We introduce a novel amortised resource analysis couched in a type-and-effect system. Our analysis is formulated in terms of the physicist’s method of amortised analysis and is potentialbased. The type system makes use of logarithmic potential functions and is the first such system to exhibit logarithmic amortised complexity. With our approach, we target the automated analysis of self-adjusting data structures, like splay trees, which so far have only manually been analysed in the literature. In particular, we have implemented a semi-automated prototype, which successfully analyses the zig-zig case of splaying, once the type annotations are fixed.
Being able to argue about the performance of self-adjusting data structures such as splay trees has been a main objective, when Sleator and Tarjan introduced the notion of amortised complexity.Analysing these data structures requires sophisticated potential functions, which typically contain logarithmic expressions. Possibly for these reasons, and despite the recent progress in automated resource analysis, they have so far eluded automation. In this paper, we report on the first fully-automated amortised complexity analysis of self-adjusting data structures. Following earlier work, our analysis is based on potential function templates with unknown coefficients.We make the following contributions: 1) We encode the search for concrete potential function coefficients as an optimisation problem over a suitable constraint system. Our target function steers the search towards coefficients that minimise the inferred amortised complexity. 2) Automation is achieved by using a linear constraint system in conjunction with suitable lemmata schemes that encapsulate the required non-linear facts about the logarithm. We discuss our choices that achieve a scalable analysis. 3) We present our tool $$\mathsf {ATLAS}$$ ATLAS and report on experimental results for splay trees, splay heaps and pairing heaps. We completely automatically infer complexity estimates that match previous results (obtained by sophisticated pen-and-paper proofs), and in some cases even infer better complexity estimates than previously published.
We introduce a novel amortised resource analysis couched in a type-and-effect system. Our analysis is formulated in terms of the physicist's method of amortised analysis, and is potential-based. The type system makes use of logarithmic potential functions and is the first such system to exhibit logarithmic amortised complexity. With our approach we target the automated analysis of self-adjusting data structures, like splay trees, which so far have only manually been analysed in the literature. In particular, we have implemented a semi-automated prototype, which successfully analyses the zig-zig case of splaying, once the type annotations are fixed.
The importance of subsumption testing for redundancy elimination in first-order logic automatic reasoning is well-known. Although the problem is already NP-complete for first-order clauses, the meanwhile developed test pipelines efficiently decide subsumption in almost all practical cases. We consider subsumption between first-oder clauses of the Bernays-Schönfinkel fragment over linear real arithmetic constraints: BS(LRA). The bottleneck in this setup is deciding implication between the LRA constraints of two clauses. Our new sample point heuristic preempts expensive implication decisions in about 94% of all cases in benchmarks. Combined with filtering techniques for the first-order BS part of clauses, it results again in an efficient subsumption test pipeline for BS(LRA) clauses.
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