### Table 1. Axioms of INES constraints over non-empty sets of in nite trees

"... In PAGE 6: ...Table 1. Axioms of INES constraints over non-empty sets of in nite trees Table1 contains ve rules A1-A5 representing sets of axioms.1 The union of these sets is denoted by A.... ..."

### Table 3. Axioms for inclusion constraints over (possibly empty) sets of nite trees

"... In PAGE 19: ...Table 3. Axioms for inclusion constraints over (possibly empty) sets of nite trees In Table3 , we present the set of axioms B, which adapts the set A for the new constraints. The axiom sets A20, A30, and A60 are changed to make implicit non- emptiness premises explicit, and B7 and B8 have been added.... In PAGE 20: ... Proof. The axioms in Table3 again induce a xed point algorithm for the sat- is ability test. By carrying over the techniques for the complexity results from Section C, we obtain the same complexity bound.... ..."

### Table 1: Axioms for inclusion constraints over non-empty sets of nite trees

1996

"... In PAGE 6: ... Then implies a x _ x y, but it implies neither a x nor x y. We will now refer to the axioms in Table1 . Axiom 1 needs to come in two versions since a ground term may be the lower bound of a term.... In PAGE 6: ... We say that a term t occurs implicitly in if x t occurs implicitly in for some variable x. Remark 3 The axioms in Table1 are valid over non-empty sets of nite trees. Proof.... In PAGE 6: ... Satis ability can be tested in cubic time [25]. Remark 5 Let 0 be the closure of the constraint under of consequences of the axioms in Table1 . Then j= x y i 0 j= x y.... In PAGE 7: ... We de ne an algorithm by xed point iteration. At each iteration step, we add the consequences under the axioms in Table1 of the set of constraints derived so far. After termination, the obtained set of constraints is equivalent to the initial one and it is closed.... ..."

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### Table 1. Classi cation of changes in a process. The symbol ; denotes the case where the set of constraints is empty; static | there are no changes in time; var | there are some changes in time

1999

"... In PAGE 3: ... Note that all problems represented in the model can be divided further into six categories, since there are two categories of objective function and three categories of the set of constraints: { the objective function may or may not depend on the time variable t, and { the set of constraints C can be empty, non-empty and time independent, and non- empty and time dependent. Table1 provides a classi cation of all possible cases. The rst and the second class of problems are the stationary cases, i.... ..."

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### Table 4. The axioms for conjunctions of inclusion constraints and non-emptiness con- straints over sets of nite trees (1{7) and over sets of in nite/rational trees (1{5 and 7)

"... In PAGE 20: ... We use as a meta-variable for variables x or terms of the form f( u). ::= x j f( u) apos; ::= x6 0 j 1 2 j apos;1 ^ apos;2 j 9x apos; j ? We next modify the axiom system in Table 1 according to the new interpreta- tion domain and add one axiom (Axiom 7) which handles non-emptiness (see Table4 ). Here, u6 0 stands for the conjunction u16 0 ^ : : :un6 0 where n 0 is the length of the tuple u.... In PAGE 21: ... Theorem17. The constraint system of inclusion constraints 1 2 and non- emptiness constraints 6 0 over (empty and non-empty) sets of nite trees sat- is es Axioms 1-7 in Table4 . Every closed clash-free and occurs-free set of con- straints is satis able over P(TFin ).... In PAGE 21: ... Namely, f( u)6 0 62 apos; and, hence, ui6 0 62 apos; for at least one component ui of u. 2 The axioms in Table4 induce a xed point algorithm for the satis ability test. In order to ensure termination, we consider consequences of Axiom 7 only if they yield a non-emptiness constraint 6 0 for a term which already occurs in the derived set of constraints.... In PAGE 21: ...mptiness constraints, i.e., atomic set constraints in the terminology of [23]. The axiom system in Table4 can be reduced to the one in Table 5 for characterizing the satis ability in this special case. The axioms can be read as rules for adding... In PAGE 22: ... Proof. Since 6 0 conjuncts can be added only through Axiom 2 and 4 (namely as consequences of t for some ground term t), the consequences under Ax- ioms 2 and 5 in Table4 are subsumed by the ones under Axiom 1 and Axiom 4. This yields the above statement for the case of rational, or in nite, trees.... ..."

### Table 4: Constraints generating function. Here AD is implicitly assumed to range over non-empty sequences of field and method names and ending in a method name, AB over non-empty sequences of method names and AC over those AD containing at least one field name.

"... In PAGE 5: ... It is used to shuffle the variables into the correct slots for method invocation. Table4 contains the definition of function Cons which maps a vec- tor of variables DIDC and a term CP involving DIDC to a set of logical for- mulas involving the previously introduced predicate variables for CP. The idea is that ConsB4DIDCBN CPB5 summarises the constraints arising when trying to construct a proof in AL with subject CP.... ..."

### Table 1. Constraints for Phase 1

"... In PAGE 3: ... No set ever changes from {T] back to empty because all constraints are inclusion constraints. Table1 Iists the rules to construct the constraints between alias sets. Our implementation applies these rules, ignoring control flow, to each statement within a method, starting with the main method.... ..."

### Table 1. Constraints for Phase 1

"... In PAGE 3: ... No set ever changes from {T} back to empty because all constraints are inclusion constraints. Table1 lists the rules to construct the constraints between alias sets. Our implementation applies these rules, ignoring control flow, to each statement within a method, starting with the main method.... ..."

### Table 2 - Intersection of coincident constraints

"... In PAGE 18: ... All constraints specified under a quot;[namespace:] limit-scope quot; function SHALL be considered as if they were individual constraints under a single AND operator at the policy framework level. Except as specified in the preceding list, the FunctionId attribute in an lt;Apply gt; element SHALL be one of those listed in Table2 of Section 6. 2.... In PAGE 21: ...2 Intersection of constraints The intersection of two coincident constraints is either the empty set, the two constraints, or a single constraint. The result of taking the intersection of two constraints, and the FunctionId and lt;AttributeValue gt; of any single resulting constraint are specified in Table2 in Section 6. If the intersection of the two constraints is the empty set, then the constraints are incompatible.... In PAGE 25: ... If two coincident constraints are compatible according to the Compatibility test column, then their intersection is the constraint specified in the Replacement constraint column of Table 2. Table2 is to be interpreted according to the following key. Columns one, two and four contain shorthand versions of an XACML lt;Apply gt; element.... ..."

### Table 1. The Subcomponent Dynamic Constraints

2004

"... In PAGE 11: ... The Subcomponent Dynamic Constraints The result of step 2 is a set of allowed states for the subcomponents, automatically extracted from the result of the bound analysis. For each subcomponent we note the visited state and the mailbox contents, for instance, see Table1 , the CL2 state for Client is activated with an empty buffer (a24 ), with a fail or with a ticket message. From these constraints a simple specialise algorithm, see Table 2, computes the minimal STS for the subcomponent which realises the constraints.... ..."

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