### Table 2: Lower bounds of termination delay (gossiping) in communication steps

1996

"... In PAGE 12: ... Therefore, we can use the gossiping results for edge-colored graphs to compare with the above lower bounds. In the following, if the gossiping time in some colored graph is optimal with respect to the corresponding lower bound in Table2 , then the termination delay due to our algorithm using the same coloring scheme is optimal or near optimal. Precisely, if we let g to be the gossiping time, then the termination delay is less than g + 2 communication steps.... In PAGE 14: ... By comparing Theorem 5.1 and Table2 , we nd that our termination detection algorithm is optimal for the chain and the even ring (d ? g is a constant 2 to 4 time steps).... In PAGE 15: ... Moreover, the edge-coloring technique used in the algorithm allows e cient communication while avoiding the possibility of message collisions and congestions. From Table2 , we note that the numbers of time steps for the optimal cases under the 1-port model is actually equal to the respective absolute lower bounds for information dissemination in these structures regardless of the communication model . Therefore, if we use the all-port communication model, our algorithm performs as well as the broadcast-based algorithm or any other algorithm in these structures.... ..."

Cited by 4

### Table 2: Lower bounds of termination delay (gossiping) in communication steps

1996

"... In PAGE 11: ... Therefore, we can use the gossiping results for edge-colored graphs to compare with the above lower bounds. In the following, if the gossiping time in some colored graph is optimal with respect to the corresponding lower bound in Table2 , then the termination delay due to our algorithm using the same coloring scheme is optimal or near optimal. Precisely, if we let g to be the gossiping time, then the termination delay is less than g + 2 communication steps.... In PAGE 13: ... By comparing Theorem 5.1 and Table2 , we nd that our termination detection algorithm is optimal for the chain and the even ring (d ? g is a constant 2 to 4 time steps).... In PAGE 14: ... Moreover, the edge-coloring technique used in the algorithm allows e cient communication while avoiding the possibility of message collisions and congestions. From Table2 , we note that the numbers of time steps for the optimal cases under the 1-port model is actually equal to the respective absolute lower bounds for information dissemination in these structures regardless of the communication model. Therefore, if we use the all-port communication model, our algorithm performs as well as the broadcast-based algorithm or any other algorithm in these structures.... ..."

Cited by 4

### Table 1: Consistency checking for theorem provers and model builders

1999

"... In PAGE 10: ... Model building o ers a partial solution to this problem: as well as calling the theorem prover with input : , simultaneously call the model builder with input . In practice, this should successfully deal with many of the formulas the theorem prover can apos;t handle, as is shown in Table1 . Here the top row lists possible responses from the theorem prover to : , while the left hand column lists possible responses of the model builder to .... ..."

Cited by 16

### Table 1 Stochastic automata for (X = p 2 E)

1998

"... In PAGE 12: ... To associate a stochastic automaton to a given term, we need to de ne the di erent parts of the stochastic automaton. We start by de ning the clock setting function and the set of edges - as the least relations satisfying the rules in Table1 . However, not all the processes can have a straightforward stochastic automaton as a semantic interpretation.... In PAGE 12: ... Consider the process p fjxGjg(a; fxGg7!(fjxG; yHjgfyHg7!b; stop)) (1) The second occurrence of xG is intended to be bound to the outermost clock setting as shown by the grey arrow. Using the rules in Table1 , the following... In PAGE 13: ... q q0jjAq00, q0jj Aq00, or q0jAq00 implies bv(q0)\var(q00) = var(q0)\bv(q00) = ; De nition 12 Let p be a process without con ict of variables. The stochastic automaton associated to p is de ned by [[p]] def= ( ck; p; C; A; -; ; F), where - and are de ned in Table1 , and C, A and F are de ned as for the syntax of . 2 The reader is invited to check that the processes of the switch system de- ned in Example 3 do not have con ict of variables, and that the stochastic automaton associated to the process System is the one depicted in Figure 1 modulo the identi cation of ck(ck(p)) and ck(p), for all p.... In PAGE 15: ... The proof of Theorem 15 is quite involved since it has to be done in a traditional way: a relation is given for each case and it is proven to be a potential bisimulation (up to $P ). Instead, the proof that $ is a congruence uses the results of [3] since rules in Table1 can be easily rewritten into path format.Another important result that we would like to highlight is that proper renaming of variables preserves potential bisimulation.... ..."

Cited by 36

### Table 1 Stochastic automata for (X = p 2 E)

1998

"... In PAGE 12: ... To associate a stochastic automaton to a given term, we need to de ne the di erent parts of the stochastic automaton. We start by de ning the clock setting function and the set of edges - as the least relations satisfying the rules in Table1 . However, not all the processes can have a straightforward stochastic automaton as a semantic interpretation.... In PAGE 12: ... Consider the process p fjxGjg(a; fxGg7!(fjxG; yHjgfyHg7!b; stop)) (1) The second occurrence of xG is intended to be bound to the outermost clock setting as shown by the grey arrow. Using the rules in Table1 , the following... In PAGE 13: ... q q0jjAq00, q0jj Aq00, or q0jAq00 implies bv(q0)\var(q00) = var(q0)\bv(q00) = ; De nition 12 Let p be a process without con ict of variables. The stochastic automaton associated to p is de ned by [[p]] def= ( ck; p; C; A; -; ; F), where - and are de ned in Table1 , and C, A and F are de ned as for the syntax of . 2 The reader is invited to check that the processes of the switch system de- ned in Example 3 do not have con ict of variables, and that the stochastic automaton associated to the process System is the one depicted in Figure 1 modulo the identi cation of ck(ck(p)) and ck(p), for all p.... In PAGE 15: ... The proof of Theorem 15 is quite involved since it has to be done in a traditional way: a relation is given for each case and it is proven to be a potential bisimulation (up to $P ). Instead, the proof that $ is a congruence uses the results of [3] since rules in Table1 can be easily rewritten into path format.Another important result that we would like to highlight is that proper renaming of variables preserves potential bisimulation.... ..."

Cited by 36

### Table 1: Stochastic automata for L (X = p 2 E)

1997

"... In PAGE 9: ... To associate a stochastic automaton to a given term in the language, we need to de ne the di erent parts of the stochastic automaton. We start by de ning predicates and - as the least relations satisfying rules in Table1 . However, not all the processes can have a straightforward stochastic automaton as a semantic interpretation.... In PAGE 9: ...The second occurrence of xG is intended to be bound to the outermost clock setting as shown by the grey arrow. Using the rules in Table1 , the following stochastic automaton would be obtained b; fxG; yHg a; ; xG xG yH... In PAGE 13: ... We use the notion of adversaries or schedulers [29, 24] to resolve non-deterministic choices. Since parallel composition of stochastic automata can be easily de ned (actually, it is de ned just like for the process algebra, see Table1 ), the simulation algorithm can compose the complete stochastic automaton on the y, which reduces the state space explosion problem. Although (probabilistic) adversaries allow to obtain a complete probabilistic nal model, the inclusion of them as a new ingredient is not that appealing since it would require an additional e ort when modelling systems.... ..."

Cited by 13

### Table 1: Stochastic automata for L (X = p 2 E)

1997

"... In PAGE 8: ... To associate a stochastic automaton to a given term in the language, we need to de ne the di erent parts of the stochastic automaton. We start by de ning predicates and - as the least relations satisfying rules in Table1 . However, not all the processes can have a straightforward stochastic automaton as a semantic interpretation, as we see as follows.... In PAGE 8: ... Consider the process p1 fjxGjg (a; fxGg7!(fjxG; yHjg fyHg7!b; stop)) (2) The second occurrence of xG is intended to be bound to the outermost clock setting as shown by the grey arrow. Using the rules in Table1 , the following stochastic automaton would be obtained b; fxG; yHg a; ; xG xG yH In this sense, xG would be captured by the innermost clock setting as shown by the black arrow in formula (2). Therefore, we consider that clocks are di erent if they are set in di erent places, although they may have the same name.... In PAGE 12: ... We use the notion of adversaries or schedulers [29, 24] to resolve non-deterministic choices. Since parallel composition of stochastic automata can be easily de ned (actually, it is de ned just like for the process algebra, see Table1 ), the simulation algorithm can compose the complete stochastic automaton on the y, which reduces the state space explosion problem. Although (probabilistic) adversaries allow to obtain a complete probabilistic nal model, the inclusion of them as a new ingredient is not that appealing since it would require an additional e ort when modelling systems.... ..."

Cited by 13

### Table 8: Timed automata for the parallel operator

1996

"... In PAGE 27: ... So, for the sake of correctness in our de nitions, we choose a wide enough set of bound clocks in ck(p). We give the rules for the timed automaton in Table8 . Operators jj A and jA are the left-merge and the communicating versions of the parallel operator, respectively.... In PAGE 54: ...4, pjjAq a; - p0jjAck(q) and j= (v[ (pjjAq) a0] + d)( ) ^ @(pjjAq)). By rules in Table8 , a = 2 A and p a; - p0, (pjjAq) = (p) [ (q) and @(pjjAq) = @(p) ^ @(q). Since ncv(pjjAq), j= (v[ (p) a0] + d)( ) and j= (v[ (p) a0] + d)(@(p)).... In PAGE 55: ...4, pjjAq a; - ck(p)jjAq0 and j= (v[ (pjjAq) a0] + d)( ^ @(pjjAq)). By rules in Table8 , a = 2 A and q a; - q0, (pjjAq) = (p) [ (q) and @(pjjAq) = @(p) ^ @(q). Again, by rules in Table 8, p0jjAq a; - ck(p0)jjAq0, (p0jjAq) = (p0) [ (q) and @(p0jjAq) = @(p0) ^ @(q).... In PAGE 56: ...4, pjjAq a; ^ 00 - p0jjAq0 and j= (v[ (pjjAq) a0] + d)(( ^ 00) ^ @(pjjAq)). By rules in Table8 , a 2 A and p a; - p0, q a; 00 - q0, (pjjAq) = (p) [ (q) and @(pjjAq) = @(p) ^ @(q). Since ncv(pjjAq), j= (v[ (p) a0] + d)( ^ @(p)).... In PAGE 57: ...4, ck(p)jjAq a; - p0jjAck(q) and j= (v[ (ck(p)jjAq) a0] + d)( ^ @(ck(p)jjAq)). By rules in Table8 , a = 2 A and p a; - p0, (ck(p)jjAq) = (q) and @(ck(p)jjAq) = @(p) ^ @(q). By de nition of S1, there exists v, v0 and d0 such that v var(p) = (v[ (p) a0] + d0) var(p), v0 var(p0) = (v0[ (p0) a0] + d0) var(p0) and (p; v)Rvar(q)(p0; v0).... In PAGE 59: ...4, ck(p)jjAq a; - ck(ck(p))jjAq0, and j= (v[ (ck(p)jjAq) a0] + d)( ^ @(ck(p)jjAq)). By rules in Table8 , a = 2 A and q a; - q0, (ck(p)jjAq) = (q) and @(ck(p)jjAq) = @(p) ^ @(q). Again, by rules in Table 8, ck(p0)jjAq a; - ck(ck(p0))jjAq0, (ck(p0)jjAq) = (q) and @(ck(p0)jjAq) = @(p0) ^ @(q).... In PAGE 60: ...4, ck(p)jjAq a; ^ 00 - p0jjAq0 and j= (v[ (ck(p)jjAq) a0]+d)(( ^ 00)^@(ck(p)jjAq)). By rules in Table8 , a 2 A and p a; - p0, q a; 00 - q0, (ck(p)jjAq) = (q) and @(ck(p)jjAq) = @(p) ^ @(q). By de nition of S1, there exists v, v0 and d0 such that v var(p) = (v[ (p) a0] + d0) var(p), v0 var(p0) = (v0[ (p0) a0] + d0) var(p0) and (p; v)Rvar(q)(p0; v0).... ..."

Cited by 48

### Table 1: Comparison of Parametric Model Checking Methods proposed by model property decidable? time complexity

1998

"... In PAGE 1: ... One of the considerable approaches to such problems is restricting the expressing power of Timed Automata. Comparison between existing parametric model checking methods are summarized in Table1 . For example, [6] proposed a parametric model checking algorithm in that both a model and a property are written in finite parametric timed automata (PTA) with one integer clock variable.... In PAGE 3: ... Finally, we perform the extension of DDFS to derive the weakest parameter condition in order that the given model never executes invalid runs specified by the given property. Class hierarchies for models and properties in Table1 are shown in Figs. 1, 2, and 3.... ..."

Cited by 2

### Table 7. Algorithm to check for EMB

1998

"... In PAGE 37: ...Table 7. Algorithm to check for EMB The algorithm, which is shown in Table7 , is an adaptation to our framework of the algorithm described in [13] (which could be applied to the functional semantic models of finite-state terms of G), which is in turn a variant of the algorithm proposed in [29] to solve the relational coarsest partition problem. Given a labeled transition system with state space S representing the union of the integrated semantic models of two finite-state terms of G to be checked for EMB, or the integrated semantic model of a finite-state term of G to be minimized with respect to EMB, the idea of the algorithm is to repeatedly refine the current partition until this is a strong EMB.... In PAGE 38: ...By following the proposal of [29], this algorithm can be implemented in O(m log n) time and O(m + n) space where n is the number of states and m is the number of transitions. It is worth noting that a variant of the algorithm in Table7 can be used to compute the coarsest ordinary lumping [37] of the Markovian semantics of a given term, hence allowing for the determination of performance measures by solving a smaller Markov chain which is equivalent to the original one. Definition 5.... ..."

Cited by 74