Big-Stop Semantics: Small-Step Semantics in a Big-Step Judgment

David M Kahn; Jan Hoffmann; Runming Li

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arXiv · 2508.15157

Big-Stop Semantics: Small-Step Semantics in a Big-Step Judgment

David M Kahn·Jan Hoffmann·Runming Li

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Abstract

As is 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 jumps directly to the ever-important result of the computation. Big-step semantics also typically involves fewer inference rules than their small-step counterparts. However, in exchange for ergonomics, big-step semantics gives 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 that sacrifice ergonomics by introducing many additional inference rules, global state, and/or less-commonly-understood reasoning principles like coinduction. The ergonomics of big-stop semantics is exemplified via concise Agda proofs for some key results and compilation theorems.

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BibTeX RIS

David M Kahn, Jan Hoffmann, Runming Li. 2025-10-22. Big-Stop Semantics: Small-Step Semantics in a Big-Step Judgment. https://arxiv.org/abs/2508.15157

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