Timed automata are objects that model the behavior of systems with clocks. They have states and clocks, and the values of the clocks grow with a unit slope. Timed automata can transition between states based on guards that are satisfied by the values of the clocks. By assigning letters to the states and fixing initial and final states, timed automata can be used as generators/acceptors of timed languages.

A timed automaton is a tuple $A = (Q, C,\Delta,\Sigma, \lambda,S, F)$ consisting of a finite set of states (Q), a finite set of clocks (C), an output alphabet (), a transition relation (), an output map ( Q ), an initial set (S), and an accepting set (F).

The transition relation is a set of tuples, where each tuple consists of a current state (q), a next state (q'), and a guard (a Boolean combination of formulas of the form c I ), where c is a clock in C and I is a time interval.

In summary, a timed automaton is a mathematical model that defines the behavior of a system with clocks, consisting of states, clocks, transitions, and input/output symbols.

A timed regular expression is defined recursively as either:

• a single symbol $a$ from the alphabet $\Sigma$
• the empty expression $\epsilon$
• the timed delay expression $t$, where $t$ is a non-negative real number
• the union of two timed regular expressions $E_1 \cup E_2$, where $E_1$ and $E_2$ are timed regular expressions
• the concatenation of two timed regular expressions $E_1 \cdot E_2$, where $E_1$ and $E_2$ are timed regular expressions
• the Kleene star of a timed regular expression $E^*$, where $E$ is a timed regular expression

The "timed" part of the expressions comes from the timed delay expression $t$, which specifies a minimum amount of time that must pass before the next symbol in the language can be recognized.

For example, the timed regular expression $(a\cdot t\cdot b)^*$ represents the language consisting of all strings that start and end with the symbols $a$ and $b$, respectively, and where there is a delay of at least $t$ time units between the $a$ and $b$ symbols.

The definition also specifies that the alphabet $\Sigma$ can contain any set of symbols, and that the timed delay expression $t$ can be any non-negative real number.

Overall, timed regular expressions provide a way to describe languages that take into account not only the symbols in the language, but also the timing constraints between those symbols. This makes them useful for modeling and analyzing systems with time-dependent behavior, such as real-time systems and communication protocols.

A timed automaton is defined by a tuple $A = (S, s_0, \Sigma, \Delta, F)$, where:

• $S$ is a finite set of states