Local reasoning about programs exploits the natural local behaviour common in programs by focussing on the footprint - that part of the resource accessed by the program. We address the problem of formally characterising and analysing the footprint notion for abstract local functions introduced by Calcagno, O Hearn and Yang. With our definition, we prove that the footprints are the only essential elements required for a complete specification of a local function. We formalise the notion of small specifications in local reasoning and show that for well-founded resource models, a smallest specification always exists that only includes the footprints, and also present results for the non-well-founded case. Finally, we use this theory of footprints to investigate the conditions under which the footprints correspond to the smallest safe states. We present a new model of RAM in which, unlike the standard model, the footprints of every program correspond to the smallest safe states, and we also identify a general condition on the primitive commands of a programming language which guarantees this property for arbitrary models.

• 68N30 - Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)