From Small-Step Semantics to Big-Step Semantics, Automatically

Abstract

Small-step semantics and big-step semantics are two styles for operationally defining the meaning of programming languages. Small-step semantics are given as a relation between program configurations that denotes one computational step; big-step semantics are given as a relation directly associating to each program configuration the corresponding final configuration. Small-step semantics are useful for making precise reasonings about programs, but reasoning in big-step semantics is easier and more intuitive. When both small-step and big-step semantics are needed for the same language, a proof of the fact that the two semantics are equivalent should also be provided in order to trust that they both define the same language. We show that the big-step semantics can be automatically obtained from the small-step semantics when the small-step semantics are given by inference rules satisfying certain assumptions that we identify. The transformation that we propose is very simple and we show that when the identified assumptions are met, it is sound and complete in the sense that the two semantics are equivalent. For a strict subset of the identified assumptions, we show that the resulting big-step semantics is sound but not necessarily complete. We discuss our transformation on a number of examples.