Encoding many-valued logic in $\lambda$-calculus

We will extend the well-known Church encoding of Boolean logic into $\lambda$-calculus to an encoding of McCarthy's $3$-valued logic into a suitable infinitary extension of $\lambda$-calculus that identifies all unsolvables by $\bot$, where $\bot$ is a fresh constant. This encoding refines to $n$-valued logic for $n\in\{4,5\}$. Such encodings also exist for Church's original $\lambda\mathbf{I}$-calculus. By way of motivation we consider Russell's paradox, exploiting the fact that the same encoding allows us also to calculate truth values of infinite closed propositions in this infinitary setting.

From Reduction-Based to Reduction-Free Normalization

We present a systematic construction of a reduction-free normalization function. Starting from a reduction-based normalization function, i.e., the transitive closure of a one-step reduction function, we successively subject it to refocusing (i.e., deforestation of the intermediate reduced terms), simplification (i.e., fusing auxiliary functions), refunctionalization (i.e., Church encoding), and direct-style transformation (i.e., the converse of the CPS transformation). We consider two simple examples and treat them in detail: for the first one, arithmetic expressions, we construct an evaluation function; for the second one, terms in the free monoid, we construct an accumulator-based flatten function. The resulting two functions are traditional reduction-free normalization functions.<br /> <br />The construction builds on previous work on refocusing and on a functional correspondence between evaluators and abstract machines. It is also reversible.

On One-Pass CPS Transformations

We bridge two distinct approaches to one-pass CPS transformations, i.e., CPS transformations that reduce administrative redexes at transformation time instead of in a post-processing phase. One approach is compositional and higher-order, and is due to Appel, Danvy and Filinski, and Wand, building on Plotkin's seminal work. The other is non-compositional and based on a syntactic theory of the lambda-calculus, and is due to Sabry and Felleisen. To relate the two approaches, we use Church encoding, Reynolds's defunctionalization, and an implementation technique for syntactic theories, refocusing, developed in the second author's PhD thesis.

A Unifying Approach to Goal-Directed Evaluation

Goal-directed evaluation, as embodied in Icon and Snobol, is built on the notions of backtracking and of generating successive results, and therefore it has always been something of a challenge to specify and implement. In this article, we address this challenge using computational monads and partial evaluation.<br /> <br />We consider a subset of Icon and we specify it with a monadic semantics and a list monad. We then consider a spectrum of monads that also fit the bill, and we relate them to each other. For example, we derive a continuation monad as a Church encoding of the list monad. The resulting semantics coincides with Gudeman's continuation semantics of Icon.<br /> <br />We then compile Icon programs by specializing their interpreter (i.e., by using the first Futamura projection), using type-directed partial evaluation. Through various back ends, including a run-time code generator, we generate ML code, C code, and OCaml byte code. Binding-time analysis and partial evaluation of the continuation-based interpreter automatically give rise to C programs that coincide with the result of Proebsting's optimized compiler.