### Table 3: The rewrite system R0

"... In PAGE 5: ...Table 3: The rewrite system R0 Table3 contains a rewrite system R0, which reduces sequential composition to its pre x counterpart (rules 1-5), and which eliminates expressions of the form x and x (rules 6 and 7), and which reduces occurrences of pre x iteration in the context of alternative composition and of pre x iteration (rules 8 and 9). The rewrite rules are to be interpreted modulo AC of the +.... In PAGE 11: ...2, we may conclude that R=E is terminating. Hence, the rewrite system R0 in Table3 is terminating modulo AC of the +. 4 Completeness of the Axioms In this section we present the proof of the completeness theorem for the axioms of BPAp , with respect to bisimulation equivalence.... ..."

### Table 3: The rewrite system R 0

"... In PAGE 5: ...Table 3: The rewrite system R 0 Table3 contains a rewrite system R 0 , which reduces sequential composition to its pre#0Cx counterpart #28rules 1-5#29, and which eliminates expressions of the form #0E #03 x and #0F #03 x #28rules 6 and 7#29, and which reduces occurrences of pre#0Cx iteration in the context of alternative composition and of pre#0Cx iteration #28rules 8 and 9#29. The rewrite rules are to be interpreted modulo AC of the +.... In PAGE 11: ...2, wemay conclude that R=E is terminating. Hence, the rewrite system R 0 in Table3 is terminating modulo AC of the +. 4 Completeness of the Axioms In this section we present the proof of the completeness theorem for the axioms of BPA p#03 #0E#0F , with respect to bisimulation equivalence.... ..."

### Table 3: The rewrite system R

1996

"... In PAGE 10: ...7 For each term p there exists a basic term q with g(q) g(p) and p = q. Proof: The rewrite system R in Table3 reduces each occurrence of sequential compo- sition in which the left-hand side is not an atomic action. It applies to process terms modulo AC of the + (but not modulo pre xing with quot;).... ..."

Cited by 1

### Table 1: Inverse Table

1998

"... In PAGE 4: ...e., truncn(sd; 8), page 11) in Table1 . The table maps each of the 128 8-bit non-0 signi cands to an 8-bit approximation of its reciprocal.... In PAGE 7: ... At line 6 the variable sd2 is assigned a 32,,17 oating point number that (we will prove) is 1=d with a relative error less than 2?28. This is done by obtaining an initial approximation via Table1 and then re ning it with two iterations of an easily computed variation of the Newton-Raphson method, sdi+1 = sdi(2 ? sdi d) (0 i 1): The variation is obtained by making the following transformations on the equation above. Instead of d we use the oating point number obtained by rounding d with the mode [away 32], i.... In PAGE 26: ...26 It is helpful to generalize away from the particulars of Table1 . Therefore, consider any table mapping keys to values.... In PAGE 26: ... Thus, if a table is quot;-ok and it contains a value v for truncn(d; 8) then jdv ? 1j lt; quot;. It is easy to con rm by computation that Table1 is quot;-ok for quot; = 3=512 and that it contains an entry assigning a value for the 8-bit truncation of every 1 d lt; 2 (e.g.... In PAGE 26: ...roved. Q.E.D. Perhaps the most interesting aspect of checking this proof mechanically is the quot;-ok prop- erty of Table1 . Just as described above, we de ned this property as an ACL2 (Common Lisp) predicate and proved the general lemma stating that any table satisfying that predicate gives su ciently accurate answers.... In PAGE 26: ... Just as described above, we de ned this property as an ACL2 (Common Lisp) predicate and proved the general lemma stating that any table satisfying that predicate gives su ciently accurate answers. When the general lemma is applied to our particular lookup, the system executes the predicate on Table1 to con rm that it has the required property.... In PAGE 27: ...27 var = value error bounds sd0 = (1=d)(1 + quot;sd0(d)) j quot;sd0(d)j lt; 2?8 + 2?9 sdd0 = 1 + quot;sdd0(d) quot;sd0(d) quot;sdd0(d) quot;sd0(d) + 2?30 sd1 = (1=d)(1 ? quot;sd1(d)) 0 quot;sd1(d) quot;sd0(d)2 + sdd1 = (1 ? quot;sdd1(d)) quot;sd1(d) ? 2?30 quot;sdd1(d) quot;sd1(d) sd2 = (1=d)(1 ? quot;sd2(d)) 0 quot;sd2(d) quot;sd1(d)2 + Table 2: Error Analysis for Lines 1-6 ( = 2?29 + 2?31 + (9=512)2?31) the particulars of Table1 are involved in the proof is when the predicate is executed. This example illustrates the value of computation in a general-purpose logic.... ..."

Cited by 27

### Table 21: Term rewriting system for BPA drt.

1996

"... In PAGE 46: ...Table 21: Term rewriting system for BPA drt. Proof The term rewriting system of Table21 is associated to BPA drt by assigning a direction to the axioms. With the method of the lexicographical path ordering it is easily proven that this term rewriting system is strongly normalizing.... ..."

Cited by 7

### Table 2. Basic priority rewrite system.

1996

"... In PAGE 11: ... In our PRS, these rules will be given the highest priority. For the kernel objects of Fig 1, this gives the rewrite rules (1-3) in Table2 . Because the invocation of the continuation K will discard the current continuation, for simplicity we chose this one to be IK, the identity continuation.... In PAGE 12: ... We need three di erent priorities to cope with the previous requirements. The Table2 uses labels l; m; h to suggest lower, medium and higher priority respectively. We now give a simple example of the rewriting process where the current program con guration contains an object o that has a method m, which... In PAGE 15: ... We distinguish two subsets in C: S is the set of objects representing slot names and O = CnS contains the remaining objects. F should rst contain the prede ned functors used as the set F0 in Table2 . For all other objects o 2 O and s 2 S not represented in F0, F can be completed with functors such as o1, o2, : : : and s1, s2, : : : respectively.... In PAGE 17: ... Consider the program con guration depicted in Figure 2. The following table gives the rewrite rules added to the one of Table2 by (we omit certain rules which are not used for the moment). h : eval(o; ;[];K) ! eval(K; ;[mo];ik) (19) h : eval(mo; ;[];K) ! eval(K; ;[BMO];ik) (20) h : eval(lmo; ;[];K) ! eval(K; ;[BMO];ik) (21) h : blf( ;mo) ! lmo (22) h : blf( ;lmo) ! BA (23) The Table 4 illustrates the reduction process for the lookup phase assuming a message (o apos;s a) and current continuation k.... In PAGE 26: ... Terms of the form blf(s; o) are I- reduced either to a term m representing a method if o represents an object in C, or to the term doesNotUnderstand, and in both cases no further I-reduction is possible. Terms of the form eval(o; s; a; k) can match many rules in (C), but in the most general case, they are I-reduced using rule (14) ( Table2 ) to eval(o; ; []; c( ; [s; o]; c( ; [o; a; k]; IK))). Since o is an object in the kernel, by de nition its meta-object is BMO and by Def.... In PAGE 26: ... Proof: The proof proceeds by induction on the number of objects in the program con guration C. Basic case: applied to the kernel produces a PRS whose rewrite rules are rules (4) to (14) in Table2 augmented with meta-object fetching rules and rules whose... ..."

Cited by 7

### Table 3: Maximal or-matchings for query graph G

"... In PAGE 12: ...Thus, whenever x is bound in a weak matching, then it is bound to the chairman of the department. Table3 shows the set of maximal or-matchings for the query graph in Figure 3. For terminal nodes, wehave listed the attached atoms instead of the identity of the object.... ..."

### Table 5. Box rewrite rules

"... In PAGE 9: ... Table 7 gives the usual cut-elimination steps, whereas Table 6 gives the extra cut-elimination steps for the \unordered n-ary quot; treat- ment we give for the contraction links. Table5 lists additional rewrites for storage boxes needed for the categorical semantics. In (Blute et al 1992) a considerable e ort was spent in making the rewiring of thinning links as \local quot; as possible (see the discussion there of rules of surgery).... In PAGE 36: ... the reductions and expansions of Tables 4, 6, 7, are just those of (Danos 1990), and so form a con uent system for which we have strong normalization. The \box-expansion quot; rules of Table5... In PAGE 40: ... A terminal storage link may be handled by a method that depends on the type of thinning link involved. For an exponential thinning, the box rewrite rule in Table5 allows us to move the thinning link outside the box. For a unit thinning, the thinning link and the empire of the unit lie either completely inside or completely outside the storage box|in either case the induction assumption is easily applied.... ..."

### Table 8: Rewrite rules for 6 m Grids

"... In PAGE 43: ...rewrite rules. The rules used for rewriting strings representing a 6 m grid are given in Table8 . The basis for this rewrite system is the SMT for the 6 3 grid which is AC.... ..."