In the following we give a pseudo-formal semantics for UPPAAL. The semantics defines a timed transition system (S, s0, ->) describing the behaviour of a network of extended timed automata. The set of states S is defined as {(L, v) | v satisfies Inv(L)}, where L is a location vector, v is a function (called a valuation) mapping integer variables and clocks to their values, and Inv is a function mapping locations and location vectors to invariants. The initial state s0 is the state where all processes are in the initial location, all variables have their initial value, and all clocks are zero. The transition relation, ->, contains two kinds of transitions: delay transitions and action transitions. We will describe each type below.
Given a valuation v and an expression e, we say that v satifies e if e evaluates to non-zero for the given valuation v.
If during a successor computation any expression evaluation is invalid (consult the section on expressions for further details about invalid evaluations), the verification is aborted.
Delay transitions model the passing of time without changing the current location. We have a delay transition (L, v) --(d)--> (L, v'), where d is a non-negative real, if and only if:
For action transtions, the synchronisation label of edges is important. Since UPPAAL supports arrays of channels, we have that the label contains an expression evaluating to a channel. The concrete channel depends on the current valuation. To avoid cluttering the semantics we make the simplifying assumption that each synchronisation label refers to a channel directly.
Priorities increase the determinism of a system by letting a high priority action transition block a lower priority action transition. Note that delay transitions can never be blocked, and no action transition can be blocked by a delay transition.
For action transitions, there are three cases: Internal transitions, binary synchronisations and broadcast synchronisations. Each will be described in the following.
We have a transition (L, v) --*--> (L', v') if there is an edge e=(l,l') such that:
We have a transition (L, v) --*--> (L', v') if there are two edges e1=(l1,l1') and e2=(l2,l2') in two different processes such that:
Assume an order p1, p2, ... pn of processes given by the order of the processes in the system declaration statement. We have a transition (L, v) --*--> (L', v') if there is an edge e=(l,l') and m edges ei=(li,li') for 1<=i<=m such that: