It is known that a Sleptsov net, with multiple firing of a transition at a step, runs exponentially faster than a Petri net, opening prospects for its application as a graphical language of concurrent programming. We provide a classification of place-transition nets based on firability rules considering general definitions and their strong and weak variants. We introduce and study a strong Sleptsov net, where a transition with the maximal firing multiplicity fires at a step, and prove that it is Turing complete. We follow the proof pattern of Peterson which he applied to prove that an inhibitor Petri net is Turing complete by simulating a Shepherdson and Sturgis register machine. The central construct of our proof is a strong Sleptsov net that checks whether a register value (place marking) equals zero.