If you have an interactive UI, you have a state. A button is either pressed or it is not pressed, active or disabled or hidden. A text form has a text value. Your layout has state (position, color, etc.).
If your UI is interactive, you have events and actions that mutate that state. A keyboard event modifies the state of your text form field. A button press signals an action. Etc.
A state machine is `State0 + Event -> Action + State1`. Since your UI's entire functionality can be described by the state it is in combined by the Events and resulting Actions and changes in States, your UI is a state machine. This describes the superset of all UIs. Non-interactive UIs can still be described by this, although the benefits are slim because the state isn't really mutated without interactivity, so your State0 is always equal to your State1.
Note that I'm saying UIs are state machines...not like state machines. They literally are state machines. Whether you model them as such is up to you, but my claim here is that modeling your UIs as state machines has tangible benefits because they are state machines.
Funnily enough, there are plenty of articles out there that claim to make stateless UIs. These claims are false...any critical look at the model shows that the state exists, but either is modeled implicitly (keeping state in the DOM and outside of javascript, or within a rendered object, or within a closure attached to an observer somewhere, etc.), or explicitly moved somewhere else. There might be some merits to the latter model (as well as some drawbacks), but the former model is just delusion...a perfectly leaky abstraction where you have state, but have lost your ability to work with it.
The person you're replying to didn't specify whether UIs are represented as state machines which are finite or infinite, but I'll assume finite since infinite automata are more powerful than Turing machines so they presumably can represent something which is made by a computer.
UIs in general can not be represented by finite state machines. An example is a page with a button which when pressed adds something to the page. Since there are an infinite number of things that could potentially be added, this can't be represented by a finite number of states.
On the other hand, I think many UIs could be represented by finite state machines, although perhaps sometimes only roughly. You can start by trying to specify that each instance of a specific page a user can see is presented by a state, and any interaction with the user that leads to a different rendering of the same or a new page is a transition. However, this fails to account for things where a user can do something which doesn't change the current page but might change a later page. You can account for this by having multiple states which show the same exact UI but have different transitions to future pages. Of course, it isn't just the user who can interact with your system to produce a different UI for other people, so you'd have to model those transitions and states too..
I'm not a front end developer so I don't know how many UIs can actually be modeled this way, but I can see how it might be a useful way to think about things, even if it isn't necessarily always correct.
I didn't specify finite or infinite, and you're right that FSMs alone aren't sufficient to describe all UIs. But they are sufficient to describe nearly all interactive UIs.
Your infinite button example still has a finite state, but it's not modeled explicitly, and would probably be a good idea, for very practical reasons, to do so. If it really goes on forever, you'll eventually run out of memory, whether that memory is managed by the rendering engine (appending dom elements?) or by the language runtime.
> However, this fails to account for things where a user can do something which doesn't change the current page but might change a later page.
A Hierarchical State Machine model covers this scenario quite nicely...which is exactly what React is.
I've certainly never found a common UI pattern that isn't describable with FSMs, but you could probably think of a few toy examples of non-terminating state. Anything involving recursion, exponential, or asymptotic behavior for example could theoretically involve infinite states. Again, likely of no practical use.
Read any real-world user interface specification like Apple's classic Macintosh Human Interface Guidelines [1]. It specifies all kinds of complex behaviors, like how pull-down submenus don't pop up immediately as you move the cursor over them, and how dragging and dropping text handles the white space at the beginning and end of the selection, or how adjusting the end of the selection in mixed right-to-left and left-to-right text works, which is much too complex and nuanced and ad-hoc hacky to describe with finite state machines, and require a Turing complete machine to implement.
Even if it were technically possible to restrict yourself to a FSM, you would end up going insane and producing exponentially explosive unmaintainable finite state spaghetti machines.
There's a huge uncanny gap between what you can elegantly describe with a clean mathematical abstraction like finite state machines, and what real human beings expect and perceive to be easy to use in a real world user interface.
The polish of a good user interface is the result of millions of tiny little scratches, and in practice that requires the power and flexibility of a Turing machine (and not one stuck in the Turing tarpit like Brainfuck) to make all of those tiny little scratches and cover all the edge cases and special circumstances, because finite state machines are just too clumsy and not powerful enough to conveniently cover that much ad-hoc messy fractal detail.
I'm not sure an "infinite state machine" [1] is actually a "thing" the same way a "finite state machine" [2] is a precisely defined mathematical concept in Automata Theory [3]: one level of the classes of automata more powerful than combinational logic and less powerful than pushdown automata and Turing machines.
In my mind, there is no question that finite state machines alone are not powerful enough to define practical real world user interfaces, which require the power of a Turing machine. The fact that most user interfaces incorporate many finite state machines at different levels doesn't mean that finite state machines alone are sufficient.
That's why we program browsers with JavaScript, not FSMML or FSMON.
That's playing fast and loose with the literal every-day definition of the words "finite", "state" and "machine".
There is actually something in computer science with a precise mathematical definition called a "finite state machine," [1] that is not simply any machine that has a finite number of states.
When you use that term, you're not just talking about what it can do, but also what it can't do, because "finite state machine" implies that it's just a finite state machine, not a pushdown automata, and not a Turing machine.
With a Turing machine, you can implement a finite state machine or a pushdown automata, but Turing machines and pushdown automata are more than just finite state machines.
That said, user interfaces typically use lots of finite state machines at different levels (hierarchical, and unrelated), but they also require more powerful machines like pushdown automata and Turing machines (for validating email addresses, for examples).
If all you had to program user interfaces were finite state machines, but no Turing machines, you would be a very frustrated programmer, and your users would be unhappy to have such a brittle inflexible user interface.
It's not playing fast and loose at all - at least not with the precise mathematical definition (perhaps it is playing fast and loose with the "literal every-day" definition which might mean a state machine with few enough states to fit on a whiteboard). A push down automata is defined as having infinite stack and a Turing machine is defined as having infinite tape. A finite stack push down automata permits an equivalent (much larger) finite state machine. A finite tape Turing machine also permits an equivalent (much much larger) finite state machine.
The comment I replied to was "My experience shows it is a state machine, but I have not see any formal study of it." - I was simply replying "of course it's a state machine".
Lots of situations in GUIs can be modelled as quite compact state machines - but it depends on the level you model it at. If you need to model the state of a text box then your state machine will grow very large since you need a new state for every possible input string. Given modelling simplifications where we abstract away some parts with large state spaces by collapsing states, there are very few pieces of software which don't benefit from being modelled with state machines.
Personally I would be a very frustrated programmer and if I had to program user interfaces with Turing machines since writing a transition table would feel awfully low level and since the Turing machine model doesn't have any facilities for user interaction I'm not sure any users would be terribly happy with the result (so I'm not quite sure what you're trying to say here).
