Dimitri Hendriks

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}

On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic

-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

Circular coinduction in coq using bisimulation-up-to techniques

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

Highlights in infinitary rewriting and lambda calculus

-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.

A Coinductive Framework for Infinitary Rewriting and Equational Reasoning

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.

Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting

We propose to combine interactive proof construction with proof automation for a fragment of first-order logic called Coherent Logic (CL). CL allows enough existential quantification to make Skolemization unnecessary. Moreover, CL has a constructive proof system based on forward reasoning, which is easy to automate and where standardized proof objects can easily be obtained. We have implemented in Prolog a CL prover which generates Coq proof scripts. We test our approach with a case study: Hessenberg’s Theorem, which states that in elementary projective plane geometry Pappus’ Axiom implies Desargues’ Axiom. Our CL prover makes it possible to automate large parts of the proof, in particular taking care of the large number of degenerate cases.

The Degree of Squares is an Atom

We investigate methods for proving equality of infinite objects using circular coinduction, a combination of coinduction with term rewriting, in the Coq proof assistant. In order to ensure productivity, Coq requires the corecursive construction of infinite objects to be guarded. However, guardedness forms a severe confinement for defining infinite objects, and this includes coinductive proof terms. In particular, circular coinduction is troublesome in Coq, since rewriting usually obstructs guardedness. Typically, applications of transitivity are in between the guard and the coinduction hypothesis. Other problems concern the use of lemmas, and rewriting under causal contexts. We show that the method of bisimulation-up-to allows for an elegant rendering of circular coinduction, and we use this to overcome the troubles with guardedness.

Proof Reflection in Coq

We present some highlights from the emerging theory of infinitary rewriting, both for first-order term rewriting systems and @l-calculus. In the first section we introduce the framework of infinitary rewriting for first-order rewrite systems, so without bound variables. We present a recent observation concerning the continuity of infinitary rewriting. In the second section we present an excursion to the infinitary @l-calculus. After the main definitions, we mention a recent observation about infinite looping @l-terms, that is, terms that reduce in one step to themselves. Next we describe the fundamental trichotomy in the semantics of @l-calculus: Bohm trees, Levy-Longo trees, and Berarducci trees. We conclude with a short description of a new refinement of Bohm tree semantics, called clocked semantics.

From Outermost to Context-Sensitive Rewriting

We present a coinductive framework for defining infinitary analogues of equational reasoning and rewriting in a uniform way. The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.