Signal automaton

In automata theory, a field of computer science, a signal automaton is a finite automaton extended with a finite set of real-valued clocks. During a run of a signal 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. Further, clocks can be reset. 
Example.
Before formally defining what a signal automaton is, an example will be given. Let one consider the language formula_1 of signals, over a binary alphabet formula_2, which contains signals formula_3 such that:
This language can be accepted by the automaton pictured nearby.
As for finite automaton, incoming arrows represents initial locations and double circle represents accepting locations. However, contrary to finite automata, letters occurs in locations and not in transition. This is because letters are emitted continuously and transitions are taken discretely. The symbol formula_7 represents a clock. This clock allow to measure the time since the last time where formula_4 was emitted. Thus formula_9 ensures that formula_4 is emitted discretely. And formula_11 ensures that no more than a unit of time can pass without formula_4 occurring.
Formal definition.
Signal automaton.
Formally, a signal automaton is a tuple formula_13 that consists of the following components:
An edge formula_29 from formula_30 is a transition from locations formula_31 to formula_32 which reset the clocks of formula_33.
Extended state.
A pair with a location formula_31 and a clock valuation formula_35 is called either an "extended state" or a "state".
Note that the word state is thus ambiguous, since, depending on the author, it may means either a pair or an element of formula_16. For the sake of the clarity, this article will use the term location for element of formula_16 and the term extended location for pairs.
Here lies one of the biggest difference between signal-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 now in which location you are, and the clock valuation.
Run.
As for finite automata, a run is essentially a sequence of locations, such that there exists a transition between two locations. However, two differences must be emphasized. The letter is not determined by the transition but by the locations; this is due to the fact that the letters are emitted continuously while transitions are taken discretely. Some time elapse while in a location; the clock constraints labelling a location or its successor may constraint the time spent in a single location.
A "run" is a sequence of the form formula_38 satisfying some constraints. Before stating those constraints, some notations are introduced. The sequences are discrete but represents continuous events. A continuous verson of the sequences formula_39, formula_40, formula_41 are now introduced. Let formula_42 integral and formula_43, then
The constraints satisfied by run are, for each formula_42 integral and formula_52 real:
The signal defined by this run is the function formula_57 defined above. It is said that the run defined above is a run for the signal formula_57.
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 formula_59 is finite of length formula_60, then the run is accepting if formula_61. If the word is infinite, then the run is accepting if and only if there exists an infinite number of position formula_62 such that formula_63.
Accepted signals and language.
A signal formula_3 is said to be accepted by a signal automaton formula_15 if there exists a run of formula_15 on formula_3 accepting it. The set of signals accepted by formula_15 is called the language accepted by formula_15 and is denoted by formula_70.
Deterministic signal automaton.
As in the case of finite and Büchi automaton, a signal-automaton may be deterministic or non-deterministic. Intuitively, being deterministic as the same meaning in each of those case. It means that the set of start locations is a singleton, and that, given an extended state formula_71, and a letter formula_72, there is only one possible extended state which can be reached from formula_71 by reading formula_72. More precisely, either it is possible to stay in the location longer, or there is at most one possible successor location.
Formally, this can be defined as follows:
Simplified signal automata.
Depending on the authors, the exact definition of signal automata may be slightly different. Two such definitions are now given.
Half-open intervals.
In order to simplify the definition of a run, some authors requires that each interval of a run is right-closed and left-open. This restrict automata to accept only signals whose underlying partition satisfies the same property. However, it ensures that at each time formula_52, formula_88 for formula_89 representing any of the function formula_57, formula_91 or formula_32 introduced above.
Bipartite signal automaton.
A bipartite signal automaton is a signal automaton in which the run alternates between open intervals and singular intervals (i.e. intervals which are singletons). It ensures that the graph underlying the automaton is a bipartite graph, and thus that the set of locations can be partitionned into formula_93, the set of open locations and of singular locations. Since the first interval contains 0, it can not be an open location, it follows that formula_94. In order to ensure that each singular location is indeed singular, for each location formula_31, there must be a clock formula_96 which is reset when entering formula_31 and such that the clock constraint of formula_31 contains formula_9.
Any signal automaton can be transformed into an equivalent bipartite signal automaton. It suffices to replace each location formula_31 by a pair of locations formula_101 and introduce a new clock formula_7, such that for each formula_31, formula_104.
Nearby is pictured a bipartite automaton equivalent to the signal automaton from the example section. Rectangle states represents singular locations. 
Synchronization of automata.
The notion of product of finite automaton is extended to signal automaton. However, such a product is called a synchronization of automaton to emphasize the fact that the time should pass similarly in both automata considered. The main difference between synchronization and product is that, when two finite automata read the same word, they take transition simultaneously. This is not the case anymore for signal automata, since they can take transition at arbitrary time. Thus, the transition relation of a signal automaton may allow transition to be taken in one or two automata.
Let formula_105 and formula_106 two signal automata, their synchronization is the signal automaton formula_107, where formula_30 contains the following transitions:
Difference with timed automata.
Timed automata is another extension of finite automata, which adds a notion of time to words. We now state some of the main differences between timed automata and signal automata.
In timed automata, letters are emitted on the transitions and not in the locations. As explained above when comparing signal automata to finite automata, letters are emitted on transitions when the words are emitted discretely, as for words and timed-words while they are emitted on locations when letters are emitted continuously, as for signals.
In timed automata, guards are only checked on transitions. This simplifies the definition of deterministic automaton, since it means that the constraint must be satisfied before the clocks are restarted.