Big-Stop Semantics: A Simple Way to Get the Benefits of Small-Step Semantics in a Big-Step Judgment

As evident in the programming language literature, many practitioners favor specifying dynamic program behavior using big-step over small-step semantics. Unlike small-step semantics, which must dwell on every intermediate program state, big-step semantics conveniently jump directly to the ever-important result of the computation. Big-step semantics also typically involve fewer inference rules than their small-step counterparts. However, in exchange for ergonomics, big-step semantics give up power: Small-step semantics describes program behaviors that are outside the grasp of big-step semantics, notably divergence. This work presents a little-known extension of big-step semantics with inductive definitions that captures diverging computations without introducing error states. This big-stop semantics is illustrated for typed, untyped, and effectful variants of PCF, as well as a while-loop-based imperative language. Big-stop semantics extends the standard big-step inference rules with a few additional rules to define an evaluation judgment that is equivalent to the reflexive-transitive closure of small-step transitions. This simple extension contrasts with other solutions in the literature which sacrifice ergonomics by introducing many additional inference rules, global state, and/or less-commonly-understood reasoning principles like coinduction.

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