Digital Bifurcation Theory

What do these items have in common?

physical (continuous) components

digital (discrete) components

interactions

interactions

Cyber-physical systems

Cyber-physical systems are used at global scale in safety critical applications.

Wrong assumptions about a cyber-physical system can cost lives! (and millions of dollars)

Problem: System's behaviour can change with parameters.

Friction 

Wind speed

Traction control on/off

Turbocharger on/off

Emission avoiding on/off

Fuel efficiency

Problems in cyber-physical systems:

Verification: given a system, validate that the system satisfies given specification.

(Regardless of wind speed, the airplane won't crash)

Synthesis: given a family of systems, find one that satisfies given specification.

(Across all possible fuels, this turbocharger setting provides best efficiency)

Control: given a system's state, adjust the system to satisfy the specification.

(For current wheel rotation speed, breaks have to be turned on to avoid skid)

How to solve them?

Reductionist approach: Analysis in isolation

Continuous (physical) Systems:

  • differential equations
  • behaviour patterns
  • bifurcation analysis
  • analytical or numerical methods

In continuous systems small (smooth) change in parameters usually produces a small change in behaviour.

Problem: Usually

Bifurcation: Big (qualitative) change of behaviour 

Behaviour patterns:

     Stability ➔ 

➔ Oscillation ➔ 

➔ Stability

Reductionist approach: Analysis in isolation

Discrete (digital) Systems:

  • transition systems
  • formal verification
  • temporal properties
  • model checking

If I insert a coin, will I eventually get a beverage?

\textbf{G} (coin \Rightarrow \textbf{F} (soda \lor beer))
G(coin⇒F(soda∨beer))\textbf{G} (coin \Rightarrow \textbf{F} (soda \lor beer))

In discrete systems, there is no small parameter change. Any change in parameters can have unexpected consequences.

Temporal properties are often too restrictive: 

System stabilises in state X.

What if we don't know X?

??

??

Is there a more holistic approach?

Behavioural Patterns

Stable equilibrium

Cycle (oscillation)

Unstable equilibrium

General attractor

\textbf {bind \textrm{s}: AX \textrm{s}}
bind s: AX s\textbf {bind \textrm{s}: AX \textrm{s}}
\textbf{bind} \text{s}: \textbf{EX}\ \text{s}\ \land\ \neg\textbf{AX}\ s
binds:EX s ∧ ¬AX s\textbf{bind} \text{s}: \textbf{EX}\ \text{s}\ \land\ \neg\textbf{AX}\ s
\textbf{bind} \text{s}: \textbf{AF}\ \text{s}\ \land\ \neg\textbf{AX}\ s
binds:AF s ∧ ¬AX s\textbf{bind} \text{s}: \textbf{AF}\ \text{s}\ \land\ \neg\textbf{AX}\ s
\textbf{bind} \text{s}: \textbf{AG EF}\ \text{s}
binds:AG EF s\textbf{bind} \text{s}: \textbf{AG EF}\ \text{s}

HUCTL: Logic for behavioural pattern specification

Behavioural patterns

Parametric model checking

Pattern validity regions

Summary

  • Cyber-physical systems appear in safety critical applications across various scientific and industrial fields.
  • Holistic, global analysis techniques are not properly developed.
  • Our technique:
    • Extends the notion of bifurcation to discrete systems.
    • Provides an automated method based on formal verification to solve the global discrete bifurcation problem.

Interested? Contact us at sybila.fi.muni.cz