### Table 1. Composition of basic relations in partially-ordered time (left gure) and branching time (right gure). lt; gt; = k

1999

Cited by 7

### Table of partial permutivities of some elementary cellular automata.

### Table 7. Average execution time (s)

1997

Cited by 3

### Table 1 Performance of the Branch-and-Bound algorithm improving system slackness averaged across 50 runs in the complete and partial allocation scenarios

2007

"... In PAGE 11: ... In the rectangle above each bar, the top cor- responds to the initial upper bound for that run and the bottom corresponds to the upper bound tightened with the Branch-and- Bound algorithm. Table1 compares the efficiency of the max- first, min-first, and arbitrary orderings for the system slackness improvement and upper bound tightening. The results were averaged across 50 runs for each of the orderings.... In PAGE 12: ...omponent of the objective metric, i.e., system slackness. As Table1 shows, compared to the complete allocation scenario, all three variants of the Branch-and-Bound heuristic succeeded in system slackness improvement over the results of the Per- mutation Space Genitor-Based heuristic in approximately the same number of runs, but their relative improvement on system slackness was higher. As indicated before, the Branch-and-Bound heuristic re- quires a relatively long execution time.... ..."

### Table 1: Timed automata for L

1996

"... In PAGE 8: ...3 (Associated timed automaton) Let p 2 L. ncv, the predicate of non-con ict of variables is de ned inductively according to rules in Table1 . For all process p such that ncv(p) the timed automaton associated to p is de ned by [[p]]T = (L; A; C; p; - ; @; ) where -, @ and are de ned as the least sets satisfying the rules of Table 1.... In PAGE 10: ... But it can be straightforwardly proven by induction on the depth of the proof tree taking into account that if ; 0 2 (C) then ^ 0; _ 0 2 (C). 2 Rules in Table1 capture the behaviour described in Section 3.1 in terms of timed automata.... In PAGE 10: ... For instance, consider the term p (x 2) (fjxjg (x = 1)7!a; stop). Clearly, x is free in the invariant (x 2), however, using rules in Table1 , we derive @(p) = (x 2) and (p) = fxg. Thus, according to De nition 2.... In PAGE 11: ... De nition 3.6 (Associated timed automaton) Let E be a recursive speci cation such that ncv(E) holds according to rules in Table1 and Table 2, i.... In PAGE 11: ...able 1 and Table 2, i.e., E does not have con ict of variables. The timed automaton associated to p 2 Lv is de ned by [[p]]T = (L; A; C; p; -; @; ) where -, @ and are de ned as the least set satisfying rules in Table1 and rules in Table 2. 2 Table 2: Timed automata for recursion The following rules are de ned for all X = p 2 E ncv(X) ncv(p) ncv(X = p) 8X = p 2 E: ncv(X = p) ncv(E) (p[p=X]) = C (X) = C @(p[p=X]) = @(X) = p[p=X] a; - p0 X a; - p0 De nition 3.... ..."

Cited by 48

### Table 1. Timed automata for L

"... In PAGE 6: ... But it can be straightforwardly proven by induction on the depth of the proof tree taking into account that if ; 0 2 (C) then ^ 0; _ 0 2 (C). Rules in Table1 capture the behaviour above described in terms of timed automata. In particular, it deserves to notice that a process p + q can idle as long as one of them can.... In PAGE 6: ... For instance, consider the term p (x 2) (fjxjg (x = 1)7!a; stop). Clearly, x is free in the invariant (x 2), however, using rules in Table1... In PAGE 8: ...-; @; ) where -, @ and are de ned as the least set satisfying rules in Table1 and rules in Table 2. u t Table 2.... ..."

### Table 1: Tractable classes of the point algebra for partially ordered time [5].

"... In PAGE 19: ... xky i neither x y nor y x The point algebra for partially ordered time has been throughly investigated earlier and a total classi cation with respect to tractability has been given in Broxvall and Jonsson [4]. In Broxvall and Jonsson [5] the sets of relations in Table1 are de ned and it is proven that ?A _ A, ?B _ B, ?C _ C and D are the unique maximal tractable disjunctive classes of relations for partially ordered time. The proofs of tractability for those sets relied on a series of handmade independence proofs.... ..."

### Table 3: Growing Automata with the Capability and the Size of an Electronic Computer # of ops. # of steps total time time per step size

1995

"... In PAGE 13: ... In order to t in the same space as the electronic computer, each element can take up at most 0:09 n m3. Results for problem sizes of 109, 1012, 1015, and 1018 operations are calculated in Table3 . The number of operations column shows the total number of operations n which are required to solve the problem.... In PAGE 13: ... The size column shows the cube side of one element of the growing automaton, assuming that each element occupies a cube. The numbers in Table3 provide an estimate for the performance of growing automata to equal electronic computers in execution speed. The most interesting column is time per step which gives the time available for the creation of new elements.... In PAGE 13: ... It can be concluded that, with the increasing problem size, time constraints considerably favor growing automata over electronic computers. It is interesting to compare the numbers in Table3 to cells in living organisms. Data are taken from Dyson (1978).... In PAGE 13: ... The rate of cell growth varies between twenty minutes and one day, approximately, for one cell division. The row that corresponds to these numbers in Table3 is the row with around 1015 operations. It follows that an implementation of growing automata with cells similar to cells in living organisms would be better than electronic computers at tasks that require at least 1015 operations.... ..."

Cited by 1

### Table 3: Growing Automata with the Capability and the Size of an Electronic Computer # of ops. # of steps total time time per step size

1995

"... In PAGE 13: ... In order to t in the same space as the electronic computer, each element can take up at most 0:09 n m3. Results for problem sizes of 109, 1012, 1015, and 1018 operations are calculated in Table3 . The number of operations column shows the total number of operations n which are required to solve the problem.... In PAGE 13: ... The size column shows the cube side of one element of the growing automaton, assuming that each element occupies a cube. The numbers in Table3 provide an estimate for the performance of growing automata to equal electronic computers in execution speed. The most interesting column is time per step which gives the time available for the creation of new elements.... In PAGE 13: ... It can be concluded that, with the increasing problem size, time constraints considerably favor growing automata over electronic computers. It is interesting to compare the numbers in Table3 to cells in living organisms. Data are taken from Dyson (1978).... In PAGE 13: ... The rate of cell growth varies between twenty minutes and one day, approximately, for one cell division. The row that corresponds to these numbers in Table3 is the row with around 1015 operations. It follows that an implementation of growing automata with cells similar to cells in living organisms would be better than electronic computers at tasks that require at least 1015 operations.... ..."

Cited by 1

### Table 1. Tractable classes of the point algebra for partially ordered time.

2000

Cited by 2