### Table 1. Automata state variables Automata class State Variable

"... In PAGE 5: ... Resident automata are endowed with a set of state variables, including economic status, ethnicity, preferences for housing choice, and with the ability to sense their neighboring environment (beyond the fixed and symmetric neighborhoods classically employed in urban CA models). The organization of automata in code is illustrated in Figure 1; state variables (and their abbreviations in the following equations) for each automaton are listed in Table1 . State variables that have a value range from 0 to 1 are normalized.... ..."

### Table 1: (Un)decidability of the equational theories of classes of cylindric algebras.

1999

"... In PAGE 6: ... In fact, the expansion Crs+ 3 of Crs3 with a single operation k1 1 has an undecidable equational theory. We summarized these results in Table1 . We note that, similarly to the relation algebra case, the unde- cidable equational theories are in fact r.... ..."

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### 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.... ..."

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### 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.... ..."

### Table2. Complexity classes of automata with logarithmically space-bounded tape and empty alternation

1994

"... In PAGE 5: ... In the following, for X 2 fLOG, PDA{TIME(pol), PDA, P, PSPACEg and a function g, where we again admit the cases that g is a constant or that g is unbounded, let EA log g X denote the set of all languages recognized by logspace Turing machines augmented with storage of type X, which make g(n) 1 empty alternations. The main results of this chapter are collected in Table2 , which is the \empty quot; analogue of Table 1. 3.... ..."

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### Table 4: Class for nondeterministic finite automata

2005

"... In PAGE 2: ... But we provide methods to test for epsilon1-transitions and to convert an epsilon1-NDFA to a NDFA. See Table4 , Appendix A, for more details. 2.... ..."

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### Table 1. Decidability of the model property for timed automata with behavioural re- strictions wrt. speci cations from di erent fragments of Duration Calculus. The frag- ments are named after the shapes of allowed atomic formulae.

"... In PAGE 9: ... To the best of our knowledge, these are the rst e ective procedures available for a dense-time Duration Calculus with metric time, the chop modality, and unrestricted negation. Table1 provides an overview over the decidability results for model-checking timed automata against various fragments of dense-time Duration Calculus. A particular implication of these ndings is that even extremely abstract real-time formalisms can be integrated into the design process of embedded controllers through key-press techniques.... ..."

### Table 5: Timed automata for the hiding operator

1996

"... In PAGE 24: ... Free and bounded variables are de ned as follows. fv(hide A in p) def= fv(p) bv(hide A in p) def= bv(p) Rules for the timed automata and TTS are given in Table5 and Table 6 respectively. The axiomatic de nition is given in Table 7.... ..."

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### 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).... ..."

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