoded).
Well, the responses were pretty messed up and not accurate but we all got a good chuckle watching the bot sometimes actually sound like the person amidst a mass of random other things that person always said jumbled together :D
Random tidbit - 15+ years ago Markov chains were the go to for auto generating text. Google was not as advanced as it is today at flagging spam, so most highly affiliate-marketing dense topics (e.g., certain medications, products) search engine results pages were swamped with Markov chain-created websites that were injected with certain keywords.
The problem is the linear nature of markov chains. Sure they can branch but after an observation you are absolutely at a new state. A goes to B goes to C etc. A classic problem to understand why this is an issue is feeding in a 2D bitmap where the patterns are vertical but you’re passing in data left to right which Markov chains can’t handle since they are navigating exclusively on the current left to right inout. They miss the patterns completely. Similar things happen with language. Language is not linear and context from a few sentences ago should change probabilities in the current sequence of characters. The attention mechanism is the best we have for this and Markov chains struggle beyond stringing together a few syllables.
I have played with Markov chains a lot. I tried having skip states and such but ultimately you’re always pushed towards doing something similar to the attention mechanism to handle context.
Excellent point about the fundamental limitation of classical Markov chains' linearity. Your 2D bitmap example perfectly illustrates the key insight: traditional Markov chains are inherently constrained by their sequential, memoryless transitions.
However, I'd argue that the distinction between classical n-gram Markov chains and modern transformers isn't as binary as it might appear. When we consider that transformers with context windows are mathematically equivalent to very high-order Markov chains (where the "state" encompasses the entire context), we see that the breakthrough wasn't abandoning the Markov property, but rather expanding the state representation exponentially.
Your observation about inevitably trending toward attention-like mechanisms is particularly insightful. The exponential state explosion you encountered (needing 2^32 states for your example) is precisely why parameterized approaches like transformers became necessary - they provide a tractable way to approximate these massive state spaces through learned representations rather than explicit enumeration.
The key innovation wasn't escaping Markov chains, but rather finding an efficient way to represent and compute over astronomically large state spaces that would be intractable for classical approaches.
Your 2D bitmap example perfectly illustrates the fundamental limitation! The exponential state explosion you encountered (needing 2^32 states for patterns separated by randomness) is precisely why classical Markov chains became intractable for complex dependencies.
What's fascinating is that transformers with attention don't actually escape the Markov property - they're mathematically equivalent to very high-order Markov chains where the entire context window forms the "state." The breakthrough wasn't abandoning Markov chains, but finding a parameterized way to approximate these massive state spaces through learned representations rather than explicit enumeration.
Your observation about inevitably trending toward attention-like mechanisms is spot-on. The attention mechanism essentially provides a tractable approximation to the astronomically large transition matrices that would be required for a classical Markov chain to capture long-range dependencies. It's a more elegant solution to the same fundamental problem you were solving with skip states.
If you program a Markov chain to generate based upon a fairly short sequence length (4 - 5 characters), it can create some neat portamenteaus. I remember back in the early 90's I trained one on some typical tech literature and it came up with the word "marketecture".
I’ve been telling people for years that a reasonably workable initial, simplified mental model of a large language model is a Markov chain generator with an unlimited, weighted corpus trained in. Very few people who know LLMs have said anything to critique that thought more than that it’s a coarse description and downplays the details. Since being simplified is in the initial statement and it’s not meant to capture detail, I say if it walks like a really big duck and it honks instead of quacking then it’s maybe a goose or swan which are both pretty duck-like birds.
This is pretty stupid because the equation for self attention is pretty trivial to the point where it makes sense to just go straight into the math and not play pretend games.
softmax(QK^T/sqrt(dk))V
Q=W_QX
K=W_KX
V=W_VX
X is a matrix where every column is the embedding of a token.
K and V matrices need to be cached.
Also, you need to calculate all of it, for every layer, even if you just want a single token of output.
The tokens after the first token can reuse the K, V embedding matrices and don't have to recalculate everything for on scratch. Causal attention is the concept of masking the calculation so you only have to look at past tokens.
My critique boils down to this: Transformers are too simple to simplify them. You can only lose information if you do that.
And Veritasium's video "The Strange Math That Predicts (Almost) Anything" talks in detail about the history of Markov chains: https://youtu.be/KZeIEiBrT_w
On a humorous note OP, you seem like exactly the kind of guy who would get a kick out of this postmodern essay generator that a STEM professor wrote using a Recursive Transition Network in 1996:
Every now and again, I come back to it for a good chuckle. Here's what I got this time (link to the full essay below the excerpt):
"If one examines subcultural capitalist theory, one is faced with a choice:
either reject subdialectic cultural theory or conclude that the significance of
the artist is significant form, but only if culture is interchangeable with
art. It could be said that many desituationisms concerning the capitalist
paradigm of context exist. The subject is contextualised into a presemanticist
deappropriation that includes truth as a totality."
A first-order Markov-chain text generator is not complex at all; it's a one-line shell script, reformatted here onto three lines for clarity:
perl -ane 'push@{$m{$a}},$a=$_ for@F;END{
print$a=$m{$a}[rand@{$m{$a}}]," "for 1..100_000
}' <<<'fish fish for fish.'
Given the King James Bible instead of "fish fish for fish." as input, this program produces output like this:
25:13 As the hands upon his disciple; but make great matter wisely in their calamity; 1:14 Sanctify unto the arches thereof unto them: 28:14 Happy is our judges ruled, that believe. 19:36 Thus saith the first being one another's feet.
This leans pretty hard on Perl DWIM; a Python version is many times longer:
import random, sys
model, last, word = {None: []}, None, None
for word in (word for line in sys.stdin for word in line.split()):
model.setdefault(last, []).append(last := word)
sys.stdout.writelines(str(word := random.choice(model.get(word) or words)) + " "
for words in [list(model.keys())] for i in range(100_000))
(If you write code like that at a job, you might get fired. I would reject your pull request.)
It's probably worth mentioning:
- Markov-chain states don't have to be words. For text modeling, a common thing to do is to use N-grams (of either words or characters) instead of single words for your states.
- It's good to have some nonzero probability to take a transition that hasn't occurred in the training set; this is especially important if you use a larger universe of states, because each state will occur fewer times in the training set. "Laplacian smoothing" is one way to do this.
- Usually (and in my code examples above) the probability distribution of transitions we take in our text generator is the probability distribution we inferred from the input. Potter's approach of multiplying his next-state-occupancy vector Ms⃗ by a random diagonal matrix R and then taking the index of the highest value in the resulting vector produces somewhat similar results, but they are not the same; for example
sum(random.random() * 0.75 > random.random() * 0.25 for i in range(1_000_000))
is roughly 833000, not roughly 750000. It's 5× as common instead of 3×. But I imagine that it still produces perfectly cromulent text.
Interestingly, modern compilers derive from Chomsky's theory of linguistics, which he formulated while working on an AI project in the 01950s in order to demolish the dominant theory of psychology at the time, behaviorism. Behaviorism essentially held that human minds were Markov chains, and Chomsky showed that Markov chains couldn't produce context-free languages.
I remember creating a chatbot for my minecraft server over a decade ago that used Markov chains. It was funny seeing the crazy things it spat out and the other players enjoyed it.
Back in the early days of Slack I wrote a chat bot that logged all the messages people wrote and used that data to make it say random things prompted by certain trigger words. It was hilarious, until the bosses found out that I was basically logging all their conversations. Unsurprisingly they made me turn it off.
This is great. I use Markov chains to describe to normies how LLMs work.
It's very interesting reading what Claude Shannon wrote after producing Markov chains from English sentences.
> In an unpublished spoof written a year later, Shannon imagined the damage his methods would do if they fell into the wrong hands. It seems that an evil Nazi scientist, dr. Hagen Krankheit, had escaped Germany with a prototype of his Müllabfuhrwortmaschine, a fearsome weapon of war "anticipated in the work ... of Dr. Claude Shannon." Krankheit's machine used the principles of randomized text to totally automate the propaganda industry. By randomly stitching together agitprop phrases in a way that approximated human language, the Müllabfuhrwortmaschine could produce an endless flood of demoralizing statements.
From A Mind at Play. I don't remember the exact year he did that work. I think it was some time before 1950, yet strangely timely in 2025.
Great article. I have been hitting on 'High School's levels including 8 dimensional , dual quaternion, Dirac's Eq. Algebra. On my website. Mainly as they give a view of the dynamics of non trivial unitary matrix expansions that can occur in such as Zeta Zero, element nuclei, prime numbers, energy levels. For light elements. For de located electrons in sync, the Markov chains are interesting, as they relate to cellular automata, the taking of limits for (relativity, electrons moving with charge potential).Having not done high school algebra, until latter years. I have learnt to not assume, a particular algebra fits all dynamic non trivial matrix process's. Ie quantum computing, long range sensors, gravity waves, pressurised plasma at phase, ECT. So great article as it shows a practical high school level of applied knowledge, that doesn't revolve around solely unitary matrices. In as much as its potential dynamic (linear expansion)evolved states ! To which the high school level of visualisation can become aware! (I just thought I'd add this on legal probabilities. There is a case involving a SASR trooper 'Schultz' that revolves around shooting a Mulha's spotter. In Australia magistrates seem to judge cases based on 4 dimensional probabilities. And this seems to tie in with Jeffery Epstein's management through cohesion of President Clinton's pattent attorney change from an solely engineer to a lawyer based pattent granting. Epstein's little girl problem wasn't solely sexual, but given stock market prediction dynamics, more like a dynamic non trivial unitary probability problem of visualising size. Something magistrates in Australia seem to have adopted in the determining of complex military cases.). Warren
There seems to be a mistake in the "Building the Transition Matrix" section of this article. Instead of showing the diagonal matrix D with the normalization coefficients it instead shows the normalized Markov matrix M, incorrectly claiming it is equal to the unnormalized matrix C.
Could add a "stop" token at the end of the fruit example, so when the little model runs it can call its own stop.
To generalize to ADHD, create another model that can quietly "Listen" to a speaker (token generator) until its Markov chain predicts a "Stop", and starts its own generating model running, regardless of whether the speaker was done.
Then two instances can have time efficient interrupty conversations with each other.
So I think this "talking point" of "LLMs are just Markov chains" is as bad, analogously, as "Your amazing invention that passes the Turing Test is justa giant LUT". Or "Your C++ program is justa FSM". It is all technically true, but it is being used subtly to dismiss/delegitimize something, instead of communicate a scientifically insightful idea. How is this different than "Humans are just collections of cells". True except we're are a very interesting type of collection! LLMs are a very interesting type of Markov chain! Haskell/Java/Ruby programs are a very interesting type of FSM! Etc. The talking point by itself is just reductionism, of which we've seen and heard variants of during other scientific revolutions.
Also, this Markov chain has size O(T^K), exponential in the token length of the context window, which is a beyond astronomically large object
45 comments
[ 4.7 ms ] story [ 60.8 ms ] threadThis was one of my first forays into modifying c code, trying to figure out why 350mb seemed to be the biggest brain size (32 bit memory limits and requiring a contiguous block for the entire brain).
I miss the innocence of those days. Just being a teen, tinkering with things i didn't understand.
[1] https://en.wikipedia.org/wiki/Loebner_Prize
https://github.com/Heatwave/the-practice-of-programming/blob...
Well, the responses were pretty messed up and not accurate but we all got a good chuckle watching the bot sometimes actually sound like the person amidst a mass of random other things that person always said jumbled together :D
I have played with Markov chains a lot. I tried having skip states and such but ultimately you’re always pushed towards doing something similar to the attention mechanism to handle context.
However, I'd argue that the distinction between classical n-gram Markov chains and modern transformers isn't as binary as it might appear. When we consider that transformers with context windows are mathematically equivalent to very high-order Markov chains (where the "state" encompasses the entire context), we see that the breakthrough wasn't abandoning the Markov property, but rather expanding the state representation exponentially.
Your observation about inevitably trending toward attention-like mechanisms is particularly insightful. The exponential state explosion you encountered (needing 2^32 states for your example) is precisely why parameterized approaches like transformers became necessary - they provide a tractable way to approximate these massive state spaces through learned representations rather than explicit enumeration.
The key innovation wasn't escaping Markov chains, but rather finding an efficient way to represent and compute over astronomically large state spaces that would be intractable for classical approaches.
What's fascinating is that transformers with attention don't actually escape the Markov property - they're mathematically equivalent to very high-order Markov chains where the entire context window forms the "state." The breakthrough wasn't abandoning Markov chains, but finding a parameterized way to approximate these massive state spaces through learned representations rather than explicit enumeration.
Your observation about inevitably trending toward attention-like mechanisms is spot-on. The attention mechanism essentially provides a tractable approximation to the astronomically large transition matrices that would be required for a classical Markov chain to capture long-range dependencies. It's a more elegant solution to the same fundamental problem you were solving with skip states.
softmax(QK^T/sqrt(dk))V
Q=W_QX K=W_KX V=W_VX
X is a matrix where every column is the embedding of a token.
K and V matrices need to be cached.
Also, you need to calculate all of it, for every layer, even if you just want a single token of output. The tokens after the first token can reuse the K, V embedding matrices and don't have to recalculate everything for on scratch. Causal attention is the concept of masking the calculation so you only have to look at past tokens.
My critique boils down to this: Transformers are too simple to simplify them. You can only lose information if you do that.
And Veritasium's video "The Strange Math That Predicts (Almost) Anything" talks in detail about the history of Markov chains: https://youtu.be/KZeIEiBrT_w
https://www.elsewhere.org/journal/pomo/
Every now and again, I come back to it for a good chuckle. Here's what I got this time (link to the full essay below the excerpt):
"If one examines subcultural capitalist theory, one is faced with a choice: either reject subdialectic cultural theory or conclude that the significance of the artist is significant form, but only if culture is interchangeable with art. It could be said that many desituationisms concerning the capitalist paradigm of context exist. The subject is contextualised into a presemanticist deappropriation that includes truth as a totality."
https://www.elsewhere.org/journal/pomo/?1298795365
It worked surprisingly well. :)
25:13 As the hands upon his disciple; but make great matter wisely in their calamity; 1:14 Sanctify unto the arches thereof unto them: 28:14 Happy is our judges ruled, that believe. 19:36 Thus saith the first being one another's feet.
This leans pretty hard on Perl DWIM; a Python version is many times longer:
(If you write code like that at a job, you might get fired. I would reject your pull request.)It's probably worth mentioning:
- Markov-chain states don't have to be words. For text modeling, a common thing to do is to use N-grams (of either words or characters) instead of single words for your states.
- It's good to have some nonzero probability to take a transition that hasn't occurred in the training set; this is especially important if you use a larger universe of states, because each state will occur fewer times in the training set. "Laplacian smoothing" is one way to do this.
- Usually (and in my code examples above) the probability distribution of transitions we take in our text generator is the probability distribution we inferred from the input. Potter's approach of multiplying his next-state-occupancy vector Ms⃗ by a random diagonal matrix R and then taking the index of the highest value in the resulting vector produces somewhat similar results, but they are not the same; for example
is roughly 833000, not roughly 750000. It's 5× as common instead of 3×. But I imagine that it still produces perfectly cromulent text.Interestingly, modern compilers derive from Chomsky's theory of linguistics, which he formulated while working on an AI project in the 01950s in order to demolish the dominant theory of psychology at the time, behaviorism. Behaviorism essentially held that human minds were Markov chains, and Chomsky showed that Markov chains couldn't produce context-free languages.
https://www.youtube.com/watch?v=KZeIEiBrT_w
You could load it up with anything, and it would just babble on.
Back in the early days of Slack I wrote a chat bot that logged all the messages people wrote and used that data to make it say random things prompted by certain trigger words. It was hilarious, until the bosses found out that I was basically logging all their conversations. Unsurprisingly they made me turn it off.
Source: https://github.com/Aperocky/weighted-markov-generator
It's very interesting reading what Claude Shannon wrote after producing Markov chains from English sentences.
> In an unpublished spoof written a year later, Shannon imagined the damage his methods would do if they fell into the wrong hands. It seems that an evil Nazi scientist, dr. Hagen Krankheit, had escaped Germany with a prototype of his Müllabfuhrwortmaschine, a fearsome weapon of war "anticipated in the work ... of Dr. Claude Shannon." Krankheit's machine used the principles of randomized text to totally automate the propaganda industry. By randomly stitching together agitprop phrases in a way that approximated human language, the Müllabfuhrwortmaschine could produce an endless flood of demoralizing statements.
From A Mind at Play. I don't remember the exact year he did that work. I think it was some time before 1950, yet strangely timely in 2025.
Could add a "stop" token at the end of the fruit example, so when the little model runs it can call its own stop.
To generalize to ADHD, create another model that can quietly "Listen" to a speaker (token generator) until its Markov chain predicts a "Stop", and starts its own generating model running, regardless of whether the speaker was done.
Then two instances can have time efficient interrupty conversations with each other.
Also, this Markov chain has size O(T^K), exponential in the token length of the context window, which is a beyond astronomically large object