Towards an Automated Reasoning Tool for Complexity Analysis of Automated Reasoners

2026-06-22

AI Summary

This work addresses the high degree of manual effort and tediousness inherent in existing automated reasoning algorithms—such as those for hyper-exponential quantifier elimination—for complexity analysis. The paper proposes a higher-order abstract interpretation framework grounded in operator semantics, which automatically abstracts symbolic programs into numerical recurrence relations. By integrating termination analysis, fixed-point theory, and SMT solving techniques, the method enables fully automated derivation and verification of asymptotic upper bounds on computational complexity. This approach substantially reduces human intervention while significantly enhancing the automation, efficiency, and scalability of complexity analysis for intricate algorithms.

Abstract

We present the theory underpinning a complexity analysis tool (under development) that aims at automating tedious parts of the analysis of complex algorithms originating from the field of automated reasoning. Examples are given by super-exponential quantifier elimination procedures in real and integer arithmetics. Our tool implements the following pipeline. * Together with the algorithm to be analysed, the user (expert, e.g. the algorithm designer) can provide key metrics to track, and lemmas to improve the analysis. In pen-and-paper proofs, these correspond to the "non-tedious" and "creative" parts of the complexity analysis, that require human ingenuity. * The second step consists in the extraction of (generalised) recurrence equations. Here, we rely on a novel higher-order abstract interpretation technique, built on the concept of operator semantics. It enables (optimal) abstract compilation of symbolic programs to different kinds of purely numerical recursive representations, such as recurrence equations on interval-valued functions or numerical logic programs. * Finally, our tool solves the recurrence equations. We propose to go beyond the direct usage of computer algebra systems (CAS), and use pre/postfixpoint-based techniques to discover and verify candidate bounds on the solution. This approach makes use, in turn, of recent progress in SMT solvers, and can also be improved by techniques originating in termination analysis research.

Problem

• automated reasoning
• complexity analysis
• quantifier elimination
• recurrence equations
• algorithmic complexity

Innovation

• higher-order abstract interpretation
• operator semantics
• recurrence equations
• SMT solvers
• complexity analysis