We introduce a simple functional language foetus (lambda calculus with tuples, constructors and pattern matching) supplied with a termination checker. This checker tries to find a well-founded structural order on the parameters on the given function to prove termination. The components of the check algorithm are: function call extraction out of the program text, call graph completion and finding a lexical order for the function parameters. The HTML version of this paper contains many ready-to-run Web-based examples.

Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finitepath) trees of a certain kind. The underlying idea can be traced back to Brouwer’s justification of bar-induction, or to Kreisel and Troelstra’s elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity.

The coinductive -calculus co arises by a coinductive interpretation of the grammar of the standard -calculus and contains non-wellfounded -terms. An appropriate notion of reduction is analyzed and proven to be conuent by means of a detailed analysis of the usual Tait/MartinL of style development argument. This yields bounds for the lengths of those joining reduction sequences that are guaranteed to exist by conuence. These bounds also apply to the wellfounded -calculus.

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

In previous papers we have proposed an elementary discipline of strong functional programming (ESFP), in which all computations terminate. A key feature of the discipline is that we introduce a type distinction between data which is known to be finite, and codata which is (potentially) infinite. To ensure termination, recursion over data must be wellfounded, and corecursion (the definition schema for codata) must be productive, and both of these restrictions must be enforced automatically by the compiler. In our previous work we used abstract interpretation to establish the productivity of corecursive definitions in an elementary strong functional language. We show here that similar ideas can be applied in the dual case to check whether recursive function definitions are strongly normalising. We thus exhibit a powerful termination analysis technique which we demonstrate can be extended to partial functions.

In the calculus of possibly non-wellfounded -terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluence statement. Similarly,

Vol. 5 (3:9) 2009, pp. 1–17 www.lmcs-online.org