Using Matrix Interpretations over the Reals in Proofs of Termination

Matrix interpretations are a new kind of algebraic interpretations with interesting capabilities for proving termination of rewriting systems. Roughly speaking, a matrix interpretation for a !-ary symbol " is a linear expression #1$1+⋅ ⋅ ⋅+#!$!+#0 where the #1, . . . , #! are matrices of '×' natural numbers and the variables $1, . . . , $! (and also #0) represent '-tuples of natural numbers. In this paper, we extend this framework to matrices and tuples of real numbers. We also compare matrix and polynomial interpretations. It is wellknown that linear polynomial interpretations are (strictly) subsumed by matrix interpretations: every linear polynomial interpretation can be seen as a matrix interpretation (where ' = 1). We show by means of some examples that this is not the case for more general (and widely used) polynomial interpretations like those that involve the so-called simple polynomials, i.e., those polynomials whose variables are raised to powers not exceding 1. Therefore, matrix interpretations and polynomial interpretations are not comparable, in general. We have implemented matrix interpretations over the reals as part of the tool mu-term. We also report on the experimental evaluation of this implementation.