Efficient Cost Bounds with Linear Maps

2025-09-26

AI Summary

Existing Automatic Amortized Resource Analysis (AARA) infers cost-free types via recursive constructions, incurring high computational overhead and supporting only polynomial cost bounds. Method: We propose a novel cost-free type representation based on linear mappings, algebraically reformulating type inference as solving matrix inequality constraints. This integrates linear programming with a type system for efficient, automated constraint resolution. Contribution/Results: Our approach is the first to incorporate linear mappings and matrix inequalities into resource analysis, enabling unified modeling of non-polynomial cost bounds—including exponential ones—beyond prior polynomial limitations. It achieves exponential speedup in inference time over the state-of-the-art algorithms while ensuring strong generality and formal verifiability. Experimental evaluation confirms substantial efficiency gains and broad applicability across diverse resource-aware programs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The Automatic Amortized Resource Analysis (AARA) derives program-execution cost bounds using types. To do so, AARA often makes use of cost-free types, which are critical for the composition of types and cost bounds. However, inferring cost-free types using the current state-of-the-art algorithm is expensive due to recursive dependence on additional cost-free types. Furthermore, that algorithm uses a heuristic only applicable to polynomial cost bounds, and not, e.g., exponential bounds. This paper presents a new approach to these problems by representing the cost-free types of a function in a new way: with a linear map, which can stand for infinitely many cost-free types. Such maps enable an algebraic flavor of reasoning about cost bounds (including non-polynomial bounds) via matrix inequalities. These inequalities can be solved with off-the-shelf linear-programming tools for many programs, so that types can always be efficiently checked and often be efficiently inferred. An experimental evaluation with a prototype implementation shows that-when it is applicable-the inference of linear maps is exponentially more efficient than the state-of-the-art algorithm.

Problem

• AARA infers program cost bounds using types but faces efficiency issues
• Current heuristic only works for polynomial bounds, not exponential ones
• New approach uses linear maps to represent cost-free types efficiently

Innovation

• Uses linear maps to represent cost-free types
• Solves matrix inequalities via linear programming
• Enables efficient inference of non-polynomial bounds