8 research outputs found
Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited
Polynomial interpretations are a useful technique for proving termination of
term rewrite systems. They come in various flavors: polynomial interpretations
with real, rational and integer coefficients. As to their relationship with
respect to termination proving power, Lucas managed to prove in 2006 that there
are rewrite systems that can be shown polynomially terminating by polynomial
interpretations with real (algebraic) coefficients, but cannot be shown
polynomially terminating using polynomials with rational coefficients only. He
also proved the corresponding statement regarding the use of rational
coefficients versus integer coefficients. In this article we extend these
results, thereby giving the full picture of the relationship between the
aforementioned variants of polynomial interpretations. In particular, we show
that polynomial interpretations with real or rational coefficients do not
subsume polynomial interpretations with integer coefficients. Our results hold
also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201
Polynomial Interpretations over the Reals do not Subsume Polynomial Interpretations over the Integers
Polynomial interpretations are a useful technique for proving termination
of term rewrite systems. They come in various flavors:
polynomial interpretations with real, rational and integer coefficients.
In 2006, Lucas proved that there are rewrite systems that can be shown
polynomially terminating by polynomial interpretations with
real (algebraic)
coefficients, but cannot be shown polynomially terminating using
polynomials with rational coefficients only.
He also proved a similar theorem with respect to the use of
rational coefficients versus integer coefficients.
In this paper we show that polynomial interpretations with real or
rational coefficients do not subsume polynomial interpretations with
integer coefficients, contrary to what is commonly believed.
We further show that polynomial interpretations with real
coefficients subsume polynomial interpretations with rational
coefficients
Revisiting Matrix Interpretations for Proving Termination of Term Rewriting
Matrix interpretations are a powerful technique for proving termination of term rewrite systems, which is based on the well-known paradigm of interpreting terms into a domain equipped with a suitable well-founded order, such that every rewrite step causes a strict decrease. Traditionally, one uses vectors of non-negative numbers as domain, where two vectors are in the order relation if there is a strict decrease in the respective first components and a weak decrease in all other components. In this paper, we study various alternative well-founded orders on vectors of non-negative numbers based on vector norms and compare the resulting variants of matrix interpretations to each other and to the traditional approach. These comparisons are mainly theoretical in nature. We do, however, also identify one of these variants as a proper generalization of traditional matrix interpretations as a stand-alone termination method, which has the additional advantage that it gives rise to a more powerful implementation