Institute of Informatics
Acta Cybernetica
Past Issues
Volume 19, Number 2, 2009
Weighted Automata Define a Hierarchy of Terminating String Rewriting Systems
# Weighted Automata Define a Hierarchy of Terminating String Rewriting Systems

**Andreas Gebhardt and Johannes Waldmann**

### Abstract (in LaTeX format)

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).

**Keywords: ** string rewriting, relative termination, weighted automaton, matrix interpretation, monotone algebra.

### Full text

Available electronic editions: PDF.

### DOI

DOI is not available for this article.

### BibTeX entry
`
@article{Gebhardt:2009:ActaCybernetica,`

author = {Andreas Gebhardt and Johannes Waldmann},

title = {Weighted Automata Define a Hierarchy of Terminating String Rewriting Systems},

journal = {Acta Cybernetica},

volume = {19},

number= {2},

pages = {295--312},

year = {2009},

abstract = {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).},

keywords = {string rewriting, relative termination, weighted automaton, matrix interpretation, monotone algebra}

}