We are concerned with demonstrating productivity of specifications of infinite streams of data, based on orthogonal rewrite rules. In general, this property is undecidable, but for restricted formats computable sufficient conditions can be obtained. The usual analysis, also adopted here, disregards the identity of data, thus leading to approaches that we call data-oblivious. We present a method that is provably optimal among all such data-oblivious approaches. This means that in order to improve on our algorithm one has to proceed in a data-aware fashion. 3

We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called 'productive' if it can be evaluated continuously in such a way that a uniquely determined stream is obtained as the limit. Whereas productivity is undecidable for stream definitions in general, we show that it can be decided for 'pure' stream definitions. For every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements, called 'pebbles', in a finite 'pebbleflow net(work)'. And the production of a pebbleflow net associated with a pure stream definition, that is, the amount of pebbles the net is able to produce at its output port, can be calculated by reducing nets to trivial nets. This research has been partially funded by the Netherlands Organisation for Scientific Research (NWO) under FOCUS/BRICKS grant number 642.000.502.

a b s t r a c tWe present a procedure for transforming strongly sequential constructor-based term rewriting systems (TRSs) into context-sensitive TRSs in such a way that productivity of the input system is equivalent to termination of the output system. Thereby automated termination provers become available for proving productivity. A TRS is called productive if all its finite ground terms are constructor normalizing, and all 'inductive constructor paths' through the resulting (possibly non-wellfounded) constructor normal form are finite. To our knowledge, this is the first complete transformation from productivity to termination.The transformation proceeds in two steps: (i) The strongly sequential TRS is converted into a shallow TRS, where patterns do not have nested constructors. (ii) The shallow TRS is transformed into a context-sensitive TRS, where rewriting below constructors and in arguments not 'consumed from' is disallowed.Furthermore, we show how lazy evaluation can be encoded by strong sequentiality, thus extending our transformation to, e.g., Haskell programs.Finally, we present a simple, but fruitful extension of matrix interpretations to make them applicable for proving termination of context-sensitive TRSs.

We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called 'productive' if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit. Whereas productivity is undecidable for stream definitions in general, we show that it can be decided for 'pure' stream definitions. For every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements, called 'pebbles', in a finite 'pebbleflow net(work)'. And the production of a pebbleflow net associated with a pure stream definition, that is, the amount of pebbles the net is able to produce at its output port, can be calculated by reducing nets to trivial nets.

We answer an open question in the theory of degrees of infinite sequences with respect to transducibility by finite-state transducers. An initial study of this partial order of degrees was carried out in [5], but many basic questions remain unanswered. One of the central questions concerns the existence of atom degrees, other than the degree of the 'identity sequence' 10 0 10 1 10 2 10 3 • • •. A degree is called an 'atom' if below it there is only the bottom degree 0, which consists of the ultimately periodic sequences. We show that also the degree of the 'squares sequence' 10 0 10 1 10 4 10 9 10 16 • • • is an atom. As the main tool for this result we characterise the transducts of 'spiralling' sequences and their degrees. We use this to show that every transduct of a 'polynomial sequence' either is in 0 or can be transduced back to a polynomial sequence for a polynomial of the same order.