Product and Join Sample Clauses

Product and Join. In this section we give a definition of product and join of two ABAR’s. Definition 5.10 Let ⟨B1, l1⟩ and ⟨B2, l2⟩ be two ABAR over the same alphabet, say RecN (D). Their product is defined as the ABAR ⟨B, l⟩, where B = B1×B2 and l (⟨q, p⟩) = l1(q ).l2(p). Similarly, we define the join of two ABARs in terms of the join of their underlying BAR’s. Definition 5.11 Let ⟨B1, l1⟩ and ⟨B2, l2⟩ be two ABAR over the same alphabet, say RecN (D). Their join ⟨B1, l1⟩ ◃▹ ⟨B2, l2⟩ is defined as the ABAR ⟨B, l⟩, where B = B1 ◃▹ B2 and l (⟨q, p⟩) = l1(q ).l2(p). r1 It is easy to check that the join of ABARs is again an ABAR. In fact, if q1 −→ p1 is a r2 ⟨ ⟩ ∪ ⟨ ⟩ −→ transition in ⟨B1, l1⟩ and dom(r1) has no name in common with those used by another ABAR B2, l2 , then l1(q1).l2(q2) implies wp(r1) for all state q2 of B2. Similarly, if q2 p2 is another transition in B2, l2 such that comp(r1, r2), then l1(q1).l2(q2) implies wp(r1 r2). As for BAR’s, the join of two ABAR’s with the same alphabet coincides with their prod-