How does Uppaal clock evolve? I have two locations 1 and 2 without invariant. What is clock value?

Question

How does Uppaal clock evolve? I have two locations 1 and 2 without invariant, a clock reset to zero (0) on the transition into location 1. On the edge between location 1 and 2, how do I know the value of the clock at this time? (That is, the value of the clock in between the two locations before location 2). Does the clock continue to evolve from location 1 to location 2 and beyond or an automatic reset takes place at the entrance to a new location?

Answer 1

TLDR; answer. If the automaton starts with x==0 and moves through the locations without invariants, then automaton may delay, say 5 time units, then move to another location with x==5, then delay again, say 3.141 time units, which moves x to value 8.141, and so on. Notice that clock x can get to arbitrary real value (hence my arbitrary choices) by delaying and taking transitions, which means that all those possibilities need to be analysed. Uppaal captures all those possible values in the form of constraints (or the lack of them in this case when there are no invariants or guards, the simulator may show just x==y because all clocks are synchronized).

Some Examples. Suppose we have a timed automaton as described:

Then the value of clock x can evolve like this:

If we make 1000 simulations then the trajectories of x value will look like this:

As you can see it slowly fills the lower half of the plane -- a zone. UPPAAL uses Difference Bound Matrices to describe and analyze such zones symbolically.

The plots above were drawn using the following queries:

simulate [<=50] { x }
simulate [<=50; 1000] { x }

Some context. Uppaal implements timed automata with clock variables whose values change continuously with a rate (time derivative) of 1. So if the clock is reset to 0 and automaton arrives to a location without invariants (also not urgent, nor committed), then clocks are free to evolve, therefore can have arbitrary value from 0 to infinity. Uppaal represents such valuations symbolically using constraints (intervals) packed into difference bound matrices (DBM). If an automaton takes a transition, then Uppaal will analyse all possible transitions satisfying the constraints at once. For example, if a location has an invariant x<=5 and edge has a guard x>=2, then transition is available when x is anywhere between 2 and 5, therefore Uppaal will take a symbolic transition with constraints 2<=x && x<=5 which captures all those possible transitions at once. This allows analysis of infinitely many transitions in finite data structures and finite time.

Some common cases which may confuse the novice. If there are multiple automata in the system, then the passage of time is analysed globally, i.e. an invariant in one automaton will have an effect on other clocks in other automata, because all clocks are synchronized through global time.

Timed automata allow only integers in the guards and invariants, which in principle can be scaled to accomodate models with rational numbers. Uppaal also extends the timed automata with urgent, committed locations, select statements, broadcast synchronization, integer variables, function calls etc, which are still analysable under the same timed automata theory, but makes modeling more expressive and succint.

You can read more in Tutorial on Uppaal, under the documentation section of http://uppaal.org : http://www.it.uu.se/research/group/darts/uppaal/documentation.shtml#tutorials