String-rewriting systems
Semirings and formal power series: their relevance to formal languages and automata
Handbook of formal languages, vol. 1
Termination of term rewriting using dependency pairs
Theoretical Computer Science - Trees in algebra and programming
Handbook of Weighted Automata
Weighted automata for proving termination of string rewriting
Journal of Automata, Languages and Combinatorics
Matrix interpretations for proving termination of term rewriting
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Termination of string rewriting with matrix interpretations
RTA'06 Proceedings of the 17th international conference on Term Rewriting and Applications
On the domain and dimension hierarchy of matrix interpretations
LPAR'12 Proceedings of the 18th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
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The "matrix method" (Hofbauer and Waldmann 2006) proves termination of string rewriting via linear monotone interpretation into the domain of vectors over suitable semirings. Equivalently, such an interpretation is given by a weighted finite automaton. This is a general method that has as parameters the choice of the semiring and the dimension of the matrices (equivalently, the number of states of the automaton). We consider the semirings of non-negative integers, rationals, algebraic numbers, and reals; with the standard operations and ordering. Monotone interpretations also allow to prove relative termination, which can be used for termination proofs that consist of several steps. The number of steps gives another hierarchy parameter. We formally define the hierarchy and we prove that it is infinite in both directions (dimension and steps).