An alternating timed automaton (ATA) is a modeling formalism that combines features of timed automaton[1] and an alternating finite automaton[2] to succinctly express sets of timed event sequences. Classical timed automata only allow existential nondeterministic branching in their transitions, while alternating finite automata model discrete untimed behaviors. Unlike timed automata, alternating timed automata are closed under complementation. However, this increased expressive power comes at the cost of undecidability in their emptiness problem. A one clock alternating timed automaton (OCATA) is a restricted version of an ATA, limited to the use of a single clock. OCATAs can express timed languages that cannot be expressed using standard timed automata.[3]
In automata theory, a timed automaton is a finite automaton extended with a finite set of real-valued clocks. During a run of a timed automaton, clock values increase all with the same speed. Along the transitions of the automaton, clock values can be compared to integers. These comparisons form guards that may enable or disable transitions and by doing so constrain the possible behaviors of the automaton are more expressive than timed automata.
An alternating timed automaton is defined as a timed automaton, where the transitions are more complex.
Given a set
X
lB+(X)
X
X
\phi\land\psi
\phi\lor\psi
\phi,\psi\inlB+(X)
For each letter
a
\ell
lBa,\ell
| |X| | |
| R | |
| \ge0 |
|X|
\nu
c(a,\ell,\nu)
lBa,\ell
\nu
An alternating timed-automaton contains a transition function, which associates to a 3-tuple
(\ell,a,c)
c\inlBa,\ell
lB+(L x lP(C))
For example,
(\ell1,\emptyset)\lor((\ell2,\{x\})\land(\ell2,\{y\}))
lB+(L x lP(C))
\ell1
\ell2
x
y
Formally, an alternating timed automaton is a tuple
lA=\langle\Sigma,L,L0,C,F,E\rangle
lA
L
L
lA
C
lA
L0\subseteqL
F\subseteqL
E\subseteqL x \Sigma x lB(C)\tolB+(L x lP(C))
lA
Any Boolean expression can be rewritten into an equivalent expression in disjunctive normal form. In the representation of an ATA, each disjunction is represented by a different arrow. Each conjunct of a disjunction is represented by a set of arrows with the same tail and multiple heads. The tail is labelled by the letter and each head is labelled by the set of clocks it resets.
A run of an alternating timed automaton over a timed word
w=(\sigma1,t1),(\sigma2,t2),...
In this definition, a run is not merely a list of pairs, but a rooted tree. The nodes of the rooted tree are labelled by pairs with a location and a clock valuation. The tree is defined as follows:
(\ell0,\nu0)
\ell0\inL0
n
i
(\ell,\nu)
E(ai+1,\ell,c(a,\ell,\nu))
| mi | |
| vee | |
| j=1 |
(\elli,j,ri,j)
n
mi
1\lei\len
j
(\elli,j,(\nu+ti+1-ti)[ri,j\to0]
The definition of an accepting run differs depending on whether the timed word is finite or infinite. If the timed word is finite, then the run is accepting if the label of each leaf contains an accepting location. If the timed word is infinite, then a run is accepting if each branch contains an infinite number of accepting locations.
A run can also be defined as a two player game
GA,w
Each state of the game is a tuple composed of a location, a clock valuation, a position in the word, and potentially an element of
lB+(L x lP(C))
(\ell,\nu,i,b)
i
\ell
\nu
b
(\ell0,\nu0,0)
\ell0\inL0
(\ell,\nu,i)
i
(\ell,\nu,i,c(ai+1,\ell,\nu))
(\ell,\nu,i,(\ell',r))
(\ell',\nu+ti-ti-1[r\to0],i+1)
(\ell,\nu,i,\phi\lor\psi)
(\ell,\nu,i,\phi)
(\ell,\nu,i,\psi)
(\ell,\nu,i,\phi\land\psi)
(\ell,\nu,i,\phi)
(\ell,\nu,i,\psi)
The set of successive states starting in a state of the form
(\ell,\nu,i)
The definition of an accepting run is the same as that for timed automata.
A one clock alternating timed automaton (OCATA) is an alternating timed automaton using a single clock.
The expressivity of OCATAs and of timed-automata are incomparable.
For example, the language over the alphabet
\{a\}
x
x
An ATA is said to be purely-universal (respectively, purely-existential) if its transition function does not use disjunction (respectively, conjunction).
Purely-existential ATAs are as expressive as non-deterministic timed-automaton.
The class of language accepted by ATAs and by OCATAs is closed under complement. The construction is explained for the case where there is a single initial location.
Given an ATA
lA=\langle\Sigma,L,\{q0\},C,F,E\rangle
L
Lc
Ac
\langle\Sigma,L,\{q0\},C,L\setminusF,E'\rangle
E'(\ell,a,c)
E(\ell,a,c))
E'(q0,a,c)
L0
It follows that the class of language accepted by ATAs and by OCATAs are accepted by unions and intersection. The union of two languages is constructed by taking disjoint copies of the automata accepting both languages. The intersection can be constructed from union and concatenation.
The emptiness problem, the universality problem and the containability problem are undecidable for ATAs, but decidable for OCATAs, though it is a nonelementary problem.