{ rjmh, pareto, sabry We have designed and implemented a type-based analysis for proving some baaic properties of reactive systems. The analysis manipulates rich type expressions that contain in-formation about the sizes of recursively defined data struc-tures. Sized types are useful for detecting deadlocks, non-termination, and other errors in embedded programs. To establish the soundness of the analysis we have developed an appropriate semantic model of sized types. 1 Embedded Functional Programs In a reactive system, the control software must continu-ously react to inputs from the environment. We distin-guish a class of systems where the embedded programs can be naturally expressed as functional programs manipulat-ing streams. This class of programs appears to be large enough for many purposes [2] and is the core of more ex-pressive formalisms that accommodate asynchronous events, non-determinism, etc. The fundamental criterion for the correctness of pro-grams embedded in reactive systems is Jwene.ss. Indeed, before considering the properties of the output, we must en-sure that there is some output in the first place: the program must continuous] y react to the input streams by producing elements on the output streams. This latter property may fail in various ways: e the computation of a stream element may depend on itself creating a “black hole, ” or e the computation of one of the output streams may demand elements from some input stream at different rates, which requires unbounded buffering, or o the computation of a stream element may exhaust the physical resources of the machine or even diverge.

We introduce the lambda-coiteration schema for a distributive law lambda of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final F-coalgebra, generalising the basic coiteration schema as given by finality. The duals of primitive recursion and course-of-value iteration, which are known extensions of coiteration, arise as instances of our framework. One can furthermore obtain schemata justifying recursive specifications that involve operators such as addition of power series, regular operators on languages, or parallel and sequential composition of processes. Next...

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.

This paper describes the language 8 1/2 , an embedding of data-parallelism in a declarative framework.

81/2 is a data-parallel language that relies on the notions of stream and collection in a high-level declarative framework. We describe in this research report semantics and compilation of sequential streams of collections for this language. Firstly, a denotational semantics is associated with recursively defined sequential 81/2 streams. Furthermore, we explain how the fixed point calculus corresponding to the foregoing semantics is implemented. Secondly, we describe an effective code generation scheme targetted towards either sequential, vector or SIMD architectures. Then we present four optimization processes for the generated code: the sharing of common control expressions, the optimization of delay copies, the loop fusion and the concatenation optimization. Next, some elements for the evaluation of the generated code are given. As a conclusion, we recall the overall effectiveness of the stream compilation and draw the future work. Key-words: stream, compilation of data-flow graphs...

Abstract. Most of the standard pleasant properties of term rewriting systems are undecidable; to wit: local confluence, confluence, normalization, termination, and completeness. Mere undecidability is insufficient to rule out a number of possibly useful properties: For instance, if the set of normalizing term rewriting systems were recursively enumerable, there would be a program yielding “yes ” in finite time if applied to any normalizing term rewriting system. The contribution of this paper is to show (the uniform version of) each member of the list of properties above (as well as the property of being a productive specification of a stream) complete for the class Π 0 2. Thus, there is neither a program that can enumerate the set of rewriting systems enjoying any one of the properties, nor is there a program enumerating the set of systems that do not. For normalization and termination we show both the ordinary version and the ground versions (where rules may contain variables, but only

Abstract. In functional programming languages the use of infinite structures is common practice. For total correctness of programs dealing with infinite structures one must guarantee that every finite part of the result can be evaluated in finitely many steps. This is known as productivity. For programming with infinite structures, productivity is what termination in well-defined results is for programming with finite structures. Fractran is a simple Turing-complete programming language invented by Conway. We prove that the question whether a Fractran program halts on all positive integers is Π 0 2-complete. In functional programming, productivity typically is a property of individual terms with respect to the inbuilt evaluation strategy. By encoding Fractran programs as specifications of infinite lists, we establish that this notion of productivity is Π 0 2-complete even for some of the most simple specifications. Therefore it is harder than termination of individual terms. In addition, we explore generalisations of the notion of productivity, and prove that their computational complexity is in the analytical hierarchy, thus exceeding the expressive power of first-order logic. 1

Recursive definition of streams (infinite lists of values) have been proposed as a fundamental programming structure in various fields. A problem is to turn such expressive recursive definitions into an efficient imperative code for their evaluation. One of the main approach is to restrict the stream expressions to interpret them as a temporal sequence of values. Such sequential stream rely on a clock analysis to decide at what time a new stream value must be produced. In this paper we present a denotational semantics of recursively defined sequential streams. We show how an efficient implementation can be derived as guarded statements wrapped into a single imperative loop.

ing out and studying those patterns of computation as useful objects in their own right leads to further insights into the nature of computation. The list operators studied later in these notes follow this approach. ffl Functional programs are usually an order of magnitude more concise than their imperative counterparts. Besides being shorter, they can be much more readable. The functional community like to cite studies that show the number of bugs per line is more or less constant, independent of the level of the language in use. Higher-level languages encode more concept per line, and therefore have relatively fewer bugs. ffl Functional programs are often more akin to formal specifications than their conventional counterparts. A good notation goes a long way towards solving the problem[47]. ffl Lazy evaluation permits a new approach to some algorithms. It is a simple but powerful idea that can remove the need for explicit backtracking, and can allow the programmer to manipulate in...

8 ½, an experimental language combining features of collection and stream oriented languages...