A syntactic and functional correspondence between reduction semantics and reduction-free full normalisers

Abstract

Olivier Danvy and others have shown the syntactic correspondence between reduction semantics (a small-step semantics) and abstract machines, as well as the functional correspondence between reduction-free normalisers (a big-step semantics) and abstract machines. The correspondences are established by program transformation (so-called interderivation) techniques. A reduction semantics and a reduction-free normaliser are interderivable when the abstract machine obtained from them is the same. However, the correspondences fail when the underlying reduction strategy is hybrid, i.e., relies on another sub-strategy. Hybridisation is an essential structural property of full-reducing and complete strategies. Hybridisation is unproblematic in the functional correspondence. But in the syntactic correspondence the refocusing and inlining-ofiterate-function steps become context sensitive, preventing the refunctionalisation of the abstract machine. We show how to solve the problem and showcase the interderivation of normalisers for normal order, the standard, full-reducing and complete strategy of the pure lambda calculus. Our solution makes it possible to interderive, rather than contrive, full-reducing abstract machines. As expected, the machine we obtain is a variant of Pierre Crégut's full Krivine machine KN.

Deriving the full-reducing Krivine machine from the small-step operational semantics of normal order

“…Our operational semantics are 'single-stage' [17,18], i.e., they define a single but hybrid normal order strategy that relies on a subsidiary call-byname strategy (Section 3). Consequently, we can use single-layer CPS without control delimiters, as opposed to a two-layer CPS or a single-layer CPS with control delimiters, as found in other works (Section 10).…”

“…Using the correlation between explicit control and the continuation stack [17,18], we finally obtain the open-terms version of KN by applying further standard derivation steps.…”

“…The unique decomposition of C" 0 [ ] is proven by induction on terms [17,18]. The Ap calculus of Biernacka and Danvy [5] extends the pure lambda calculus with definitions for closures C and environments p. This calculus is itself an extension of Curien's calculus of closures X p .…”

“…In a recent work [17,18] we have refined the current interderivation techniques to inter-derive semantic artefacts for fullreducing strategies, and have showcased the inter-derivation of 'We prefer 'full-reducing' to 'strong-' or 'strongly-reducing', as used by some authors, because the latter can be confused with 'stronglynormalising' which means something different. Strong normalisation is a property of a calculus.…”

Deriving interpretations of the gradually-typed lambda calculus

“…We think two important ingredients in this regard are hybrid or layered calculi [16,17], and two-level continuation-passing style [12]. On the one hand hybrid or layered calculi depend on subsidiary sub-calculi, which ought to be turned into a parameter.…”