Thursday, February 7, 2013

Clarke-Emerson Theorem

The theorem gives CTL formula an iterative interpretation of their denotational semantics. \begin{align*} [\mathbf{EG} \phi] &= \nu z. ([\phi] \cap [\mathbf{EX} z]) \\ [\mathbf{E}(\phi \mathbf{U} \psi)] &= \mu z. ([\psi] \cup ([\phi] \cap [\mathbf{EX}z])) \\ [\mathbf{EF} \phi] &= \mu z. ([\phi] \cup [\mathbf{EX} z] \end{align*}

Appendix

  • $[\mathbf{EX} \phi]$ is the existential pre-image of $[\phi]$.
  • $\nu z. F z$ calculates the greatest fixed point.
  • $\mu z. F z$ calculates the least fixed point.
  • $\mathbf{EF} \phi = \mathbf{E}(\top \mathbf{U} \phi)$

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