A meta-theory for big-step semantics

TL;DR

This paper develops a meta-theory for big-step semantics, enabling it to distinguish between stuck and diverging computations, thus facilitating reasoning about infinite computations and properties like type soundness.

Contribution

It introduces an abstract framework and proof technique to extend big-step semantics to characterize both stuck and diverging computations, overcoming longstanding limitations.

Findings

Extended semantics can distinguish stuck and diverging computations.

The proof technique applies even when subject reduction fails.

The approach simplifies reasoning about infinite computations.

Abstract

It is well-known that big-step semantics is not able to distinguish stuck and non-terminating computations. This is a strong limitation as it makes very difficult to reason about properties involving infinite computations, such as type soundness, which cannot even be expressed. We show that this issue is only apparent: the distinction between stuck and diverging computations is implicit in any big-step semantics and it just needs to be uncovered. To achieve this goal, we develop a systematic study of big-step semantics: we introduce an abstract definition of what a big-step semantics is, we define a notion of computation by formalising the evaluation algorithm implicitly associated with any big-step semantics, and we show how to canonically extend a big-step semantics to characterise stuck and diverging computations. Building on these notions, we describe a general proof technique to show that a predicate is sound, that is, it prevents stuck computation, with respect to a big-step semantics. One needs to check three properties relating the predicate and the semantics and, if they hold, the predicate is sound. The extended semantics are essential to establish this meta-logical result, but are of no concerns to the user, who only needs to prove the three properties of the initial big-step semantics. Finally, we illustrate the technique by several examples, showing that it is applicable also in cases where subject reduction does not hold, hence the standard technique for small-step semantics cannot be used.