Theoretical Computer Science
Conditional rewrite rules: Confluence and termination
Journal of Computer and System Sciences
Information and Computation
Handbook of logic in computer science (vol. 2)
Termination of term rewriting: interpretation and type elimination
Journal of Symbolic Computation - Special issue on conditional term rewriting systems
Total termination of term rewriting is undecidable
Journal of Symbolic Computation
Omega-termination is undecidable for totally terminating term rewriting systems
Journal of Symbolic Computation
Termination of term rewriting using dependency pairs
Theoretical Computer Science - Trees in algebra and programming
Relative undecidability in term rewriting: I. the termination Hierarchy
Information and Computation
Relative undecidability in term rewriting: II. the confluence hierarchy
Information and Computation
Termination of Rewriting is Undecidable in the One-Rule Case
MFCS '88 Proceedings of the Mathematical Foundations of Computer Science 1988
RtA '99 Proceedings of the 10th International Conference on Rewriting Techniques and Applications
Undecidable Properties of Syntactic Theories
RTA '91 Proceedings of the 4th International Conference on Rewriting Techniques and Applications
Decidable Approximations of Term Rewriting Systems
RTA '96 Proceedings of the 7th International Conference on Rewriting Techniques and Applications
Reachability and confluence are undecidable for flat term rewriting systems
Information Processing Letters
A new decidability technique for ground term rewriting systems with applications
ACM Transactions on Computational Logic (TOCL)
Equality of streams is a Π0 over 2-complete problem
Proceedings of the eleventh ACM SIGPLAN international conference on Functional programming
Normalization of Infinite Terms
RTA '08 Proceedings of the 19th international conference on Rewriting Techniques and Applications
RTA '09 Proceedings of the 20th International Conference on Rewriting Techniques and Applications
Degrees of undecidability in term rewriting
CSL'09/EACSL'09 Proceedings of the 23rd CSL international conference and 18th EACSL Annual conference on Computer science logic
On the complexity of equivalence of specifications of infinite objects
Proceedings of the 17th ACM SIGPLAN international conference on Functional programming
Highlights in infinitary rewriting and lambda calculus
Theoretical Computer Science
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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 @P"2^0-complete. The particular problem of local confluence turns out to be @P"2^0-complete for ground terms, but only @P"2^0-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 @P"1^1-complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.