Origin Tracking for Higher-order Term Rewriting Systems 1 Origin Tracking

Origin Tracking is a technique which, in the framework of rst-order term rewriting systems, establishes relations between each subterm t of a normal form and a set of subterms, the origins of t, in the initial term. Origin tracking is based on the notion of residuals. It has been used successfully for the generation of error handlers and de-buggers from algebraic speciications of programming languages. Recent experiments with the use of higher-order algebraic speciications for the deenition of programming languages, revealed a need to extend origin tracking to higher-order term rewriting systems. This extension is discussed, covering a deenition and some alternatives, as well as an assessment with respect to existing speciications. When algebraic speciications are being executed as term rewriting systems (TRSs), computations are performed by reducing an initial term to its result value | its normal form. Often, it is enough just to compute this result value, but in many cases it can be useful to have some more information. For instance, one may wish to know how the initial term innuenced the normal form; are there perhaps parts of the initial term that were copied without a change to the result term? Or, if a subterm of the normal form does not literally recur in the initial term, can it be possible to identify a set of subterms in the initial term which in some sense were responsible for its the creation? Trying to capture how intermediate and nal terms originate from the initial term is formalized in a notion called origin tracking Ber91, Ber92, DKT93]. Origin tracking is based on so-called residuals, which have been used successfully in more theoretically oriented papers HL91, Mar91] for reasoning about optimal reduction strategies in TRSs. 1.1 Applications Our motivation to work on origin tracking was that we needed it for the automatic generation of tools from algebraic speciications of programming languages. As an example,