### Table 2: Complete axiomatizations for the equivalences

2001

"... In PAGE 5: ... The results obtained for the equivalences are extended to the associated preorders as well. Acknowledgment My thanks to Tony Hoare for suggesting that the axioms of Table2 could be... In PAGE 61: ...61 17.2 Axiomatizing the equivalences In Table2 , complete axiomatizations can be found for twelve of the fteen semantic equivalences of this paper that di er on BCCSP. Axioms for singleton-failures, 2-nested simulation and possible- futures semantics are more cumbersome, and the corresponding testing notions are less plausible.... In PAGE 61: ... I is a unary operator that calculates the set of initial actions of a process expression, coded as a process expression again. Theorem 8 For each of the semantics O 2 fT; S; CT; CS; F; R; F T; RT; P W; RS; Bg two process expressions p; q 2 T(BCCSP) are O-equivalent i they can be proved equal from the axioms marked with \+ quot; in the column for O in Table2 . The axioms marked with \v quot; or \! quot; are valid in O-semantics but not needed for the proof.... In PAGE 62: ... So assume p vO q and (3) has been proven for all pairs of smaller expres- sions p0; q0 2 T(BCCSP). Provided TO contains at least the rst four axioms of Table2 , one has TO ` q = q + p i TO ` q = q + ap0 for every summand ap0 of p. Take O = B, so p vB q.... In PAGE 63: ... This law falls outside conditional equational logic, but it can be reformulated equationally by considering the two cases I(x) = 0 = I(y) and I(x) 6 = 0 6 = I(y). In the rst case it must be that TB ` x = 0 = y and hence the law follows from the third and fourth axiom of Table2 . In the second, observe that I(p) 6 = 0 i p has the form bq +r with b 2 Act.... In PAGE 65: ...2 gives TB ` == nnhnn == = == nnU(h)nn == for h 2 IH. 2 In Theorem 8 the fth and seventh axioms of Table2 may be replayed by a n X i=1 (bixi + biyi) = a n X i=1 (bixi + biyi) + a n X i=1 biyi and a n X i=1 bixi + a n X i=1 biyi = a n X i=1 (bixi + biyi): These laws derive the same closed substitution instances. Thus none of the axiomatizations require the operator I, or conditional equations.... In PAGE 66: ...The linear time { branching time spectrum I from the axioms marked with `+ apos; or `! apos; in the column for O in Table2 . It follows that the axioms marked with `v apos; are derivable.... In PAGE 66: ... 17.3 Axiomatizing the preorders In Table 3, complete axiomatizations can be found for the eleven preorders corresponding to the equivalences axiomatized in Table2 (there is no preorder for tree semantics (U)). This time prov- ability is de ned according to the standards of either rst-order logic with inequality or conditional inequational logic, i.... In PAGE 66: ... In the latter case, the axioms of Table 3 also constitute complete axiomatizations of the equivalences. The three axioms in Table 3 in which the inequality is written \v quot; represent strengthenings of the corresponding axioms in Table2 . The axioms in which the inequality is written \w quot; are merely slick reformulations of the corresponding axioms in Table 2, and could be replaced by them.... In PAGE 66: ... The three axioms in Table 3 in which the inequality is written \v quot; represent strengthenings of the corresponding axioms in Table 2. The axioms in which the inequality is written \w quot; are merely slick reformulations of the corresponding axioms in Table2 , and could be replaced by them. Unlike in Table 2, the characteristic axiom for the readiness preorder (the ninth) is now a substitution instance of the characteristic axiom for the failures preorder (the tenth).... In PAGE 66: ... The axioms in which the inequality is written \w quot; are merely slick reformulations of the corresponding axioms in Table 2, and could be replaced by them. Unlike in Table2 , the characteristic axiom for the readiness preorder (the ninth) is now a substitution instance of the characteristic axiom for the failures preorder (the tenth). Note that the characteristic axiom for the ready simulation preorder (the fth) derives all closed instances of I(x) = I(y) ) ax v a(x + y), which gives the fth axiom of Table 2.... In PAGE 66: ... Unlike in Table 2, the characteristic axiom for the readiness preorder (the ninth) is now a substitution instance of the characteristic axiom for the failures preorder (the tenth). Note that the characteristic axiom for the ready simulation preorder (the fth) derives all closed instances of I(x) = I(y) ) ax v a(x + y), which gives the fth axiom of Table2 . Hence all closed instances of the characteristic axiom for the ready trace preorder (the seventh) are derivable from the fth and eighth axioms.... In PAGE 68: ... Thus requirements 1 and 2(a) are ful lled. As (the closed instances of) the axioms for the respective equivalences from Table2 are easily derivable from the ones for the corresponding preorders from Table 3, requirement 2(b) is ful lled as well. Requirement 3, which used to follow from Theorem 6 and Propositions 16.... In PAGE 71: ...1 and the soundness of the axioms for BCCSP. \First ) quot; (completeness of the axioms for the equivalences): Let T 0 O be the set of axioms marked with \+ quot; in the column for O in Table2 , but using a Pn i=1 bixi + a Pn i=1 biyi = a Pn i=1(bixi + biyi) and a Pn i=1(bixi + biyi) = a Pn i=1(bixi + biyi) + a Pn i=1 biyi instead of the axioms involving the operator I. As Theorem 8 establishes completeness for closed terms only, it holds for T 0 O as well.... In PAGE 71: ... As Theorem 8 establishes completeness for closed terms only, it holds for T 0 O as well. Claim: If T 0 O ` p = Pm j=1 aqj for p; qj 2 T(BCCSP), then, modulo applications of the rst three axioms of Table2 , p has the form p = Pn i=1 api. Proof of the claim: As all axioms in T 0 O are equations, I may use induction on the proof of p = Pm j=1 aqj in equational logic.... In PAGE 78: ... Furthermore, the predicate p T(BCCSP quot;) is generated by the rules of Table 6. Now the complete axiomatizations of Table2 apply to BCCSP quot; p( quot;) p(p) p(p + q) p(q) p(p + q) Table 6: Rules for the termination predicate as well, provided that the occurrences of 0 are changed into , an axiom I( quot;) = quot; is added, and the characteristic axioms for CS and CT also get variants in which by + z resp. cy + v is replaced by quot;.... ..."

Cited by 61

### Table 1: Linear Temporal Logic Operators

"... In PAGE 1: ... If more than one proposition is true, they are written between parentheses. Table1 lists the usual LTL operators with example and counter example traces. Table 1: Linear Temporal Logic Operators ... ..."

### Table 4. Comparison of tools and supported temporal logics.

"... In PAGE 11: ...) Which tools may be applied at all for a given type of question depends on the temporal logic it provides. Table4 gives a comparison of the model checkers in use and the versions of logics provided. Which tools should be applied in which order depends on the analytical methods they are based on.... ..."

### Table 1. Model checking the main temporal logics

"... In PAGE 34: ...Ph. Schnoebelen 6 The two parameters of model checking Table1 that concludes Section 3.3 shows that model checking CTL formulae is much easier than model checking LTL formulae, so that it seems CTL is perhaps the better choice when it comes to picking a temporal logic in which to state behavioral properties.... In PAGE 36: ...TL formulae. There is a parallel here with the results from Section 3.1 where we saw that model checking for these three logics can be done in time that linearly depends on jSj. Finally, Table1 is not a fair comparison of the relative merits of CTL and LTL 25. 7 The complexity of symbolic model checking We already observed that, in practical situations, model checking mostly has to deal with large Kripke structures and small temporal formulae.... ..."

### Table 1. A Taxonomy for CMC Technologies Temporality Anonymity Modality Spatiality

"... In PAGE 8: ... TESTING THE EFFECTS OF DIFFERENT CMC TECHNOLOGIES In these studies, we focused on two common CMC technologies: the Palace and the ChatNet. As outlined in Table1 , the Palace and ChatNet, two popular CMC applications, can be contrasted across the attributes of Modality and Spatiality. The Palace is a multimedia two-dimensional chat program, while ChatNet is a text-only chat program.... ..."

### Table 2. Principle of Translating Frame Logic to Predicate Logic

"... In PAGE 10: ... We adopted that approach: the input is processed and translated in several stages: The first step (besides the necessary parser) is the Frame-Logic-translator, that translates the Frame-Logic expressions to first-order logic expressions. Table2 gives an idea of how this translation is performed, but it does not catch the complete translation: e.g.... ..."

### Table 1. Horn temporal linear logic

1999

Cited by 6

### Table 1. Horn temporal linear logic

1999

Cited by 6

### Table 1. Horn temporal linear logic

1999

Cited by 6