arXiv subjects

Matthias Hetzenberger

Publications and source records attributed to Matthias Hetzenberger.

5 recordsLinked to original sources

arXiv2026

A Cost-Aware Probability Monad for Liquid Haskell

Probabilistic algorithms and data structures are widely used to obtain favourable expected performance guarantees. While their mathematical analysis is often well understood, mechanising expected-cost analyses remains challenging, requiring reasoning about probability distributions, expectations, and recursive stochastic behaviour. Existing formal approaches frequently require substantial manual proof effort, since expected costs are often encoded separately from probabilistic computations and must therefore be propagated explicitly throughout proofs. In this paper, we present a cost-aware probability monad for \LH/ that supports reasoning about probabilistic programs together with their expected costs. Our approach combines executable probabilistic programs with refinement-type-based verification and SMT-supported automation. The monad intrinsically tracks probability mass, expected values, and expected costs through refinement types, enabling many quantitative properties of probabilistic computations to be inferred compositionally from program structure. We evaluate our approach on several classical probabilistic algorithms and data structures, including meldable heaps, randomised quicksort and quickselect, randomised splay trees, random permutations, and the hiring problem. The case studies demonstrate different points along the spectrum between automated and interactive verification.

cs.LO↗

arXiv2025

Optimistic Higher-Order Superposition

The $λ$-superposition calculus is a successful approach to proving higher-order formulas. However, some parts of the calculus are extremely explosive, notably due to the higher-order unifier enumeration and the functional extensionality axiom. In the present work, we introduce an "optimistic" version of $λ$-superposition that addresses these two issues. Specifically, our new calculus delays explosive unification problems using constraints stored along with the clauses, and it applies functional extensionality in a more targeted way. The calculus is sound and refutationally complete with respect to a Henkin semantics. We have yet to implement it in a prover, but examples suggest that it will outperform, or at least usefully complement, the original $λ$-superposition calculus.

cs.LO↗

arXiv2025

Term Orders for Optimistic Lambda-Superposition

We introduce $λ$KBO and $λ$LPO, two variants of the Knuth-Bendix order (KBO) and the lexicographic path order (LPO) designed for use with the $λ$-superposition calculus. We establish the desired properties via encodings into the familiar first-order KBO and LPO.

cs.LO↗

arXiv2025

The Vampire Diary

During the past decade of continuous development, the theorem prover Vampire has become an automated solver for the combined theories of commonly-used data structures. Vampire now supports arithmetic, induction, and higher-order logic. These advances have been made to meet the demands of software verification, enabling Vampire to effectively complement SAT/SMT solvers and aid proof assistants. We explain how best to use Vampire in practice and review the main changes Vampire has undergone since its last tool presentation, focusing on the engineering principles and design choices we made during this process.

cs.LO↗

arXiv2025

To Zip Through the Cost Analysis of Probabilistic Programs

Probabilistic programming and the formal analysis of probabilistic algorithms are active areas of research, driven by the widespread use of randomness to improve performance. While functional correctness has seen substantial progress, automated reasoning about expected runtime remains comparatively limited. In this work, we address this challenge by introducing a refinement-typed probability monad in Liquid Haskell. Our monad enables automated reasoning about expected values and costs by encoding probabilistic behaviour directly in types. Initially defined for discrete distributions over finite support, it is extended to support infinite distributions via an axiomatic approach. By leveraging Liquid Haskell's SMT-based refinement type checking, our framework provides a high degree of automation. We evaluate our approach through four case studies: meldable heaps, coupon collector, randomised quicksort, and zip trees. The first two demonstrate automation with minimal annotation overhead. The latter two showcase how our monad integrates with interactive proofs, including the first formal verification of the expected runtime of zip trees.

cs.LO↗