### Table 1. The tree automata A, B, and C.

"... In PAGE 7: ...utomata share no other states. Hence L(Gnv) = !k Rnv. The recognizability of !k Rs is obtained by replacing B by the tree automaton C that accepts in state i all terms. ut Table1 shows the tree automata A, B, and C used in the proof of the above lemma for the following TRS R: 1: f(g(x); a) ! f(h(h(x)); x) 2: h(a) ! h(b) 3: h(f(x; b)) ! x The underlinings are to ensure that only states 1, 2, and 3 are shared between A and B (C). Consider the tree automaton A.... ..."

### 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: 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. The tree automata A, B, and C. a ! a ! a !

"... In PAGE 7: ...utomata share no other states. Hence L(Gnv) = !k Rnv. The recognizability of !k Rs is obtained by replacing B by the tree automaton C that accepts in state i all terms. u t Table1 shows the tree automata A, B, and C used in the proof of the above lemma for the following TRS R: 1: f(g(x); a) ! f(h(h(x)); x) 2: h(a) ! h(b) 3: h(f(x; b)) ! x The underlinings are to ensure that only states 1, 2, and 3 are shared between A and B (C). Consider the tree automaton A.... ..."

### Table 1: Size of few reactive automata

in The steam boiler controller problem in ESTEREL and its verification by means of symbolic analysis

"... In PAGE 10: ... Experimental results We present results obtained using symbolic bisimulation minimisation with the fc2symbmin processor on a selection of automata produced from esterel programs together with com- puting time on an Ultra Sparc, 320 Mbytes memory size. Table1 provides gures before reduction, and table 2 the corresponding number after re- duction. Memory size is in bytes.... ..."

### Table 9: Mapping between the automata approach and the biological immune system. Automata Identification Immune System

"... In PAGE 5: ...xamples, OS/IN: One Shot/Incremental, SY: Symbolic, NT: Noise Tolerant, XOR: XOR problem. ..................36 Table9 : Mapping between the automata approach and the biological immune system.... In PAGE 40: ... The proliferation of an automaton (clone) was function of two factors: the degree of recognized examples (antigens), and the interactions among the automata (clones). Table9 presents the trade-off between their immune algorithm and the biological immune system. The fitness function, responsible for controlling the immune algorithm, took into account the fitness of the automaton at each generation, the antigen score, the automaton turnover, the network... ..."

### Table 1 shows the tree automata A, B, and C used in the proof of the above lemma for the following TRS R:

"... In PAGE 8: ...Table1 . The tree automata A, B, and C.... ..."

### Table 3.1: The signature of DIOA operators of DIOA speci es the same transition trees as of the corresponding operators for I/O automata.Table 3.1 presents the signature for DIOA. The sort symbols associated with the opera- tors range over all possible action signatures with a single internal action if no additional restrictions are mentioned. Thus, rather than a single operator (e.g. parallel, renaming, etc.) we actually have a family of operators parameterized on the sorts of the operands. To avoid heavy notation we will drop the sort indexes from the operators whenever the sorts are evident. Indeed all non-constant operators are uniquely determined by the sorts of their operands. As additional simpli cation we will represent action signatures as pairs (in; out) since the set of internal actions is xed to be f g. In choosing the operators we had in mind two major goals: representing the three main operators of I/O automata (i.e., parallel, hiding and renaming) and expressing a su cient number of transition trees. The second goal is achieved through pre x- ing, external choice and recursion; the internal choice operator will turn out to be useful for proving completeness of axioms. Recursion is obtained in a De Simone style [De 84, De 85b]. 23

### Table 1: The results from applying the bisimulation minimization algorithm to tree automata that arose in the verification of protocols Perculate and Leader.

"... In PAGE 14: ... The protocol is further described in [4]. Table1 shows the execution time, and the size of the tree automata before and after running our minimization algorithm.... ..."