Sure many things can be modelled as Markov chains, which is why they're useful. But it's a mathematical model so there's no bound on how big the state is allowed to be. The only requirement is that all you need is the current state to determine the probabilities of the next state, which is exactly how LLMs work. They don't remember anything beyond the last thing they generated. They just have big context windows.

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sigbottle 6 hours ago | parent | next [-]

The etymology of the "markov property" is that the current state does not depend on history.

And in classes, the very first trick you learn to skirt around history is to add Boolean variables to your "memory state". Your systems now model, "did it rain The previous N days?" The issue obviously being that this is exponential if you're not careful. Maybe you can get clever by just making your state a "sliding window history", then it's linear in the number of days you remember. Maybe mix the both. Maybe add even more information .Tradeoffs, tradeoffs.

I don't think LLMs embody the markov property at all, even if you can make everything eventually follow the markov property by just "considering every single possible state". Of which there are (size of token set)^(length) states at minimum because of the KV cache.

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chpatrick 5 hours ago | parent [-]

The KV cache doesn't affect it because it's just an optimization. LLMs are stateless and don't take any other input than a fixed block of text. They don't have memory, which is the requirement for a Markov chain.

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sigbottle 4 hours ago | parent [-]

Have you ever actually worked with a basic markov problem?

The markov property states that your state is a transition of probabilities entirely from the previous state.

These states, inhabit a state space. The way you encode "memory" if you need it, e.g. say you need to remember if it rained the last 3 days, is by expanding said state space. In that case, you'd go from 1 state to 3 states, 2^3 states if you needed the precise binary information for each day. Being "clever", maybe you assume only the # of days it rained, in the past 3 days mattered, you can get a 'linear' amount of memory.

Sure, a LLM is a "markov chain" of state space size (# tokens)^(context length), at minimum. That's not a helpful abstraction and defeats the original purpose of the markov observation. The entire point of the markov observation is that you can represent a seemingly huge predictive model with just a couple of variables in a discrete state space, and ideally you're the clever programmer/researcher and can significantly collapse said space by being, well, clever.

Are you deliberately missing the point or what?

	▲	chpatrick 4 hours ago \| parent [-]
		> Sure, a LLM is a "markov chain" of state space size (# tokens)^(context length), at minimum. Okay, so we're agreed.

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famouswaffles 6 hours ago | parent | prev [-]

>Sure many things can be modelled as Markov chains

Again, no they can't, unless you break the definition. K is not a variable. It's as simple as that. The state cannot be flexible.

1. The markov text model uses k tokens, not k tokens sometimes, n tokens other times and whatever you want it to be the rest of the time.

2. A markov model is explcitly described as 'assuming that future states depend only on the current state, not on the events that occurred before it'. Defining your 'state' such that every event imaginable can be captured inside it is a 'clever' workaround, but is ultimately describing something that is decidedly not a markov model.

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chpatrick 5 hours ago | parent [-]

It's not n sometimes, k tokens some other times. LLMs have fixed context windows, you just sometimes have less text so it's not full. They're pure functions from a fixed size block of text to a probability distribution of the next character, same as the classic lookup table n gram Markov chain model.

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famouswaffles 4 hours ago | parent [-]

1. A context limit is not a Markov order. An n-gram model’s defining constraint is: there exists a small constant k such that the next-token distribution depends only on the last k tokens, full stop. You can't use a k-trained markov model on anything but k tokens, and each token has the same relationship with each other regardless. An LLM’s defining behavior is the opposite: within its window it can condition on any earlier token, and which tokens matter can change drastically with the prompt (attention is content-dependent). “Window size = 8k/128k” is not “order k” in the Markov sense; it’s just a hard truncation boundary.

2. “Fixed-size block” is a padding detail, not a modeling assumption. Yes, implementations batch/pad to a maximum length. But the model is fundamentally conditioned on a variable-length prefix (up to the cap), and it treats position 37 differently from position 3,700 because the computation explicitly uses positional information. That means the conditional distribution is not a simple stationary “transition table” the way the n-gram picture suggests.

3. “Same as a lookup table” is exactly the part that breaks. A classic n-gram Markov model is literally a table (or smoothed table) from discrete contexts to next-token probabilities. A transformer is a learned function that computes a representation of the entire prefix and uses that to produce a distribution. Two contexts that were never seen verbatim in training can still yield sensible outputs because the model generalizes via shared parameters; that is categorically unlike n-gram lookup behavior.

I don't know how many times I have to spell this out for you. Calling LLMs markov chains is less than useless. They don't resemble them in any way unless you understand neither.

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chpatrick 4 hours ago | parent [-]

I think you're confusing Markov chains and "Markov chain text generators". A Markov chain is a mathematical structure where the probabilities of going to the next state only depend on the current state and not the previous path taken. That's it. It doesn't say anything about whether the probabilities are computed by a transformer or stored in a lookup table, it just exists. How the probabilities are determined in a program doesn't matter mathematically.

	▲	saithound an hour ago \| parent [-]
		Just a heads-up: this is not the first time somebody has to explain Markov chains to famouswaffles on HN, and I'm pretty sure it won't be the last. Engaging further might not be worth it.