By Olaf Burkart
A universal process in software program engineering is to use through the layout section a number of based concepts like top-down layout, decomposition and abstraction, whereas in simple terms in this case, within the implementation section, is the layout demonstrated to make sure reliability. yet this strategy neglects that imperative features of software program layout and application improvement have a robust formal personality which admits device help for the development of trustworthy and proper desktops in keeping with formal reasoning. This monograph offers a lot details either for theoreticians drawn to algebraic theories, and for software program engineers construction essentially suitable instruments. the writer provides the theoretical foundations wanted for the verification of reactive, sequential infinite-state structures. new algorithms are brought taking into consideration computerized verification of significant features similar to security or liveness properites of a given infinite-state process. The formal framework constructed contains contemporary effects from quite a few theoretical components like approach algebras, fixpoint conception, modal logics and version checking.
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Additional info for Automatic Verification of Sequential Infinite-State Processes
Finally, we close this section by proving the additional properties that ∼ =R is transitive, as well as a right-congruence with respect to sequential composition. 4. 5. Let R be a binary relation between processes. Then we have 1. 6 Context-Free Processes 31 2. p1 ∼ =R p2 implies p1 q ∼ =R p2 q. Proof. To prove the first part of the lemma let p1 ∼ =R p2 and p2 ∼ =R p3 . a Assume that p1 → p1 for some action a and some process p1 . Due to p1 ∼ =R p2 a we know that p2 → p2 for some p2 such that p1 ↔∗R p2 .
5. Let R be a binary relation between processes. Then we have 1. 6 Context-Free Processes 31 2. p1 ∼ =R p2 implies p1 q ∼ =R p2 q. Proof. To prove the first part of the lemma let p1 ∼ =R p2 and p2 ∼ =R p3 . a Assume that p1 → p1 for some action a and some process p1 . Due to p1 ∼ =R p2 a we know that p2 → p2 for some p2 such that p1 ↔∗R p2 . Moreover, from p2 ∼ =R a p3 we deduce the existence of some process p3 which satisfies p3 → p3 and p2 ↔∗R p3 . Now the transitivity of ↔∗R yields p1 ↔∗R p3 .
1 (Cancellation rules for normed BPA). Let α, β and γ be normed. Then 1. γα ∼ γβ implies α ∼ β and 2. αγ ∼ βγ implies α ∼ β. Note however that both implications of the cancellation lemma are invalid for unnormed processes as demonstrated by the following examples: 1. Let X = a + aX + aY and Y = bY then we have XY ∼ XXY but Y ∼ XY . 2. Let X = a and Y = aY then we have XY ∼ XXY but X ∼ XX. 2. b Y a a XY a a a XXY a ... ... XXY XX a a XY a a YY YXY Y b b a Example 1 X ε Example 2 Fig. 2. Examples that cancellation does not hold for unnormed BPA processes.
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