I am noob at electronics but in short I am interested in stable states of circuit with
no inputs which represents a boolean formula.
Let \$f\$ be some hash function, say md5. Restrict the size of input
to be equal to the output (128 bits for md5).
Cryptographer told be it is open problem if there is
solution to \$f(x)=x\$ (treated as sequence of bits).
Implement \$f\$ as pure combinatorial circuit (NO CLOCK),
with 128 \$i_k\$ inputs and 128 outputs \$o_k\$.
For all \$j\$, connect \$i_j\$ with \$o_j\$ so the circuit
doesn't have inputs(basically
loop the circuit \$f\$).
Power on.
Measure (possibly with oscilloscope) \$i_j=o_j\$.
If you are lucky to hit stable state (logically consistent), you have solved
\$f(x)=x\$.
In an unstable stable (no solution), maybe one will measure very
high frequency.
Probably this is well studied, though an
engineer couldn't find reference for the general case.
Are there references for this?
If it fails why will it fail?
If a solution exists is it known for how long it
will be found?If this makes sense, would please someone try it on something
simple, though not entirely trivial?
Partial experimental results:
It wasn't easy, but me convinced an engineer to test it
on bare metal.
Take \$n\$ inverters (logical negation) and loop them
in a cycle (basically this corresponds to the cycle
graph \$C_n\$).
There are no input and no outputs.
For even \$n\$ the circuit has exactly 2 stable
states.
For odd \$n\$, there is no stable state since
the boolean formula is unsatisfiable.
The engineer worked with NAND gates on a stand.
Experimentally, for \$n=4\$, the engineer reached
stable state. Depending on which NAND gates he chose,
the stable state differed.
For \$n=3\$ there was no stable state.
The voltage was in the middle between logical 0-1,
possibly with high frequency which his oscilloscope
couldn't measure.
I wouldn't trust a software simulation for this,
might be wrong.
The experiments were too trivial, so it is
possible electric current to solve easy problems
and not solve hard problems.
Someone suggested this is "adiabatic bruteforce", though
still can't find the right search terms.
Best Answer
The general field of digital logic without a clock is asynchronous logic. Research has been done in this area but it hasn't really been commercialised.
Your cryptographic example might fail for two reasons:
it may find a cycle, possibly a long one, that produces a sequence of values but does not converge on the desired solution. This is a mathematical aspect of the problem rather than an implementation detail.
If you use regular logic and don't have some system for keeping signal propagation in sync, you won't compute the right thing at all, as some parts of the answer will be ready before others, which in turn will perturb the input. See the "Muller C-element" for a solution to this.
Your engineer should have told you that you were reinventing the "ring oscillator". These are often used in conjunction with phase-locked-loops to produce high frequency digital clocks that are some multiple of an input clock without using a high-frequency crystal.