Origin tracking in term rewriting

Abstract

The notion of descendants or residuals with its inverse notion of ancestors is classical in the theory of rewriting, both in first order term rewriting and in higher-order rewriting, such as lambda calculus. Recently this classical notion has been given much attention. On the one hand, the notion has been studied in an abstract, axiomatic way, in order to isolate the essential properties of the descendant concept. On the other hand descendants were studied in a very concrete way, inspired by practical considerations such as error recovery in program executions and program slicing.

In the latter endeavour the emphasis is on tracing back the symbols constituting an expression to their ’causes’ in an earlier expression in the rewrite sequence. This method is also known as origin tracking. The corresponding descendant notion is a refinement of the classical one.

In our talk we present some basic properties of origin tracking. It will be apparent that various labeled versions of the rewrite systems concerned are important to establish such properties.

We treat three (theoretical) applications of the origin tracking technique: first, a simplified proof of the classical theorem of Huet and Le'vy about needed reduction; second, another proof of the Genericity Lemma in pure lambda calculus; third, a simple proof of Berry's sequentially theorem for lambda calculus.

The third application actually takes place in the setting of infinitary lambda calculus, where Boehm trees are normal forms; and Berry's the-orem will easily generalize to lazy lambda trees (or Levy-Longo trees), and to Berarducci trees.