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Complexity of Fractran and productivity
 In Proceedings of the 22th Conference on Automated Deduction (CADE’09
, 2009
"... 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 prog ..."
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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 welldefined results is for programming with finite structures. Fractran is a simple Turingcomplete programming language invented by Conway. We prove that the question whether a Fractran program halts on all positive integers is Π 0 2complete. 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 2complete 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 firstorder logic. 1
Degrees of undecidability in term rewriting
 Proceedings of Computer 30 Logic (CSL09), volume 5771 of Lecture Notes in Computer Science
, 2009
"... Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy cl ..."
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Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Most of the standard properties are Π 0 2complete, that is, of the same level as uniform halting of Turing machines. In this paper we show two exceptions. Weak confluence is Σ 0 1complete, and therefore essentially easier than ground weak confluence which is Π 0 2complete. The most surprising result is on dependency pair problems: we prove this to be Π 1 1complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. A minor variant, dependency pair problems with minimality flag, turns out be Π 0 2complete again, just like the original termination problem for which dependency pair analysis was developed. 1
PROVING PRODUCTIVITY IN INFINITE DATA STRUCTURES
"... Abstract. For a general class of infinite data structures including streams, binary trees, and the combination of finite and infinite lists, we investigate the notion of productivity. This generalizes stream productivity. We develop a general technique to prove productivity based on proving context ..."
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Abstract. For a general class of infinite data structures including streams, binary trees, and the combination of finite and infinite lists, we investigate the notion of productivity. This generalizes stream productivity. We develop a general technique to prove productivity based on proving contextsensitive termination, by which the power of present termination tools can be exploited. In order to treat cases where the approach does not apply directly, we develop transformations extending the power of the basic approach. We present a tool combining these ingredients that can prove productivity of a wide range of examples fully automatically. 1.
TuringCompleteness of Polymorphic Stream Equation Systems
"... Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a sys ..."
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Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a system of equations using only stream constructors and destructors and recursive calls. Using methods reminiscent of prior results in the field, we first show this class consists of exactly the computable polymorphic stream functions. Using much more intricate techniques, our main result states this holds true even for unary equations free of mutual recursion, yielding an elegant model of Turingcompleteness in a severely restricted environment and allowing us to recover previous complexity results in a much more restricted setting.