### Table 4 is complete for BPA if and only if the axiomatisation in Table 2 is complete for BPA .

1994

"... In PAGE 6: ...Table 3: Action rules for proper iteration PI1 x(x y + y) = x y PI2 (x y)z = x (yz) PI3 x (y((x + y) z + z) + z) = x((x + y) z + z) Table4 : Axioms for proper iteration 3.3 One rule for axiom PI2 Now that we have replaced SEI by PI, we can continue to de ne rewrite rules for this new... ..."

Cited by 33

### Table 1. Complete equational axiomatisations of BCCS and the parallel composition

2005

"... In PAGE 11: ... Proposition 3. Let L be a GSOS language extending BCCS, and Ax a head normalising equational axiomatisation, respecting $, and containing the axioms A1{4 of Table1 . Then Ax is sound and complete for $ on nite processes.... In PAGE 12: ... However, Bergstra amp; Klop [3] gave such an axiomatisation on the language obtained by adding two auxiliary operators, the left merge k and the communi- cation merge j, with rules x1 a ! y1 x1k x2 a ! y1kx2 and x1 a ! y1 x2 a ! y2 x1jx2 ! y1ky2 , provided the alphabet Act of actions is nite. The axioms are CM1{9 of Table1 , in which + binds weakest and a: strongest, and a; b range over Act. Aceto, Bloom amp; Vaandrager [1] generalise this idea to arbitrary GSOS languages with nitely many rules, each with nitely many premises, and assum- ing a nite alphabet Act.... In PAGE 13: ... First of all, the operators 0, a: and + are added, if not already there. The corresponding axioms are A1{4 of Table1 . If all other operators are smooth and distinctive, for each of them the axioms just described are taken, which nishes the job.... In PAGE 14: ... Nevertheless, such an axiomatisation was found by Bergstra amp; Klop [3], using a variant of the communication merge that is not a GSOS operator. Their axiomatisation of k is obtained from the one in Table1 by requiring a; b 6 = in CM6, and adding the axioms :xjy = xj :y = xjy. Here I generalise their approach to arbitrary RWB cool (or RDB cool) GSOS languages.... ..."

Cited by 2

### Table 1. Complete equational axiomatisations of BCCS and the parallel composition

2005

"... In PAGE 11: ... Proposition 3. Let L be a GSOS language extending BCCS, and Ax a head normalising equational axiomatisation, respecting $, and containing the axioms A1{4 of Table1 . Then Ax is sound and complete for $ on nite processes.... In PAGE 12: ... However, Bergstra amp; Klop [3] gave such an axiomatisation on the language obtained by adding two auxiliary operators, the left merge k and the communi- cation merge j, with rules x1 a ! y1 x1k x2 a ! y1kx2 and x1 a ! y1 x2 a ! y2 x1jx2 ! y1ky2 , provided the alphabet Act of actions is nite. The axioms are CM1{9 of Table1 , in which + binds weakest and a: strongest, and a; b range over Act. Aceto, Bloom amp; Vaandrager [1] generalise this idea to arbitrary GSOS languages with nitely many rules, each with nitely many premises, and assum- ing a nite alphabet Act.... In PAGE 13: ... First of all, the operators 0, a: and + are added, if not already there. The corresponding axioms are A1{4 of Table1 . If all other operators are smooth and distinctive, for each of them the axioms just described are taken, which nishes the job.... In PAGE 14: ... Nevertheless, such an axiomatisation was found by Bergstra amp; Klop [3], using a variant of the communication merge that is not a GSOS operator. Their axiomatisation of k is obtained from the one in Table1 by requiring a; b 6 = in CM6, and adding the axioms :xjy = xj :y = xjy. Here I generalise their approach to arbitrary RWB cool (or RDB cool) GSOS languages.... ..."

Cited by 2

### Table 1: AX.

"... In PAGE 2: ... Thus R1 _ R2 is again a test, that succeeds when we can proceed with either an R1- or an R2-step. Table1 contains a nite set of axioms, AX, that is intended to completely axiomatise equational validity in dynamic relation algebras. We write ` t1 = t2 if this equation is derivable from the equations in AX and the rules of equational logic.... ..."

### Table 1: Modal systems of knowledge and belief and Weak connection axiomatisation.

2003

Cited by 2

### Table 4: Syntactical systems of knowledge and belief and Weak connection axiomatisation.

2003

Cited by 2

### Table 2: Axioms for BPA

1994

"... In PAGE 3: ...quivalence is a congruence with respect to all the operators, i.e. if p $ p0 and q $ q0, then p + q $ p0 + q0 and pq $ p0q0 and p q $ p0 q0. Table2 contains an axiom system for BPA , which originates from [BBP93]. It consists of the axioms A1-5 for BPA together with three axioms SEI1-3 for iteration.... In PAGE 4: ... They are obtained from the action rules and axioms for SEI, using the obvious equivalence x y $ x y+y. Clearly, the axiomatisation in Table 4 is complete for BPA if and only if the axiomatisation in Table2 is complete for... ..."

Cited by 33

### Table 2: Modal systems of knowledge and belief and Strong s1 connection axiomatisation.

2003

Cited by 2

### Table 3: Modal systems of knowledge and belief and Strong s2 connection axiomatisation.

2003

Cited by 2