## A Complete Deductive System for the µ-Calculus (1995)

Citations: | 13 - 0 self |

### BibTeX

@MISC{Walukiewicz95acomplete,

author = {Igor Walukiewicz},

title = {A Complete Deductive System for the µ-Calculus},

year = {1995}

}

### OpenURL

### Abstract

The propositional µ-calculus as introduced by Kozen in [12] is considered. In that paper

### Citations

888 | Dynamic logic
- Harel
- 1984
(Show Context)
Citation Context ...lity of syntactic model construction which is provided by collapsed model theorem or a result of similar kind. For several more expressive logics of programs like: PDL∆, PAL, temporal µ-calculus (see =-=[10, 22]-=-) the completeness problem remains open. One of the reasons for this is that these logics, as well as the µ-calculus, do not enjoy the collapsed model property (see for example [10]), hence known meth... |

648 | L.J.: Symbolic model checking: 10 20 states and beyond. Inf Comput 98(2
- Burch, Clarke, et al.
- 1992
(Show Context)
Citation Context ... it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixpoints is bounded =-=[3, 1]-=-. One of lacking elements in this picture was finitary complete axiomatisation of the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker proposit... |

326 | Symbolic Model Checking: 10 States and Beyond - Burch, Clarke, et al. - 1992 |

273 |
Results on the propositional mu-calculus
- Kozen
- 1983
(Show Context)
Citation Context ... the µ-Calculus Igor Walukiewicz 1,2 BRICS 3 Department of Computer Science University of Aarhus Ny Munkegade DK-8000 Aarhus C, Denmark Abstract The propositional µ-calculus as introduced by Kozen in =-=[12]-=- is considered. In that paper a finitary axiomatisation of the logic was presented but its completeness remained an open question. Here a different finitary axiomatisation of the logic is proposed and... |

239 |
Efficient model checking in fragments of the propositional mu-calculus
- Emerson, Lei
- 1986
(Show Context)
Citation Context ... it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixpoints is bounded =-=[3, 1]-=-. One of lacking elements in this picture was finitary complete axiomatisation of the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker proposit... |

233 | Tree automata, mu-calculus and determinacy
- Emerson, Jutla
- 1991
(Show Context)
Citation Context ...n Computer Science, Centre of the Danish National Research Foundation. 1can be encoded into the µ-calculus. On binary trees the logic is as expressive as monadic second order logic of two successors =-=[18, 6]-=-. On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME-complete [15, 23, 4] which means that it is of the same complexity as for many much less expre... |

201 |
Modal and temporal logics
- Stirling
- 1992
(Show Context)
Citation Context ...lity of syntactic model construction which is provided by collapsed model theorem or a result of similar kind. For several more expressive logics of programs like: PDL∆, PAL, temporal µ-calculus (see =-=[10, 22]-=-) the completeness problem remains open. One of the reasons for this is that these logics, as well as the µ-calculus, do not enjoy the collapsed model property (see for example [10]), hence known meth... |

156 |
The Complexity of Tree Automata and Logics of Programs
- Emerson, Jutla
- 1988
(Show Context)
Citation Context ...es the logic is as expressive as monadic second order logic of two successors [18, 6]. On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME-complete =-=[15, 23, 4]-=- which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixp... |

147 | Decision procedures and expressiveness in the temporal logic of branching time
- Emerson, Halpern
- 1985
(Show Context)
Citation Context ...ete axiomatisation of the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker propositional logics of programs like PDL [8, 19, 14, 11, 16], CTL∗ =-=[5]-=- or Process Logic [9]. Completeness proofs for these logics use the so called Henkin method which consists of constructing a model for a non refutable formula. The use of this method depends on the ab... |

111 |
On the complexity of ω-automata
- Safra
- 1988
(Show Context)
Citation Context ... simpler will be our procedure. This is why we do not use any standard determinization constructions but define Aϕ0 from the scratch. The construction is nevertheless based on Safra’s determinization =-=[20]-=-. 16From Aϕ0 we construct an appropriate Rabin automaton over one letter alphabet TAϕ0 whose runs closely correspond to refutations of ϕ0. This allows us to conclude that there is a small graph Gϕ0 ,... |

108 |
Local model checking in the modal mu-calculus
- Stirling, Walker
- 1989
(Show Context)
Citation Context ...7] which gives a characterisation of the validity of the µ-calculus formulas by means of infinite tableaux. We will briefly recall the result here. First we introduce the concept of a definition list =-=[21]-=- which will name the fixpoint subformulas of a given formula in order of their nesting. We extend vocabulary of the µ-calculus by a countable set Dcons of fresh symbols that will be referred to as def... |

67 | On the complexity of !-automata - Safra - 1988 |

54 | Games for the µ-calculus
- Niwiński, Walukiewicz
- 1996
(Show Context)
Citation Context ...s closely related to this technique we hope that it can be also generalised to other logics. 2The outline of the paper is as follows. We begin by giving basic definitions and recalling a result from =-=[17]-=- which we will use. Next we present our axiomatisation, prove some of its properties and relate it to Kozen’s system. Finally we present the completeness proof. Acknowledgements I am very grateful to ... |

45 | Process Logic: expressiveness, decidability, completeness
- Harel, Kozen, et al.
- 1983
(Show Context)
Citation Context ... the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker propositional logics of programs like PDL [8, 19, 14, 11, 16], CTL∗ [5] or Process Logic =-=[9]-=-. Completeness proofs for these logics use the so called Henkin method which consists of constructing a model for a non refutable formula. The use of this method depends on the ability of syntactic mo... |

31 |
Fixed points vs. infinite generation
- Niwiński
- 1988
(Show Context)
Citation Context ...n Computer Science, Centre of the Danish National Research Foundation. 1can be encoded into the µ-calculus. On binary trees the logic is as expressive as monadic second order logic of two successors =-=[18, 6]-=-. On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME-complete [15, 23, 4] which means that it is of the same complexity as for many much less expre... |

30 | Games for -calculus - Niwinski, Walukiewicz - 1996 |

29 |
An elementary proof of the completeness of PDL
- Kozen, Parikh
- 1981
(Show Context)
Citation Context ...picture was finitary complete axiomatisation of the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker propositional logics of programs like PDL =-=[8, 19, 14, 11, 16]-=-, CTL∗ [5] or Process Logic [9]. Completeness proofs for these logics use the so called Henkin method which consists of constructing a model for a non refutable formula. The use of this method depends... |

28 |
Propositional modal logic of programs
- Fischer, Ladner
- 1977
(Show Context)
Citation Context ...tableau for ϕ0 with µ-trace on them. Let us first observe that the set of formulas which can appear in a refutation for ϕ0 is finite. Let us call this set FL(ϕ0) as it is almost Fisher-Ladner closure =-=[7]-=- of ϕ0. It is not exactly the closure because we have definition constants around. Definition 4.1 For any formula ϕ, possibly with definition constants, we define the FL-closure of ϕ, FL(ϕ) as the set... |

28 |
and Jerzy Tiuryn. Logics of programs
- Kozen
- 1990
(Show Context)
Citation Context ...picture was finitary complete axiomatisation of the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker propositional logics of programs like PDL =-=[8, 19, 14, 11, 16]-=-, CTL∗ [5] or Process Logic [9]. Completeness proofs for these logics use the so called Henkin method which consists of constructing a model for a non refutable formula. The use of this method depends... |

21 |
The completeness of propositional dynamic logic
- PARIKH
- 1978
(Show Context)
Citation Context ...picture was finitary complete axiomatisation of the logic (infinitary axiomatisation was given in [13]). There exist finitary axiomatisations for many weaker propositional logics of programs like PDL =-=[8, 19, 14, 11, 16]-=-, CTL∗ [5] or Process Logic [9]. Completeness proofs for these logics use the so called Henkin method which consists of constructing a model for a non refutable formula. The use of this method depends... |

19 | A model theorem for the propositional -calculus - Kozen - 1988 |

19 |
An automata theoretic procedure for the propositional mu-calculus
- Streett, Emerson
- 1989
(Show Context)
Citation Context ...es the logic is as expressive as monadic second order logic of two successors [18, 6]. On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME-complete =-=[15, 23, 4]-=- which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixp... |

17 |
temporal logics
- Automata
- 1985
(Show Context)
Citation Context ...round given a refutation of ϕ0 we can run the path automaton Aϕ0 on each path of the refutation separately and because it is deterministic we will obtain a tree which is an accepting run of TAϕ0 . In =-=[2]-=- the following theorem is (implicitly) stated Theorem 4.8 (Emerson) Suppose A is a Rabin automaton over a single letter alphabet. If A accepts some tree then there is a graph G with states of A as nod... |

13 | A finite model theorem for the propositional µ–calculus - Kozen - 1988 |

11 |
A decision procedure for the propositional mu-calculus
- Kozen, Parikh
- 1984
(Show Context)
Citation Context ...es the logic is as expressive as monadic second order logic of two successors [18, 6]. On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME-complete =-=[15, 23, 4]-=- which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixp... |

2 | On models for propositional dynamic logic - Knijnenburg, Leeuwen - 1991 |

1 |
Axiomatizations of logics of programs. Unpublished manuscript, Bar-Ilan Univ
- Gabbay
- 1977
(Show Context)
Citation Context |