### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### Cite this

*Electronic Notes in Theoretical Computer Science*,

*319*, 289-313. https://doi.org/10.1016/j.entcs.2015.12.018

}

*Electronic Notes in Theoretical Computer Science*, vol. 319, pp. 289-313. https://doi.org/10.1016/j.entcs.2015.12.018

**Alternation-free weighted mu-calculus : decidability and completeness.** / Larsen, Kim G.; Mardare, Radu; Xue, Bingtian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Alternation-free weighted mu-calculus

T2 - decidability and completeness

AU - Larsen, Kim G.

AU - Mardare, Radu

AU - Xue, Bingtian

PY - 2015/12/21

Y1 - 2015/12/21

N2 - 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.

AB - 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.

KW - complete axiomatization

KW - non-compact modal logics

KW - satisfiability

KW - weighted modal mu-calculus

KW - weighted transition systems

UR - http://www.scopus.com/inward/record.url?scp=84951842098&partnerID=8YFLogxK

U2 - 10.1016/j.entcs.2015.12.018

DO - 10.1016/j.entcs.2015.12.018

M3 - Article

AN - SCOPUS:84951842098

VL - 319

SP - 289

EP - 313

JO - Electronic Notes in Theoretical Computer Science

JF - Electronic Notes in Theoretical Computer Science

SN - 1571-0661

ER -