Orthogonal Term Rewriting Systems Research Articles

Infinitary Term Rewriting for Weakly Orthogonal Systems: Properties and Counterexamples

We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property fails by an example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that this property also fails for the infinitary lambda-beta-eta-calculus. As positive results we obtain the following: Infinitary confluence, and hence the infinitary unique normal forms property, holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we establish the triangle and diamond properties for infinitary multi-steps (complete developments) in weakly orthogonal TRSs, by refining an earlier cluster-analysis for the finite case.

Formalizing the Confluence of Orthogonal Rewriting Systems

Orthogonality is a discipline of programming that in a syntactic manner guarantees determinism of functional specifications. Essentially, orthogonality avoids, on the one side, the inherent ambiguity of non determinism, prohibiting the existence of different rules that specify the same function and that may apply simultaneously (non-ambiguity), and, on the other side, it eliminates the possibility of occurrence of repetitions of variables in the left-hand side of these rules (left linearity). In the theory of term rewriting systems (TRSs) determinism is captured by the well-known property of confluence, that basically states that whenever different computations or simplifications from a term are possible, the computed answers should coincide. Although the proofs are technically elaborated, confluence is well-known to be a consequence of orthogonality. Thus, orthogonality is an important mathematical discipline intrinsic to the specification of recursive functions that is naturally applied in functional programming and specification. Starting from a formalization of the theory of TRSs in the proof assistant PVS, this work describes how confluence of orthogonal TRSs has been formalized, based on axiomatizations of properties of rules, positions and substitutions involved in parallel steps of reduction, in this proof assistant. Proofs for some similar but restricted properties such as the property of confluence of non-ambiguous and (left and right) linear TRSs have been fully formalized.

On Constructor Rewrite Systems and the Lambda Calculus

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by- value beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in a term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.

An Abstract Concept of Optimal Implementation

In previous works, we introduced Stable Deterministic Residual Structures (SDRSs), Abstract Reduction Systems with an axiomatized residual relation which model orthogonal term and graph rewriting systems, and Deterministic Family Structures (DFSs), which add an axiomatized notion of redex-family to capture known sharing concepts in the λ-calculus and other orthogonal rewrite systems. In this paper, we introduce and study a concept of implementation of DFSs. We show that for any DFS F, its implementation FI is a non-duplicating DFSs with zig-zag as the family relation, where zig-zag is simply the symmetric and transitive closure of the residual relation on redexes with histories. Further, we show that sharing is compositional: the sharing in a DFS F can be decomposed into a weaker sharing F' (such as zig-zag) and a sharing in the implementation F'I of F' stronger than zig-zag. These results require study of the family relation in non-duplicating SDRSs. We show that zig-zag forms a family-relation in every non-duplicating SDRS, and that it is the only separable family relation in such SDRSs.

A sequential reduction strategy

Kennaway proved the remarkable result that every (almost) orthogonal term rewriting system admits a computable sequential normalizing reduction strategy. In this paper we present a computable sequential reduction strategy similar in scope, but simpler and more general. Our strategy can be thought of as an outermost-fair-like strategy that is allowed to be unfair to some redex of a term when contracting the redex is useless for the normalization of the term. Unlike the strategy of Kennaway, our strategy does not rely on syntactic restrictions that imply confluence. On the contrary, it can easily be applied to any term rewriting system, and we show that the class of term rewriting systems for which our strategy is normalizing properly includes all (almost) orthogonal systems. Our strategy is more versatile; in case of (almost) orthogonal term rewriting systems, it can be used to detect certain cases of non-termination. Our normalization proof is more accessible than Kennaway's. We also show that our sequential strategy sometimes succeeds where the parallel-outermost strategy fails.

Infinitary Rewriting and Cyclic Graphs

Infinitary rewriting allows infinitely large terms and infinitely long reduction sequences. There are two computational motivations for studying these: the infinite data structures implicit in lazy functional programming, and the use of rewriting of possibly cyclic graphs as an implementation technique for functional languages.We survey the fundamental properties of infinitary rewriting in orthogonal term rewrite systems, and its relation to cyclic graph rewriting.

Sequentiality in orthogonal term rewriting systems

For orthogonal term rewriting systems Q. Huet and J.-J. Lévy have introduced the property of ‘strong sequentiality’. A strongly sequential orthogonal term rewriting system admits an efficiently computable normalizing one-step reduction strategy. As shown by Huet and Lévy, strong sequentiality is a decidable property. In this paper we present an alternative analysis of strongly sequential term rewriting systems, leading to two simplified proofs of the decidability of this property. We also compare some related notions of sequentiality that recently have been proposed.