Let $L$ be a first order alphabet with only a binary function symbol, $*$.
I am aware that one can express the associative law formula $\phi$, defined by $a*(b*c)=(a*b)*c$, with less than $4$ occurrence of the symbol $*$.
Although, I can't seem to find any formula which uses less than $4$ symbols. Looks like there is even a way to express this using a single $*$.
Can any of you find these formulas, or give some kind of hint$?$
Thank you very much
EDIT: I forgot to mention that I allow for quantifiers to be used in the alternate such formula $\psi$. Thus, we can let $\phi$ be $\forall a \forall b \forall c(a*(b*c)=(a*b)*c)$.Just to be clear, one can use any quantity of any logical symbol in the language except for the function symbol $*$ which has to be used at most $3$ times. Sorry for the mistake.