Well-formed or illegal mutual dependency between discrete-valued variables?

[henrikt-ma]

In appendix C, equation (C.1b), it is said that the discrete-valued variables m should be possible to determine by evaluation of fm(v, c). Even though this is some sort of high level informal description, the fact that this equation is given in the form of an assignment seems to suggest that given the other variables, one should be able to determine m in a straight-forward manner.

Does this mean that the following model does not fit this description, and therefore is invalid?

model MutualDiscreteValued
  Integer x;
  Integer y;
algorithm /* Use of algorithm instead of equation will become clear below. */
  x := y;
equation
  y = x;
end MutualDiscreteValued;

(SystemModeler currently rejects this model.)

If the model above is invalid, then what about this one?

model MutualDiscreteValuedUnused
  Integer x;
  Integer y;
algorithm
  x := y; /* Use of 'y' has no impact on the algorithm outputs. */
  x := 1;
equation
  y = x;
end MutualDiscreteValuedUnused;

The reason I ended up wondering is because of this model, where pre(x) takes the role of y during initialization:

model AlgorithmWithUnusedPre
  Integer x;
initial algorithm
  x := 1; /* Overwrites the value pre(x) formally assigned at the start of the algorithm. */
equation
  /* When-equation for x, resulting in
   *   x = pre(x)
   * during initialization, expected to be solved for pre(x).
   */
  when false then
    x = pre(x);
  end when;
end AlgorithmWithUnusedPre;

Of course, one of the assumptions in this last example was that the default equation for the inactive when-equation is really allowed to be solved for pre(x), which would actually need clarification in the specification considering that x is a discrete-time variable. Perhaps a clarification would be good either way?

[henrikt-ma]

For background, the problem I was trying to solve using the structure of AlgorithmWithUnusedPre was to use a variable to enforce an order between algorithm sections in the initialization problem. Maybe the correct way of doing this would be to use a parameter instead of a discrete-time variable? Like so:

model InitialAlgorithmOrdering
protected
  parameter Boolean initialAlgorithmComplete(fixed = false, start = false);
initial algorithm
  initializeExternalStuff();
  initialAlgorithmComplete := true;
algorithm
  assert(initialAlgorithmComplete, "Algorithms evaluated in the wrong order!");
  doStuffAssumingInitialized();
end InitialAlgorithmOrdering;

[HansOlsson]

I would say that the first two models are illegal. The fact that y has no impact in the algorithm in the 2nd example isn't something we would expect a tool to find.
Algorithms and equations for discrete variables can be sorted, but the algorithm-part rules out cycles. (Would be good if this was specified in more detail - similarly to https://specification.modelica.org/master/class-predefined-types-and-declarations.html#acyclic-bindings-of-constants-and-parameters

The InitialAlgorithmOrdering is on the other hand correct, as parameters with fixed=false are just unknowns for the initialization.

The AlgorithmWithUnusedPre should also be legal. The initialization problem becomes:

algorithm x:=1;
equation  x=pre(x);

where pre(x) has guess-value x.start. I admit that it's a bit weird to determine pre(x) in this backwards way, but at least it's consistent.

[henrikt-ma]

I would say that the first two models are illegal. The fact that y has no impact in the algorithm in the 2nd example isn't something we would expect a tool to find.
Algorithms and equations for discrete variables can be sorted, but the algorithm-part rules out cycles. (Would be good if this was specified in more detail - similarly to https://specification.modelica.org/master/class-predefined-types-and-declarations.html#acyclic-bindings-of-constants-and-parameters

Yes, that's the point of this issue. I can see how writing it nicely will be tricky, however, as one probably still want some kinds of simplifications to happen before analyzing the algorithm for dependencies, such as evaluation of if-equations and if-expressions with constant conditions, when-statement simplification for the initialization problem, etc…

[henrikt-ma]

The AlgorithmWithUnusedPre should also be legal. The initialization problem becomes:

algorithm x:=1;
equation  x=pre(x);

where pre(x) has guess-value x.start. I admit that it's a bit weird to determine pre(x) in this backwards way, but at least it's consistent.

Does this mean solving for a discrete-valued variable on the right hand side of an equation is OK in general? I've had the impression that all discrete-valued variables should have their own equation in solved form (with the variable on the left hand side of an equation or statement), but I'm not aware of this being explicitly stated anywhere…

[HansOlsson]

Does this mean solving for a discrete-valued variable on the right hand side of an equation is OK in general? I've had the impression that all discrete-valued variables should have their own equation in solved form (with the variable on the left hand side of an equation or statement), but I'm not aware of this being explicitly stated anywhere…

Good point. The restriction is not explicitly stated, but since we want to handle alias equations x=y due to connections without specifying alias-handling in detail it would make sense to allow it in solved form with the variable either on the left or the right hand side. I also noticed that the support in Dymola is somewhat inconsistent.

[eshmoylova]

By the way, in appendix C Henrik linked above I see some strange \cref@addtoreset...:

Is it just my browser or can somebody else see it? Is that a known issue or do we need to file it?

[HansOlsson]

By the way, in appendix C Henrik linked above I see some strange \cref@addtoreset...:

Is it just my browser or can somebody else see it? Is that a known issue or do we need to file it?

Known issue #2629

[HansOlsson]

Something like the following:

It shall be possible to compute m without any cycles, provided that the rest of v is known (thus there might be cyclic dependencies between m and y that require the solution of systems of equations). The variables in m need to occur explicitly solved on one of the sides of an equation, or assigned in an algorithm.

[henrikt-ma]

Something like the following:

It shall be possible to compute m without any cycles, provided that the rest of v is known (thus there might be cyclic dependencies between m and y that require the solution of systems of equations). The variables in m need to occur explicitly solved on one of the sides of an equation, or assigned in an algorithm.

Almost. I'd like to split m into its continuous-valued (discrete-time Real) and discrete-valued parts (all the rest), and then apply the above to just the discrete-valued part. (While I see no reason to have this sort of constraint on discrete-time Real variables in general, it is of course a given for any discrete Real variable due to the current requirement of being assigned in a when-clause.)

[HansOlsson]

Something like the following:
It shall be possible to compute m without any cycles, provided that the rest of v is known (thus there might be cyclic dependencies between m and y that require the solution of systems of equations). The variables in m need to occur explicitly solved on one of the sides of an equation, or assigned in an algorithm.

Almost. I'd like to split m into its continuous-valued (discrete-time Real) and discrete-valued parts (all the rest), and then apply the above to just the discrete-valued part. (While I see no reason to have this sort of constraint on discrete-time Real variables in general, it is of course a given for any discrete Real variable due to the current requirement of being assigned in a when-clause.)

Being assigned in a when-clause does not prevent loops:

  when sample(2, 2) then
     x=A*pre(x)+B*u+time;
  end when;
  when sample(3, 3) then
     u=A*pre(u)+B*x; 
  end when;

[henrikt-ma]

Being assigned in a when-clause does not prevent loops:

  when sample(2, 2) then
     x=A*pre(x)+B*u+time;
  end when;
  when sample(3, 3) then
     u=A*pre(u)+B*x; 
  end when;

Right, being assigned in a when-clause just means that there is no point requiring that the variable appears "explicitly solved on one side of the sides of an equation". I don't think we need to forbid loops like this, at least not for the same reasons why we don't want mutually dependent discrete-valued variables?

[HansOlsson]

Being assigned in a when-clause does not prevent loops:

  when sample(2, 2) then
     x=A*pre(x)+B*u+time;
  end when;
  when sample(3, 3) then
     u=A*pre(u)+B*x; 
  end when;

Right, being assigned in a when-clause just means that there is no point requiring that the variable appears "explicitly solved on one side of the sides of an equation". I don't think we need to forbid loops like this, at least not for the same reasons why we don't want mutually dependent discrete-valued variables?

I agree that part is automatically true for when-clauses, but the proposed text:

It shall be possible to compute m without any cycles, provided that the rest of v is known (thus there might be cyclic dependencies between m and y that require the solution of systems of equations). The variables in m need to occur explicitly solved on one of the sides of an equation, or assigned in an algorithm.

have two parts: solvable without cycles (which this examples fail), and they need to be explicitly solvable from an equation (which when-clauses must satisfy).

Boolean equations can fail the first and/or the second part:

It could be that it is not in one of the allowed forms (this example could actually be solved):
(b == (x>2)) = time>=1;
Or it just form a boolean system of equations:

   b1=if b2 then time>=1 else time>=2;
   b2=if b1 then x>2 else false;

Note that the restriction to allowed forms says that the variables should occur explicitly, which is stronger than saying that discrete-valued equations must be on this form, consider:

   b1=if b2 then time>=1 else time>=2;
   b1=(b2==(x>=2);

Here b2 cannot be solved from any equation, and thus we don't have to figure out whether there is a cycle or not.

I don't see the point of making an exception for the second part for when-clauses merely because when-clauses automatically fulfill that.

[henrikt-ma]

It shall be possible to compute m without any cycles, provided that the rest of v is known (thus there might be cyclic dependencies between m and y that require the solution of systems of equations). The variables in m need to occur explicitly solved on one of the sides of an equation, or assigned in an algorithm.

have two parts: solvable without cycles (which this examples fail), and they need to be explicitly solvable from an equation (which when-clauses must satisfy).

Yes, and of the two parts, it's the cycles part that I don't find relevant for the discrete-time Real variables. (It was probably wrong to bring up that variables assigned in when-clauses already fulfill the other part.)

Boolean equations can fail the first and/or the second part:

Sure.

Here b2 cannot be solved from any equation, and thus we don't have to figure out whether there is a cycle or not.

Right.

I don't see the point of making an exception for the second part for when-clauses merely because when-clauses automatically fulfill that.

Agreed, there would be no point to make an exception for that reason. The reason for making an exception would be to not mix up the requirements needed for discrete-valued variables with any requirements needed for discrete-time variables. To me, the explicitly solvable and not-in-a-cycle has only to do with being discrete-valued, as it is the foundation for the "standard" fixed-point iteration method for solving systems of equations involving both discrete-valued and continuous-valued variables.