I was wondering if a computational problem exists with the following properties:It should be solvable only having $K$ bytes of memory, or solvable with $K' < K$ bytes of memory only in exponential time on $K - K'$ (or such an high degree polynomial in $K - K'$ that for any practical purpose the computation is unfeasible).The input to the problem should not be bigger than $O(\ln(K))$.The output to the problem should not be bigger than $O(\ln(K))$.The time complexity of the problem should be less or equal than $O(K \ln (K))$.I have thought of many possible solutions but could always find a counterexample. Any sub-optimal suggestion or pointer is also very appreciated.
Problem that is only solvable in a given space
complexity theory;space complexity
This kind of question is studied in the cryptography literature under the name memory-bound computation or memory-hard function.  Such a function is one that can be computed efficiently using $K$ bytes of memory but takes much longer to compute if you have significantly less than $K$ bytes of memory.  So, I suggest that you take a look at cryptographic constructions for memory-hard functions.  It's an active area of research, and researchers focus more on practical complexity (concrete running time, not asymptotic running time) than on complexity theory, but you might find it useful.See https://en.wikipedia.org/wiki/Memory_bound_function#Using_memory_bound_functions_to_prevent_spam for a high-level overview.