## Model checking and boolean graphs.(English)Zbl 0798.03017

Summary: We describe a method for translating a satisfaction problem of the modal $$\mu$$-calculus into a problem of finding a certain marking of a Boolean graph. By giving algorithms to solve the graph problem, we present a global model checking algorithm for a subset of the modal $$\mu$$-calculus, which has time-complexity $$O(| A| | T|)$$, where $$| A|$$ is the size of the assertion and $$| T|$$ is the size of the model (a labelled transition system). This algorithm is extended to an algorithm for the full modal $$\mu$$-calculus running in time $$O(| A|^{\text{ad}} | S|^{\text{ad}-1} | T|)$$, where ad is the alternation depth and $$| S|$$ is the number of states in the transition system, improving on earlier presented algorithms. Moreover, a local algorithm is presented for alternation depth one. This algorithm runs in time $$O(| A| | T|\log (| A| | T|))$$ and is also an improvement on earlier algorithms.

### MSC:

 03B45 Modal logic (including the logic of norms) 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) 68Q25 Analysis of algorithms and problem complexity 68R10 Graph theory (including graph drawing) in computer science
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### References:

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