TITLE Specifiability and computability of functions by equations on partial algebras. ABSTRACT The aim of this research is to compare and contrast specifiability and computability of functions on many-sorted partial algebras A by systems of equations and conditional equations. As our model of computability, we take the system muPR*(A) of primitive recursive schemes over A with added array sorts and the "mu" (least number) operator. We show: (1) Any muPR*-computable function is specifiable (i.e. uniquely definable) by a finite set of conditional equations over A, using either Kleene's semantics, or a "strict" semantics, for the equality relation between partially defined terms; but not conversely, i.e, not all conditionally equationally specifiable functions are computable. (2) If however we replace "unique definability" by "definability as a minimal solution" in Kleene equational logic, and if we consider only equations, not conditional equations, then we obtain the class of functions ED*(A), which is shown to be equal to muPR*(A). This equivalence provides added support for a Generalised Church-Turing Thesis. However the class CED*(A) of minimal solutions of conditional equations goes beyond muPR*(A) computability. In fact such functions are in muPR*(A,eq), i.e., muPR* over A extended by the equality operator at all sorts.