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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
Levels of Undecidability in Rewriting
, 2011
"... 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 classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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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 classification 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 quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
On Dehn functions of finitely presented biautomatic monoids
 J. Autom. Lang. Comb
, 1998
"... For each automatic monoid the word problem can be solved in quadratic time (Campbell et al 1996), and hence, the Dehn function of a finitely presented automatic monoid is recursive. Here we show that this result on the Dehn function cannot be improved in general by presenting finitely presented bia ..."
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For each automatic monoid the word problem can be solved in quadratic time (Campbell et al 1996), and hence, the Dehn function of a finitely presented automatic monoid is recursive. Here we show that this result on the Dehn function cannot be improved in general by presenting finitely presented biautomatic monoids the Dehn functions of which realize arbitrary complexity classes that are sufficiently rich. Keywords: automatic monoid, stringrewriting system, derivation, Dehn function, recursive function, sufficiently rich complexity class. 1 Introduction It is wellknown that in general it is not possible to extract much information on the algebraic structure of a monoid or a group from a given finite presentation for that monoid or group. On the other hand various methods have been developed that can be applied successfully in certain restricted instances. One of the classical methods is the ToddCoxeter method of enumerating the cosets of a group G with respect to a finitely gener...