Adapting Big-Step Semantics to Small-Step Style: Coinductive Interpretations and "Higher-Order" Derivations

We adapt Kahn-style (\big-step") natural semantics to take on desirable aspects of small-step and denotational semantics forms, more precisely: (i) the ability to express divergent computations; (ii) the ability to reason about the (length of a) computation of a derivation; and (iii) the ability to compute upon and reason about higher-order values. To accomplish these results, we extend the classical, inductive interpretation of natural semantics with coinduction mechanisms and use \negative" rules to express divergence. A simple reformatting of the syntax of derivations allows a simple description of the \length" of a derivation. Finally, the recoding of closure values into denotational-semantics-like functions lets one embed derivations within closures that embed within derivations; in this sense, the semantics becomes \higher order." Examples are given to support the deenitional developments. Kahn-style (\big-step") natural semantics 9] captures a program's entire computation within a single derivation. This representation is more compact than what one obtains from state-transition, \small-step," and denotational deenitions; in addition, natural semantics provides several additional attractions: When the natural semantics is deened in \environment form," that is, the semantics' propositions are ternary relations of the form, ` e + v, where 2 Environment, e 2 Expression, and v 2 Value, the resulting derivations are (almost always) semi-compositional. That is, when one draws a physical derivation tree starting from an initial source program e 0 and initial (empty) environment 0 , then every node within the tree has the form 0 ` e 0 + v 0 , where e 0 is a subphrase of e 0. See Figure 1 for an example. One can prove concisely properties of convergent derivations by induction on the height of the derivation tree. Static analyses are simple to formalize. The key concept, a program's collecting semantics 2, 13, 15], is deened as a simple function of the program's derivation: Given a