SEE-Reach-py

graph LR
    InputFile([Target Language]) --> Compiler[HL Language Compiler]
    Compiler --> SymbolicExecutor
    SymbolicExecutor --> HybridSystemModel([Hybrid System Model])
    HybridSystemModel --> ReachabilityAnalysis
    HybridSystemModel --> TestGenerator
       subgraph "SEE-Reach"
        Compiler
        SymbolicExecutor
    end

SEE-Reach Architecture

SEE-Reach is an experimental prototype of a symbolic execution engine (SEE) for closed-loop control software verification and analysis using reachability analysis (Reach). The tool is designed to extract models from control functions, determining whether a given state of a system is reachable from a start state under certain conditions.

The symbolic executor executes on a high-level target language that resembles a small subset of Rust (the intended implementation target), tailored specifically for defining and manipulating the hybrid systems. Important to note, the current version of SEE-Reach does not support loops. The target language has been kept minimalistic for the simplicity and to focus more on the proof of concept.

Example

The Program

fn pendulum_dynamics(theta: real, omega: real, kp: real, kd: real) -> tuple {
    // param theta: pendulum angle
    // param omega: angular rate
    // param kp: proportional gain
    // param kd: derivative gain
    
    // get the torque output
    let u: real = controller(theta, omega, kp, kd);
    
    // gravity and length parameters
    let g: real = -9.81;
    let l: real = 2.0;
    
    // state derivative
    let thetap: real = omega;
    let omegap: real = u + g / l * sin(theta);
    return (thetap, omegap)
}

fn controller(x: real, omega: real, kp: real, kd: real) -> real {
    // signal contributions from proportional and derivative
    let up: real = -1.0 * kp * x;
    let ud: real = -1.0 * kd * omega;
    let u: real = up + ud;
    
    // we are torque limited--u must be in [-5.0, 5.0]
    if u < -5.0 {
        return -5.0
    } else {
        if u > 5.0 {
            return 5.0
        } else {
            return u
        }
    }
}

Produces the following model for $k_p=1.0, k_d=0.2$,

$0.2 \omega + 1.0 \theta > 5.0$,

$$ \left[\begin{matrix}\theta \ \omega\end{matrix}\right]\rightarrow\left[\begin{matrix}\omega \ -7.405 \sin{\left(\theta \right)}\end{matrix}\right] $$

$0.2 \omega + 1.0 \theta < -5.0$,

$$ \left[\begin{matrix}\theta \ \omega\end{matrix}\right]\rightarrow\left[\begin{matrix}\omega \ -2.405 \sin{\left(\theta \right)}\end{matrix}\right] $$

$0.2 \omega + 1.0 \theta \geq -5.0 \wedge 0.2 \omega + 1.0 \theta \leq 5.0$,

$$ \left[\begin{matrix} \theta \ \omega \end{matrix} \right] \rightarrow \left[ \begin{matrix} \omega \ \left(-0.5 \theta -4.905 \right) \sin{\left( \theta \right)}\end{matrix}\right] $$

See the notebook for more.

Installation

Install

pip install z3-solver ply jupyter

Run

jupyter notebook SEE-Reach.ipynb

Current Limitations

While SEE-Reach is a functional prototype, it's under ongoing development. The current version does not support loops and some other advanced features of Rust. Future versions may include a more comprehensive support for the language and additional symbolic execution strategies.

This implementation is confusing and fragile---it will be reimplemented in a different language (e.g., Rust or OCaml) and with a better architecture.

TODO: fill this out a bit more