Symbolic Model Construction for Saturated Constrained Horn Clauses

Clause sets saturated by hierarchic ordered resolution do not offer a model representation that can be effectively queried, in general. They only offer the guarantee of the existence of a model. We present an effective symbolic model construction for saturated constrained Horn clauses. Constraints are in linear arithmetic, the first-order part is restricted to a function-free language. The model is constructed in finite time, and non-ground clauses can be effectively evaluated with respect to the model. Furthermore, we prove that our model construction produces the least model

Type-Based Analysis of Logarithmic Amortised Complexity

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.Comment: 35 pages. arXiv admin note: text overlap with arXiv:1807.0824

Automatisierte Analyse der Amortisierten Komplexität von Selbst-modifzierenden Datenstrukturen

Arbeit an der Bibliothek noch nicht eingelangt - Daten nicht geprüftAbweichender Titel nach Übersetzung der Verfasserin/des VerfassersBeing 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 thesis, we report on the first fully automatedamortised complexity analysis of self-adjusting data structures.We make the following contributions:1. 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.2. 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.3. 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.4. We present our tool 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.6

Symbolic Model Construction for Saturated Constrained Horn Clauses

Clause sets saturated by hierarchic ordered resolution do not offer a model representation that can be effectively queried, in general. They only offer the guarantee of the existence of a model. We present an effective symbolic model construction for saturated constrained Horn clauses. Constraints are in linear arithmetic, the first-order part is restricted to a function-free language. The model is constructed in finite time, and non-ground clauses can be effectively evaluated with respect to the model. Furthermore, we prove that our model construction produces the least model