Alternating timed automaton

In automata theory, an alternating timed automaton (ATA) is a mix of both timed automaton and alternating finite automaton. That is, it is a sort of automata which can measure time and in which there exists universal and existential transition.
ATAs are more expressive than timed automaton. one clock alternating timed automaton (OCATA) is the restriction of ATA allowing the use of a single clock. OCATAs allow to express timed languages which can not be expressed using timed-automaton.
Definition.
An alternating timed automaton is defined as a timed automaton, where the transitions are more complex.
Difference from a timed-automaton.
Given a set formula_1, let formula_2 the set of positive Boolean combination of elements of formula_1. I.e. the set containing the elements of formula_1, and containing formula_5 and formula_6, for formula_7.
For each letter formula_8 and location formula_9, let formula_10 be a set of clock constraints such that their zones partition formula_11, with formula_12 the number of clocks. Given a clock valuation formula_13, let formula_14 be the only clock constraint of formula_10 which is satisfied by formula_13.
An alternating timed-automaton contains a transition function, which associates to a 3-tuple formula_17, with formula_18, to an element of formula_19.
For example, formula_20 is an element of formula_19. Intuitively, it means that the run may either continue by moving to location formula_22, and resetting no clock. Or by moving to location formula_23 and should be successful when either formula_24 or formula_25 is reset.
Formal definition.
Formally, an alternating timed automaton is a tuple formula_26 that consists of the following components:
Any Boolean expression can be rewritten into an equivalent expression in disjunctive normal form. In the representation of a 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.
Run.
We now define a run of an alternating timed automaton over a timed word formula_37. There are two equivalent way to define a run, either as a tree or as a game.
Run as a tree.
In this definition of a run, a run is not anymore a list of pairs, but a rooted tree. The node of the trooted tree are labelled by pairs with a location and a clock valuation. The tree is defined as follows:
The definition of an accepting runs 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 location.
Run as a game.
A run can also be defined as a two player game formula_50. Let us call the two players "player" and "opponent". The goal of the player is to create an accepting run and the goal of the opponent is to create a rejecting (non-accepting) run.
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 formula_19. Intuitively, a tuple formula_52 means that the run has read formula_41 letters, is in location formula_9, with clock value formula_13, and that the transition will be as described by formula_56. The run is defined as follow:
The set of successive states starting in a state of the form formula_59 and ending in before the next such state is called a phase.
The definition of an accepting run is the same than for timed automata.
Subclass of ATA.
One clock alternating timed automaton.
A one clock alternating timed automaton (OCATA) is an alternating timed automaton using a single clock.
The expressivity of OCATAs and of timed-automaton are incomparable.
For example, the language over the alphabet formula_71 such that there is never exactly one time unit between two letters can not be recognized by a timed-automaton. However, the OCATA pictured nearby accepts it. In this alternating timed automaton, two branches are started. A branch restarts the clock formula_24, and ensures that each time in the future when a letter is emitted, the clock formula_24 is distinct from 1. This ensure that between this letter and the next ones, the time elapsed is not one. The second branch only waits for other letters to be emitted and do the same checking.
Purely-Universal and Purely-Existential ATA.
An ATA is said to be purely-universal (respectively, purely-exisential) if its transition function does not use disjunction (respectively, conjunction).
Purely-existential ATAs are as expressive as non-deterministic timed-automaton.
Closure.
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 formula_74 accepting a timed language formula_28, its complement language formula_76 is accepted by an automaton formula_77 which is essentially formula_78, where formula_79 is defined as formula_80 where disjunction and conjunctions are reversed and formula_81 simulates the run from each of the locations of formula_82 simultaneously.
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 language. The intersection can be constructed from union and concatenation.
Complexity.
The emptiness problem, the universality problem and the containability problem for OCATA is decidable but is a nonelementary problem.
Those three problems are undecidable over ATAs.