Joerg Endrullis

Complexity of Fractran and Productivity

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

Data-Oblivious Stream Productivity

{\rm \Pi}^{0}_{2}

Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting

-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

Proving equality of streams automatically

{\rm \Pi}^{0}_{2}

On the complexity of equivalence of specifications of infinite objects

-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.

On periodically iterated morphisms

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.

Automatic Sequences and Zip-Specifications

We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property (UNinf) fails by a simple example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that UNinf also fails for the infinitary lambda-beta-eta-calculus. As positive results we obtain the following: Infinitary confluence, and hence UNinf, holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we consider the triangle and diamond properties for infinitary developments in weakly orthogonal TRSs, by refining an earlier cluster-analysis for the finite case.

A coinductive treatment of infinitary rewriting

Streams are infinite sequences over a given data type. A stream specification is a set of equations intended to define a stream. In this paper we focus on equality of streams, more precisely, for a given set of equations two stream terms are said to be equal if they are equal in every model satisfying the given equations. We investigate techniques for proving equality of streams suitable for automation. Apart from techniques that were already available in the tool CIRC from Lucanu and Rosu, we also exploit well-definedness of streams, typically proved by proving productivity. Moreover, our approach does not restrict to behavioral input format and does not require termination. We present a tool Streambox that can prove equality of a wide range of examples fully automatically.

Let's Make a Difference!

We study the complexity of deciding the equality of infinite objects specified by systems of equations, and of infinite objects specified by λ-terms. For equational specifications there are several natural notions of equality: equality in all models, equality of the sets of solutions, and equality of normal forms for productive specifications. For λ-terms we investigate Böhm-tree equality and various notions of observational equality. We pinpoint the complexity of each of these notions in the arithmetical or analytical hierarchy. We show that the complexity of deciding equality in all models subsumes the entire analytical hierarchy. This holds already for the most simple infinite objects, viz. streams over {0,1}, and stands in sharp contrast to the low arithmetical ϖ02-completeness of equality of equationally specified streams derived in [17] employing a different notion of equality.

Arithmetic Self-Similarity of Infinite Sequences.

We investigate the computational power of periodically iterated morphisms, also known as D0L systems with periodic control; we call them PerD0L systems for short. These systems give rise to a class of one-sided infinite sequences, called PerD0L words. We construct a PerD0L word with exponential subword complexity, thereby answering a question raised by Lepistö [23] on the existence of such words. We solve another open problem concerning the decidability of the first-order theories of PerD0L words [24]; we show it is already undecidable whether a certain letter occurs in a PerD0L word.