Can all programs with mutable state be refactored in programs without?

Given the space A of all programs that could be possibly written in a given language and given the space B of all programs that could be possibly written in the same language but without any mutable state: can we be sure that with just B we can solve all the same problems we could solve with A ?

There is no general notion of "mutable state" for a generic language. I believe we need to restrict this question to some specific language.

For instance, consider a basic imperative language with arithmetic expressions ($*,+,-$), read input, write output, assignment, composition, conditional ($e=0$ guards), and while loop. No user-defined functions, hence no recursion. Unbounded-precision integer variables. Forbidding mutability would greatly affect the computational power. Unrestricted, it is Turing powerful, but after restriction becomes a very weak language which is able (I believe) to compute only piecewise polynomials / divergence.

If the language is instead a higher order functional language with recursion and assignment (e.g. Scheme), forbidding mutability will not affect its computational power: it was and remains Turing powerful. Mutable state can still be simulated without mutability. A way to do it is to encode state-mutating functions $f$ as pure functions $f({\sf input},{\sf old\_ state})=({\sf output},{\sf new\_state})$ (see the so-called "state monad" in functional programming).

Further, Turing & Church proved that the (pure) $\lambda$-calculus has the same power of the Turing machine. So this is a pretty established result! It is one of the very first discoveries that started computer science in the 30s.

StackExchange Computer Science Q#77319, answer score: 5