CIF 3

State (exclusion) invariantsΒΆ

The lesson on Discrete variables used the following CIF specification:

automaton counter:
  event increment, decrement;

  disc int count = 3;

  location:
    initial;
    edge decrement when count > 0 do count := count - 1;
    edge increment when count < 5 do count := count + 1;
 end

The counter can repeatedly be incremented and decremented by one, as long as the count remains at least one and at most five. To keep the count in the allowed range of values, guards were used to limit the occurrence of the increment and decrement events.

Instead of using guards, it is also possible to use state (exclusion) invariants, also called state invariants, or just invariants:

automaton counter:
  event increment, decrement;

  disc int count = 3;

  invariant count >= 0;                   // Added invariants
  invariant count <= 5;

  location:
    initial;
    edge decrement do count := count - 1; // No more guards
    edge increment do count := count + 1;
 end

The guards on the edges have been replaced by the two invariants. The first invariant specifies that the value of variable count must always be at least zero. The second invariant specifies that the value must also be at most five.

Invariants specify conditions that must always hold. Invariants must hold in the initial state, and all states reached via transitions. If a transition results in a state where an invariant doesn’t hold, the transition is not allowed and can’t be taken.

For the counter example, initially the count is 3. The edge for the increment event can be taken, leading to a state where the count is 4. Taking another transition for the increment event leads to a state where the count is 5. If we then were to take another transition for the increment event, the count would then become 6. However, that violates the invariant. Therefore, in the state where the count is 5, no transition for the increment event is possible. In other words, the invariant disables the transition for the increment event for that state.

The two invariants can be specified in various ways:

// Multiple invariants, each with a single predicate.
invariant count >= 0;
invariant count <= 5;

// Single invariant, with multiple predicates.
invariant count >= 0, count <= 5;

// Single invariant, with single predicate.
invariant count >= 0 and count <= 5;

Each of these variants leads to the exact same behavior, and which variant to use depends mostly on the modeler’s own preference.

The benefit of guards over invariants is that they more explicitly state the condition under which an edge can lead to a transition. If a guard doesn’t hold, the edge can’t be part of a transition. It is thus immediately clear when the edge can lead to a transition. For invariants, the update has to be calculated first, after which the invariants can be evaluated for the state resulting from the transition. If one of the invariants doesn’t hold, the transition is not allowed. In the case of the invariants, it is not as immediately clear from the edge alone, when that specific edge can or can not lead to a transition.

The benefit of invariants over guards is that they apply to all edges. If several edges in an automaton have updates to the same variable, then the invariants need to be specified only once, and apply to all transitions, for all edges. Using guards, all the edges that modify the variable would need their own guards, and if the updates are different, the different edges usually require different guards. Furthermore, if new edges with updates to the same variables are added, the invariant is already present, but guards have to be added, which can easily be forgotten. In those cases, invariants can thus help keep the specification consistent.

Another benefit of invariants is that they explicitly state the conditions that must hold in relation to the variables, while guards specify the condition under which the update is allowed. Consider the following CIF specification:

// Using invariants.
automaton a:
  disc int x;

  invariant 0 <= x, x <= 100;

  location:
    initial;
    edge do x := 2 * x + 3;
end

// Using guards.
automaton a:
  disc int x;

  location:
    initial;
    edge when x <= 48 do x := 2 * x + 3;
end

The goal is to keep the value of in the range [0..100]. The invariant is simple and direct. The guard however, has to state the condition under which the update does not violate the goal. That is, the upper bound has to be decreased by three, and the result has to be divided by two, to get the highest value (48) for which the update is still within the valid range of values. That is, for value 48 the update results in value 99 (2 * 48 + 3 = 99), and for value 49 the update results in value 101 (2 * 48 + 3 = 99). The more complex the update, the harder it is to figure out the guard to use to keep satisfy the goal.

You can of course also use 2 * x + 3 <= 100 as guard, instead of x <= 48. However, this duplicates part of the update in the guard.

So far, all invariants have been specified in automata. They may however also be specified outside of the automata, similar to initialization predicates. It is generally recommended to place an invariant inside an automaton if the condition only applies to declarations from that automaton, and to place it outside of the automata if the condition applies to declarations of multiple automata.

Furthermore, invariants can be placed in a location. Such an invariant only has to hold while the location in which it is specified is the current location of its automaton.