Abstract
In this chapter, we consider modeling and verification of reactive systems. We discuss why the correctness statement formalized for sequential programs is inadequate for reactive systems. We then formalize a different correctness statement based on refinements and derive a deductive recipe for proving such a correctness statement.
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Notes
- 1.
Any nondeterministic system that accepts a sequence of external inputs can be modeled as a deterministic system if any of the external inputs to the system do not depend somehow on any intermediate output produced by the system. This is because we can model the system with all the external inputs as being stored initially in one of the state component; whenever an external input is required that component is consulted for the appropriate stimulus.
- 2.
Throughout this chapter, we will make use of well-foundedness to formalize several arguments. We assume that we use a fixed well-founded structure ⟨o-p ≺, ≺ ⟩. In general, our framework may be thought to be parameterized over the well-founded structure used.
- 3.
One can extend GZ with an axiom positing the enumerability of the ACL2 universe, that is, the existence of a bijection between all ACL2 objects and the natural numbers. It is easy to see that the extended theory is consistent: there are models of GZ which are enumerable. The restriction of the legal inputs to only constitute good objects for the purpose of formalizing fairness is necessary if such an axiom is added. However, we did not go along that path since adding axioms incurs a certain amount of logical burden for soundness and also induces practical problems, for example, introduction of axioms cannot be local (to avoid risk of inconsistency), and hence, a book in which an axiom is defined cannot be locally included.
- 4.
This deficiency was pointed out to the author by John Matthews in a private conversation on October 20, 2005. The author is grateful to him for the contribution.
- 5.
Manolios [152] achieves this one-sided abstraction for branching time, by introducing proof rules for stuttering simulation.
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Ray, S. (2010). Reactive Systems. In: Scalable Techniques for Formal Verification. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5998-0_7
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