### Table 4: Weak-head normal forms

1999

"... In PAGE 23: ... We have not built any Vclosed assumptions into the de nition, as this will always be used in a context in which all terms are Vclosed. The reader may validate such a de nition by considering the weak- head normal forms ( Table4 ), and various lemmas relating !wh and... ..."

Cited by 31

### Table 4: Weak-head normal forms

909

"... In PAGE 23: ... We have not built any Vclosed assumptions into the de nition, as this will always be used in a context in which all terms are Vclosed. The reader may validate such a de nition by considering the weak- head normal forms ( Table4 ), and various lemmas relating !wh and... ..."

### Table 4: Performance Under an Enhanced Form of Head Normalization

2004

"... In PAGE 24: ... We, once again, experimented with its use within the Teyjus system. Referring to the new approach as enhanced head normalization, Table4 contrasts performance under it to that under the head normalization with lazy substitution regime. The new procedure is designed to avoid the multiple walks over the bodies of embedded redexes that a ected adversely the head normalization with eager substitution approach as well as the redundant walks over already normalized structures that mark the full normalization scheme.... ..."

Cited by 6

### Table 3: Conditional Axioms. The rst axiom in Table 3, says that the encapsulation operator can be pushed inside when the encapsulated actions are not possible vehicles of communication between x and y: The axiom CA2 can be explained in a similar way. The other axioms speak for themselves. The major advantage of the axioms given in Table 3 is that they allow to prove important properties of processes without having to expand processes to their (head) normal form which can be very tiresome in case of parallel compositions with more than two components. For instance, showing that three connected bags form a bag without using such axioms may be very involved (see Section 8).

"... In PAGE 5: ...2 A more interesting axiom is CA2 (see Table3 ), which allows pushing the I operator inside parallel compositions (in appropriate circumstances). Milner uses similar rules in the veri cation of the scheduler, although he uses another terminology (see [Mil89]).... In PAGE 7: ...possible to compute the alphabet of a process in an e ective way. In Table3 , seven conditional axioms taken from [BBK87] are presented6 which use the alphabets of processes. As usual in process algebra, the binary operator denotes the communication function (see e.... In PAGE 8: ... That CAr1 is derivable is motivated as follows. By substituting @H1(x) for x and @H2(y) for y in CA1 (see Table3 ), we obtain ( (@H1(x)); (@H2(y))\H) H ! @H(@H1(x) k @H2(y)) = @H(@H1(x) k @H@H2(y)) (1) Obviously, H1 can not be part of the alphabet of the process @H1(x): In for- mula: (@H1(x)) H1: Similarly we have (@H2(y)) H2, and hence also ( (@H1(x)); (@H2(y)) \ H) (H1; H2 \ H). Thus, @H(@H1(x) k @H2(y)) = @H(@H1(x) k @H@H2(y)) if (H1; H2 \ H) H (2) Then, by using the fact that H2 \ H is equal to the set H ? H2; we have transformed CA1 into CAr1.... ..."

### Table 2: Summary of Domain for Amounts of Evaluation of Lists Although it is too di cult to draw even an approximation of the domain diagram, it can be thought of as the coalesced sum of an in nite sequence of nite subdomains, each corresponding to the amounts of evaluation for a nite list containing exactly n elements. The least element of each subdomain is the element ?L (hence the coalesced sum). Corresponding to each of the other three points of List4 is a hypercube of dimension n. Coalesced with this domain for nite lists is a hypercube of in nite dimension corresponding to the amounts of evaluation for in nite lists. 2.3 Amounts of Evaluation for Functions We take the view that expressions of function type may be evaluated only to weak head normal form, similar to integers, so the elements of E ! are ? ! , ? ! , gt; ! and ? ! , and the domain is isomorphic to that for integers.

### Table 5: The Frequency of Renumbering with the de Bruijn Representation

2004

"... In PAGE 27: ... Cases of this kind can be identi ed as those in which rule (r5) is used where the skeletal term is non-atomic and an immediate simpli cation by rule (r12) is not possible. Table5 tabulates the data gathered towards this end for the two viable and comparable approaches that was identi ed by the analysis in Section 4, namely head normalization with lazy substitution and the enhanced form of eager head normalization. An interesting observation is that no renumbering is actually involved in the case of L style program- ming.... ..."

Cited by 6

### Table 3: Comparison of Laziness and Eagerness in Reduction

2004

"... In PAGE 23: ... However, once a generalized head normal form has been exposed, the same procedure is invoked on each of the argu- ments. Table3 contrasts the heap usage, term traversal and running time observed under this interpretation of full normalization with those obtained when only head normalization is performed and a lazy approach to substitution is employed. The data in Table 3 indicate a near parity in terms of the objects allocated on the heap between the lazy and the eager reduction strategies being considered.... In PAGE 23: ... Table 3 contrasts the heap usage, term traversal and running time observed under this interpretation of full normalization with those obtained when only head normalization is performed and a lazy approach to substitution is employed. The data in Table3 indicate a near parity in terms of the objects allocated on the heap between the lazy and the eager reduction strategies being considered. This actually re ects, as we had anticipated, a substantial reduction in the creation of new term structures if a full normalization strategy is used instead of a head reduction approach that also carries out substitution eagerly.... ..."

Cited by 6

### Table 1: Informal relationships to Notions in Process Algebra

1998

"... In PAGE 30: ... This enables us to handle open expressions, but the proof technique fails to be complete (for example, it cannot prove that any pair of weak head normal forms are equivalent). We summarise some informal correspondences between the notions in process calculus and the relations defined in this article in Table1 . Our attempts to complete this picture find a more exact correspondence between the proof techniques relating to bisimulation up to improvement and contexthave so far been unsuccessful.... ..."

Cited by 10

### Table 1: Informal relationships to Notions in Process Algebra

"... In PAGE 30: ... This enables us to handle open expressions, but the proof technique fails to be complete (for example, it cannot prove that any pair of weak head normal forms are equivalent). We summarise some informal correspondences between the notions in process calculus and the relations defined in this article in Table1 . Our attempts to complete this picture find a more exact correspondence between the proof techniques relating to bisimulation up to improvement and contexthave so far been unsuccessful.... ..."

### Table 1: The system CDV

"... In PAGE 4: ... Finally, the system with ! also allows, as we will see, a simple type characterization of terms possessing head normal forms [33]. The complete type assignment system, which is known in the literature as the system CDV, is reported, for the reader ease, in Table1 , along with the syntax of type expressions. The system without rule (!) will be referred to as the system CDV 6 ! .... ..."