Analysis, Verification, and Synthesis for Infinite-State Systems

Project Information

AVS-ISS

Grant agreement ID: 648701

DOI

10.3030/648701

Project closed

EC signature date 24 June 2015

Start date 1 August 2015

End date 31 January 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 834 975,00

EU contribution

€ 1 834 975,00

Coordinated by

MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Germany

Periodic Reporting for period 4 - AVS-ISS (Analysis, Verification, and Synthesis for Infinite-State Systems)

Reporting period: 2019-08-01 to 2021-01-31

Summary of the context and overall objectives of the project

The main focuses of this project is on theoretical problems arising out of automated-verification research, broadly construed, with a particular emphasis on algorithmic questions. During the reporting period, our efforts have revolved around the analysis of discrete and continuous linear dynamical systems. Such systems are widely used as abstractions of components of computer programs and embedded systems, including cyber-physical systems, that are ubiquitous in our modern society. The problems considered include reachability, invariant synthesis, model-checking, and controllability questions -- such questions relate intimately to the correct and dependable workings of many of the computer systems around us.

The overall objective of the project is to comprehensively map the algorithmic landscape of verification problems for both discrete and continuous linear dynamical systems, and attendant extensions.

Some 41 peer-reviewed articles were published throughout the course of the action. Our work has substantially enhanced the algorithmic understanding of both discrete and continuous linear dynamical systems; in particular, we have provided algorithms for invariant synthesis and model checking that we expect will be incorporated within program-analysis tools and automated-verification engines in the years to come.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Since the start of the project, we mainly tackled fundamental scientific questions relating to reachability, invariant generation, and model checking for linear dynamical systems. Our main highlights results are the following:

1) The solution of a 30-year old problem of Kannan and Lipton, published in our JACM 2016 paper (Pub#7).

2) The solution of the hyperplane-hitting problem for continuous linear dynamical systems (or equivalently the zero problem for linear differential equations), subject to Schanuel's Conjecture, as published in our LICS 2016 (Pub#2) and ICALP 2016 (Pub#4) papers.

3) An algorithm to decide the existence of, and synthesise, semialgebraic invariants for discrete-time linear dynamical systems, for the point-to-point reachability problem, published in a STACS 2017 paper (Pub#8), with a subsequent invited longer version in the journal Theory of Computing Systems (2019) (Pub#21).

4) The decidability of the longstanding question of structural liveness for linear hybrid systems is decidable, published in our HSCC 2017 paper (Pub#10).

5) The computability of strongest polynomial invariants for affine programs, solving a 40-year-old open problem, published in our LICS 2018 paper (Pub#12). An extension to linear hybrid automata was published in CONCUR 2020 (Pub#30).

6) The computability of strongest families of o-minimal invariants for discrete linear dynamical systems (ICALP 2018) (Pub#13).

7) The solution of the termination problem for linear loops over the integers, answering a famous question of Ashish Tiwari from 2003 (ICALP 2019) (Pub#18).

8) The solution to the "Monniaux Problem" (on automated invariant generation) in our SAS 2019 paper (Pub#24).

9) The most general results on model checking for discrete linear dynamical systems (MFCS 2020 (Pub#26), POPL 2021 (Pub#39), and POPL 2022 (Pub#41)).

10) Advances on the Skolem Problem for linear recurrence sequences (ISSAC 2020 (Pub#27) and LICS 2021 (Pub#35) [which received "distinguished paper award"]).

11) The computability of strongest families of o-minimal invariants for continuous linear dynamical systems (ICALP 2020) (Pub#28).

12) Algorithmic results on reachability under small perturbations for discrete linear dynamical systems (FSTTCS 2020 (Pub#29) and MFCS 2021 (Pub#33)).

13) Moving beyond linear systems: algorithmic results on holonomic systems (MFCS 2021 (Pub#32) and ICALP 2021 (Pub#36)).

14) Algorithmic advances on invariant-generation and ranking-function synthesis for affine and polynomial loops (CONCUR 2020 (Pub#30), MFCS 2021 (Pub#34), CAV 2021 (Pub#38)). The CAV paper also featured a tool, POROUS, for the efficient calculations of invariants.

15) Algorithmic advances on reachability problems for parametric linear dynamical sytems (CONCUR 2021) (Pub#37).

Dissemination has taken place mainly via published peer-reviewed articles, conference presentations, and invited talks at various institutions. As the work is foundational in nature, it has not (yet) led to (industrial) exploitation at the present time.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Although the project has now ended, research in ongoing, especially focussing on:

* Continuous dynamics
* Multimodal systems (and in particular hybrid systems)
* Stochastic aspects
* Robustness aspects

Fractal image depicting the behaviour of a dynamical system