state explosion is a very common problem in the control of large scale systems due to the interconnection of numerous subsystems .therefore , it is usually desired to reduce the state space of subsystems while overapproximating or preserving their external behavior important for their interconnection to surrounding components . +this mechanism is also used to reduce the complexity of verification problems in the theoretical computer science community . here, systems are usually modeled by so called transition systems , a subclass of discrete time state space models . for these models ,the notion of bisimilarity plays an important role .this concept was introduced by milner in the context of concurrent processes to describe how state trajectories of two transition systems mimic each other while producing the same `` external '' behavior , i.e. , using the same transition symbols .if such a bisimulation relation exists , it was shown that many interesting properties expressible in temporal logics , in particular reachability , are preserved when replacing a system by a bisimilar one .+ the use of bisimulation relations for other system models was discussed in the survey paper . here , special classes of hybrid systems are rewritten into a transition system and it was shown that they allow for purely discrete abstractions bisimilar to the constructed transition system .pappas adapted this method for linear time - invariant continuous state space models with finite observation maps , still using both a rewriting and an abstraction step . to remove the rewriting step ,van der schaft introduced a notion of bisimulation directly applicable to continuous systems .he showed that this equivalence interpretation unifies the concepts of state space equivalence and reduction using controlled invariant subspaces .these results where generalized by van der schaft and coworkers to hybrid systems , switched linear systems and behavioral systems .+ recently , davoren and tabuada presented simulation and bisimulation relations using general flow systems , preserving properties formulated in the so called general flow logic .general flow systems are able to model continuous , discrete , hybrid or even `` meta - hybrid '' autonomous state dynamics also allowing equivalence relations between systems with different time scales .this feature extends all previous approaches where only relations between systems with unique time scales are possible .although davoren and moor discussed in how general flow systems can be equipped with input and output maps , the simulation relations in do not incorporate the feature of ensuring identical external signals of bisimilar systems . in a comparison between simulation relations on transition systems and simulation relations on general flow systems is presented .+ tabuada and coworkers extended the work of alur et.al . towards finite state abstraction methods ensuring similarity or bisimilarity between the original and the abstracted system . independently from this work , the notion of -complete abstraction evolved as a discrete abstraction technique in the framework of behavioral systems theory . in both frameworks a finite state abstraction of a possibly continuous or hybrid dynamical system is obtained if the external signal space is finite and the trajectories of external signals evolve on the discrete time axis . in the context of bisimilarity relations ,these external signals should be preserved during abstraction .this raises the problem of deriving a bisimilarity notion that ensures equivalence of discrete external signals while comparing state trajectories that evolve on possibly continuous or hybrid time lines .this issue has up until now not been explicitly addressed , neither in the context of -complete approximations nor in the work by tabuada and coworkers . in the latter , as in and ,the original system is first rewritten into a transition system , previous to the abstraction step .the bisimulation relation is then only ensured to hold between the transition system and its abstraction . + to also incorporate the rewriting step into the exploration of equivalence , we introduce a system model with distinct external and internal signals possibly evolving on a different time axis in section [ sec : sys ] . to cover a very general class of systems , we use behavioral systems theory to formalize our notion .we note that this restricts each time axis to be either continuous or discrete .it is future research to also incorporate hybrid time scales for the internal signals as formalized , for example , in .inspired by the the work in and , we derive a simulation relation for the newly introduced system model in section [ sec : simrel ] .we show that the introduced simulation and bisimulation relations are preorders and equivalence relations , respectively , for the system class under consideration .+ this work is a first step towards the comparison of different existing approaches to construct ( bi)similar finite state abstractions .due to page limitations this comparison is only shortly touched in various remarks and will be explored in more detail in subsequent publications .a _ dynamical system _ is given by , consisting of the right - unbounded time axis , the signal space and the behavior of the system , where is the set of all _ signals _ evolving on and taking values in . slightly abusing notation , we also write if is a _ partial function_. this is understood to be shorthand for , where is the _ domain _ of .furthermore , is the _ identity map _ s.t . '' , meaning that all statements after the dot hold for all variables in front of the dot .`` '' is interpreted analogously . ]now let be a product space .then the _ projection _ of a signal to is given by and denotes the projection of all trajectories in the behavior .given two signals and two points in time , the _ concatenation _ is given by where we denote by .when reasoning about similarity and bisimilarity of systems one has to distinguish between `` external '' signals , which are required to match or satisfy an inclusion property , and the remaining `` internal '' signals .depending on the chosen system representation and/or the real world problem at hand , this distinction may differ . to incorporate a wide range of possibilities ,we define a so called -dynamical system , where is a set - valued map which describes the relation between internal and external signals .+  [ def : dynsysinducesfinitev ] let be a dynamical system . then is a * -dynamical system * if where is an external signal space , is a right - unbounded time axis ,      \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \end{tikzpicture}}}\right\}}\ ] ] is a set of time scale transformations and is the external behavior .furthermore , denotes the inverse time scale transformation , by slightly abusing notation , we denote the unique element by itself and write .] , i.e. , .the construction of in definition [ def : dynsysinducesfinitev ] was inspired by the deterministic map in ( * ? ? ?note , that the map in ( * ? ? ?12 ) is required to be strictly causal . in analogy, one would typically require that the map is non - anticipating , i.e. , }}}={\ensuremath{w}}'{\ensuremath{|_{[0,t ] } } } \end{propconja } } { { \{\ensuremath{\left(\exists { \ensuremath{\exists \;\;.\;.}}\;}\right)}}{\tilde{\gamma}\in\gamma^{\ensuremath{{t_e } } } , \tilde{\tau}\in{\mathcal{t } } } { \begin{propconja } { { (   \firsttrue   \\i:=\tilde{\gamma},\tilde{\tau}\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}}')\\  \tau{\ensuremath{|_{[0,t]}}}=\tilde{\tau}{\ensuremath{|_{[0,t]}}}\\ \gamma{\ensuremath{|_{[0,{{\tau}}(t)]}}}=\tilde{\gamma}{\ensuremath{|_{[0,{{\tau}}'(t ) ] } } } \end{propconja}. } } { \{\ensuremath{\left({\ensuremath{\ifthenelse{\isempty{{\ensuremath{\ifthenelse{\isempty{}}}{\cdot}{}}\rightarrow\ifthenelse{\isempty{}}}{\cdot}{}}}}{\cdot}{{\ensuremath{\ifthenelse{\isempty{}}}{\cdot}{}}\rightarrow\ifthenelse{\isempty{\rightarrow}}{\cdot}{\rightarrow}}}\ifthenelse{\isempty{}}{\cdot}{}}}}\right ) } } { \begin{propconja } { { (   \firsttrue   \\i:=\gamma,\tau\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}})\\ { { (   \firsttrue   \\i:=\gamma',\tau'\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}}')\\ { \ensuremath{w}}{\ensuremath{|_{[0,t]}}}={\ensuremath{w}}'{\ensuremath{|_{[0,t ] } } } \end{propconja } } { { \{\ensuremath{\left(\exists { \ensuremath{\exists \;\;.\;.}}\;}\right)}}{\tilde{\gamma}\in\gamma^{\ensuremath{{t_e } } } , \tilde{\tau}\in{\mathcal{t } } } { \begin{propconja } { { (   \firsttrue   \\i:=\tilde{\gamma},\tilde{\tau}\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}}')\\  \tau{\ensuremath{|_{[0,t]}}}=\tilde{\tau}{\ensuremath{|_{[0,t]}}}\\ \gamma{\ensuremath{|_{[0,{{\tau}}(t)]}}}=\tilde{\gamma}{\ensuremath{|_{[0,{{\tau}}'(t ) ] } } } \end{propconja}. } } \forall\;w , w'\in{\ensuremath{\mathcal{b}}},\gamma,\gamma'\in\gamma^{\ensuremath{{t_e}}},\tau,\tau'\in{\mathcal{t}},t\in{\ensuremath{t}}\;. ( e.south west)+(1ex,-1ex)+(3ex+.5\sfs,-\sfs) { \{\ensuremath{\left({\ensuremath{\ifthenelse{\isempty{{\ensuremath{\ifthenelse{\isempty{}}}{\cdot}{}}\rightarrow\ifthenelse{\isempty{}}}{\cdot}{}}}}{\cdot}{{\ensuremath{\ifthenelse{\isempty{}}}{\cdot}{}}\rightarrow\ifthenelse{\isempty{\rightarrow}}{\cdot}{\rightarrow}}}\ifthenelse{\isempty{}}{\cdot}{}}}}\right ) } } { \begin{propconja } { { (   \firsttrue   \\i:=\gamma,\tau\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}})\\ { { (   \firsttrue   \\i:=\gamma',\tau'\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}}')\\ { \ensuremath{w}}{\ensuremath{|_{[0,t]}}}={\ensuremath{w}}'{\ensuremath{|_{[0,t ] } } } \end{propconja } } { { \{\ensuremath{\left(\exists { \ensuremath{\exists \;\;.\;.}}\;}\right)}}{\tilde{\gamma}\in\gamma^{\ensuremath{{t_e } } } , \tilde{\tau}\in{\mathcal{t } } } { \begin{propconja } { { (   \firsttrue   \\i:=\tilde{\gamma},\tilde{\tau}\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}({\ensuremath{w}}')\\  \tau{\ensuremath{|_{[0,t]}}}=\tilde{\tau}{\ensuremath{|_{[0,t]}}}\\ \gamma{\ensuremath{|_{[0,{{\tau}}(t)]}}}=\tilde{\gamma}{\ensuremath{|_{[0,{{\tau}}'(t ) ] } } } \end{propconja}. } } \forall\;w , w'\in{\ensuremath{\mathcal{b}}},\gamma,\gamma'\in\gamma^{\ensuremath{{t_e}}},\tau,\tau'\in{\mathcal{t}},t\in{\ensuremath{t}}\;. ( e.east)+(1ex,0ex) ( e.east)+(1ex,0ex) ( e.south east)+(1ex,-0.5ex) ( e.south west)+(1ex,-0.5ex) ( e.south west)+(1ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) ] and iff is continuouse .using , and the sets ,  \end{aligned}\ ] ] the external signals are constructed via the discretization s.t . so far , this discretization does not include any information about its timing , i.e. , the formal construction of . out of the many different options ,we discuss two possible maps and as depicted in figure [ fig : timescale1 ] and figure [ fig : timescale2 ] .+ first , consider a signal map s.t .for all and , it holds that  and for all , where denotes the greatest lower bound and .this generates the * point to point time scale transformation * depicted in figure [ fig : timescale1 ] ( middle ) , where different points in are mapped to different points in , and an external event is triggered when leaving the interval .the generated external signal is depicted in figure [ fig : timescale2 ] .this map can be extended to generate a * set to point time scale transformation * by defining where every point in is in the domain of .this time scale transformation is depicted in figure [ fig : timescale1 ] ( bottom ) . combining the construction of with the construction of in defines a signal map .+ now assume , that we have a signal that stays in for all .this signal would only generate one external event at time but not an infinite sequence of events , where is right unbounded .therefore , the signal maps and map to the empty set .obviously , one could repeat the symbol infinitely often to generate a signal in from .however , if one has to know that will never leave to do so , as suggested in ( * ? ? ?* def.7.2 ) , this generates an anticipating signal map .a non - anticipating version is , for example , obtained , if a symbol is repeated after a fixed time , if the quantization interval is not left . this would combine event triggered with slow time triggered discretization .  background ( 0,0 )node ( o ) ) and set to point ( ) time scale transformations as constructed in example [ exp:1].,title="fig : " ] ;  foreground ( o.north west)+(-0.1,-0.2 ) node ; ( o.north west)+(-0.1,-0.7 ) node ; ( o.north west)+(-0.1,-1.2 ) node ; ( o.north west)+(-0.1,-1.7 ) node ; ( o.north west)+(-0.1,-2.2 ) node ; ( o.north west)+(0,-2.7 ) node ( b ) ; ( o.north east)+(0.2,-2.3 ) node ; ( o.north east)+(-0.1,-3.1 ) node ( t2 ) ; ( b)+(0,-0.7 ) node ; ( b)+(0,-1.2 ) node ; ( b)+(0,-1.7 ) node ; ( b)+(0,-2.2 ) node ; ( b)+(0,-2.7 ) node ; ( b)+(0,-3.2 ) node ( c ) ; ( t2)+(0.3,-0.9 ) node ; ( t2)+(0,-3.2 ) node ( t3 ) ; ( c)+(0,-0.8 ) node ; ( c)+(0,-1.5 ) node ; ( c)+(0,-2.1 ) node ; ( c)+(0,-2.6 ) node ; ( c)+(0,-3.1 ) node ; ( c)+(0,-3.6 ) node ( c ) ; ( t3)+(0.3,-1.2 ) node ; ( t3)+(0,-3.5 ) node ;  background ( 0,0 ) node ( o )  corresponding to the internal signal depicted in figure [ fig : timescale1 ] ( top).,title="fig : " ] ;  foreground ( o.north west)+(0,-0.2 ) node ( a ) ; ( o.north west)+(0,-0.8 ) node ( b ) ; ( o.north west)+(0,-1.2 ) node ( b ) ; ( o.north west)+(0,-1.55 ) node ( b ) ; ( o.north west)+(0,-1.95 ) node ( b ) ; ( o.north east)+(-0.1,-2.7 ) node ( t1 ) ; ( o.north east)+(0,-1.8 ) node ( t1 ) ; ( o.north west)+(0.3,-2.7 ) node ( b ) ; ( o.north west)+(1.9,-2.7 ) node ( b ) ; ( o.north west)+(3.4,-2.7 ) node ( b ) ; ( o.north west)+(5,-2.7 ) node ( b ) ; ( o.north west)+(6.55,-2.7 ) node ( b ) ;  example [ exp:1 ] shows that in general is indeed a function of when using an event - triggered discretization scheme .of course , using time - triggered discretization would result in a unique time scale transformation independent from .furthermore , the signal maps used in example [ exp:1 ] are deterministic in the sense that every signal generates a one element set or the empty set . however , nondeterministic maps occur for example if is constructed from a cover of with overlaps of more than two sets .states are internal variables for which the axiom of state holds , i.e. , all relevant information on the past of the system is captured by those variables . in the literature two concepts of the state property exist for behavioral systems .firstly , the well known version by willems , where state trajectories and can be concatenated , if they exhibit the same value at the same time ( i.e. , is also a state trajectory ) . andsecondly , a generalized version that allows state trajectories to be also concatenated if they reach the same value at different times ( i.e. , is also a state trajectory ) , as used in the context of state maps by julius and van der schaft in . to clearly differentiate both notions we call the first one synchronous and the second one asynchronous . using these two state properties , we construct state space -dynamical systems such that the discussed state property is preserved by the signal map .+  [ def : ssdynsysinducesfinitev ] let be a -dynamical system , be a set and .+ then is an * asynchronous state space -dynamical system * if      \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture}}}}\ ] ] where .furthermore , is an * externally synchronous * state space -dynamical system if holds for and a * synchronous * state space -dynamical system if holds for and .it is easy to see that every asynchronous state space -dynamical system is also an externally synchronous and a synchronous one , because we can always pick and in . with the same argument, every externally synchronous state space -dynamical system is also a synchronous one . for the asynchronous and the synchronous case in definition [ def : ssdynsysinducesfinitev ] ,the implication is equivalent to the asynchronous and for to the synchronous state property for the system .the additional requirement in ensures , that this concatenation property also holds for the external behavior . note that for the externally synchronous case , synchronization is only required on the external time axis .+ in the remainder of this paper , we refer to a system as introduced in definition [ def : ssdynsysinducesfinitev ] simply as _ state space -dynamical system _ , if the respective adjective ( asynchronous , externally synchronous , synchronous ) is irrelevant . + sincepossibly not all states are reachable by a state trajectory in , we define the following reachable subsets of the state space ( comp .* def.5.37 ) ) .+  [ def : timeindexedstatespace1 ] let be a state space -dynamical system . then      \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } }  \end{small } \right\}}\end{aligned}\ ] ] are the internal and external time - indexed state spaces and , respectively.  obviously , the internal and external time - indexed state spaces are equivalent if is a total function .one system simulates another one , if its external behavior contains the external behavior of the latter , while ensuring that the state trajectories generated by both systems only visit states , at each instant of time , that are associated by a relation . to formalize this property , a special relation , called _simulation relation _ , is constructed between both state spaces .+ in the behavioral framework signals are usually right - unbounded . it is well known that a local ( i.e. , on a finite time interval ) evaluation of properties is only possible , if the system is complete . inspired by ( * ? ? ?5.21 ) , we therefore define a concatenation based simulation relation for -dynamical systems .in contrast to the locally defined simulation relation used for transition systems ( e.g. , in ) or general flow systems ( in ) , it also relates not necessarily complete systems .+  [ def : simrel ] let and be state space -dynamical systems .+ then a relation is an * asynchronous simulation relation * from to ( written ) , i.e. , asynchronously simulates , if  [ equ : def : simrel ] and       \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t< t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \exists\ ; { { (   \firsttrue   \\i:=w_2,x_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\ensuremath{\ifthenelse{\isempty{2}}{{\ensuremath{\mathcal{b}}}_{s}}{{\ensuremath{\mathcal{b}}}_{s,2 } } } } , { { (   \firsttrue   \\i:=\gamma_2,\tau_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}_2(w_2)\;. ( e.south west)+(1ex,-1ex)+(3ex+.5\sfs,-\sfs) \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } }\\ x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \exists\ ; { { (   \firsttrue   \\i:=w_2,x_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\ensuremath{\ifthenelse{\isempty{2}}{{\ensuremath{\mathcal{b}}}_{s}}{{\ensuremath{\mathcal{b}}}_{s,2 } } } } , { { (   \firsttrue   \\i:=\gamma_2,\tau_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}_2(w_2)\;. ( e.east)+(1ex,0ex) ( e.east)+(1ex,0ex) ( e.south east)+(1ex,-0.5ex) ( e.south west)+(1ex,-0.5ex) ( e.south west)+(1ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) }     \settowidth{\sfs}{ \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \exists\ ; { { (   \firsttrue   \\i:=w_2,x_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\ensuremath{\ifthenelse{\isempty{2}}{{\ensuremath{\mathcal{b}}}_{s}}{{\ensuremath{\mathcal{b}}}_{s,2 } } } } , { { (   \firsttrue   \\i:=\gamma_2,\tau_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}_2(w_2)\;. ( e.south west)+(1ex,-1ex)+(3ex+.5\sfs,-\sfs) \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\\end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \exists\ ; { { (   \firsttrue   \\i:=w_2,x_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\ensuremath{\ifthenelse{\isempty{2}}{{\ensuremath{\mathcal{b}}}_{s}}{{\ensuremath{\mathcal{b}}}_{s,2 } } } } , { { (   \firsttrue   \\i:=\gamma_2,\tau_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}_2(w_2)\;. ( e.east)+(1ex,0ex) ( e.east)+(1ex,0ex) ( e.south east)+(1ex,-0.5ex) ( e.south west)+(1ex,-0.5ex) ( e.south west)+(1ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \exists\ ; { { (   \firsttrue   \\i:=w_2,x_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\ensuremath{\ifthenelse{\isempty{2}}{{\ensuremath{\mathcal{b}}}_{s}}{{\ensuremath{\mathcal{b}}}_{s,2 } } } } , { { (   \firsttrue   \\i:=\gamma_2,\tau_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}_2(w_2)\;. ( e.south west)+(1ex,-1ex)+(3ex+.5\sfs,-\sfs) \begin{propconja } \gamma_2={\gamma'\wedge^{k_2}_{k_1}\gamma_1}\\ { \{\ensuremath{\left(\forall { \ensuremath{\forall \;\;.\;.}}\;}\right)}}{t\in{\ensuremath{t}}_2 , t < t_2 } { \begin{propconja } w_2(t)=w'(t)\\ x_2(t)=x'(t)\\ \tau_2(t)=\tau'(t)\\ \end{propconja } } \\x_2(t_2)=x'(t_2)\\ { {     \settoheight{\sfs}{ }     \settowidth{\sfs}{ }     \mbox{\begin{tikzpicture}[baseline=(current bounding box.center ) ]     \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}. \exists\ ; { { (   \firsttrue   \\i:=w_2,x_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\ensuremath{\ifthenelse{\isempty{2}}{{\ensuremath{\mathcal{b}}}_{s}}{{\ensuremath{\mathcal{b}}}_{s,2 } } } } , { { (   \firsttrue   \\i:=\gamma_2,\tau_2\do {   \iffirst\listop{\i}\firstfalse\else,\listop{\i}\fi   }   )   \let\listop\relax } } \in{\phi}_2(w_2)\;. ( e.east)+(1ex,0ex) ( e.east)+(1ex,0ex) ( e.south east)+(1ex,-0.5ex) ( e.south west)+(1ex,-0.5ex) ( e.south west)+(1ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) ( e.south west)+(2.5ex,-1ex)-(0,\sfs) } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture}}}}\ ] ]  it is an * externally synchronous simulation relation * from to ( written ) if  [ equ : def : simrelsync ]   and holds for . + furthermore , if , then is a * synchronous simulation relation * from to ( written ) if  [ equ : def : simrelsync:2 ]   and holds for and .the construction of the externally synchronous simulation relation in definition [ def : simrel ] is inspired by the so called _ synchronized simulation relation _ defined in ( * ? ? ?* def . 5.38 ) .however , the latter does not restrict to hold only for .the intuitive interpretation of the terms asynchronous , synchronous and externally synchronous is strongly related to the ones used in definition [ def : ssdynsysinducesfinitev ]. however , in definition [ def : simrel ] the synchronization takes place between signals of _ different _ systems that are related . + in contrast to definition [ def : ssdynsysinducesfinitev ] , it is not true that every asynchronous simulation relation is an ( externally ) synchronous one , since does generally not imply and .intuitively , if is an asynchronous simulation relation , we know that holds for and . however , we can generally not ensure that for every state in reachable at external time and internal time , there exists a related state in that is reachable at the same external and internal time .we can therefore possibly not relate the whole state space in a synchronous or externally synchronous fashion , implying that may formally not be an ( externally ) synchronous simulation relation .+ to generate some intuition for the simulation relation constructed in definition [ def : simrel ] , we will discuss using some graphical illustrations .for this purpose assume that we have signals and such that the states and , with and , are related . to simulate , the system must be able to continue from time with the same external signal as produced by after .this is expressed in by requiring the existence of an external signal which is constructed from the concatenation of the signals and , as depicted in figure [ fig : simrel_gamma ] .background ( 0,0 ) node ( o )  in ., title="fig : " ] ;  foreground  ( o.north west)+(0,-0.2 ) node ( a ) ; ( o.north east)+(-0.3,-1.9 ) node ( t1 ) ; ( o.north east)+(-1.4,-1.9 ) node ( t2 ) ; ( o.north east)+(-1.4,-2.5 ) node ( t2 ) ; ( o.north east)+(0,-3.5 ) node ( t2 ) ; ( o.south west)+(2.9,0 ) node ( k2 ) ; ( o.south west)+(4.1,0 ) node ( k2 ) ;  to ensure that is non - anticipating , this concatenation is not allowed to change the past , which is why we require that the past of and match the past of and . moreover, we have to ensure , that the state trajectories and match at time , expressed by .+  background ( 0,0 ) node ( o )  and , with , , and .gray lines connect related states.,title="fig : " ] ;  foreground  ( o.north west)+(0,-0.2 ) node ( a ) ; ( o.north west)+(0,-0.8 ) node ( b ) ; ( o.north west)+(0,-2.5 ) node ( c ) ; ( o.north west)+(0,-3.1 ) node ( d ) ; ( o.north east)+(-0.4,-0.7 ) node ; ( o.north east)+(-0.1,-1.4 ) node ( t1 ) ; ( o.north east)+(-0.1,-2.1 ) node ( k1 ) ; ( o.north east)+(-0.4,-2.9 ) node ( t2 ) ; ( o.north east)+(-0.1,-3.9 ) node ( t2 ) ; ( o.north east)+(-0.1,-4.5 ) node ( k2 ) ;  ( 0.9,1.8 ) node ( xa ) ; ( o.north west)+(0.4,-1.45 ) node ( d ) ; ( o.north west)+(0.4,-2.05 ) node ( d ) ; ( 1,0.8 ) node ( xa ) ; ( 1.3,0.2 ) node ( xa ) ; ( o.south west)+(0.4,0.6 ) node ( k2 ) ; ( o.south west)+(0.4,-0.1 ) node ( k2 ) ; ( o.south west)+(2.7,0.6 ) node ( k2 ) ; ( o.south west)+(3.1,-0.1 ) node ( k2 ) ; ( o.south west)+(2.3,1.2 ) node ( k2 ) ;  background ( 0,0 ) node ( o )  and , with and .gray lines connect related states.,title="fig : " ] ;  foreground  ( o.north west)+(0,-0.2 ) node ( a ) ; ( o.north west)+(0,-0.8 ) node ( b ) ; ( o.north west)+(0,-2.5 ) node ( c ) ; ( o.north west)+(0,-3.1 ) node ( d ) ; ( o.north east)+(-0.4,-0.7 ) node ; ( o.north east)+(-0.1,-1.4 ) node ( t1 ) ; ( o.north east)+(-0.1,-2.1 ) node ( k1 ) ; ( o.north east)+(-0.4,-2.9 ) node ( t2 ) ; ( o.north east)+(-0.1,-3.9 ) node ( t2 ) ; ( o.north east)+(-0.1,-4.5 ) node ( k2 ) ;  ( 0.9,1.8 ) node ( xa ) ; ( o.north west)+(2.5,-1.4 ) node ( d ) ; ( o.north west)+(0.4,-1.45 ) node ( d ) ; ( o.north west)+(0.4,-2.05 ) node ( d ) ; ( 1,0.8 ) node ( xa ) ; ( 1.3,0.2 ) node ( xa ) ; ( o.south west)+(1.3,0.6 ) node ( k2 ) ; ( o.south west)+(0.4,0.6 ) node ( k2 ) ; ( o.south west)+(0.4,-0.1 ) node ( k2 ) ; ( o.south west)+(2.7,0.6 ) node ( k2 ) ; ( o.south west)+(3.1,-0.1 ) node ( k2 ) ; ( o.south west)+(2.3,1.2 ) node ( k2 ) ;  background ( 0,0 ) node ( o ) .,title="fig : " ] ;  foreground  ( o.north west)+(0,-0.2 ) node ( a ) ; ( o.north west)+(0,-1 ) node ( b ) ; ( o.north west)+(0,-2 ) node ( c ) ; ( o.north west)+(0,-2.7 ) node ( d ) ; ( o.north east)+(-0.6,-1 ) node ( t1 ) ; ( o.north east)+(-0.1,-1.6 ) node ( t1 ) ; ( o.north east)+(-0.6,-2.6 ) node ( t2 ) ; ( o.north east)+(-0.1,-3.4 ) node ( t2 ) ; ( o.south west)+(0.4,1.7 ) node ( k2 ) ; ( o.south west)+(0.4,0 ) node ( k2 ) ;  the last line of basically says that the state trajectories and need to stay related for all future external time instants .however , the nature of significantly influences how restrictive this requirement is .for example , having a point to point time scale transformation in both systems only requires state trajectories to be related at sampling points ( figure [ fig : r - simulation_hscc ] ) , while a set to point time scale transformation , for example , requires state trajectories to be related at all future times ( figure [ fig : p - simulation_hscc ] ) .however , as clearly visible in figure [ fig : r - simulation_hscc ] and [ fig : p - simulation_hscc ] , both cases allow for a stretching or shrinking of time between related state trajectories . if both systems have an identity time scale transformation ( and therefore ) this stretching or shrinking of time is no longer allowed , as shown in figure [ fig : t - simulation_hscc ] .note that the latter case only implies that the constructed asynchronous simulation relation is also a synchronous one , if we additionally require , which immediately implies .+  [ rem : gfs ] the intuitive interpretation of the different simulation relations depicted in figure [ fig : r - simulation_hscc ] - [ fig : t - simulation_hscc ] is very similar to the idea behind the - , - and -simulation relations constructed in for general flow systems .this suggests that for the subclass of discrete and continuous systems , our simulation relation can reproduce the relations in by choosing different time scale transformations .however , our relation extends the constructions in by allowing to include the simulation of external trajectories . furthermore, relating two systems with different time scale transformations gives an even richer variety of relations .[ rem : ts ] recall that -dynamical systems capture the dynamics of transition systems and linear time - invariant continuous systems ( see remark [ rem:0 ] ) , if . relating two systemsimplies a state trajectory matching requirement as depicted in figure [ fig : t - simulation_hscc ] .additionally , the external signal , which , in the case of transition systems is the output , and , in the case of linear time - invariant continuous systems is the pair , needs to satisfy the requirement depicted in figure [ fig : simrel_gamma ] . observe that for complete systems this interpretation coincides with the locally defined simulation relation for transition systems , e.g. , in .the same is true for the simulation relation constructed for linear time invariant systems in .this suggests that both notions of simulation relations can be captured by our notion .  using the simulation relations constructed in definition [ def : simrel ], we can define similarity and bisimilarity for the class of state space -dynamical systems in the usual fashion .+  [ def : bisimrel_behaviour ] is * asynchronously simulated * by , denoted by , if there exists an asynchronous simulation relation from to . and are * asynchronously bisimilar * , denoted by , if there exists a relation s.t. and are asynchronous simulation relations from to and from to , respectively .+ is * externally synchronously simulated * by , denoted by , if there exists an externally synchronous simulation relation from to . and are * externally synchronously bisimilar * , denoted by , if there exists a relation s.t. and are externally synchronous simulation relations from to and from to , respectively .+ is * synchronously simulated * by , denoted by , if there exists a synchronous simulation relation from to . and are * synchronously bisimilar * , denoted by , if there exists a relation s.t. and are synchronous simulation relations from to and from to , respectively .before proving the soundness of our construction we introduce another simulation relation to discuss the connection between behavioral equivalence and bisimilarity of two systems .+  [ def : initsimrel ] let and be state space -dynamical systems and let .+ then is an * -initial simulation relation * from to ( written ) if  [ equ : def : initsimrel ]   and holds .  for this simulation relation , -initially similar and bisimilar systemsare defined analogously to definition [ def : bisimrel_behaviour ] and are denoted by and , respectively .+ observe that in definition [ def : initsimrel ] , the statement in still needs to hold for arbitrary and ( as for the asynchronous simulation relation ) .however , we require in that states reached at external time are related to states also reachable at external time . observe that this does in general not imply that holds . due to the iterative nature of , intuitively , relating states reached at external time leads to a relation between all states reachable for external time ( explaining the name for this relation ) .in particular , if the external time axis has a minimal element ( e.g. , or with ) , a -initial simulation relation will imply that all reachable states are related in an externally synchronized fashion .the following lemma formalizes this intuition by proving various connections between the different relation types .+  [ lem : linitsimimpliesothersims ] let and be state - space -dynamical systems s.t. has the minimal element .then   ,   , and    pick and observe the following facts :  [ item : proof : lem : linitsimimpliesothersims:1 ] holds for : + as holds for ( using definition [ def : timeindexedstatespace1 ] ) we can fix and s.t . . since , implies that there exist s.t .using definition [ def : timeindexedstatespace1 ] and the fact that is the minimal element of , this implies that holds .[ item : proof : lem : linitsimimpliesothersims:2 ] implies since for from definition [ def : timeindexedstatespace1 ] .[ item : proof : lem : linitsimimpliesothersims:3 ] let . if and then from definition [ def : timeindexedstatespace1 ] implies .[ item : proof : lem : linitsimimpliesothersims:4 ] if holds , it also holds for and .  now  follows from and ,  follows from and , and  follows from , and .the inverse implication in ( i ) does not hold , as does not imply that holds for arbitrary .recall from remark [ rem:0 ] that -dynamical systems can represent transition systems using an external time axis ( with minimal element ) . for this system class ,simulation relations are usually defined by requiring that the initial states are related and a local property , similar to , holds ( see , e.g. , ) .this suggests , that simulation relations defined for transition systems are -initial simulation relations in our sense .  as the main result of this sectionwe generalize the results in ( * ? ? ?* thm . 5.41 )to state space -dynamical systems with external time axis having the minimal element and show that the existence of a -initial simulation relation from one system to another one implies that the behavior of the first is a subset of the second one . as an immediate consequence ,behavioral equivalence is obtained if two systems are -initially bisimilar .+  [ thm : bisimimplextbehinclusion ] let and be state - space -dynamical systems s.t. has the minimal element .then      using , the statement is equivalent to      \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture}}}}}\ ] ] where is the set of valid time scale transformations from to . fix s.t. .since , holds for .therefore , we can pick s.t . . using for this implies that , which proves statement ( i ) . part ( ii ) follows immediately from ( i ) and definition [ def : bisimrel_behaviour ] .does not extend to the asynchronous simulation case , since here we can not ensure finding pairs and s.t .their initial states are related .as the main result of this paper we show that the simulation relations in definition [ def : simrel ] are well defined by proving that they are preorders for their respective class of state space -dynamical systems .+  [ thm : simaspreorder ] the relations , , and are preorders for the class of * asynchronous * state space -dynamical systems .  to simplify notation , we denote the conjunction on the right hand side of by , i.e.      \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture } } } } \end{propconja}.\ ] ] a relation is a preorder , if it is reflexive and transitive . + * 1 .reflexivity : * + to prove reflexivity , pick an arbitrary , construct s.t . and show that , , and hold :  , , and hold by construction .remember from fact ( d ) in the proof of that if holds , it also holds for and .  to show that holds , fix s.t .the left side of is true , pick , , , s.t . and show that the right side of is true .+  observe that the first three lines of follow directly from and from the construction of implying .+  now using definition [ def : ssdynsysinducesfinitev ] we can conclude that and since and .+  to show that the last line of is true , observe that implies . from the construction of implies .* 2 . transitivity * + to prove transitivity , pick arbitrary as their unique representative . ] s.t .this implies that there exist simulation relations and from to and to , respectively .now construct s.t . and show that , , and hold for , implying .observe that , , and hold for and , implying    respectively .using the construction of this implies that , , and hold for .remember from fact ( d ) in the proof of that if holds , it also holds for and .  to show , fix s.t . . +  from the construction of we know that there exists some , , , s.t . and+  this implies that we can fix some , s.t . holds and therefore and .+  this implies that we can fix some , s.t . holds .  with this choice of signals , we show that also holds : +  observe , that the second and third line of are equivalent to the second and third line of , respectively .+  using the first line of and we get implying that the first line of holds . +  finally , to show that the last line of holds , observe that it is equivalent to      \node [ ] ( e ) at ( 0,0 ) { } ;     \node[inner sep=0pt ] ( f ) at ( ) { } ;     \node [ ] ( e ) at ( 0,0 ) { \phantom{ } } ;     \draw[fill ] ( ) circle ( .2ex ) ;     \draw[- ] ( ) -- ( ) -- ( ) -- ( ) -- ( ) ;     \draw[fill ] ( ) circle ( .2ex ) ;     \end{tikzpicture}}}}\ ] ] to show that holds , fix and pick . +  with this choice it follows immediately from the last line of that .+  if we now pick , we have , since .+  now it follows from that and from that .+  using the last line of this implies that   .[ thm : simaspreorder : wsync ] the relations and are preorders for the class of * externally synchronous * state space -dynamical systems .this proof is identical to the proof of by using in all statements .this substitution is applicable since is also restricted to for and .[ thm : simaspreorder : sync ] the relation is a preorder for the class of * synchronous * state space -dynamical systems .this proof is identical to the proof in by using and in all statements .this substitution is applicable since is also restricted to and for .[ thm : bisimasequivalencerelation ] the relations , , and are equivalence relations for the class of * asynchronous * state space -dynamical systems .furthermore , the relations and are equivalence relations for the class of * externally synchronous * state space -dynamical systems , and the relation is an equivalence relation for the class of * synchronous * state space -dynamical systems .a relation is an equivalence relation , if it is reflexive , transitive and symmetric . from definition [ def : bisimrel_behaviour ], it follows that all relations are defined by two simulation relations. therefore reflexivity and transitivity follows from - [ thm : simaspreorder : sync ] .  to prove symmetry , pick arbitrary and show .observe that it follows immediately from definition [ def : bisimrel_behaviour ] that for any bisimulation relation between and we can pick as a bisimulation relation between and , implying .we have proposed a behavioral system model with distinct external and internal signals possibly evolving on different time scales . for this new system model different notions of simulation and bisimulationwere derived and their soundness was proven . in remarks[ rem:0 ] , [ rem : gfs ] and [ rem : ts ] , we discussed in an intuitive manner that our notion can capture a broad selection of similarity concepts available in the literature .the formal proofs of these intuitive connections will be presented in a subsequent paper .it is our goal for the near future to use the presented framework to compare existing abstraction techniques in the control systems community .j. davoren , v. coulthard , n. markey , and t. moor .non - deterministic temporal logics for general flow systems . in r.alur and g. pappas , editors , _ hybrid systems : computation and control _ , volume 2993 of _ lecture notes in computer science _ ,pages 280295 .springer berlin heidelberg , 2004 .j. davoren and t. moor .non - deterministic reactive systems , from hybrid systems and behavioural systems perspectives . in _ proc .2nd ifac conference on analysis and design of hybrid systems ( adhs06 ) , ifac _ , pages 409416 , 2006 .

Abstract: this paper introduces a new behavioral system model with distinct external and internal signals possibly evolving on different time scales . this allows to capture abstraction processes or signal aggregation in the context of control and verification of large scale systems . for this new system model different notions of simulation and bisimulation are derived , ensuring that they are , respectively , preorders and equivalence relations for the system class under consideration . + these relations can capture a wide selection of similarity notions available in the literature . this paper therefore provides a suitable framework for their comparison .


echocardiography plays an important part in diagnostic aid in cardiac diseases .it relies on the use of ultrasound images to reflect the information about the heart of a patient . in the process of diagnosis, images are captured from specific views of heart , which are commonly named _heart views _ or _standard planes_. standard planes tend to provide promising views to present different heart structures that are benefit to diagnosis .the task for standard plane location ( spl ) exactly refers to selecting 2d planes from a candidate list considering both static and dynamic characteristics from different views of heart .automatic recognition of standard planes have been extensively studied in recent works .most of them focus on ultrasound images with 2d static images , 3d static volumes and 2d - temporal sequences . for 2d static images , ebadollahi et.al  deal with view classification of echocardiograms based on spatial layout of 2d static frames . propose an approach by applying local detectors sequentially on the pre - selected locations with selective search .for 3d volumes , lu et.al  proposed an automated supervised learning method to detect standard multiplanar reformatted planes ( mprs ) from a 3d echocardiographic volume with coarse - to - fine strategy .chykeyuk et.al  build a class - specific random forest regression model for standard plane classification based on 3d static volumes . for 2d - temporal sequences , qian et.al  employed bag of visual words based on spatial - temporal features .sparse coding and max pooling are used in a codebook to generate the representation .previous works usually regard the standard plane location as a classification problem based on individual frames or a set of sequences . in this paper , the data used in our work are 4d stic volumes where all slices have been captured including both 3d position and temporal information .the task for standard plane location is to select a specific or small set of 2d planes from 4d volumes according to different preferences of views .different from the previous works , the spl task on 4d image introduces the following problems :  1 .  restricted by the facility performance , the image quality of 4d stic volume is often poor. the spatial plane is generated by interpolation and the noise is inevitably introduced .the standard plane of abnormal cases can be quite similar to the plane which is not the standard plane in normal cases .3 .  given the 4d stic data , the searching space for * 2d plane * is considerably large .the real - time location is in urgent demand .considering the issues above , we propose a novel framework for spl task in 4d stic volume .  * different from the disease diagnosis, it is observed that the spl tasks focus more on the boundary characteristic but not the fine - grained information .therefore , we use an smoothing algorithm  which removes the noise and preserves the boundary simultaneously ( for problem 1 ) . *static sift  and spatial - temporal sift ( stip )  are deployed to model the feature of standard plane ( for problem 2 ) . *based on sift and stip , we propose a supervised method to learn the discriminative subspace and to generate the compact feature which can obviously reduce the searching consumption ( for problem 2&3 ) .the main framework of the proposed method is shown in fig [ fig::framework ] . giventhe 4d stic volume data , a large set of candidate 2d planes are generated to identify the standard plane .each 2d plane is featured with a sequence of grey - scale images indicating the cardiac motion .an smoothing filter is firstly deployed to remove the noise in each image .sift and stip features are extracted to capture both static and dynamic information . in order to learn compact representation , a low - dimensional subspace is obtained by embedding sift and stip features via supervised semantic correlation . finally a support vector machine ( svm ) classifies the candidates into different views .we deploy the fast smoothing algorithm  to generate the filtered image and detect the sift points as shown in figure [ fig::imagevis ] .as shown in figure [ fig::imagevis ] , the boundary of original frame is perfectly preserved while the noise is largely reduced in the filter image .we can also find the sift points detected in the filtered image mostly distributed along the boundary and the number of points is much smaller than the number in the original image .as mentioned above , it is observed that the contour is more discriminative to locate the standard planes other than fine grained local features .the 4d stic volume data naturally involves the spatial - temporal information . in order to present both dynamic and static features ,sift and stip are used in our work . for each frame , we extract sift points as static features that are used to indicate local features and invariant to scale and rotation transforms . demonstrates that the dynamic feature ( stip ) is powerful in standard plane location which takes both spatial invariant and temporal structure into consideration . however , the classification accuracy of standard plane is not promising enough according to our experiments on real datasets .stip prefers to select the interesting points based on spatial - temporal structure .although stip performs well in action recognition based on video captured in natural scenes , the quality and sampling rate of ultrasound image sequence is poor without preprocessing .moreover , the stip feature can not fully capture the static information which is also important for standard plane location .therefore , we aim at learning the common space between dynamic and static features .   many strategies are designed for learning the representation from multiple features including multiple kernel learning  , subspace learning  , etc . as mentioned in section [ sec::introduction ] ,the searching space is considerably large in standard plane location .therefore , we tend to learn the feature that integrates both dynamic and static information as well as in compact form . in this section ,we follow a supervised embedding scheme to generate the subspace feature based on sift and stip .  given a 2d plane with grey - scale frames , the plane is represented by bag - of - words feature with sift ( denoted by ) and stip ( denoted by ) .each plane is labeled with semantic information including plane description and diagnosis result . is a vector with elements where if current plane belongs to type of standard plane , otherwise , . if current plane is not a standard plane , the last element .the diagnosis result label indicates whether current plane supports the diagnosis of specific disease . following the similar scheme of plane description ,the first elements shows the existence of specific disease ( i.e. supports disease .otherwise , ) . for normal cases ,the last element is set to be .based on the labels , the semantic similarity between two planes can be measured as follows : according to the definition of , when two samples belong to different types of standard planes , the similarity is zero .however , the diagnosis information is taken into consideration when samples belong to the same standard plane .based on the pair - wise similarity , the task for multi - view feature embedding is to learn and mapping and into common latent space that maximizes the semantic similarity as defined in eq.([eq::semantic_sim ] ) .therefore , we have the objective function as follows : where is the length of the compact codes to learn . and are set of features of all training samples .after learning and , the original feature and are transformed into and which is compact ( ) and more discriminative than original feature . in order to obtain the solution of eq.([eq::c_loss ] ) , we impose orthogonality constraints , i.e. and , and then expand the original problem into the following form : problem in eq.([eqn::conver1 ] ) with orthogonality constraints is equivalent to a generalized eigenvalue problem that has closed - form solution .the embedded feature or is finally fed into a support vector machine ( svm ) to determine whether current plane belongs to specific standard plane . in order to address the problem of nonlinear cases , a histogram intersection kernel ( hik )is adopted in this paper . for each class of standard plane, we use one - vs - rest strategy to train binary svm classifier .to evaluate the performance of the proposed embedded feature , a 4d stic volume dataset is collected by xinhua hospital from ge e8 and the data is exported and stored in the dicom format by 4dview software from ge .the dicom image has the size of four dimensions with three - dimensional structure and frame numbers ( with 40 frames ) . to simplify the cases of disease ,the volume data are divided into two parts with 92 normal cases and 93 abnormal cases . as shown in figure [ fig::standardplane ] ,three standard planes including four chamber section ( a4c ) , three vessel view ( tvv ) and transverse abdominal section ( tas ) are selected for evaluations . for each volume, we uniformly generate 400 planes as candidate list and , for each standard plane , three candidates are labeled by expert as ground truth . in training step ,we randomly select 46 normal cases and 46 abnormal cases from the full dataset . in testing step , we construct two evaluation settings :  * * synthetic data : * for standard planes , we randomly select 75 planes from abnormal cases and 184 from normal cases . for non - standard planes ,the numbers of samples from abnormal and normal cases are 1392 and 1304 respectively .therefore , the evaluation on synthetic data is equivalent to a conventional classification problem .the performance can be directly measured by classification accuracy . ** real volume : * we regard spl task as a retrieval problem . for each volume , the query of specific standard plane is conducted by classifying all candidate planes in the same volume .the final performance can be measured by f1-score of standard planes averaged over all volumes .  for visual representation , we extract sift and stip interesting points from the sequential image series of planes where the final feature is 5000d bag - of - words sift and 1000d bag - of - words stip .the dimension of the proposed multi - view feature is 32d .three baseline methods are implemented for comparison :  * * single view feature*. only sift ( denoted by `` sift only '' ) or stip ( denoted by `` stip only '' ) feature is deployed for visual representation for each plane . ** direct concatenation*. the sift and stip feature are concatenated into a new vector as visual representation ( denoted by `` direct concate '' ) .the dimension of feature is 6000d . * * unsupervised cca*. sift and stip are mapped into a latent space with unsupervised canonical correlation analysis ( denoted by `` cca '' )  .the dimension of feature is 32d .all experiments are conducted on a pc with intel 2.4ghz cpu and 16 gb ram .methods are implemented with matlab .we use libsvm package  for svm classifier ..classification accuracy on synthetic data [ cols="^,^,^,^,^,^",options="header " , ]   table [ tab::runningtime ] shows that the running time of training and testing steps . for cca and the proposed method ,the training part is comprised with the learning of multi - view embedding matrix and svm parameters .we can observe that the time consumption is much smaller than learning svm parameters .since the dimension of the embedded feature is 32d which is much smaller than original feature , the testing time is only 0.14s which is much faster than the rest of approaches while the training time is also acceptable .the real - time detection of standard plane can be implemented .this paper presents a spatial - temporal embedding framework to extract the standard view planes from 4d stic volumes .we evaluate the proposed method with 92 normal cases and 93 abnormal cases in three standard planes .it demonstrates that our method outperforms the baselines in both classification accuracy and computational efficiency .the future work includes the use of more powerful features ( e.g. deep convolutional network ) to obtain higher accuracy .chykeyuk , k. , yaqub , m. , noble , j.a . : class - specific regression random forest for accurate extraction of standard planes from 3d echocardiography . in : medical computer vision .large data in medical imaging , pp . 5362 .springer ( 2013 )  ebadollahi , s. , chang , s.f ., wu , h. : automatic view recognition in echocardiogram videos using parts - based representation . in : computer vision and pattern recognition , 2004 .cvpr 2004 .proceedings of the 2004 ieee computer society conference on .vol . 2 , pp .ii2 . ieee ( 2004 )      lu , x. , georgescu , b. , zheng , y. , otsuki , j. , comaniciu , d. : autompr : automatic detection of standard planes in 3d echocardiography . in : biomedical imaging : from nano to macro , 2008 .isbi 2008 .5th ieee international symposium on .. 12791282 .ieee ( 2008 )  ni , d. , li , t. , yang , x. , qin , j. , li , s. , chin , c.t . ,ouyang , s. , wang , t. , chen , s. : selective search and sequential detection for standard plane localization in ultrasound . in : abdominal imaging . computation and clinical applications , pp .springer ( 2013 )  qian , y. , wang , l. , wang , c. , gao , x. : the synergy of 3d sift and sparse codes for classification of viewpoints from echocardiogram videos . in : medical content - based retrieval for clinical decision support ,springer ( 2013 )  rasiwasia , n. , costa pereira , j. , coviello , e. , doyle , g. , lanckriet , g.r . , levy , r. , vasconcelos , n. : a new approach to cross - modal multimedia retrieval . in : proceedings of the 18th acm international conference on multimedia .. 251260 .acm ( 2010 )

Abstract: echocardiography plays an important part in diagnostic aid in cardiac diseases . a critical step in echocardiography - aided diagnosis is to extract the standard planes since they tend to provide promising views to present different structures that are benefit to diagnosis . to this end , this paper proposes a spatial - temporal embedding framework to extract the standard view planes from 4d stic ( spatial - temporal image correlation ) volumes . the proposed method is comprised of three stages , the frame smoothing , spatial - temporal embedding and final classification . in first stage , an smoothing filter is used to preprocess the frames that removes the noise and preserves the boundary . then a compact representation is learned via embedding spatial and temporal features into a latent space in the supervised scheme considering both standard plane information and diagnosis result . in last stage , the learned features are fed into support vector machine to identify the standard plane . we evaluate the proposed method on a 4d stic volume dataset with 92 normal cases and 93 abnormal cases in three standard planes . it demonstrates that our method outperforms the baselines in both classification accuracy and computational efficiency .


recently , fractional calculus , a classical branch of mathematical analysis that studies non - integer powers of differentiation operators , has proved a huge potential in solving complicated problems from science and engineering . in this framework ,the fractional calculus of variations is a research area under strong current development . for the state of the art, we refer the reader to the recent book , for models and numerical methods we refer to . a fractional variational problemconsists in finding the extremizer of a functional that depends on fractional derivatives and/or integrals subject to some boundary conditions and possibly some extra constraints . in this workwe consider the following minimization problem : =\int_a^b l(t , x(t ) , \lda x(t))dt \longrightarrow \min,\\ x(a)=x_a , \quad x(b)=x_b , \end{gathered}\ ] ] that depends on the left riemann liouville derivative , , which is defined as follows .let be an absolutely continuous function in  ] is then transformed into a function of the values of the unknown function on mesh points .assuming , and , one has \simeq \psi(x_1,x_2,\ldots , x_{n-1 } ) = h\sum_{i=1}^{n}l\left(t_i , x_i,\frac{x_{i}-x_{i-1}}{h}\right),\\ x_0=x_a , \quad x_n = x_b.\end{gathered}\ ] ] the desired values of , , give the extremum to the multi - variable function and satisfy the system   the fact that only two terms in the sum , and , depend on makes it rather easy to find the extremum of solving a system of algebraic equations . for each , we obtain a polygonal line , which is an approximate solution to the original problem . it has been shown that passing to the limit , as , the linear system corresponding to finding the extremum of is equivalent to the euler lagrange equation for the original problem .in classical theory , given a derivative of certain order , there is a finite difference approximation of the form where this is generalized to derivatives of arbitrary order and gives rise to the grnwald letnikov fractional derivative .let . the left grnwald letnikov fractional derivative is defined as here is the generalization of binomial coefficients to real numbers .similarly , the right grnwald letnikov derivative is given by   the series in and converges absolutely and uniformly if is bounded .the definition is clearly affected by the non - local property of fractional derivatives .the arbitrary order derivative of a function at a time depends on all values of that function in  ] . to overcome this difficulty, we take ,\\ 0 & \text { if } \ t\notin[ a , b ] . \end{cases}\ ] ] thus , one can assume and for  ] .suppose also that is integrable on  ] as , where for some , we approximate the left riemann liouville derivative as where .similarly , one can approximate the right riemann liouville derivative by as it is stated in , this approximation is of first order , i.e. ,   in , it has been shown that the implicit euler method is unstable for certain fractional partial differential equations with grnwald letnikov approximations . therefore ,discretizing fractional derivatives , shifted grnwald letnikov derivatives are used . despite the slight difference with respect to standard grnwald letnikov derivatives , they exhibit , at least for certain cases , a stable performance .the shifted grnwald letnikov derivative is defined by   other finite difference approximations can be found in the literature , e.g. , diethelm s backward finite difference formulas for fractional derivatives .as mentioned earlier , we consider a simple version of the fractional variational problem where the fractional term consists of a riemann liouville derivative on a finite time interval  ] with and define , for .we remark that and that is a variation of , with , for some fixed . therefore ,since is a minimizer for , proceeding with taylor s expansion , we deduce that \eta_i + \frac{\partial l}{\partial { _ ad_t^{\a}}}[i ] \frac{1}{h^\a } \sum_{k=0}^i(\omega^\a_k ) \eta_{i - k } \right]+\mathcal{o}(\epsilon),\end{gathered}\ ] ] where here , and in what follows , =\left(t_i , x_i,\frac{1}{h^\a } \sum_{k=0}^i(\omega^\a_k ) x_{i - k } \right).\ ] ] since takes any value , we can write that \eta_i + \frac{\partial l}{\partial { _ ad_t^{\a}}}[i ] \frac{1}{h^\a}\sum_{k=0}^i(\omega^\a_k ) \eta_{i - k } \right]=0.\ ] ] on the other hand , since , reordering the terms of the sum , it follows immediately that \sum_{k=0}^i(\omega^\a_k ) \eta_{i - k}= \sum_{i=1}^n \eta_i \sum_{k=0}^{n - i}(\omega^\a_k )  \frac{\partial l}{\partial { _ ad_t^{\a}}}[i+k].\ ] ] substituting this relation into equation , we obtain + \frac{1}{h^\a } \sum_{k=0}^{n - i}(\omega^\a_k)\frac{\partial l}{\partial { _ ad_t^{\a}}}[i+k ] \right]=0.\ ] ] since is arbitrary , for , we deduce that +\frac{1}{h^\a } \sum_{k=0}^{n - i}(\omega^\a_k)\frac{\partial l}{\partial { _ ad_t^{\a}}}[i+k]=0 , \quad i=1,\ldots , n-1.\ ] ] let us study the case when goes to infinity .let , b[ ] satisfying is uniformly continuous , we have , we deduce that ={_td^{\a}_b } \frac{\partial l}{\partial { _ ad_t^{\a}}}(\overline{t } , \overline{x } ( \overline{t}),{_ad_{\overline{t}}^{\a } } \overline{x}(\overline{t})).\ ] ] for sufficiently large ( and therefore also sufficiently large ) , = \frac{\partial l}{\partial x}(\overline{t } , \overline{x}(\overline{t } ) , { _ ad_{\overline{t}}^{\a } } \overline{x}(\overline{t})).\ ] ] in conclusion , using the continuity condition , we prove that the fractional euler lagrange equation holds for all values on the closed interval .in this section we try to solve test problems using what has been presented in section [ dirmethsec ] . for the sake of simplicity ,we restrict ourselves to the interval  ] . using the approximation for a fixed , and following the routine discussed in section [ dirmethsec ] , we approximate problem by since the lagrangian in this example is quadratic , system is linear and therefore easy to solve .other problems may end with a system of nonlinear equations .simple calculations lead to in which ,\ ] ] where , and with the linear system is easily solved for different values of .as illustrated in figure [ ex1fig ] , by increasing the value of we get better approximations ..,width=302 ]  although we constructed our theory for problems in the form , other operators can be included in the lagrangian .let us now move to another example that depends also on the first derivative and the solution is obtained via the fractional euler lagrange equation .suppose that the objective functional in the fractional variational problem depends on the left riemann liouville fractional derivative , , and on the first derivative , .if is an extremizer of such a problem , then it satisfies the following fractional euler lagrange equation ( for a proof see ) :   [ example2 ] consider the following minimization problem : =\int_0 ^ 1 \left(\ldz x(t)-\dot{x}^2(t)\right ) dt \longrightarrow \min,\\ x(0)=0 , \quad x(1)=1 .\end{gathered}\ ] ] in this case the euler lagrange equation gives . since , the fractional euler lagrange equation turns to be an ordinary differential equation : which subject to the given boundary conditions has solution discretizing problem , with the same assumptions of example [ example1 ] , ends in the linear system \left[\begin{array}{c}x_1\\x_2\\x_3\\\vdots\\x_{n-1}\end{array}\right ] = \left[\begin{array}{c}b_1\\b_2\\b_3\\\vdots\\b_{n-1}\end{array}\right],\ ] ] where and the linear system can be solved efficiently for any to reach the desired accuracy .the analytic solution together with some approximations are shown in figure [ ex2fig ] ..,width=302 ]  both examples [ example1 ] and [ example2 ] end with linear systems , and their solvability is simply dependent to the matrix of coefficients .now we try our method on a more complicated problem , yet analytically solvable with an oscillating solution .[ example3 ] consider the minimization problem with the lagrangian and subject to the boundary conditions and . the functional , with nonnegative , attains its minimum value for the appearance of the fourth power in the lagrangian , results in a nonlinear system when we apply the euler - like direct method to this problem .for we have where system is solved for different values of and the results are given in figure [ ex3fig ] , where we compare the obtained approximations with the exact solution .roughly speaking , an euler - like direct method reduces a variational problem to the solution of a system of algebraic equations .when the system is linear , we can freely increase the number of mesh points , , and obtain better solutions as long as the resulted matrix of coefficients is invertible .the method is very fast in this case and the execution time is of order for examples [ example1 ] and [ example2 ] .it is worth , however , to keep in mind that the grnwald letnikov approximation is of first order , , and even a large can not result in a high precision . actually , by increasing , the solution slowly converges and in example [ example2 ] , a grid of points has the same order of error , , as a points grid .the situation is completely different when the problem ends with a nonlinear system . in example[ example3 ] , a small number of mesh points , , results in a poor solution with the error .the matlab built in function takes seconds to solve the problem .as one increases the number of mesh points , the solution gets closer to the analytic solution and the required time increases drastically . finally , by we have and the time is seconds .table [ table ] summarizes the results ..number of mesh points , , with corresponding run time in seconds , , and error , . [ cols="^,^,^,^",options="header " , ]   in practice , we have no idea about the solution in advance and the worst case should be taken into account . comparing the results of the three examples considered , reveals that for a typical fractional variational problem , the euler - like direct method needs a large number of mesh points and most likely a long running time .the euler - like direct method for fractional variational problems here presented can be improved in some stages .one can try different approximations for the fractional derivative that exhibit higher order precisions .better quadrature rules can be applied to discretize the functional and , finally , we can apply more sophisticated algorithms for solving the resulting system of algebraic equations .further works are needed to cover different types of fractional variational problems .work partially presented at the 5th symposium on fractional differentiation and its applications ( fda12 ) , held in hohai university , nanjing , may 14 - 17 , 2012 . supported by _ feder _funds through _ compete _ operational programme factors of competitiveness ( `` programa operacional factores de competitividade '' ) and by portuguese funds through the _ center for research and development in mathematics and applications _ ( university of aveiro ) and the portuguese foundation for science and technology ( `` fct fundao para a cincia e a tecnologia '' ) , within project pest - c / mat / ui4106/2011 with compete number fcomp-01 - 0124-feder-022690 .pooseh was also supported by fct through the ph.d .fellowship sfrh / bd/33761/2009 .s. bhalekar , v. daftardar - gejji , d. baleanu and r. magin , generalized fractional order bloch equation with extended delay , internat . j. bifur. chaos appl .* 22 * ( 2012 ) , no . 4 , 1250071 , 15 pp .t. odzijewicz , a. b. malinowska and d. f. m. torres , fractional calculus of variations in terms of a generalized fractional integral with applications to physics , abstr .2012 * ( 2012 ) , art .i d 871912 , 24 pp . arxiv:1203.1961

Abstract: finite differences , as a subclass of direct methods in the calculus of variations , consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem . this article generalizes the same idea for fractional variational problems . we consider a minimization problem with a lagrangian that depends on the left riemann  liouville fractional derivative . using the grnwald  letnikov definition , we approximate the objective functional in an equispaced grid as a multi - variable function of the values of the unknown function on mesh points . the problem is then transformed to an ordinary static optimization problem . the solution to the latter problem gives an approximation to the original fractional problem on mesh points .   fractional calculus , fractional calculus of variations , direct methods .  26a33 , 49m25 .


many quantum physical systems are modelled approximately as closed systems .however , in most practical cases the interaction of a given system with its environment can not be neglected and this makes it necessary to use the theory of open quantum systems  . in the past , it has been very common to neglect quantum memory effects and to resort to a markovian approximation . in many applications the time evolution of the open system is described by a quantum dynamical semigroup represented by a generator of the lindblad - gorini - kossakowski - sudarshan form .  however , when the environment has a nontrivial structure , one has to account for memory effects influencing the dynamics of the open system . during recent years, there has been significant conceptual , theoretical , and experimental progress dealing with non - markovian processes  . in particular ,the very definition of non - markovianity and quantification of quantum memory effects in the dynamics of open systems has received a lot of interest . several measures for quantum non - markovianityhave been proposed which are based on different mathematical and physical concepts  .  in the present paperwe investigate the non - markovianity measure proposed in which expresses the degree of memory effects in terms of the amount of information exchanged between the open system and its environment . within this approachthe degree of non - markovianity is connected to certain optimal initial pairs of quantum states , leading to a maximal flow of information from the environment to the open system . here, we will study the mathematical and physical properties of such optimal state pairs .  throughoutthis paper refers to the hilbert space of the open quantum system and the corresponding set of physical states , i.e. the set of positive trace class operators with unit trace is denoted by .we assume that the dynamics of the open quantum system can be represented in terms of a one - parameter family of completely positive and trace preserving ( cpt ) linear maps , where is equal to the identity map .we will use the quantity as a measure for the degree of memory effects in the open system dynamics . here , where denotes the trace distance defined by by definition , the non - markovianity measure is a positive functional of the process .it is zero if and only if the trace distance between any pair of initial states is a monotonically decreasing function of time signifying a continuous loss of information from the open system to its environment .a nonzero value for the measure means that there is an initial state pair for which the trace distance increases over a certain time interval which can be interpreted as a flow of information from the environment back to the open system implying the presence of memory effects .the time integral in determines the total backflow of information for a certain pair of initial states and .the quantity is then found by taking the maximum over all initial state pairs , i.e. represents the maximal possible backflow of information .a pair of states is said to be an _optimal state pair _ if the maximum in eq .is attained for this pair . in the present paperwe are interested in the mathematical and physical features of the maximization procedure involved in expression .in particular , we will obtain general statements about the physical properties of optimal state pairs which give rise to the maximal possible degree of memory effects in their dynamics . to this end , we first show in sec .[ sec : boundary ] that optimal pairs of states must lie on the boundary of the state space .we then proceed in sec .[ orthogonality ] to demonstrate that the states of any optimal pair must be orthogonal .this is our central result which is physically very plausible since it implies that an optimal pair must have the largest possible initial trace distance .thus , the maximal flow of information from the environment back to the open system emerges if the initial state pair is distinguishable with certainty , i.e. has a maximal information content .we discuss specific features of the case of infinite - dimensional state spaces in sec .[ infinite ] , and present in sec .[ translatable ] an alternative proof for the orthogonality of optimal state pairs which employs the joint translatability of nonorthogonal states .  in the simple case of a qubit, the orthogonality of the optimal pair implies that both states of the pair must be pure . in sec .[ sec : counterex1 ] we study the question of whether this statement holds true for higher dimensional systems .the answer is negative .in fact , we will construct in this section an example for the dynamics of a three - level system ( -system ) for which the optimal pair is not a pure state pair .thus , optimal state pairs exhibiting maximal backflow of information can indeed be mixed states . finally , we draw some conclusions from our results in sec . [sec : conclusion ] .the state space of an open quantum system is given by the set of positive trace class operators on its hilbert space with trace one , on the basis of the convex structure of the state space one can define the boundary of as follows .a point is defined to be an interior point if and only if for all there is a real number such that denoting the set of all interior points by , one defines the boundary of the state space by thus , if and only if there exists such that for all the operator does not belong to the state space .it is easy to show that a density matrix belongs to the boundary of the state space if it has a zero eigenvalue .in fact , let be a normalized eigenvector of with eigenvalue zero , and the corresponding projection operator .then , for all the operator has the negative eigenvalue with corresponding eigenvector , and thus does not belong to the state space .  for finite - dimensional systems the boundary actually contains only states with zero eigenvalues .this can be seen by setting with in which yields the operator .if all eigenvalues of are strictly positive , it follows from the continuity of the roots of the characteristic polynomial that for sufficiently small also all eigenvalues of are positive .this shows that states with strictly positive eigenvalues are in the interior such that for finite - dimensional systems .as has been mentioned already in the introduction a pair of states is said to be an optimal state pair if the maximum in eq .is attained for this pair , i.e. , if holds for this pair .if we speak of an optimal pair we will always assume that the quantum process under consideration is non - markovian , i.e. , since otherwise any pair of quantum states would be trivially optimal .moreover , we note that the optimality property implies of course that .  [the : boundary ] let be an optimal state pair . then both states lie on the boundary of the states space , .suppose that ( at least ) one state of the pair , say , does not belong to the boundary .hence , is an interior point and there exists such that is a quantum state ( see fig . [ fig:1 ] ) .the time evolution of the three states is given by , . by the linearity of the dynamical map we have and , hence , it follows that note that is a fixed number strictly larger than .thus , the last equation tells us that the trace distance between and is always larger by the constant factor than the trace distance between and .this implies that the quantity is larger than by the same factor ( see eq . ) .it follows that can not be an optimal pair which is a contradiction .consequently , any optimal pair of states must belong to the boundary of the state space .thus we see that the maximization over all initial state pairs in the definition of the non - markovianity measure can be restricted to the boundary of the state space , we emphasize that this result is valid for any hilbert space and for any family of linear dynamical maps .the linearity guarantees the invariance of the decomposition defined in for all times ( see fig .[ fig:1 ] ) .   in this sectionwe demonstrate that optimal state pairs must be orthogonal which strengthens the result of sec . [ sec : boundary ] .we recall that two quantum states and are said to be orthogonal , , if and only if their supports are orthogonal , , where the support is defined as the subspace spanned by the eigenvectors with nonzero eigenvalues .note that the orthogonality of two states implies that both states have a zero eigenvalue and , hence , that both states belong to the boundary of the state space .we also mention that the trace distance between any two quantum states satisfies , where the equality sign holds if and only if . orthogonality is thus equivalent to unit trace distance .[ the : orthogonality ] optimal state pairs are orthogonal .let be an optimal pair of states , and suppose that the states are not orthogonal , . according to the jordan - hahn decomposition there exist positive and orthogonal operators and such that taking the trace of this equation we see that .from we conclude that . using eq . , the orthogonality of and , and the fact that the trace distance between nonorthogonal states is always strictly smaller than , we find thus , we have .now we define the operators and .being positive and of unit trace , these operators represent quantum states .moreover we have by use of the linearity of the dynamical maps we obtain from which it follows that since we can conclude from the last equation that the pair yields a non - markovianity which is strictly larger than that of the pair , which contradicts the assumption that is an optimal pair .hence , and must be orthogonal .it follows from theorem [ the : orthogonality ] that the maximization in the definition of the measure for quantum non - markovianity can be restricted to orthogonal initial state pairs .again , this result holds for any hilbert space and any family of linear dynamical maps .it is important to emphasize that for infinite dimensional spaces an optimal state pair need not necessarily exist . to be mathematically more precise ,the maximum in the definition for the non - markovianity measure should be replaced by the supremum to account for such cases .  in theorems [ the : boundary ] and [ the : orthogonality ]we have assumed that an optimal pair of states does exist .if there is no such pair there is a sequence of state pairs such that converges to for . employing the construction used in the proof of theorem [ the : orthogonality ]one can show that the pairs can always be taken to be orthogonal .this means that the non - markovianity measure can be approximated with arbitrary precision by orthogonal state pairs and that we can write    in this section we present an alternative proof for the orthogonality of optimal state pairs , where we restrict ourselves to finite dimensional hilbert spaces , .this proof could be of interest also in other contexts since it relies on the behavior of pairs of states under parallel translations .the idea of the proof is based on the observation gained from two - level systems that nonorthogonal states on the boundary of the state space can be simultaneously translated by a traceless hermitian operator to yield a pair of mixed states , while the trace distance of the pair is invariant under such translations . to make this idea more precise we define to be the set of nonzero , hermitian and traceless operators on .[ def1:trans ] two states are called jointly translatable if and only if there exists an such that for .hence , two states are said to be jointly translatable if and only if there is a nontrivial hermitian and traceless operator which can be subtracted from the states without leaving the state space .we prove that any pair of nonorthogonal states is jointly translatable in such a way that both translated states do not belong to the boundary of the state space .[ the : trans ] if and then are jointly translatable .moreover , there exists an operator such that for .let be given in terms of their spectral decomposition , where , , denotes the eigenvalues and the corresponding eigenvectors of .the assumption implies that and have at least one eigenvector with nonzero eigenvalue which are not orthogonal . after possiblerelabeling one can assume that and holds .moreover , by an appropriate choice of the phases of the eigenstates one can assume without restriction that is real and positive , i.e.  .we consider the following superpositions of the two overlapping eigenvectors , where the normalization constants obey .the projections onto these states are denoted by . by positivity of the overlap has .now , we define \cdot \frac{1}{n}\mathbb{1}_n , \label{eq : shiftop}\end{aligned}\ ] ] where is a real number to be chosen later .clearly , , and so that is a candidate for the traceless hermitian operator which simultaneously shifts both states .furthermore , as one recognizes that the two last terms are negative semidefinite while the first one is positive semidefinite . in order to show positivity of for an appropriate choice of we consider the quantity for an arbitrary normalized vector find \nonumber\\  = & ~b_k(|\chi\rangle)~+p_1^{(k)}\bigl|\langle\chi|\psi_1^{(k)}\rangle\bigr|^2\nonumber\\  & + 4c_+^2\epsilon\bigl\{\frac{\alpha}{n}-\mathrm{re}\bigl(\langle\psi_1^{(1)}|\chi  \rangle~ \langle\chi|\psi_1^{(2)}\rangle\bigr)\bigr\}\nonumber\\  \geq&~b_k(|\chi\rangle)~+p_1^{(k)}\bigl|\langle\chi|\psi_1^{(k)}\rangle\bigr|^2  \nonumber\\  & + 4c_+^2\epsilon\bigl\{\frac{\alpha}{n}-|\langle\chi|\psi_1^{(k)}\rangle|\bigr\}~,\end{aligned}\ ] ] where and since by definition of . in the last step oneuses the fact that for all complex numbers , and that for all normalized vectors by the cauchy - schwarz inequality .consider now the quadratic function one can show that this function is strictly positive for if satisfies thus , if we choose any satisfying we obtain for all normalized vectors and which leads to .this demonstrates that is positive and has no zero eigenvalue , i.e. due to the characterization of the boundary in terms of the eigenvalues given in sec .[ sec : boundary ] .in particular , and are jointly translatable .we remark that the converse of the statement of theorem [ the : trans ] holds , too .thus , two states are jointly translatable if and only if they are not orthogonal .theorem [ the : orthogonality ] can now be proven as follows .let be an optimal pair , and suppose that .then by theorem [ the : trans ] there exists an operator such that . applying theorem [ the : boundary ]we conclude that the states are not optimal .since it follows that also the states are not optimal , which represents a contradiction .therefore , the optimal pair has to be orthogonal .as a simple application of theorem [ the : orthogonality ] one obtains the result that for all non - markovian quantum processes of a two - dimensional system ( qubit ) the maximal backflow of information occurs for a pair of pure , orthogonal initial states , corresponding to antipodal points on the surface of the bloch sphere .this follows immediately from the fact that for qubits the set of pure states is identical to the boundary of the state space . for higher - dimensional systemsthis is no longer true , i.e. , the set of pure states represents a proper subset of the boundary in this case . in this sectionwe will construct an explicit example for an open system dynamics in a three - dimensional hilbert space for which the optimal pair is not a pair of pure states .we consider a -system which interacts with an off - resonant cavity field .the weak - coupling master equation of this model is given by -i\lambda_2(t)[|a \rangle\langle a|,\rho(t)]\nonumber\\ & + & \gamma_1(t)\big[|b\rangle\langle a|\rho(t)|a\rangle\langle b|-\frac{1}{2}\{\rho ( t),|a\rangle\langle a|\}\big]\nonumber\\ & + & \gamma_2(t)\big[|c\rangle\langle a|\rho(t)|a\rangle\langlec|-\frac{1}{2}\{\rho(t),| a\rangle\langle a|\}\big]\label{eq:6},\end{aligned}\ ] ] where refers to the excited and to the two ground states . the coefficients and are determined by the spectral density of the cavity field . introducing the functions where we find that the solution of the master equation yields the dynamical map the functions , and have to obey the following relations which guarantee that is trace preserving and completely positive ,   we consider now , for simplicity , the case where the lamb - shifts are equal to zero , while the decay rates in the dissipator of the master equation ( [ me ] ) are chosen to obey one period of small oscillation , it is easy to check that for this choice the conditions and are satisfied .we have carried out numerical simulations , drawing random pairs of pure , orthogonal initial states and determining the corresponding increase of the trace distance for each initial pair .the results are shown in fig .[ fig : distrpure2 ] .we see from the figure that there is a finite gap between the maximal possible increase of the trace distance for pure , orthogonal initial pairs , and the increase of the trace distance corresponding to the initial pair which consists of the excited state and the uniform mixture of the two ground states ( indicated by the arrow in fig .[ fig : distrpure2 ] ) .thus , this example clearly demonstrates that for hilbert space dimensions larger than two , the optimal initial state pair can indeed contain a mixed state for certain non - markovian dynamical maps .  ) including a mixed state .the sample size of pairs of pure orthogonal states is equal to .,scaledwidth=35.0% ]in this paper we have shown that optimal pairs of initial states for non - markovian quantum dynamics must be orthogonal which implies that these states can be distinguished with probability by a single measurement .optimal state pairs thus have the maximal possible amount of initial information and are therefore capable of emitting and reabsorbing the maximal amount of information during the non - markovian dynamics .we emphasize that the proof of these statements only relies on the convexity of the state space and on the linearity of the dynamical maps .our results are thus very general and can be applied to any quantum process describable by a family of linear dynamical maps in any hilbert space . strictly speaking, even the complete positivity of the dynamics is not needed .in fact , it suffices to assume that the maps are positive since even trace - preserving positive maps are contractions for the trace distance .we have further demonstrated that for hilbert spaces with dimensions of at least optimal state pairs need not consist of pure states , in contrast to the case of a qubit where optimal pairs are always antipodal points on the bloch sphere , and as such pure .the example constructed here leads to an optimal pair consisting of a pure and a mixed quantum state .we conjecture that in hilbert spaces of dimension and higher one can also construct quantum processes for which both states of the optimal pair are true mixtures .finally , we mention that on the basis of the present results further , more specific statements could be proven if one assumes additional properties of the quantum process , such as invariance under certain symmetry groups , or the existence of invariant states .in addition , the notion of joint translatability provides new insights into the structure of the state space and might serve for further applications .we gratefully acknowledge financial support by the german academic exchange service ( daad ) , the magnus ehrnrooth foundation , the jenny and antti wihuri foundation , the graduate school of modern optics and photonics , and the academy of finland ( mobility from finland 259827 ) .sw and hpb would like to thank bruno leggio for helpful discussions .

Abstract: we study a recently proposed measure for the quantification of quantum non - markovianity in the dynamics of open systems which is based on the exchange of information between the open system and its environment . this measure relates the degree of memory effects to certain optimal initial state pairs featuring a maximal flow of information from the environment back to the open system . we rigorously prove that the states of these optimal pairs must lie on the boundary of the space of physical states and that they must be orthogonal . this implies that quantum memory effects are maximal for states which are initially distinguishable with certainty , having a maximal information content . moreover , we construct an explicit example which demonstrates that optimal quantum states need not be pure states .