Timed automaton

Formal definition

Timed automaton

Formally, a timed automaton is a tuple ${\displaystyle \mathcal{A}=⟨\mathrm{\Sigma },L,L_0,C,F,E⟩}$ that consists of the following components:

• ${\displaystyle \mathrm{\Sigma }}$ is a finite set called the alphabet or actions of ${\displaystyle \mathcal{A}}$.
• ${\displaystyle L}$ is a finite set. The elements of ${\displaystyle L}$ are called the locations or states of ${\displaystyle \mathcal{A}}$.
• ${\displaystyle L_0\subseteq L}$ is the set of start locations.
• ${\displaystyle C}$ is a finite set called the clocks of ${\displaystyle \mathcal{A}}$.
• ${\displaystyle F\subseteq L}$ is the set of accepting locations.
• ${\displaystyle E\subseteq L\times \mathrm{\Sigma }\times \mathcal{B}(C)\times \mathcal{P}(C)\times L}$ is a set of edges, called transitions of ${\displaystyle \mathcal{A}}$, where
  • ${\displaystyle \mathcal{B}(C)}$ is the set of clock constraints involving clocks from ${\displaystyle C}$, and
  • ${\displaystyle \mathcal{P}(C)}$ is the powerset of ${\displaystyle C}$.

An edge ${\displaystyle (\ell ,\sigma ,g,r,{\ell }^{\prime })}$ from ${\displaystyle E}$ is a transition from locations ${\displaystyle \ell }$ to ${\displaystyle {\ell }^{\prime }}$ with action ${\displaystyle \sigma }$, guard ${\displaystyle g}$ and clock resets ${\displaystyle r}$.

Extended state

A pair with a location ${\displaystyle \ell }$ and a clock valuation ${\displaystyle \nu }$ is called either an extended state or a state.

Note that the word state is thus ambiguous, since, depending on the author, it may mean either a pair or an element of ${\displaystyle L}$. For the sake of the clarity, this article will use the term location for element of ${\displaystyle L}$ and the term extended location for pairs.

Here lies one of the biggest difference between timed automata and finite automata. In a finite automaton, at some point of the execution, the state is entirely described by the number of letter read and by a finite number of possible values, which are actually called "states". That means that, given a state and a suffix of the word to read, the remaining of the run is totally determined. Thus, the word "finite" in the name "finite automata". However, as it is explained in the section "run" below, in order to resume clocks are used to determine which transitions can be taken. Thus, in order to know the state of the automaton, you must both know in which location you are, and the clock valuation.

Run

Given a timed word ${\displaystyle w=({\sigma }_1,t_1),({\sigma }_2,t_2),\dots ,}$ with ${\displaystyle {\sigma }_i\in \mathrm{\Sigma }}$, ${\displaystyle (t_i{)}_i}$ an increasing sequence of non-negative number, and a timed automaton ${\displaystyle \mathcal{A}}$ as above, a run is a sequence of the form ${\displaystyle ({\ell }_0,{\nu }_0)\underset{t_1}{\overset{{\sigma }_1}{\to }}({\ell }_1,{\nu }_1)\dots }$ satisfying the following constraint:

• (initialization) ${\displaystyle {\ell }_0\in L_0}$
• (consecution), for all ${\displaystyle i\ge 1}$, there exists an edge in ${\displaystyle E}$ of the form ${\displaystyle ⟨{\ell }_{i-1},{\sigma }_i,g_i,r_i,{\ell }_i⟩}$ such that:
  • we assume that ${\displaystyle t_i-t_{i-1}}$ time units passed, and at this time, the guard is satisfied. I.e. ${\displaystyle {\nu }_{i-1}+t_i-t_{i-1}}$ satisfies ${\displaystyle g_i}$,
  • the new clock valuation ${\displaystyle {\nu }_i}$ corresponds to ${\displaystyle {\nu }_{i-1}}$, in which ${\displaystyle t_i-t_{i-1}}$ time units passed and in which the clocks of ${\displaystyle r_i}$ where reset. Formally, ${\displaystyle {\nu }_i=({\nu }_{i-1}+t_i-t_{i-1})[r_i\to 0]}$.

The notion of accepting run is defined as in finite automata for finite words and as in Büchi automata for infinite words. That is, if ${\displaystyle w}$ is finite of length ${\displaystyle n}$, then the run is accepting if ${\displaystyle {\ell }_n\in F}$. If the word is infinite, then the run is accepting if and only if there exists an infinite number of position ${\displaystyle i}$ such that ${\displaystyle {\ell }_i\in F}$.

Deterministic timed automaton

As in the case of finite and Büchi automaton, a timed automaton may be deterministic or non-deterministic. Intuitively, being deterministic has the same meaning in each of those case. It means that the set of start locations is a singleton, and that, given a state ${\displaystyle s}$, and a letter ${\displaystyle a}$, there is only one possible state which can be reached from ${\displaystyle s}$ by reading ${\displaystyle a}$. However, in the case of timed automaton the formal definition is slightly more complex. Formally, a timed automaton is deterministic if:

• ${\displaystyle L_0}$ is a singleton
• for each pair of transitions ${\displaystyle (\ell ,\sigma ,g,r,{\ell }^{\prime })}$ and ${\displaystyle (\ell ,\sigma ,g^{\prime },r^{\prime },{\ell }^{″})}$, the set of clocks valuations satisfying ${\displaystyle g}$ is disjoint from the set of clocks valuations satisfying ${\displaystyle g^{\prime }}$.

Closure property

The class of languages recognized by non-deterministic timed automata is:

• closed under union, indeed, the disjoint union of two timed automata recognize the union of the language recognized by those automata.
• closed under intersection.
• not closed under complement.

Problems and their complexity

The computational complexity of some problems related to timed automata are now given.

The emptiness problem for timed automata can be solved by constructing a region automaton and checking whether it accepts the empty language. This problem is PSPACE-complete.

The universality problem of non-deterministic timed automaton is undecidable, and more precisely Π¹
₁. However, when the automaton contains a single clock, the property is decidable; however, it is not primitive recursive. This problem consists of deciding whether every word is accepted by a timed automaton.