. We consider some basic problems on the decidability and complexity of trace rewriting systems. The new contribution of this paper is an O(n log(n)) algorithm for some computing irreducible normal forms in the case of certain one-rule systems. 1 Introduction The notes of this paper are based on the Font Romeu Lecture and on an invited lecture at FCT-93 conference [11] of the second author. In the first part of the paper we give some basic background about trace rewriting systems. There is some overlap with the published notes from FCT-93. However, the second part is original and constitutes a new contribution to the theory of trace rewriting systems. The theory of rewriting over free partially commutative monoids (trace rewriting) combines combinatorial aspects from string rewriting (modulo a congruence) and graph rewriting. This restriction of graph rewriting leads to feasible algorithms, but some interesting complexity questions are still open. For example, a challenging open probl...

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 0 2-complete. The particular problem of local confluence turns out to be Π 0 2-complete for ground terms, but only Σ 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 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 [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.