Andrew Polonsky

Infinitary Rewriting Coinductively

We provide a coinductive definition of strongly convergent reductions between infinite lambda terms. This approach avoids the notions of ordinals and metric convergence which have appeared in the earlier definitions of the concept. As an illustration, we prove the existence part of the infinitary standardization theorem. The proof is fully formalized in Coq using coinductive types. The paper concludes with a characterization of infinite lambda terms which reduce to themselves in a single beta step.

Clocked lambda calculus

One of the best-known methods for discriminating λ-terms with respect to β-convertibility is due to Corrado Bohm. The idea is to compute the infinitary normal form of a λ-term M, the Bohm Tree (BT) of M. If λ-terms M, N have distinct BTs, then M ≠β N, that is, M and N are not β-convertible. But what if their BTs coincide? For example, all fixed point combinators (FPCs) have the same BT, namely λx.x(x(x(. . .))). We introduce a clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps needed in the normalization to the BT. This extension is infinitary strongly normalizing, infinitary confluent and the unique infinitary normal forms constitute enriched BTs, which we call clocked BTs. These are suitable for discriminating a rich class of λ-terms having the same BTs, including the well-known sequence of Bohms FPCs. We further increase the discrimination power in two directions. First, by a refinement of the calculus: the atomic clocked λ-calculus, where we employ symbols τp that also witness the (relative) positions p of the β-steps. Second, by employing a localized version of the (atomic) clocked BTs that has even more discriminating power.

Clocks for Functional Programs

Of the current authors the oldest one remembers with fondness numerous meetings with Rinus from the ancient times of the European Basic Research Actions and from personal tutorials in Nijmegen about λ-terms, term graphs and processes on the one hand, and the practice of functional programming in the Clean environment on the other hand.

New Results on Morris's Observational Theory: The Benefits of Separating the Inseparable

Working in the untyped lambda calculus, we study Morriss lambda-theory H+. Introduced in 1968, this is the original extensional theory of contextual equivalence. On the syntactic side, we show that this lambda-theory validates the omega-rule, thus settling a long-standing open problem.On the semantic side, we provide sufficient and necessary conditions for relational graph models to be fully abstract for H+. We show that a relational graph model captures Morriss observational preorder exactly when it is extensional and lambda-Konig. Intuitively, a model is lambda-Konig when every lambda-definable tree has an infinite path which is witnessed by some element of the model. Both results follow from a weak separability property enjoyed by terms differing only because of some infinite eta-expansion, which is proved through a refined version of the Bohm-out technique.

An Ontology of States

The notion of state is ubiquitous in analysis of computational systems. State introduces intensional content into a dynamical process which cannot be directly observed from outside. Without a state, the process is defined purely by its inputoutput behaviour, and is thus expected to run itself out toward a final result, ie, compute some function. The injunction of internal data that has causal effect on the execution of a system can thus be said to be the step that extends the concept of a function to that of a process, which is no longer guaranteed to terminate.