By Roderick Bloem, Swen Jacobs, Ayrat Khalimov

Whereas the vintage version checking challenge is to come to a decision no matter if a finite approach satisfies a specification, the aim of parameterized version checking is to make a decision, given finite platforms M(n) parameterized by way of n in N, even if, for all n in N, the procedure M(n) satisfies a specification. during this booklet we contemplate the $64000 case of M(n) being a concurrent approach, the place the variety of replicated tactics is determined by the parameter n yet every one approach is self reliant of n. Examples are cache coherence protocols, networks of finite-state brokers, and platforms that remedy mutual exclusion or scheduling difficulties. additional examples are abstractions of platforms, the place the methods of the unique structures really rely on the parameter.

We literature during this zone has studied a wealth of computational types in keeping with quite a few synchronization and verbal exchange primitives, together with token passing, broadcast, and protected transitions. frequently, diverse terminology is utilized in the literature, and effects are in keeping with implicit assumptions. during this ebook, we introduce a computational version that unites the relevant synchronization and communique primitives of many versions, and unveils hidden assumptions from the literature. We survey present decidability and undecidability effects, and provides a scientific view of the elemental difficulties during this fascinating learn sector.

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**Decidability of Parameterized Verification**

Whereas the vintage version checking challenge is to determine even if a finite approach satisfies a specification, the target of parameterized version checking is to choose, given finite structures M(n) parameterized by means of n in N, even if, for all n in N, the procedure M(n) satisfies a specification. during this booklet we contemplate the real case of M(n) being a concurrent procedure, the place the variety of replicated tactics is dependent upon the parameter n yet each one method is autonomous of n.

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**Extra resources for Decidability of Parameterized Verification**

**Example text**

P ; G; F ; card/ is as follows: Input. P 2 P , and 2 F. Output. n/; card/ ˆ for all n 2 N ; and “No” otherwise. P ; G; F /. Consider the question: Is the PMCP decidable for token-passing systems on unidirectional rings, with speciﬁcations in ILTLnX? n/ is the uni-directional ring of size n. 1 COMPUTABILITY ASSUMPTIONS In parameterized model checking one usually makes the following very general assumptions. In the literature they are typically not made explicit. • Each process template P is computable, meaning there is an algorithm that computes exactly the states and transitions of P .

J; i /W j D i C 1 mod ng. i C 1 mod n; i / to ccw. Let Diri n D fundeﬁnedg, and d i ri n be the function that maps all edges to undeﬁned. Let Bsnd be the parameterized bi-directional ring with these direction labels. Let DTsnd be the set of all deterministic process templates in PTsnd . 3. 16 [Emerson and Kahlon, 2004]. DTsnd ; Bsnd / is change-bounded. Indeed, the cutoﬀ is a polynomial depending on the number of states and the transition graph of process template P . Proof idea. e proof is based on the fact that processes are deterministic, and so the token can only make a ﬁnite number of moves until the system either deadlocks or runs into a state that it has seen before, entering an inﬁnite loop.

Psi mp ; R; F / has a cutoﬀ if F is one of the the cutoﬀ is 2. i;j /2CTL nX , nX , the cutoﬀ is 3. the cutoﬀ is 4. i;j;l/2CTL nX , the cutoﬀ is 5. Proof idea. i /, and then consider the case of multiple quantiﬁed index variables. i /. 0/. n/ , except that the labeling function is restricted to atoms of the form p0 for p 2 APpr . 2/ for all n 2 N . A stuttering bisimulation implies that both system instances agree on all CTL nX formulas. , the processes with index 0 are in the same state, and thus either both have the token or both do not have the token.