### Table 1: Transition rules for BPPH. De nition 1 Behavioural Equivalences. Processes E and E0 are said to be language equivalent, E lan E0, i for every computation of E

1995

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### Table 3: Plotkin-style transitional semantics corresponding to the behaviour algebra. 3.3 Behavioural Equivalences Intuitively, two behaviours can be regarded as \equivalent quot; if they are indistinguishable through experiments (sequences of transitions) where empty actions are insigni cant. Many such equiv- alences have been proposed in the literature (branching bisimulation, observation equivalence, failure equivalence, testing equivalence, etc.). Our results will hold for most such equivalences which are congruences, ignore divergence (in nite internal computation), and respect termina- tion properties such as deadlocks. We use to range over sequences of non-empty actions, and write hi for the empty sequence. De nition 15

1993

"... In PAGE 17: ... Second, we de ne the semantics through an interpretation of terms into behaviours rather than giving a family of transition relations directly on terms. There is however only a super cial di erence between these approaches; we could instead have given a family of transition relations as in Table3 (given here only as an illustration). For any xed I the relations obtained in this way satis es A ?!B i BI[[A]] ?!BI[[B]].... ..."

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### Table 1 Structural equivalence.

"... In PAGE 8: ... 3 Operational semantics We present the operational semantics of systems by defining the structural congruence over systems and by mapping them onto labeled transition systems (LTSes). Table1 describes a relation indicating syntactic differences between systems that do not influence the behaviour of processes. More precisely, identity (i), reflexive (ii) and transitive (iii) relations hold, the order of the systems parallel composition is not relevant (iv), associative relation holds (v), portions of system can be replaced with other ones structurally equivalent (vi), null processes have no influence on the behaviour of the system (vii) and, finally, replication operator !P corresponds to an infinite parallel composition of P (vii).... ..."

### Table 6. PCPE behaviour (speedup).

"... In PAGE 8: ... Note that in our implementation, when eliminating equivalent configurations in the search tree, we still keep track of their control rules, in order to produce accurate percentages. Table6 shows the results achieved when we use speedup as a fitness function. In general, speedup values in most cases should be greater than 1.... ..."

### Table 5: Proposed description of multiplicative connectives behaviour.

"... In PAGE 16: ... Once the underlying rules were presented, the forthcoming step is to de ne how does tensor product (par) and linear negation look like. Table5 table does this task. Linear implication (| ) is de ned as the equivalence 2 on page 9 and linear negation is just the roles apos; change.... ..."

### Table 2.1: The logical connectives

1995

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### Table 2. Concurrent execution of two constructors can cause side-effects that are not present when either constructor is considered in isolation. In this example, which continues an analysis of the classes Class2 and Demon introduced in Fig. 1, interleaved executions of constructor commands results in Static.Val being set to 30 as a side-effect. Since Demon cannot interact with Class1 to generate this side-effect we conclude that Class1 and Class2 are behaviourally distinct, even though they are functionally equivalent.

### Table 2. Summary of the behaviour of MAC and FC on the different CSP encodings with respect to the behaviour of DP on the original CNF formula. For the non-binary encoding, nFC0 and nFC1 refer to certain types of generalisations of forward checking for non-binary CSPs, see [5].

"... In PAGE 10: ...orollary 1. Let F be a CNF formula and C the place encoding of F. Given equivalent branching heuristics, we have that MAC and FC, respectively, applied to C explores the same number of branches as DP on F. Combining Corollary 1 with previously known results [21], we summarise the behaviour of MAC and FC on the different CSP encodings with respect to the behaviour of DP on the original CNF formula in Table2 . For notation, consider- ing X vs Y , X = Y denotes that X and Y are equivalent in the above mentioned sense, while X gt; Y and X negationslash = Y denote that X is superior to and incomparable with Y , respectively.... ..."

### Table 7. Rules of the parallel process system

"... In PAGE 14: ... We make free use of the fact that parallel composition is associative and commu- tative with respect to any reasonable behavioural equivalence between agents, such as bisimulation equivalence [22]. We associate to a parallel ow graph system the process system (Con; Act; ) as in the sequential case, the only di erence being the rule corresponding to parallel calls, which is shown in Table7 . Observe that the left-hand side of all... ..."

### Table 2: Classi cation of Common Schema Transformations

1998

"... In PAGE 4: ... For example, we may divide an entity type into sub-types using an is-a hierarchy by utilising knowledge about the domain of an attribute associated with the entity type (see Figure 1(c)). Table2 classi es the transformations of Figures 1{3 according to which notion of equivalence they obey, where TE, ME and BE respectively denote transformational, mapping and behavioural equivalence. We illustrate how this classi cation is obtained by considering the entity/relationship equivalence of Figure 1(b).... In PAGE 6: ... In Section 7 we compare our approach with previous work on the formalisation of schema equivalence. We also show how our de nition of schema equivalence generalises the notions of transformational, mapping and behavioural equiv- alence, and we give an alternative classi cation of the transformations of Table2 in which each transformation obeys precisely one type of equivalence. Section 8 gives our concluding remarks.... In PAGE 21: ... This argument also applies in the case that the schemas S and S0 are c-equivalent with respect to a condition f, in which case the instance I above satis es f. Table2 in Section 1.2 classi ed the transformations of Figures 1{3 according to whether they obey transformational, mapping or behavioural equivalence.... In PAGE 21: ... In prac- tice, schema conformance transformations are applied bi-directionally (apart from synonym- and homonym-removal) while schema merging and restructuring ones are applied uni-directionally. For each transformation, it is easy to show formally that the properties of transformational, mapping or behavioural equivalence shown for it in Table2 indeed hold, given our discussion above of how these notions relate to our de nitions of u-equivalence or c-equivalence. Table 3 categorises these transformations according to whether they are schema-dependent (s-d), instance-dependent (i-d) or knowledge-based (k-b).... ..."

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