Reachability and confluence are undecidable for flat term rewriting systems

doi:10.1016/S0020-0190(03)00310-7

Information Processing Letters

Volume 87, Issue 5, 15 September 2003, Pages 265-270

https://doi.org/10.1016/S0020-0190(03)00310-7 Get rights and content

Abstract

Ground reachability, ground joinability and confluence are shown undecidable for flat term rewriting systems, i.e., systems in which all left and right members of rule have depth at most one.

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Cited by (16)

Some decidability results on one-pass reductions
2019, Journal of Logical and Algebraic Methods in Programming
Levels of undecidability in rewriting
2011, Information and Computation
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 $Π_{2}^{0}$ -complete. The particular problem of local confluence turns out to be $Π_{2}^{0}$ -complete for ground terms, but only $Π_{2}^{0}$ -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 $Π_{1}^{1}$ -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.
Undecidable Properties on Length-Two String Rewriting Systems
2008, Electronic Notes in Theoretical Computer Science
Length-two string rewriting systems are length-preserving string rewriting systems that consist of length-two rules. This paper shows that confluence, termination, left-most termination and right-most termination are undecidable properties for length-two string rewriting systems. These results mean that these properties are undecidable for the class of linear term rewriting systems in which depth-two variables are allowed in both sides of rules.
The reachability and related decision problems for monadic and semi-constructor TRSs
2006, Information Processing Letters
This paper shows that reachability is undecidable for confluent monadic and semi-constructor TRSs, and that joinability and confluence are undecidable for monadic and semi-constructor TRSs. Here, a TRS is monadic if the height of the right-hand side of each rewrite rule is at most 1, and is semi-constructor if all defined symbols appearing in the right-hand side of each rewrite rule occur only in its ground subterms.
Some undecidable approximations of TRSs
2005, Electronic Notes in Theoretical Computer Science
In this paper we study the decidability of reachability, normalisation, and neededness in n-shallow and n-growing TRSs. In an n-growing TRS, a variable that occurs both on the left- and right-hand side of a rewrite rule must be at depth n on the left-hand side and at depth greater than n on the right-hand side. In an n-shallow TRS, a variable that occurs both on the left- and right-hand side of a rewrite rule must be at depth n on both sides.
The n-growing and n-shallow TRSs are generalisations of the growing and shallow TRSs as introduced by Jacquemard and Comon. For both shallow and growing TRSs reachability, normalisation, and (in the orthogonal case) neededness are decidable. However, as we show, these results do not generalise to n-growing and n-shallow TRSs. Consequently, no algorithm exists that performs a needed reduction strategy in n-growing or n-shallow TRSs.
Deciding confluence of certain term rewriting systems in polynomial time
2004, Annals of Pure and Applied Logic
We present a characterization of confluence for term rewriting systems, which is then refined for special classes of rewriting systems. The refined characterization is used to obtain a polynomial time algorithm for deciding the confluence of ground term rewrite systems. The same approach also shows the decidability of confluence for shallow and linear term rewriting systems. The decision procedure has a polynomial time complexity under the assumption that the maximum arity of a function symbol in the signature is a constant.

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Reachability and confluence are undecidable for flat term rewriting systems

Abstract

Inform. and Comput.

Theoret. Comput. Sci.

On the complexity of confluence for ground rewrite systems

The confluence of ground term rewrite systems is decidable in polynomial time