Maximal Termination

Proving Termination of Integer Term Rewriting

“…Therefore, we now extend the approach of [17] to ITRSs. 12 This is more general than related previous classes of interpretations: In [17], there was no "max" and only tuple symbols could be mapped to polynomials with integer coefficients, and in [11], all ground terms had to be mapped to natural numbers.…”

“…In this paper, we consider orders based on integer 11 max-polynomial interpretations [11,17]. Such interpretations suffice for most algorithms typically occurring in practice.…”

“…10 To ease readability we use a simpler definition of DP problems than [13], since this simple definition suffices for the presentation of the new results of this paper. 11 Interpretations into the integers instead of the naturals are often needed for algorithms like sum that increase an argument y until it reaches a bound x. In [17], we already presented an approach to prove termination by bounded increase.…”

“…This conditional constraint is a conjunction of formulas like ϕ s t , ϕ s≻t , and ϕ s c for DPs s → t as well as ℓ r for usable rules ℓ → r. Moreover, one has to ensure that the I-interpretation is chosen in such a way that Pol is proper for the right-hand sides of the DPs and the usable rules. 20 Compared to our earlier work in [11], the only additional difficulty is that now we really consider arbitrary max-polynomial interpretations over the integers where [t] Pol can also be negative for any ground term t.…”

Models for Logics and Conditional Constraints in Automated Proofs of Termination

“…Furthermore, although our practical examples involve CTRSs, our development does not really depend on that and applies to arbitrary declarative languages. With regard to existing approaches to deal with conditional constraints in proofs of termination, the following result formalizes the transformational approach to deal with polynomial conditional constraints in [6,16]. [4,10] for instance).…”