Equivalence of clocks assignment

The first definition allow to easily test whether two assignments belong to the same region.

A region may be defined as an equivalence class for some equivalence relation. Two clocks assignments ${\displaystyle \nu _{1}}$ and ${\displaystyle \nu _{2}}$ are equivalent if they satisfy the following constraints:^[2]: 202

• ${\displaystyle \nu _{1}(x)\sim c}$ iff ${\displaystyle \nu _{2}(x)\sim c}$, for each ${\displaystyle x\in C}$ and ${\displaystyle 0\leq c\leq c_{x}}$ an integer, and ~ being one of the following relation =, < or ≤.
• ${\displaystyle \{\nu _{1}(x)\}\sim \{\nu _{1}(y)\}}$ iff ${\displaystyle \{\nu _{2}(x)\}\sim \{\nu _{2}(y)\}}$, for each ${\displaystyle x,y\in C}$, ${\displaystyle \nu _{1}(x)\leq c_{x}}$, ${\displaystyle \nu _{1}(y)\leq c_{y}}$, ${\displaystyle \{r\}}$ being the fractional part of the real ${\displaystyle r}$, and ~ being one of the following relation =, < or ≤.

The first kind of constraints ensures that ${\displaystyle \nu _{1}}$ and ${\displaystyle \nu _{2}}$ satisfies the same constraints. Indeed, if ${\displaystyle \nu _{1}(x)=0.5}$ and ${\displaystyle \nu _{2}(x)=1}$, then only the second assignment satisfies ${\displaystyle x=1}$. On the other hand, if ${\displaystyle \nu _{1}(x)=0.5}$ and ${\displaystyle \nu _{2}(x)=0.6}$, both assignment satisfies exactly the same set of constraint, since the constraints use only integral constants.

The second kind of constraints ensures that the future of two assignments satisfy the same constraints. For example, let ${\displaystyle \nu _{1}=\{x\mapsto 0.5,y\mapsto 0.6\}}$ and ${\displaystyle \nu _{2}=\{x\mapsto 0.5,y\mapsto 0.4\}}$. Then, the constraint ${\displaystyle y=1\land x<1}$ is eventually satisfied by the future of ${\displaystyle \nu _{1}}$ without clock reset, but not by the future of ${\displaystyle \nu _{2}}$ without clock reset.

Explicit definition of a region

While the previous definition allow to test whether two assignments belong to the same region, it does not allow to easily represents a region as a data structure. The third definition given below allow to give a canonical encoding of a region.

A region can be explicitly defined as a zone, using a set ${\displaystyle S}$ of equations and inequations satisfying the following constraints:

• for each ${\displaystyle x\in C}$, ${\displaystyle S}$ contains either:
  • ${\displaystyle x=c}$ for some integer ${\displaystyle 0\leq c\leq c_{x}}$
  • ${\displaystyle x\in (c,c+1)}$ for some integer ${\displaystyle 0\leq c<c_{x}}$,
  • ${\displaystyle x>c_{x}}$,
• furthermore, for each pair of clocks ${\displaystyle x,y\in C}$, where ${\displaystyle S}$ contains constraints of the form ${\displaystyle x\in (c,c+1)}$ and ${\displaystyle y\in (c',c'+1)}$, then ${\displaystyle S}$ contains an (in) equality of the form ${\displaystyle \{x\}\sim \{y\}}$ with ${\displaystyle \sim }$ being either =, < or ≤.

Since, when ${\displaystyle c}$ and ${\displaystyle c'}$ are fixed, the last constraint is equivalent to ${\displaystyle x\sim y+c-c'}$.

This definition allow to encode a region as a data structure. It suffices, for each clock, to state to which interval it belongs and to recall the order of the fractional part of the clocks which belong in an open interval of length 1. It follows that the size of this structure is ${\displaystyle O\left(\sum \log(c_{k})+|C|\log(|C|)\right)}$ with ${\displaystyle |C|}$ the number of clocks.

Timed bisimulation

Let us now give a third definition of regions. While this definition is more abstract, it is also the reason why regions are used in model checking. Intuitively, this definition states that two clock assignments belong to the same region if the differences between them are such that no timed automaton can notice them. Given any run ${\displaystyle r}$ starting with a clock assignment ${\displaystyle \nu }$, for any other assignment ${\displaystyle \nu '}$ in the same region, there is a run ${\displaystyle r'}$, going through the same locations, reading the same letters, where the only difference is that the time waited between two successive transition may be different, and thus the successive clock variations are different.

The formal definition is now given. Given a set of clock ${\displaystyle C}$, two assignments two clocks assignments ${\displaystyle \nu _{1}}$ and ${\displaystyle \nu _{2}}$ belongs to the same region if for each timed automaton ${\displaystyle {\mathcal {A}}}$ in which the guards never compare a clock ${\displaystyle x}$ to a number greater than ${\displaystyle c_{x}}$, given any location ${\displaystyle \ell }$ of ${\displaystyle {\mathcal {A}}}$, there is a timed bisimulation between the extended states ${\displaystyle (\ell ,\nu _{1})}$ and ${\displaystyle (\ell ,\nu _{2})}$. More precisely, this bisimulation preserves letters and locations but not the exact clock assignments.^[1]: 7

Resetting clocks

Given a region ${\displaystyle \alpha }$ defined by a set of (in)equations ${\displaystyle S}$, and a set of clocks ${\displaystyle C'\subseteq C}$, the region similar to ${\displaystyle \alpha }$ in which the clocks of ${\displaystyle C'}$ are restarted is now defined. This region is denoted by ${\displaystyle \alpha [C'\mapsto 0]}$, it is defined by the following constraints:

• each constraints of ${\displaystyle S}$ not containing the clock ${\displaystyle x}$,
• the constraints ${\displaystyle x=0}$ for ${\displaystyle x\in C'}$.

The set of assignments defined by ${\displaystyle \alpha [C'\mapsto 0]}$ is exactly the set of assignments ${\displaystyle \nu [C'\mapsto 0]}$ for ${\displaystyle \nu \in \alpha }$.

Time-successor

Given a region ${\displaystyle \alpha }$, the regions which can be attained without resetting a clock are called the time-successors of ${\displaystyle \alpha }$. Two equivalent definitions are now given.

Region graph

The region graph is a rooted directed graph which models the set of possible clock valuations during a run of a timed-autoamton. It is defined as follows:

• its nodes are regions,
• its root is the initial region ${\displaystyle \alpha _{0}}$, defined by the set of constraints ${\displaystyle \{x=0\mid x\in C\}}$,
• the set of edges are ${\displaystyle (\alpha ,\alpha '[C'\mapsto 0])}$, for ${\displaystyle \alpha '}$ a time-successor of ${\displaystyle \alpha }$.

Region automaton

Let ${\displaystyle {\mathcal {A}}=\langle \Sigma ,L,L_{0},C,F,E\rangle }$ a timed automaton. For each clock ${\displaystyle x\in C}$, let ${\displaystyle c_{x}}$ the greatest number ${\displaystyle c}$ such that there exists a guard of the form ${\displaystyle x\sim c}$ in ${\displaystyle {\mathcal {A}}}$. The region automaton of ${\displaystyle {\mathcal {A}}}$, denoted by ${\displaystyle R({\mathcal {A}})}$ is a finite or Büchi automaton which is essentially a product of ${\displaystyle {\mathcal {A}}}$ and of the region graph defined above. That is, each state of the region automaton is a pair containing a location of ${\displaystyle {\mathcal {A}}}$ and a region. Since two clocks assignment belonging to the same region satisfies the same guard, each region contains enough information to decide which transitions can be taken.

Formally, the region automaton is defined as follows:

• its alphabet is ${\displaystyle \Sigma }$,
• its set of states is ${\displaystyle L\times {\mathcal {R}}}$,
• its set of states is ${\displaystyle L_{0}\times \{\alpha _{0}\}}$ with ${\displaystyle \alpha _{0}}$ the initial region,
• its set of accepting states is ${\displaystyle F\times {\mathcal {R}}}$,
• its transition relation ${\displaystyle \delta }$ contains ${\displaystyle ((\ell ,\alpha ),a,(\ell ',\alpha '[C'\mapsto 0]))}$, for ${\displaystyle (\ell ,a,g,C',\ell ')\in E}$, such that ${\displaystyle \gamma \models \alpha '}$ and ${\displaystyle \alpha '}$ is a time-successor of ${\displaystyle \alpha }$.

Given any run ${\displaystyle r=(\ell _{0},\nu _{0}){\xrightarrow[{t_{1}}]{\sigma _{1}}}(\ell _{1},\nu _{1})\dots }$ of ${\displaystyle {\mathcal {A}}}$, the sequence ${\displaystyle (\ell _{0},[\nu _{0}]){\xrightarrow {\sigma _{1}}}(\ell _{1},[\nu _{1}])\dots }$ is denoted ${\displaystyle [r]}$, it is a run of ${\displaystyle R({\mathcal {A}})}$ and is accepting if and only if ${\displaystyle r}$ is accepting^[2]: 207. It follows that ${\displaystyle L(R({\mathcal {A}}))=\operatorname {Untime} (L({\mathcal {A}}))}$. In particular, ${\displaystyle {\mathcal {A}}}$ accepts a timed-word if and only if ${\displaystyle R({\mathcal {A}})}$ accepts a word. Furthermore, an accepting run of ${\displaystyle {\mathcal {A}}}$ can be computed from an accepting run of ${\displaystyle R({\mathcal {A}})}$.