Cyclic dependency between variable and its start attribute

[phannebohm]

According to MLS section 4.9.6

The attributes start and fixed define the initial conditions for a variable.

Consider the following model:

model CyclicStartAttribute
  parameter Real p(fixed = false);
  Real x(start = p);
initial equation
  1 = p^2 + x^2;
equation
  x = p + time;
end CyclicStartAttribute;

My interpretation is that CyclicStartAttribute is illegal because x.start and x cyclically depend on each other and have no initial guess available. Is this correct? Should it be made explicit in the MLS?

Now consider the following model:

model ReverseStartAttribute
  parameter Real p(fixed = false);
  Real x(start = p);
initial equation
  1 = p^2 + x^2;
equation
  x = time;
end ReverseStartAttribute;

Here x.start depends on x but is not part of an algebraic loop and in this case x.start is not needed because x can be computed directly from its defining equation. Should a compiler ignore x.start or should ReverseStartAttribute also be illegal?

Related Issues

This is similar but not identical to #2668 and #3278.

[phannebohm]

Adding @adrpo to the loop. We encountered this in OpenModelica issue 15433.

[henrikt-ma]

System Modeler rejects CyclicStartAttribute and happily accepts ReverseStartAttribute.

I'm afraid we don't have clear rules for this in full Modelica, but for Base Modelica we worked this out in https://github.com/modelica/ModelicaSpecification/blob/MCP/0031/RationaleMCP/0031/differences.md#implicitly-declared-guess-value-parameter:

When a guess value for 'x' is needed to solve an equation (or system of equations), this equation has an implicit dependency on guess('x'), and it is an error if guess('x') cannot be determined first.

Note the needed in this rule; this is what saves ReverseStartAttribute.

[HansOlsson]

To me the important part is that this is a guess-value, and ideally the solution shouldn't critically depend on the start-value.
Thus a practical solution is to just handle this - or even ignore the start-value (possibly with diagnostics).

Trying to formulate a rule that depends on whether the start-value is needed or not runs into the problem that it depends on what symbolic transformations tools do.

Consider:

model CyclicStartAttribute2
  parameter Real p(fixed = false);
  Real x(start = p);
  parameter Integer n=3;
initial equation 
  x=p;
equation 
  2+time = x^n;
end CyclicStartAttribute2;

Here the start-value of x is needed if:

• if n is even and tools solve this symbolically (note: the guess-value isn't needed to find a solution, but the solution closest to the guess-value).
• always if the tool views this as a general non-linear equation.

If n=1 and it was evaluated then the start-value of x would clearly not be needed.

[henrikt-ma]

Yes, this is a tricky domain to nail down in the specification, so maybe we shouldn't set the expectations on standardization too high.

As a tool vendor trying to act responsibly, I would be hesitant to just ignore guess values with a warning, and I have yet to see an example in the wild where a modeler insists on specifying a non-applicable "entangled guess".

[casella]

To me the important part is that this is a guess-value, and ideally the solution shouldn't critically depend on the start-value.

I agree that models shouldn't critically depend on start values. Homotopy-based initialization is a key tool to mitigate this issue. However, there are still models containing numerically challenging nonlinear problems for which it is very important to get the right guess-values, and it is essential that you get the same guess value with all tools, otherwise the model won't be portable. Which is exactly what is happening in the above-mentioned issue.

Thus, I believe that getting at least a warning about potential or actual cyclic dependencies between variables and their start attributes is important. If one wants to write portable models, being aware of this potential condition is essential.