# Publications on Undecidability

## 2017

- Undecidability and Finite AutomataJörg Endrullis, Jeffrey Shallit, and Tim SmithIn: Proc. Conf. Developments in Language Theory (DLT 2017), pp. 160–172, Springer (2017)paper
# Summary

Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable. In contrast, we also prove three related problems are decidable.

We apply one result to prove the undecidability of a related problem about k-automatic sets of rational numbers.

# Bibtex

@inproceedings{automata:undecidability:2017, author = {Endrullis, J\"{o}rg and Shallit, Jeffrey and Smith, Tim}, title = {{Undecidability and Finite Automata}}, booktitle = {Proc.\ Conf.\ Developments in Language Theory (DLT~2017)}, volume = {10396}, pages = {160--172}, publisher = {Springer}, series = {LNCS}, year = {2017}, doi = {10.1007/978-3-319-62809-7\_11}, keywords = {automata, undecidability}, type = {conference} }

# Digital Object Identifier

10.1007/978-3-319-62809-7_11

## 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

## 2012

- On the Complexity of Equivalence of Specifications of Infinite ObjectsJörg Endrullis, Dimitri Hendriks, and Rena BakhshiIn: Proc. Int. Conf. on Functional Programming (ICFP 2012), pp. 153–164, ACM (2012)paper
# Bibtex

@inproceedings{complexity:stream:equality:2012, author = {Endrullis, J\"{o}rg and Hendriks, Dimitri and Bakhshi, Rena}, title = {{On the Complexity of Equivalence of Specifications of Infinite Objects}}, booktitle = {Proc.\ Int.\ Conf.\ on Functional Programming (ICFP~2012)}, pages = {153--164}, publisher = {{ACM}}, year = {2012}, doi = {10.1145/2364527.2364551}, keywords = {rewriting, undecidability, streams}, type = {conference} }

# Digital Object Identifier

10.1145/2364527.2364551

## 2011

- 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

## 2009

- Complexity of Fractran and ProductivityJörg Endrullis, Clemens Grabmayer, and Dimitri HendriksIn: Proc. Conf. on Automated Deduction (CADE 2009), pp. 371–387, Springer (2009)paper
# Bibtex

@inproceedings{complexity:productivity:2009, author = {Endrullis, J\"{o}rg and Grabmayer, Clemens and Hendriks, Dimitri}, title = {{Complexity of Fractran and Productivity}}, booktitle = {Proc.\ Conf.\ on Automated Deduction (CADE~2009)}, volume = {5663}, pages = {371--387}, publisher = {Springer}, series = {LNCS}, year = {2009}, doi = {10.1007/978-3-642-02959-2\_28}, keywords = {rewriting, undecidability, productivity}, type = {conference} }

# Digital Object Identifier

10.1007/978-3-642-02959-2_28

- Degrees of Undecidability in Term RewritingJörg Endrullis, Herman Geuvers, and Hans ZantemaIn: Proc. Conf. on Computer Science Logic (CSL 2009), pp. 255–270, Springer (2009)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.

We refer to

*Levels of Undecidability in Rewriting*(Information and Computation 2011) for an extended journal version of this paper.# Bibtex

@inproceedings{rewriting:undecidability:degrees:2009, author = {Endrullis, J\"{o}rg and Geuvers, Herman and Zantema, Hans}, title = {{Degrees of Undecidability in Term Rewriting}}, booktitle = {Proc.\ Conf.\ on Computer Science Logic (CSL~2009)}, volume = {5771}, pages = {255--270}, publisher = {Springer}, series = {LNCS}, year = {2009}, doi = {10.1007/978-3-642-04027-6\_20}, keywords = {rewriting, undecidability, termination, confluence}, type = {conference} }

# Digital Object Identifier

10.1007/978-3-642-04027-6_20