A modal characterization of alternating approximate bisimilarity

Abstract

Recently, alternating transition systems are adopted to describe control systems with disturbances and their finite abstract systems. In order to capture the equivalence relation between these systems, a notion of alternating approximate bisimilarity is introduced. This paper aims to establish a modal characterization for alternating approximate bisimilarity. Moreover, based on this result, we provide a link between specifications satisfied by the samples of control systems with disturbances and their finite abstractions.

Two-thirds simulation indexes and modal logic characterization

“…Then ða,cÞ > max fða,bÞ,ðb,cÞg ¼ l leads to ða,cÞ > > l. Since u In [24], the modal logical characterization of lbisimilarity is proposed when is a general metric by adding the diamond operator < a > to Ying's logical languge [11]. Since, l-two-thirds simulation which is not an equivalence relation includes the condition that the process refuses the set X , the general logical characterization in [24] is not suitable for l-two-thirds simulation. Next, we will introduce the general modal language which is obtained by adding the operators ½X and < t > F to Definition 17 to characterize l-two-thirds simulation.…”

“…The reason is that the semantics equivalence of two processes over the logical language L l is transitive, but the relation % l 2=3 is not always an equivalence relation when is a general metric. Next, we modify the example presented in [24] to state this point.…”

“…A natural question raised at this point is: whether l-two-thirds bisimilarity coincides with L <> lequivalence when is an ultra-metric. The following example which comes from [24] gives a negative answer.…”

“…Example 6 [24] Let A ¼ fa,bg and a ¼ a, b ¼ b, l 2 ½0,1Þ. We consider ¼ ðfs,t,vg, A,fs ) a v,t ) b vgÞ and define the distance function on A: for any x,y 2 A, if x ¼ y, then ðx,yÞ ¼ 0 else ðx,yÞ ¼ l. It is easy to show that is an ultra-metric.…”

“…In order to overcome this defect, we need to find another modal logical language to characterize l-two-thirds bisimilarity. In [24], the authors provide a kind of modal logical characterization of lbisimilarity in the general metric case and demonstrate that in the ultra-metric case, this characterization degenerates into one in the regular style. Moreover, it coincides with one obtained by Ying in [11].…”