Periodic Reporting for period 1 - CARBS (Compositional Approximate Reasoning via Bialgebraic Semantics)
Période du rapport: 2019-02-01 au 2020-01-31
Programming languages with probabilistic features are used extensively in computer science and beyond, to model uncertainty, perform quantitative analysis, inference and much more. To analyse programs in such languages, it is essential to have effective tools and techniques for approximate reasoning: for instance, determining the chance of congestion in a network, or the chance of failure of a system component. CARBS proposes a general mathematical framework of compositional proof techniques for approximate reasoning, with two essential points of focus: general applicability, to deal with the wide variety of different quantitative languages and models, and compositionality, to deal with large-scale systems. A motivating case study and application for the developed proof techniques is ProbNetKAT, a probabilistic language for describing randomized protocols and analysing quantitative properties in networks such as throughput or chance of failure. Approximate reasoning about such network programs is an important but also challenging problem, and the abundance of possible case studies will allow to immediately evaluate and apply the developed proof techniques. Approximate reasoning requires to move from behavioural equivalence to behavioural metrics, formalising how far apart two programs are. CARBS is based on integrating behavioural metrics in bialgebraic semantics, a categorical approach for a systematic study of languages and calculi based on the combination of algebra and coalgebra. Coalgebra allows to define behavioural metrics, in a general manner, whereas algebra integrates compositionality in the associated proof techniques. The overall envisaged result of CARBS is an extension of bialgebraic semantics to quantitative systems, providing on the one hand fundamental insights about quantiative coalgebras and compositionality, and on the other hand concrete, effective proof techniques for approximate reasoning.
The main scientific results of this project revolve around proof methods for behavioural metrics and approximation, in line with the overall aim of the project to provide compositional proof techniques for quantitative reasoning. Two of the main results of CARBS are: (1) A general framework for expressivity of logics w.r.t. coinductive predicates, published at CSL 2020 (referring to the publication list in Section 2 of this report). This work provides the first systematic framework to cover expressivity results for coinductive predicates other than bisimilarity (such as behavioural metrics) at the general level of coalgebras. The approach applies in particular to behavioural metrics on automata. This work is expected to form a fruitful foundation for further development of expressivity results, and an essential part of compositional reasoning about quantitative systems, the overall aim of the project. It has been further extended and refined in a submitted paper [P1], which in particular addresses the issue of approximation in characterisation results for quantitative logics wrt important behavioural metrics such as the Kantorovich distance. (2) A systematic approach to learning of weighted automata, which form an indispensable quantitative model of computation. We introduced a general algorithm for learning weighted automata, and established the limits of this algorithm. This work will be presented at the conference FoSSaCS 2020.
The results and impact of this project are primarily within theoretical computer science. A main highlight of the project is formed by the results on expressivity of logics wrt coinductive predicates. These go beyond the state of the art in that they cover not only bisimilarity, but also other coinductive predicates, including quantitative properties. The results on learning weighted automata form a new step in learning quantitative systems, establishing both new negative and positive results about quantitative automata learning.