Lower Runtime Bounds for Integer Programs

Abstract

We present a technique to infer lower bounds on the worst-case runtime complexity of integer programs, where in contrast to earlier work, our approach is not restricted to tail-recursion. Our technique constructs symbolic representations of program executions using a framework for iterative, under-approximating program simplification. The core of this simplification is a method for (under-approximating) program acceleration based on recurrence solving and a variation of ranking functions. Afterwards, we deduce asymptotic lower bounds from the resulting simplified programs using a special-purpose calculus and an SMT encoding. We implemented our technique in our tool LoAT and show that it infers non-trivial lower bounds for a large class of examples.

Analyzing Program Termination and Complexity Automatically with AProVE

“…Moreover, AProVE analyzes the runtime complexity of Prolog programs and TRSs. We are currently working on extending AProVE's complexity analysis to Java as well [14,29].…”

“…, t m ) of size n are of length Θ(n 3 ). 7 Moreover, AProVE also analyzes the complexity of integer transition systems with initial states by calling the tools KoAT [14] (for upper bounds) and LoAT [29] (for lower bounds). 7 In the Proof Tree View, we do not only have complexity icons like or for problems, but proof steps also result in complexities (e.g., or ).…”

“…A graphical overview of our approach is shown below. Technical details on the techniques for transforming programs to (int-) TRSs and for analyzing rewrite systems can be found in, e.g., [10,11,12,14,17,25,26,27,28,29,30,31,32,33,34,36,37,39,40,41,43,44,51,52,57,58]. Since the current paper is a system description, we focus on the implementation of these techniques in AProVE, which we have made available as a plug-in for the popular Eclipse software development environment [23].…”

A Calculus for Modular Loop Acceleration

“…Linear metering functions can be synthesized via Farkas' Lemma and SMT solving [16]. However, many loops do not have non-trivial linear metering functions.…”

“…Here, (x 1 , x 2 ) → x 1 is not a metering function as T non-dec cannot be iterated at least x 1 times if x 2 ≤ 0. Thus, [15] proposes a refinement of [16] based on metering functions of the form…”