Vincent van Oostrom

Combinatory reduction systems: introduction and survey

Abstract Combinatory reduction systems, or CRSs for short, were designed to combine the usual first-order format of term rewriting with the presence of bound variables as in pure λ-calculus and various typed λ-calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and left-linear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power and give an outline of a short proof of confluence. This proof is a direct generalization of Aczels original proof, which is close to the well-known confluence proof for λ-calculus by Tait and Martin-Lof. There is a well-known connection between the parallel reduction featuring in the latter proof and the concept of “developments”, and a classical lemma in the theory of λ-calculus is that of “finite developments”, a strong normalization result. It turns out that the notion of “parallel reduction” used in Aczels proof gives rise to a generalized form of developments which we call “superdevelopments” and on which we will briefly comment.

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.

Normalisation in Weakly Orthogonal Rewriting

A rewrite sequence is said to be outermost-fair if every outermost redex occurrence is eventually eliminated. Outermost-fair rewriting is known to be (head-)normalising for almost orthogonal rewrite systems. We study (head-)normalisation for the larger class of weakly orthogonal rewrite systems. (Infinitary) normalisation is established and a counterexample against head-normalisation is given.

Confluence by Decreasing Diagrams

The decreasing diagrams technique is a complete method to reduce confluence of a rewrite relation to local confluence. Whereas previous presentations have focussed on the proofthe technique is correct, here we focus on applicability. We present a simple but powerful generalisation of the technique, requiring peaks to be closed only by conversions instead of valleys, which is demonstrated to further ease applicability.

Decomposition orders: another generalisation of the fundamental theorem of arithmetic

We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition order. We apply our result to establish that the commutative monoid of weakly normed processes modulo bisimulation definable in ACPe with linear communication, with parallel composition as binary operation, has unique decomposition. We also apply our result to establish that the partial commutative monoid associated with a well-founded commutative residual algebra has unique decomposition.

Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems

We study perpetuality of reduction steps, as well as perpetuality of redexes, in orthogonal rewrite systems. A perpetual step is a reduction step which retains the possibility of infinite reductions. A perpetual redex is a redex which, when put into an arbitrary context, yields a perpetual step. We generalize and refine existing criteria for the perpetuality of reduction steps and redexes in orthogonal term rewriting systems and the ?-calculus due to Bergstra and Klop and others. We first introduce context-sensitive conditional expression reduction systems (CCERSs) and define a concept of orthogonality (which implies confluence) for them. In particular, several important ?-calculi and their extensions and restrictions can naturally be embedded into orthogonal CCERSs. We then define a perpetual reduction strategy which enables one to construct minimal (w.r.t. Levys permutation ordering on reductions) infinite reductions in orthogonal fully-extended CCERSs. Using the properties of the minimal perpetual strategy, we prove 1.perpetuality of any reduction step that does not erase potentially infinite arguments, which are arguments that may become, via substitution, infinite after a number of outside steps, and 2.perpetuality (in every context) of any safe redex, which is a redex whose substitution instances may discard infinite arguments only when the corresponding contracta remain infinite. We prove both these perpetuality criteria for orthogonal fully-extended CCERSs and then specialize and apply them to restricted ?-calculi, demonstrating their usefulness. In particular, we prove the equivalence of weak and strong normalization (whose equivalence is here called uniform normalization) for various restricted ?-calculi, most of which cannot be derived from previously known perpetuality criteria.

On equal μ -terms

Abstract We consider the rewrite system R μ with μ x . M → μ M [ x : = μ x . M ] as its single rewrite rule, where the signature consists of the variable binding operator commonly denoted by μ , first-order symbols, which in this paper are restricted to a binary function symbol F , and possibly some constant symbols. This kernel system denoting recursively defined objects occurs in several contexts, e.g. it is the framework of recursive types, with F as the function type constructor. For general signatures, this rewriting system is widely used to represent and manipulate infinite regular trees. The main concern of this paper is the convertibility relation for μ -terms as given by the μ -rule, in particular, its decidability. This relation is sometimes called weak μ -equality, in contrast with strong μ -equality, which is given by the equality of the possibly infinite tree unwinding of μ -terms. While strong equality has received much attention, the opposite is the case for weak μ -equality. We present three alternative proofs for decidability of weak μ -equality. The first two proofs build upon an ingenious proof method of Cardone and Coppo. Prior to that, we prepare the way by an analysis of α -conversion. We then give a decidability proof in an ‘ α -free’ manner, essentially treating μ -terms as first-order terms, and next, a proof in higher-order style, employing α -equivalence classes and viewing R μ as a higher-order rewriting system. The third decidability proof is also derived in an α -free manner, exploiting the regular nature of the set of μ -reducts, enabling an appeal to the theory of tree automata. We conclude with additional results concerning decidability of reachability, and upward-joinability of μ -reduction, and of convertibility by α -free μ -reduction.

Diagrammatic Confluence and Completion

We give a new elegant proof that decreasing diagrams imply confluence based on a proof reduction technique, which is then the basis of a novel completion method which proof-reduction relation transforms arbitrary proofs into rewrite proofs even in presence of non-terminating reductions. Unlike previous methods, no ordering of the set of terms is required, but can be used if available. Unlike ordered completion, rewrite proofs are closed under instantiation. Examples are presented, including Kleenes and Huets classical examples showing that non-terminating local-confluent relations may not be confluent.

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.