[Paper Review] Analysis of Logarithmic Amortised Complexity

Martin Hofmann, Georg Moser|arXiv (Cornell University)|Jul 22, 2018

Markov Chains and Monte Carlo Methods5 references3 citations

TL;DR

This paper presents the first automatable type system for logarithmic amortised complexity using potential-based resource analysis, enabling precise derivation of logarithmic runtime bounds for functional programs. It formalizes the analysis of splaying in splay trees via a constraint system over logarithmic terms, leveraging Farkas' Lemma to automate inference of valid potential functions.

ABSTRACT

We introduce a novel amortised resource analysis based on a potential-based type system. This type system gives rise to logarithmic and polynomial bounds on the runtime complexity and is the first such system to exhibit logarithmic amortised complexity. We relate the thus obtained automatable amortised resource analysis to manual amortised analyses of self-adjusting data structures, like splay trees, that can be found in the literature.

Motivation & Objective

• To develop an automated, type-based amortised resource analysis system capable of deriving logarithmic complexity bounds.
• To formalize the amortized cost of splaying in splay trees using a potential-based type system.
• To reduce the verification of logarithmic bounds to a system of linear inequalities with logarithmic terms.
• To enable automation of logarithmic complexity analysis through an Ansatz with unknown coefficients and constraint solving.
• To provide a foundation for integrating logarithmic amortized analysis into existing automated program analysis tools.

Proposed method

• The method employs a potential-based type system where resource annotations include logarithmic terms of tree sizes.
• It reduces the problem of verifying logarithmic bounds to checking the validity of a system of linear inequalities involving logarithmic and linear expressions.
• The system uses an Ansatz with unknown coefficients to construct potential functions satisfying the required constraints.
• Expert knowledge—such as monotonicity of logarithms and algebraic identities—is encoded as inequalities in the form $ A\vec{x} \leqslant \vec{b} $.
• Farkas' Lemma is applied to transform the implication $ \forall\vec{x} \, A\vec{x} \leqslant \vec{b} \Rightarrow CY\vec{x} \leqslant \vec{d} $ into a solvable existential constraint $ \exists F \geqslant 0 \land U \leqslant FA \land F\vec{b} \leqslant \vec{v} $.
• The approach is parametric and generalizable to other sublinear bounds, with termination assumed via big-step semantics.

Experimental results

Research questions

• RQ1 Can a type system be designed to automatically derive logarithmic amortized complexity bounds for functional programs?
• RQ2 How can potential functions for logarithmic bounds be constructed and verified algorithmically?
• RQ3 What constraints are necessary and sufficient to ensure that logarithmic bounds on splaying are derivable via type inference?
• RQ4 Can the inference of logarithmic bounds be reduced to a solvable system of linear inequalities with logarithmic terms?
• RQ5 How can expert knowledge about logarithmic functions (e.g., monotonicity) be encoded and used in automated reasoning?

Key findings

• The proposed type system successfully derives the known logarithmic amortized cost of splaying in splay trees, confirming correctness of the analysis.
• The system reduces the verification of logarithmic bounds to a constraint satisfaction problem over logarithmic and linear terms.
• The use of Farkas' Lemma enables automated inference of potential function coefficients via existential constraints on inequality systems.
• The method is the first to achieve fully automated amortized analysis for logarithmic complexity in a type-based framework.
• The approach is generalizable to other self-adjusting data structures and sublinear bounds, with a clear path toward tool integration.
• The system assumes termination via big-step semantics, though small-step or partial semantics could be used as alternatives.