A simultaneous induction, a block of relation variables with a first-order step formula per variable and an output sentence, is read partially as follows. The partial iteration starts from the empty relations and at each stage replaces every relation by the set of tuples satisfying its step formula at the current stage: $R_i^0 = \emptyset$ and $R_i^{n+1} = \{\bar a \mid \varphi_i(\bar a) \text{ holds at stage } n\}$. Nothing is accumulated, so the stages need not converge. The induction holds on a structure, read partially, when some stage is a fixed point of the step and the output sentence is true at it; a diverging iteration holds nowhere.

A decision problem $P$ over a vocabulary $L$ is FO($\le$, PFP) definable when some simultaneous induction over $L \cup \{\le\}$ holds partially, for every nonempty finite $L$-structure $A$ and every linear order on $A$, exactly when $A$ is a yes-instance of $P$. It is order-free FO(PFP) definable, respectively order-free FO(IFP) definable, when some simultaneous induction over $L$ alone holds partially, respectively inflationarily, on every nonempty finite $L$-structure exactly when it is a yes-instance: the notions the Abiteboul–Vianu theorem compares.