Termination proofs and the length of derivations

A Dependency Pair Framework for Innermost Complexity Analysis of Term Rewrite Systems

“…a relation → is the length of the longest sequence of →-steps starting with t, i.e., dh(t, →) = sup{ n | ∃t ∈ T , t → n t }, cf. [17]. Here, for any set M ⊆ N ∪ {ω}, "sup M " is the least upper bound of M .…”

“…e.g. [7,17]. However, direct applications of orders have two main drawbacks: The obtained bounds are often far too high to be useful and there are many TRSs that cannot be oriented strictly with standard orders amenable to automation, cf.…”

“…This ensures that the mapping from constructor ground terms t ∈ T (Σ \Σ d , ∅) to their interpretations is in O(|t|), cf. [7,17]. Note that the interpretations in Ex.…”

“…Like [28], our method is modular. But in contrast to [28], which allows to investigate derivational complexity [17], we focus on innermost runtime complexity. Hence, we can inherit the modularity aspects of the DP framework and benefit from its transformation techniques, which increases power significantly.…”

Analyzing Innermost Runtime Complexity of Term Rewriting by Dependency Pairs

“…Like [30], our method is modular (i.e., we can determine the complexity of a TRS by determining the complexity of certain sub-problems and by returning the maximum of these complexities). But in contrast to [30], which allows to investigate derivational complexity [20], we focus on innermost runtime complexity. In this way, we can inherit the modularity aspects of the DP framework and benefit from its transformation techniques, which increases power significantly.…”