Roel de Vrijer

Lambda calculus with patterns

In this paper we revisit the @l-calculus with patterns, originating from the practice of functional programming language design. We treat this feature in a framework ranging from pure @l-calculus to orthogonal combinatory reduction systems.

Processes, Terms and Cycles: steps on the Road to Infinity

The Spectra of Words.- On the Undecidability of Coherent Logic.- Lobs Logic Meets the ?-Calculus.- A Characterisation of Weak Bisimulation Congruence.- Bohms Theorem, Churchs Delta, Numeral Systems, and Ershov Morphisms.- Explaining Constraint Programming.- Sharing in the Weak Lambda-Calculus.- Term Rewriting Meets Aspect-Oriented Programming.- Observing Reductions in Nominal Calculi Via a Graphical Encoding of Processes.- Primitive Rewriting.- Infinitary Rewriting: From Syntax to Semantics.- Reducing Right-Hand Sides for Termination.- Reduction Strategies for Left-Linear Term Rewriting Systems.- Higher-Order Rewriting: Framework, Confluence and Termination.- Timing the Untimed: Terminating Successfully While Being Conservative.- Confluence of Graph Transformation Revisited.- Compositional Reasoning for Probabilistic Finite-State Behaviors.- Finite Equational Bases in Process Algebra: Results and Open Questions.- Skew and ?-Skew Confluence and Abstract Bohm Semantics.- A Mobility Calculus with Local and Dependent Types.- Model Theory for Process Algebra.- Expression Reduction Systems and Extensions: An Overview.- Axiomatic Rewriting Theory I: A Diagrammatic Standardization Theorem.

Descendants and origins in term rewriting

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first-order term rewriting and ?-calculus, their finitary as well as their infinitary variants. A recurrent theme is the parallel moves lemma. Next to the classical notion of descendant, we introduce an extended version, known as origin tracking. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the genericity lemma in ?-calculus, the theorem of Huet and Levy on needed reductions in first-order term rewriting, and Berrys sequentiality theorem in (infinitary) ?-calculus.

A Direct Proof of the Finite Developments Theorem

Let M be a term of the type free λ -calculus and let be a set of occurrences of redexes in M . A reduction sequence from M which first contracts a member of and afterwards only residuals of is called a development (of M with respect to ). The finite developments theorem says that developments are always finite. There are several proofs of this theorem in the literature. A plausible strategy is to define some kind of measure for pairs ( M, ), which—if M ′ results from M by contracting a redex occurrence in and ′ is the set of residuals of in M ′— decreases in passing from ( M , ) to ( M ′, ′). This procedure is followed as a matter of fact in the proofs in Hyland [4] and in Barendregt [1] (both are covered in Klop [5]). If, as in the latter proof, the natural numbers are used as measures, then the measure of ( M , ) will actually denote an upper bound of the number of reduction steps in a development of M with respect to . In the present proof we straightforwardly define for each pair ( M , ) a natural number, which can easily be seen to indicate the exact number of reduction steps in a development of maximal length of M with respect to .

Modularity of confluence: a simplified proof

Abstract In this note we present a simple proof of a result of Toyama which states that the disjoint union of confluent term rewriting is confluent.

Local Termination

The characterization of termination using well-founded monotone algebras has been a milestone on the way to automated termination techniques, of which we have seen an extensive development over the past years. Both the semantic characterization and most known termination methods are concerned with global termination, uniformly of all the terms of a term rewriting system (TRS). In this paper we consider local termination, of specific sets of terms within a given TRS. The principal goal of this paper is generalizing the semantic characterization of global termination to local termination. This is made possible by admitting the well-founded monotone algebras to be partial. We show that our results can be applied in the development of techniques for proving local termination. We give several examples, among which a verifiable characterization of the terminating S -terms in CL.

A Calculus of Lambda Calculus Contexts

The calculus λc serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole filling, by a mechanism of delayed substitution. The context calculus λc is given in the form of an extension of the lambda calculus. Many notions of context can be represented within the framework; a particular variation can be obtained by the choice of a pretyping, which we illustrate by three examples.

Extended Term Rewriting Systems

In this paper we will consider some extensions of the usual term rewrite format, namely: term rewriting with conditions, infinitary term rewriting and term rewriting with bound variables. Rather than aiming at a complete survey, we discuss some aspects of these three extensions.

Four equivalent equivalences of reductions

Abstract Two co-initial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a different order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove that the characterisations all yield the same notion of equivalence, for the class of first-order left-linear term rewriting systems. A crucial role in our development is played by the notion of a proof term.

Iterative Lexicographic Path Orders

We relate Kamin and Levy’s original presentation of lexicographic path orders (LPO), using an inductive definition, to a presentation, which we will refer to as iterative lexicographic path orders (ILPO), based on Bergstra and Klop’s definition of recursive path orders by way of an auxiliary term rewriting sytem.