# Journal Publications

## 2020

- Decreasing Diagrams for Confluence and CommutationJörg Endrullis, Jan Willem Klop, and Roy OverbeekLogical Methods in Computer Science , Volume 16, Issue 1 (2020)paper
# Summary

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy.

The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on?

Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse.

Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems.

This paper is an extended version of Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems (FSCD 2018).

See research for an overview of my research on confluence.

# Bibtex

@article{confluence:decreasing:diagrams:2020, title = {{Decreasing Diagrams for Confluence and Commutation}}, author = {Endrullis, J{\"{o}}rg and Klop, Jan Willem and Overbeek, Roy}, doi = {10.23638/LMCS-16(1:23)2020}, journal = {{Logical Methods in Computer Science}}, volume = {{Volume 16, Issue 1}}, year = {2020}, keywords = {rewriting, confluence}, type = {journal} }

# Digital Object Identifier

10.23638/LMCS-16(1:23)2020

- Transducer Degrees: Atoms, Infima and SupremaJörg Endrullis, Jan Willem Klop, and Rena BakhshiActa Informatica , 57 (3-5) , pp. 727–758 (2020)paper
# Summary

Although finite state transducers are very natural and simple devices, surprisingly little is known about the transducibility relation they induce on streams (infinite words). We collect some intriguing problems that have been unsolved since several years. The transducibility relation arising from finite state transduction induces a partial order of stream degrees, which we call Transducer degrees, analogous to the well-known Turing degrees or degrees of unsolvability.

We show that there are pairs of degrees without supremum and without infimum. The former result is somewhat surprising since every finite set of degrees has a supremum if we strengthen the machine model to Turing machines, but also if we weaken it to Mealy machines.

# Bibtex

@article{streams:degrees:suprema:2020, author = {Endrullis, J{\"{o}}rg and Klop, Jan Willem and Bakhshi, Rena}, title = {{Transducer Degrees: Atoms, Infima and Suprema}}, journal = {Acta Informatica}, volume = {57}, number = {3-5}, pages = {727--758}, year = {2020}, doi = {10.1007/s00236-019-00353-7}, keywords = {streams, degrees, automata}, type = {journal} }

# Digital Object Identifier

10.1007/s00236-019-00353-7

## 2019

- Braids via Term RewritingJörg Endrullis, and Jan Willem KlopTheoretical Computer Science , 777 , pp. 260–295 (2019)paper
# Summary

We present a brief introduction to braids, in particular simple positive braids, with a double emphasis: first, we focus on term rewriting techniques, in particular, reduction diagrams and decreasing diagrams. The second focus is our employment of the colored braid notation next to the more familiar Artin notation. Whereas the latter is a relative, position dependent, notation, the former is an absolute notation that seems more suitable for term rewriting techniques such as symbol tracing. Artin's equations translate in this notation to simple word inversions. With these points of departure we treat several basic properties of positive braids, in particular related to the word problem, confluence property, projection equivalence, and the congruence property. In our introduction the beautiful diamond known as the permutohedron plays a decisive role.

# Bibtex

@article{rewriting:braids:2019, author = {Endrullis, J{\"{o}}rg and Klop, Jan Willem}, title = {Braids via term rewriting}, journal = {Theoretical Computer Science}, volume = {777}, pages = {260--295}, year = {2019}, doi = {10.1016/j.tcs.2018.12.006}, keywords = {rewriting,confluence}, type = {journal} }

# Digital Object Identifier

10.1016/j.tcs.2018.12.006

- Syllogistic Logic with "Most"Jörg Endrullis, and Lawrence S. MossMathematical Structures in Computer Science , 29 (6) , pp. 763–782 (2019)paper
# Summary

This paper presents a sound and complete proof system for the logical system whose sentences are of the form

- All X are Y,
- Some X are Y, and
- Most X are Y

This paper is an extended version of

*Syllogistic Logic with "Most"*presented at the International Workshop on Logic, Language, Information, and Computation (WoLLIC 2015).# Bibtex

@article{logic:most:2019, author = {Endrullis, J\"{o}rg and Moss, Lawrence S.}, title = {{Syllogistic Logic with "Most"}}, journal = {Mathematical Structures in Computer Science}, volume = {29}, number = {6}, pages = {763--782}, year = {2019}, doi = {10.1017/S0960129518000312}, keywords = {logic}, type = {journal} }

# Digital Object Identifier

10.1017/S0960129518000312

## 2018

- Degrees of Infinite Words, Polynomials and AtomsJörg Endrullis, Juhani Karhumäki, Jan Willem Klop, and Aleksi SaarelaInternational Journal of Foundations of Computer Science , 29 (5) , pp. 825–843 (2018)paper
# Summary

A finite state transducer is a finite automaton that transforms input words into output words. The transducer reads the input letter by letter, in each step producing an output word and changing its state.

While finite state transducers are very simple and elegant devices, their power in transforming infinite words is hardly understood.

In this paper we show that

**techniques from continuous mathematics can be used to reason about finite state transducers**. To be precise, we use the following methods from linear algebra and analysis:- continuity,
- Vandermonde matrices,
- invertibility of matrices, and
- the generalised mean inequality.

The main result in this paper is the existance of an infinite number of atoms in the hierarchy of streams arising from finite state transduction.

This paper is an extended version of

*Degrees of Infinite Words, Polynomials and Atoms*presented at the Conference on Developments in Language Theory (DLT), 2016.See research for an introduction to finite state transducers, an overview of my research and many open questions.

# Bibtex

@article{streams:degrees:polynomials:2018, author = {Endrullis, J\"{o}rg and Karhum{\"{a}}ki, Juhani and Klop, Jan Willem and Saarela, Aleksi}, title = {{Degrees of Infinite Words, Polynomials and Atoms}}, journal = {International Journal of Foundations of Computer Science}, volume = {29}, number = {5}, pages = {825--843}, year = {2018}, doi = {10.1142/S0129054118420066}, keywords = {streams, degrees, automata}, type = {journal} }

# Digital Object Identifier

10.1142/S0129054118420066

- Coinductive Foundations of Infinitary Rewriting and Infinitary Equational LogicJörg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, and Alexandra SilvaLogical Methods in Computer Science , 14 (1) (2018)paper
# Summary

We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform, coinductive way.

The framework lends itself to an elegant and concise definition of the infinitary rewrite relation \( \to^\infty \) in terms of the single step relation \( \to \): \[ {\to^\infty} \,=\, \mu R. \nu S. ( \to \cup \mathrel{\overline{R}} )^* \mathrel{;} \overline{S} \] Here \( \mu \) and \( \nu \) are the least and greatest fixed-point operators, respectively, and \[ \overline{R} \,=\, \{\, (\, f(s_1,\ldots,s_n),\, \,f(t_1,\ldots,t_n) \,) \mid f \in \Sigma,\, s_1\! \mathrel{R} t_1,\ldots,s_n\! \mathrel{R} t_n \,\} \cup \text{Id} \] 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 suitable for formalizations in theorem provers. To wit, we provide the first formalization of the compression lemma in Coq.

This paper is an extended version of A Coinductive Framework for Infinitary Rewriting and Equational Reasoning (RTA 2015). We build on ideas in Infinitary Rewriting Coinductively (TYPES 2012) giving a coinductive perspective on infinitary lambda calculus. We extend these ideas to rewrite sequences beyond length omega by mixing induction and coinduction (least and greatest fixed-points).

# Bibtex

@article{infintary:rewriting:coinductive:2018, author = {Endrullis, J\"{o}rg and Hansen, Helle Hvid and Hendriks, Dimitri and Polonsky, Andrew and Silva, Alexandra}, title = {{Coinductive Foundations of Infinitary Rewriting and Infinitary Equational Logic}}, journal = {Logical Methods in Computer Science}, volume = {14}, number = {1}, year = {2018}, doi = {10.23638/LMCS-14(1:3)2018}, keywords = {rewriting, infinitary rewriting, coinduction}, type = {journal} }

# Digital Object Identifier

10.23638/LMCS-14(1:3)2018

## 2017

- Clocked Lambda CalculusJörg Endrullis, Dimitri Hendriks, Jan Willem Klop, and Andrew PolonskyMathematical Structures in Computer Science , 27 (5) , pp. 782–806 (2017)paper
# Summary

We introduce the

*clocked lambda calculus*, an extension of the classical lambda calculus with a unary symbol \( \tau \) that serves as a witness of \( \beta \)-steps. This calculus consists of the following two rules: \[ \begin{aligned} (\lambda x.M) N &\to \tau( M[x = N] ) \\ \tau(M)N &\to \tau(MN) \end{aligned} \] The clocked lambda-calculus is infinitary strongly normalizing, infinitary confluent, and the unique infinitary normal forms constitute enriched Böhm trees (more precisely, Lévy-Longo trees), which we call*clocked Böhm trees*. We show that clocked Böhm trees are suitable for discriminating a rich class of lambda terms having the same Böhm trees.See research for an overview of my research on the clocked lambda claculus and fixed-point combinators.

# Bibtex

@article{clocked:lambda:calculus:2017, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Klop, Jan Willem and Polonsky, Andrew}, title = {{Clocked Lambda Calculus}}, journal = {Mathematical Structures in Computer Science}, volume = {27}, number = {5}, pages = {782--806}, year = {2017}, doi = {10.1017/S0960129515000389}, keywords = {rewriting, infinitary rewriting, lambda calculus}, type = {journal} }

# Digital Object Identifier

10.1017/S0960129515000389

## 2016

- Majority DigraphsTri Lai, Jörg Endrullis, and Lawrence S. MossProceedings of the American Mathematical Society , 144 (9) , pp. 3701–3715 (2016)paper
# Summary

A majority digraph is a finite simple digraph \( G = (V,\to) \) such that there exist finite sets \( A_v \) for the vertices \( v \in V \) with the following property: \( u \to v \) if and only if "more than half of the \( A_u \) are \( A_v \)". That is, \( u \to v \) if and only if \( | A_u \cap A_v | > \frac{1}{2} \cdot | A_u | \) . We characterize the majority digraphs as the digraphs with the property that every directed cycle has a reversal. If we change to any real number \( \alpha \in (0, 1) \), we obtain the same class of digraphs. We apply the characterization result to obtain a result on the logic of assertions "most X are Y" and the standard connectives of propositional logic.

# Bibtex

@article{logic:most:graphs:2016, author = {Lai, Tri and Endrullis, J{\"o}rg and Moss, {Lawrence S.}}, title = {Majority digraphs}, journal = {Proceedings of the American Mathematical Society}, publisher = {American Mathematical Society}, volume = {144}, number = {9}, pages = {3701--3715}, year = {2016}, doi = {10.1090/proc/13038}, keywords = {logic}, type = {journal} }

# Digital Object Identifier

10.1090/proc/13038

## 2014

- Discriminating Lambda-Terms Using Clocked Boehm TreesJörg Endrullis, Dimitri Hendriks, Jan Willem Klop, and Andrew PolonskyLogical Methods in Computer Science , 10 (2) (2014)paper
# Summary

As observed by Intrigila, there are hardly techniques available in the lambda calculus to prove that two lambda terms are not \( \beta \)-convertible. Techniques employing the usual Böhm trees are inadequate when we deal with terms having the same Böhm tree. This is the case in particular for fixed-point combinators, as they all have the same Böhm tree.

Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model \( P \omega \) of lambda calculus, and hence valid with respect to Böhm tree-equality, but nevertheless the terms are \( \beta \)-inconvertible.

To prove such beta-inconvertibilities, we refine the concept of Böhm trees: we introduce

*clocked Böhm trees's*with annotations that convey information of the tempo in which the Böhm trees are produced. Böhm trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for lambda terms. An analogous approach pertains to Levy-Longo trees and Berarducci trees.We illustrate applicability of our refined Böhm trees at the following examples:

- We show how to \( \beta \)-discriminate a large number of fixed-point combinators.
- We answer a question of Gordon Plotkin:
*Is there a fixed point combinator \( Y \) such that \[ Y ( \lambda z. f zz ) =_\beta Y ( \lambda x. Y ( \lambda y. f xy )) \]*

*atomic clocks*.This paper is an extended version of

*Modular Construction of Fixed Point Combinators and Clocked Böhm Trees*(LICS 2010).See research for an overview of my research on the clocked lambda claculus and fixed-point combinators.

# Bibtex

@article{lambda:clocks:2014, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Klop, Jan Willem and Polonsky, Andrew}, title = {{Discriminating Lambda-Terms Using Clocked Boehm Trees}}, journal = {Logical Methods in Computer Science}, volume = {10}, number = {2}, year = {2014}, doi = {10.2168/LMCS-10(2:4)2014}, keywords = {rewriting, infinitary rewriting, lambda calculus}, type = {journal} }

# Digital Object Identifier

10.2168/LMCS-10(2:4)2014

- Infinitary Term Rewriting for Weakly Orthogonal Systems: Properties and CounterexamplesJörg Endrullis, Clemens Grabmayer, Dimitri Hendriks, Jan Willem Klop, and Vincent van OostromLogical Methods in Computer Science , 10 (2:7) , pp. 1–33 (2014)paper
# Bibtex

@article{infinitary:weakly:orthogonal:2014, author = {Endrullis, J\"{o}rg and Grabmayer, Clemens and Hendriks, Dimitri and Klop, Jan Willem and van~Oostrom, Vincent}, title = {{Infinitary Term Rewriting for Weakly Orthogonal Systems: Properties and Counterexamples}}, journal = {Logical Methods in Computer Science}, volume = {10}, number = {2:7}, pages = {1--33}, year = {2014}, doi = {10.2168/LMCS-10(2:7)2014}, keywords = {rewriting, infinitary rewriting, lambda calculus}, type = {journal} }

# Digital Object Identifier

10.2168/LMCS-10(2:7)2014

- On the Complexity of Stream EqualityJörg Endrullis, Dimitri Hendriks, Rena Bakhshi, and Grigore RosuJournal of Functional Programming , 24 (2-3) , pp. 166–217 (2014)paper
# Bibtex

@article{complexity:stream:equality:2014, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Bakhshi, Rena and Rosu, Grigore}, title = {{On the Complexity of Stream Equality}}, journal = {Journal of Functional Programming}, volume = {24}, number = {2-3}, pages = {166--217}, year = {2014}, doi = {10.1017/S0956796813000324}, keywords = {rewriting, undecidability, streams}, type = {journal} }

# Digital Object Identifier

10.1017/S0956796813000324

## 2013

- De Bruijn’s Weak Diamond Property RevisitedJörg Endrullis, and Jan Willem KlopIndagationes Mathematicae , 24 (4) , pp. 1050–1072 (2013)paper
# Summary

In this paper we revisit an unpublished but influential technical report from 1978 by N.G. de Bruijn, written in the framework of the Automath project. This report describes a technique for proving confluence of abstract reduction systems, called the weak diamond property. It paved the way for the powerful technique developed by Van Oostrom to prove confluence of abstract reduction systems, called decreasing diagrams.

We first revisit in detail De Bruijn’s old proof, providing a few corrections and hints for understanding. We find that this original criterion and proof technique are still worthwhile. Next, we establish that De Bruijn’s confluence criterion can be used to derive the decreasing diagrams theorem (the reverse was already known). We also provide a short proof of decreasing diagrams in the spirit of De Bruijn. We finally address the issue of completeness of this method.

See research for an overview of my research on confluence.

# Bibtex

@article{confluence:weak:diamond:2013, author = {Endrullis, J\"{o}rg and Klop, Jan Willem}, title = {{De Bruijn’s Weak Diamond Property Revisited}}, journal = {Indagationes Mathematicae}, volume = {24}, number = {4}, pages = {1050 -- 1072}, year = {2013}, doi = {10.1016/j.indag.2013.08.005}, note = {In memory of N.G. (Dick) de Bruijn (1918–2012)}, keywords = {rewriting, confluence}, type = {journal} }

# Digital Object Identifier

10.1016/j.indag.2013.08.005

- Clocks for Functional ProgramsJörg Endrullis, Dimitri Hendriks, Jan Willem Klop, and Andrew PolonskyIn: The Beauty of Functional Code - Essays Dedicated to Rinus Plasmeijer on the Occasion of His 61st Birthday, pp. 97–126, Springer (2013)paper
# Summary

The contribution of this paper is twofold.

First, we derive a complete characterization of all

*simply-typed***fixed-point combinator (fpc) generators**using*Barendregt's Inhabitation Machines*. A fpc generator is a lambda term \( G \) such that \( Y G \) is a fpc whenever \( Y \) is. The term \( \delta = \lambda xy. y(xy) \), also known as Smullyan's Owl, is a famous fpc generator. For instance, Turing's fpc \( Y_1 \) can be obtaind from Curry's fpc \( Y_0 \) by postfixing \( \delta \): \[ Y_1 = Y_0 \delta \]Second, we present a conjecture that vastly generalises Richard Statman's question on the existance of double fixed-point combinators. Statman asked whether there is a fixed-point combinator \( Y \) such that \( Y =_\beta Y \delta \). This question remains open as the proof by Benedetto Intrigila contains a gap. We have the following conjecture about the relation of the \( \mu \)-opertator and fixed-point combinators:

ConjectureWe conjecture that for any fixed-point combinator \( Y \) and simply-typed \( \lambda\mu \)-terms \( s,t \) it holds that \[ s =_{ \beta \mu } t \iff s_Y =_{ \beta } t_Y \] where \(s_Y, t_Y\) are the*untyped*lambda terms obtained from \(s,t\), respectively, by replacing all occurrences of \( \mu \)-operators with the fixed-point combinator \( Y \).See research for an overview of my research on the clocked lambda claculus and fixed-point combinators.

# Bibtex

@inproceedings{lambda:clocks:functional:programs:2013, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Klop, Jan Willem and Polonsky, Andrew}, title = {{Clocks for Functional Programs}}, booktitle = {The Beauty of Functional Code - Essays Dedicated to Rinus Plasmeijer on the Occasion of His 61st Birthday}, pages = {97--126}, year = {2013}, doi = {10.1007/978-3-642-40355-2\_8}, series = {LNCS}, volume = {8106}, publisher = {Springer}, keywords = {rewriting, infinitary rewriting, lambda calculus}, type = {journal} }

# Digital Object Identifier

10.1007/978-3-642-40355-2_8

- Streams Are ForeverJörg Endrullis, Dimitri Hendriks, and Jan Willem KlopBulletin of the EATCS , 109 , pp. 70–106 (2013)paper
# Bibtex

@article{streams:forever:2013, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Klop, Jan Willem}, title = {{Streams are Forever}}, journal = {Bulletin of the {EATCS}}, volume = {109}, pages = {70--106}, year = {2013}, keywords = {streams, degrees, automata}, type = {journal} }

## 2012

- Highlights in Infinitary Rewriting and Lambda CalculusJörg Endrullis, Dimitri Hendriks, and Jan Willem KlopTheoretical Computer Science , 464 , pp. 48–71 (2012)paper
# Bibtex

@article{infinitary:highlights:2012, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Klop, Jan Willem}, title = {{Highlights in Infinitary Rewriting and Lambda Calculus}}, journal = {Theoretical Computer Science}, volume = {464}, pages = {48--71}, year = {2012}, doi = {10.1016/j.tcs.2012.08.018}, keywords = {rewriting, infinitary rewriting, lambda calculus}, type = {journal} }

# Digital Object Identifier

10.1016/j.tcs.2012.08.018

## 2011

- On Equal μ-TermsJörg Endrullis, Clemens Grabmayer, Jan Willem Klop, and Vincent van OostromTheoretical Computer Science , 412 (28) , pp. 3175–3202 (2011)paper
# Summary

We consider the rewrite system Rμ with μx.M → μM [x := μx.M ] as its single rewrite rule. This kernel system denoting recursively defined objects occurs in several contexts, e.g. it is the framework of recursive types. For general signatures this rewriting system is widely used to represent and manipulate infinite regular trees.

The main concern of this paper is the convertibility relation for μ-terms as given by the μ-rule, in particular its decidability. This relation is sometimes called weak μ-equality, in contrast with strong μ-equality, which is given by equality of the possibly infinite tree unwinding of μ-terms. While strong equality has received much attention, the opposite is the case for weak μ-equality.

We present three alternative proofs for decidability of weak μ-equality.

# Bibtex

@article{equal:mu:terms:2011, author = {Endrullis, J\"{o}rg and Grabmayer, Clemens and Klop, Jan Willem and van~Oostrom, Vincent}, title = {{On Equal $\mu$-Terms}}, journal = {Theoretical Computer Science}, volume = {412}, number = {28}, pages = {3175--3202}, year = {2011}, doi = {10.1016/j.tcs.2011.04.011}, keywords = {rewriting,automata}, type = {journal} }

# Digital Object Identifier

10.1016/j.tcs.2011.04.011

- Lazy Productivity via TerminationJörg Endrullis, and Dimitri HendriksTheoretical Computer Science , 412 (28) , pp. 3203–3225 (2011)paper
# Bibtex

@article{productivity:termination:2011, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri}, title = {{Lazy Productivity via Termination}}, journal = {Theoretical Computer Science}, volume = {412}, number = {28}, pages = {3203--3225}, year = {2011}, doi = {10.1016/j.tcs.2011.03.024}, keywords = {rewriting, infinitary rewriting, productivity, termination}, type = {journal} }

# Digital Object Identifier

10.1016/j.tcs.2011.03.024

- Fast Leader Election in Anonymous Rings with Bounded Expected DelayRena Bakhshi, Jörg Endrullis, Wan Fokkink, and Jun PangInformation Processing Letters , 111 (17) , pp. 864–870 (2011)paper
# Bibtex

@article{bounded:expected:delay:2011, author = {Bakhshi, Rena and Endrullis, J\"{o}rg and Fokkink, Wan and Pang, Jun}, title = {{Fast Leader Election in Anonymous Rings with Bounded Expected Delay}}, journal = {Information Processing Letters}, volume = {111}, number = {17}, pages = {864--870}, year = {2011}, doi = {10.1016/j.ipl.2011.06.003}, keywords = {protocols}, type = {journal} }

# Digital Object Identifier

10.1016/j.ipl.2011.06.003

- Degrees of StreamsJörg Endrullis, Dimitri Hendriks, and Jan Willem KlopJournal of Integers , 11B (A6) , pp. 1–40 (2011)paper
# Bibtex

@article{streams:degrees:2011, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Klop, Jan Willem}, title = {{Degrees of Streams}}, journal = {Journal of Integers}, volume = {11B}, number = {A6}, pages = {1--40}, year = {2011}, note = {Proceedings of the Leiden Numeration Conference 2010}, keywords = {streams, degrees, automata}, type = {journal} }

- Levels of Undecidability in RewritingJörg Endrullis, Herman Geuvers, Jakob Grue Simonsen, and Hans ZantemaInformation and Computation , 209 (2) , pp. 227–245 (2011)paper
# Summary

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 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 \( \Pi^0_2 \)-complete. The particular problem of local confluence turns out to be \( \Pi^0_2 \)-complete for ground terms, but only \( \Sigma^0_1 \)-complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be \( \Pi^1_1 \)-complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy.

This paper is an extended version of

*Degrees of Undecidability in Term Rewriting*(CSL 2009).# Bibtex

@article{rewriting:undecidability:levels:2011, author = {Endrullis, J\"{o}rg and Geuvers, Herman and Simonsen, Jakob Grue and Zantema, Hans}, title = {{Levels of Undecidability in Rewriting}}, journal = {Information and Computation}, volume = {209}, number = {2}, pages = {227--245}, year = {2011}, doi = {10.1016/j.ic.2010.09.003}, keywords = {rewriting, undecidability, termination, confluence}, type = {journal} }

# Digital Object Identifier

10.1016/j.ic.2010.09.003

## 2010

- Productivity of Stream DefinitionsJörg Endrullis, Clemens Grabmayer, Dimitri Hendriks, Ariya Isihara, and Jan Willem KlopTheoretical Computer Science , 411 (4-5) , pp. 765–782 (2010)paper
# Summary

We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called `productive' if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit. Whereas productivity is undecidable for stream definitions in general, we show that it can be decided for `pure' stream definitions. For every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements, called `pebbles', in a finite `pebbleflow net(work)'. And the production of a pebbleflow net associated with a pure stream definition, that is, the amount of pebbles the net is able to produce at its output port, can be calculated by reducing nets to trivial nets.

This paper is an extended version of Productivity of Stream Definitions (2007) (FCT 2007).

See research for an overview of my research on productivity.

# Bibtex

@article{productivity:streams:2010, author = {Endrullis, J\"{o}rg and Grabmayer, Clemens and Hendriks, Dimitri and Isihara, Ariya and Klop, Jan Willem}, title = {{Productivity of Stream Definitions}}, journal = {Theoretical Computer Science}, volume = {411}, number = {4-5}, pages = {765--782}, year = {2010}, doi = {10.1016/j.tcs.2009.10.014}, keywords = {rewriting, infinitary rewriting, productivity}, type = {journal} }

# Digital Object Identifier

10.1016/j.tcs.2009.10.014

- Local Termination: Theory and PracticeJörg Endrullis, Roel C. de Vrijer, and Johannes WaldmannLogical Methods in Computer Science , 6 (3) (2010)paper
# Bibtex

@article{termination:local:2010, author = {Endrullis, J\"{o}rg and de~Vrijer, Roel C. and Waldmann, Johannes}, title = {{Local Termination: Theory and Practice}}, journal = {Logical Methods in Computer Science}, volume = {6}, number = {3}, year = {2010}, doi = {10.2168/LMCS-6(3:20)2010}, keywords = {rewriting, termination, automata}, type = {journal} }

# Digital Object Identifier

10.2168/LMCS-6(3:20)2010

- Transforming Outermost into Context-Sensitive RewritingJörg Endrullis, and Dimitri HendriksLogical Methods in Computer Science , 6 (2) (2010)paper
# Summary

We define two transformations from term rewriting systems (TRSs) to context-sensitive TRSs in such a way that termination of the target system implies outermost termination of the original system. In the transformation based on `context extension', each outermost rewrite step is modeled by exactly one step in the transformed system. This transformation turns out to be complete for the class of left-linear TRSs. The second transformation is called `dynamic labeling' and results in smaller sized context-sensitive TRSs. Here each modeled step is adjoined with a small number of auxiliary steps. As a result state-of-the-art termination methods for context-sensitive rewriting become available for proving termination of outermost rewriting. Both transformations have been implemented in Jambox, making it the most successful tool in the category of outermost rewriting of the annual termination competition.

This is an extended version of the paper From Outermost to Context-Sensitive Rewriting (RTA 2009).

See research for an overview of my research on termination.

# Bibtex

@article{termination:outermost:2010, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri}, title = {{Transforming Outermost into Context-Sensitive Rewriting}}, journal = {Logical Methods in Computer Science}, volume = {6}, number = {2}, year = {2010}, doi = {10.2168/LMCS-6(2:5)2010}, keywords = {rewriting, termination, automata}, type = {journal} }

# Digital Object Identifier

10.2168/LMCS-6(2:5)2010

## 2008

- Matrix Interpretations for Proving Termination of Term RewritingJörg Endrullis, Johannes Waldmann, and Hans ZantemaJournal of Automated Reasoning , 40 (2-3) , pp. 195–220 (2008)paper
# Summary

We present a new method for automatically proving termination of term rewriting using matrix interpretation (weighted automata with natural numbers as weights). 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.

This is an extended version of the paper Matrix Interpretations for Proving Termination of Term Rewriting (IJCAR 2006).

See research for an overview of my research on termination.

# Bibtex

@article{termination:matrix:2008, author = {Endrullis, J\"{o}rg and Waldmann, Johannes and Zantema, Hans}, title = {{Matrix Interpretations for Proving Termination of Term Rewriting}}, journal = {Journal of Automated Reasoning}, volume = {40}, number = {2-3}, pages = {195--220}, year = {2008}, doi = {10.1007/s10817-007-9087-9}, keywords = {rewriting, termination}, type = {journal} }

# Digital Object Identifier

10.1007/s10817-007-9087-9