## Abstract

In this paper we introduce WMC, a weighted version of the alternation-free modal mu-calculus for weighted transition systems. WMC subsumes previously studied weighted extensions of CTL and resembles previously proposed time-extended versions of the modal mu-calculus. We develop, in addition, a symbolic semantics for WMC and demonstrate that the notion of satisfiability coincides with that of symbolic satisfiability. This central result allows us to prove two major meta-properties of WMC. The first is decidability of satisfiability for WMC. In contrast to the classical modal mu-calculus, WMC does not possess the finite model-property. Nevertheless, the finite model property holds for the symbolic semantics and decidability readily follows; and this contrasts to resembling logics for timed transitions systems for which satisfiability has been shown undecidable. As a second main contribution, we provide a complete axiomatization, which applies to both semantics. The completeness proof is non-standard, since the logic is non-compact, and it involves the notion of symbolic models.

Original language | English |
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Pages (from-to) | 289-313 |

Number of pages | 25 |

Journal | Electronic Notes in Theoretical Computer Science |

Volume | 319 |

DOIs | |

Publication status | Published - 21 Dec 2015 |

## Keywords

- complete axiomatization
- non-compact modal logics
- satisfiability
- weighted modal mu-calculus
- weighted transition systems