Jörg Endrullis

Matrix Interpretations for Proving Termination of Term Rewriting

We present a new method for automatically proving termination of term rewriting. It is based on the well-known idea of interpretation of terms where every rewrite step causes a decrease, but instead of the usual natural numbers we use vectors of natural numbers, ordered by a particular nontotal well-founded ordering. Function symbols are interpreted by linear mappings represented by matrices. This method allows us to prove termination and relative termination. A modification of the latter, in which strict steps are only allowed at the top, turns out to be helpful in combination with the dependency pair transformation. By bounding the dimension and the matrix coefficients, the search problem becomes finite. Our implementation transforms it to a Boolean satisfiability problem (SAT), to be solved by a state-of-the-art SAT solver.

Matrix interpretations for proving termination of term rewriting

We present a new method for automatically proving termination of term rewriting. It is based on the well-known idea of interpretation of terms where every rewrite step causes a decrease, but instead of the usual natural numbers we use vectors of natural numbers, ordered by a particular non-total well-founded ordering. Function symbols are interpreted by linear mappings represented by matrices. This method allows to prove termination and relative termination. A modification of the latter in which strict steps are only allowed at the top, turns out to be helpful in combination with the dependency pair transformation. By bounding the dimension and the matrix coefficients, the search problem becomes finite. Our implementation transforms it to a Boolean satisfiability problem (SAT), to be solved by a state-of-the-art SAT solver. Our implementation performs well on the Termination Problem Data Base: better than 5 out of 6 tools that participated in the 2005 termination competition in the category of term rewriting.

Automating the Mean-Field Method for Large Dynamic Gossip Networks

We investigate an abstraction method, called mean-field method, for the performance evaluation of dynamic networks with pairwise communication between nodes. It allows us to evaluate systems with very large numbers of nodes, that is, systems of a size where traditional performance evaluation methods fall short. While the mean-field analysis is well-established in epidemics and for chemical reaction systems, it is rarely used for communication networks because a mean-field model tends to abstract away the underlying topology. To represent topological information, however, we extend the mean-field analysis with the concept of classes of states. At the abstraction level of classes we define the network topology by means of connectivity between nodes. This enables us to encode physical node positions and model dynamic networks by allowing nodes to change their class membership whenever they make a local state transition. Based on these extensions, we derive and implement algorithms for automating a mean-field based performance evaluation.

Degrees of undecidability in term rewriting

Undecidability of various properties of first order term rewriting systems is well-known. 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 Π20 -complete, that is, of the same level as uniform halting of Turing machines. In this paper we show two exceptions. Weak confluence is Σ10- complete, and therefore essentially easier than ground weak confluence which is Π20-complete. The most surprising result is on dependency pair problems: we prove this to be Π11-complete, 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 Π20-complete again, just like the original termination problem for which dependency pair analysis was developed.

Infinitary Rewriting Coinductively

We provide a coinductive definition of strongly convergent reductions between infinite lambda terms. This approach avoids the notions of ordinals and metric convergence which have appeared in the earlier definitions of the concept. As an illustration, we prove the existence part of the infinitary standardization theorem. The proof is fully formalized in Coq using coinductive types. The paper concludes with a characterization of infinite lambda terms which reduce to themselves in a single beta step.

Circular coinduction in coq using bisimulation-up-to techniques

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.

Highlights in infinitary rewriting and lambda calculus

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.

Local Termination

The characterization of termination using well-founded monotone algebras has been a milestone on the way to automated termination techniques, of which we have seen an extensive development over the past years. Both the semantic characterization and most known termination methods are concerned with global termination, uniformly of all the terms of a term rewriting system (TRS). In this paper we consider local termination, of specific sets of terms within a given TRS. The principal goal of this paper is generalizing the semantic characterization of global termination to local termination. This is made possible by admitting the well-founded monotone algebras to be partial. We show that our results can be applied in the development of techniques for proving local termination. We give several examples, among which a verifiable characterization of the terminating S -terms in CL.

A Coinductive Framework for Infinitary Rewriting and Equational Reasoning

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.

The Degree of Squares is an Atom

We answer an open question in the theory of degrees of infinite sequences with respect to transducibilityby finite-state transducers. An initial study of this partial order of degrees was carried out in [1], but many basic questions remain unanswered.One of the central questions concerns the existence of atom degrees, other than the degree of the ‘identity sequence’ \(1 0^0 1 0^1 1 0^2 1 0^3 \cdots \). A degree is called an ‘atom’ if below it there is only the bottom degree \(\varvec{0}\), which consists of the ultimately periodic sequences. We show that also the degree of the ‘squares sequence’ \(1 0^0 1 0^1 1 0^4 1 0^9 1 0^{16}\cdots \) is an atom.