Verifying Temporal Properties of Systems by Julian Charles Bradfield (auth.)

By Julian Charles Bradfield (auth.)

This monograph goals to supply a robust general-purpose facts tech­ nique for the verification of platforms, even if finite or endless. It extends the belief of finite neighborhood model-checking, which was once brought by means of Stirling and Walker: instead of traversing the complete country house of a version, as is completed for model-checking within the experience of Emerson, Clarke et ai. (checking no matter if a (finite) version satisfies a formula), neighborhood model-checking asks even if a specific kingdom satisfies a formulation, and simply explores the within sight states some distance adequate to reply to that query. The procedure used used to be a tableau process, developing a tableau in accordance with the formulation and the neighborhood constitution of the version. This tableau process is right here generalized to the endless case via contemplating units of states, instead of unmarried states; as the good judgment used, the propositional modal mu-calculus, separates uncomplicated modal and boolean connectives from robust fix-point operators (which make the good judgment extra expressive than many different temporal logics), it really is attainable to provide a rela­ tively straight forward algorithm for developing a tableau. a lot of the subtlety is faraway from the tableau itself, and placed right into a relation at the kingdom area outlined via the tableau-the luck of the tableau then depends upon the well-foundedness of this relation. The generalized tableau process is exhibited on Petri nets, and diverse general notions from web thought are proven to playa half within the use of the approach on nets-in specific, the invariant calculus has a tremendous role.

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6 Variations on the theme. We consider now some variations of the system, most of which are aimed at reducing the size of tableaux. If we wish to ensure that tableau construction terminates, we can do so by forcing termination at a certain depth. We define the tableau system of degree k by adding the constraint that a node n = Sf-A U such that Un has been applied k times to U above n is terminal. 6 Variations on the theme 45 k :::: 1. In the case k = 1 we can dispense with the propositional constants entirely: such a system was presented in [Bra91].

A marking of N is a map M: 8 --+ N. Maps from a set X into N are identified with lXI-dimensional N-vectors in the natural way, and so M(s) is often written Ms. For x E 8 l:J T, ·x is the map 8 l:J T --+ N given by ·x(y) = F(y,x), and similarly x·(y) = F(x, y); sometimes ·x will be abused to mean the set {y I ·x(y) > O}, the pre-places (pre-transitions) of x, and similarly for post-places. Nets may be equipped with initial markings, usually written Mo; in this thesis we do not consider an initial marking to be part of the definition of

SO if n' is a quasi-terminal in r we are done. Otherwise, n is the companion of n', so we set S1 = s' and repeat the argument with S1 and r[V: {3ol. In this case, if the process does not terminate, giving us the desired sequence, we get an infinite sequence {30 > {31 > ... , which is impossible. 13. If S f-O 4> is the root of a successful tableau r, then S ~ 114>11. Proof. A tableau is a special case of a quasi-tableau, so by the proposition if the root is false there is a quasi-terminal; but a successful tableau has no quasi-terminals.

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