Algorithms with polynomial interpretation termination proof

Abstract

We study the effect of polynomial interpretation termination proofs of deterministic (resp. non-deterministic) algorithms defined by con uent (resp. non-con uent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of the constructors. This leads to the definition of six function classes which turn out to be exactly the deterministic (resp. non-deterministic) polynomial time, linear exponential time and linear doubly exponential time computable functions when the class is based on con uent (resp. non-con uent) rewrite systems. We also obtain a characterisation of the linear space computable functions. Finally, we demonstrate that functions with exponential interpretation termination proofs are super-elementary.

Analyzing Innermost Runtime Complexity of Term Rewriting by Dependency Pairs

“…e.g. [7,20]. 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.…”

“…While termination of term rewrite systems (TRSs) is well studied, only recently first results were obtained which adapt termination techniques in order to obtain polynomial complexity bounds automatically, e.g., [2][3][4][5]7,10,[17][18][19][22][23][24]26,29,30]. Here, [3,[17][18][19] consider the dependency pair (DP) method [1,11,12,16], which is one of the most popular termination techniques for TRSs.…”

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

“…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.…”

“…While termination of term rewrite systems (TRSs) is well studied, only recently first results were obtained which adapt termination techniques in order to obtain polynomial complexity bounds automatically, e.g., [2][3][4][5]7,9,15,16,[19][20][21]23,27,28]. Here, [3,15,16] consider the dependency pair (DP) method [1,10,11,14], which is one of the most popular termination techniques for TRSs.…”