Control Flow Analysis for SF Combinator Calculus

Programs that transform other programs often require access to the internal structure of the program to be transformed. This is at odds with the usual extensional view of functional programming, as embodied by the lambda calculus and SK combinator calculus. The recently-developed SF combinator calculus offers an alternative, intensional model of computation that may serve as a foundation for developing principled languages in which to express intensional computation, including program transformation. Until now there have been no static analyses for reasoning about or verifying programs written in SF-calculus. We take the first step towards remedying this by developing a formulation of the popular control flow analysis 0CFA for SK-calculus and extending it to support SF-calculus. We prove its correctness and demonstrate that the analysis is invariant under the usual translation from SK-calculus into SF-calculus.Comment: In Proceedings VPT 2015, arXiv:1512.0221

Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators

The symmetric interaction combinators are an equally expressive variant of Lafont's interaction combinators. They are a graph-rewriting model of deterministic computation. We define two notions of observational equivalence for them, analogous to normal form and head normal form equivalence in the lambda-calculus. Then, we prove a full abstraction result for each of the two equivalences. This is obtained by interpreting nets as certain subsets of the Cantor space, called edifices, which play the same role as Boehm trees in the theory of the lambda-calculus

The Geometry of Concurrent Interaction: Handling Multiple Ports by Way of Multiple Tokens (Long Version)

We introduce a geometry of interaction model for Mazza's multiport interaction combinators, a graph-theoretic formalism which is able to faithfully capture concurrent computation as embodied by process algebras like the $\pi$-calculus. The introduced model is based on token machines in which not one but multiple tokens are allowed to traverse the underlying net at the same time. We prove soundness and adequacy of the introduced model. The former is proved as a simulation result between the token machines one obtains along any reduction sequence. The latter is obtained by a fine analysis of convergence, both in nets and in token machines

Discriminating Lambda-Terms Using Clocked Boehm Trees

As observed by Intrigila, there are hardly techniques available in the lambda-calculus to prove that two lambda-terms are not beta-convertible. Techniques employing the usual Boehm Trees are inadequate when we deal with terms having the same Boehm Tree (BT). This is the case in particular for fixed point combinators, as they all have the same BT. Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model P-omega of lambda-calculus, and hence valid with respect to BT-equality but nevertheless the terms are beta-inconvertible. To prove such beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey information of the tempo in which the data in the BT are produced. Boehm Trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for lambda-terms. The corresponding equality is strictly intermediate between beta-convertibility and Boehm Tree equality, the equality in the model P-omega. An analogous approach pertains to Levy-Longo and Berarducci Trees. Our refined Boehm Trees find in particular an application in beta-discriminating fixed point combinators (fpc's). It turns out that Scott's equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a variety of production schemes of which the simplest was found by Boehm, stating that new fpc's are obtained by postfixing the term SI, also known as Smullyan's Owl. We prove that all these newly generated fpc's are indeed new, by considering their clocked BT's. Even so, not all pairs of new fpc's can be discriminated this way. For that purpose we increase the discrimination power by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257

A principled approach to programming with nested types in Haskell

Initial algebra semantics is one of the cornerstones of the theory of modern functional programming languages. For each inductive data type, it provides a Church encoding for that type, a build combinator which constructs data of that type, a fold combinator which encapsulates structured recursion over data of that type, and a fold/build rule which optimises modular programs by eliminating from them data constructed using the buildcombinator, and immediately consumed using the foldcombinator, for that type. It has long been thought that initial algebra semantics is not expressive enough to provide a similar foundation for programming with nested types in Haskell. Specifically, the standard folds derived from initial algebra semantics have been considered too weak to capture commonly occurring patterns of recursion over data of nested types in Haskell, and no build combinators or fold/build rules have until now been defined for nested types. This paper shows that standard folds are, in fact, sufficiently expressive for programming with nested types in Haskell. It also defines buildcombinators and fold/build fusion rules for nested types. It thus shows how initial algebra semantics provides a principled, expressive, and elegant foundation for programming with nested types in Haskell

Classical Mathematics for a Constructive World

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.Comment: v2: Final copy for publicatio

New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized

The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation fails to identify two `obviously' equal terms. If only the machine could decide a richer theory! We propose a way to decide theories which supplement evaluation with `$\nu$-rules', rearranging the neutral parts of normal forms, and report a successful initial experiment. We study a simple -calculus with primitive fold, map and append operations on lists and develop in Agda a sound and complete decision procedure for an equational theory enriched with monoid, functor and fusion laws