[Paper Review] Sapo: Reachability Computation and Parameter Synthesis of Polynomial Dynamical Systems

Tommaso Dreossi|arXiv (Cornell University)|Jul 7, 2016

Formal Methods in Verification5 citations

TL;DR

Sapo is a C++ tool for formal analysis of discrete-time polynomial dynamical systems, enabling reachability computation via over-approximated flowpipes (using boxes, parallelotopes, or parallelotope bundles) and parameter synthesis for Signal Temporal Logic (STL) specifications. It leverages Bernstein polynomial coefficients to bound polynomial dynamics over unit hypercubes, transforming them into generic domains for scalable, verified analysis of nonlinear systems with provable over-approximations.

ABSTRACT

Sapo is a C++ tool for the formal analysis of polynomial dynamical systems. Its main features are: 1) Reachability computation, i.e., the calculation of the set of states reachable from a set of initial conditions, and 2) Parameter synthesis, i.e., the refinement of a set of parameters so that the system satisfies a given specification. Sapo can represent reachable sets as unions of boxes, parallelotopes, or parallelotope bundles (symbolic representation of polytopes). Sets of parameters are represented with polytopes while specifications are formalized as Signal Temporal Logic (STL) formulas.

Motivation & Objective

• To address the lack of scalable formal tools for nonlinear dynamical systems, particularly those with polynomial dynamics.
• To provide a unified framework for reachability computation and parameter synthesis in discrete-time polynomial systems.
• To enable formal verification of safety and temporal properties through Signal Temporal Logic (STL) specifications.
• To overcome the challenges of nonlinearity in set propagation by using Bernstein coefficient-based over-approximations.
• To support practical verification tasks such as safety analysis and parameter tuning for complex biological and engineering systems.

Proposed method

• Represent reachable sets as unions of boxes, parallelotopes, or parallelotope bundles to balance precision and computational efficiency.
• Use Bernstein coefficients to bound polynomial functions over the unit hypercube [0,1]^n, enabling tight over-approximation of nonlinear transformations.
• Transform generic state and parameter domains into unit hypercubes via affine mapping, allowing Bernstein-based analysis to be applied universally.
• Reduce parameter synthesis to a sequence of linear programs by parameterizing Bernstein coefficients, enabling efficient refinement of valid parameter sets.
• Formalize system specifications using Signal Temporal Logic (STL), and integrate them into the synthesis pipeline via constraint encoding.
• Construct flowpipes iteratively by applying Bernstein-based over-approximation at each time step, ensuring all trajectories from initial sets are enclosed.

Experimental results

Research questions

• RQ1 Can Bernstein polynomial coefficient techniques be effectively adapted to over-approximate reachable sets in nonlinear polynomial dynamical systems?
• RQ2 How can parameter synthesis for polynomial systems be reduced to a sequence of linear programs using parameterized Bernstein coefficients?
• RQ3 What is the scalability of reachability and parameter synthesis in Sapo across different system dimensions and specification complexities?
• RQ4 How do different geometric representations (boxes, parallelotopes, parallelotope bundles) affect the precision and performance of flowpipe construction?
• RQ5 To what extent can Sapo handle complex STL specifications involving nested temporal operators and long time horizons?

Key findings

• Sapo successfully computes over-approximated flowpipes for three nonlinear models—SIR epidemic, honeybees site choice, and quadrotor drone—within seconds, demonstrating practical scalability.
• Reachability computation on the SIR model with 3 variables and 2 parameters completed in 0.10 seconds for a specification over [10,30] time units.
• Parameter synthesis for the Ebola model (5 variables, 4 parameters) with complex STL specifications (e.g., nested untils) completed in under 2 seconds, even for long time intervals.
• The tool scales to systems with up to 100 variables and 200 parameters, with computation times under 14 seconds for certain configurations.
• For nested STL specifications, Sapo efficiently computes valid parameter sets, with synthesis times increasing with specification nesting depth but remaining tractable for practical use.
• The use of parallelotope bundles enables tighter over-approximations than boxes or individual parallelotopes, improving precision without sacrificing scalability.