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## Structural Refinement for the Modal nu-Calculus

### Citations

1014 | Design and synthesis of synchronization skeletons using branching time temporal logic
- Clarke, Emerson
- 2008
(Show Context)
Citation Context ...n example, we refer to Fig. 1 which shows the property informally specified as “after a req(uest), no idle(ing) is allowed, but only work, until grant is executed” using the logical formalisms of CTL =-=[15]-=- and the modal µ-calculus [21] and the behavioral formalism of disjunctive modal transition systems [26]. The precise relationship between logical and behavioral specification formalisms has been subj... |

945 | Separation logic: a logic for shared mutable data structures
- Reynolds
- 2002
(Show Context)
Citation Context ...so that this translation only need be applied once in the beginning. We also note that composition and quotient operators are used in other logics such as e.g. spatial [14] or separation logics [29], =-=[33]-=-. However, in these logics they are treated as first-class operators, i.e. as part of the formal syntax. In our approach, on the other hand, they are defined as operations on logical expressions which... |

406 |
Results on the propositional µ-calculus
- Kozen
- 1983
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Citation Context ...oaches have been quite successful, with a plethora of logical formalisms available and a number of successful model checking tools. One particularly interesting such formalism is the modal µ-calculus =-=[21]-=-, which is universal in the sense that it generalizes most other temporal logics, yet mathematically simple and amenable to analysis. One central problem in the verification of formal properties is st... |

322 | Local reasoning about programs that alter data structures
- O’Hearn, Reynolds, et al.
- 2001
(Show Context)
Citation Context ...form, so that this translation only need be applied once in the beginning. We also note that composition and quotient operators are used in other logics such as e.g. spatial [14] or separation logics =-=[29]-=-, [33]. However, in these logics they are treated as first-class operators, i.e. as part of the formal syntax. In our approach, on the other hand, they are defined as operations on logical expressions... |

162 | A Spatial Logic for Concurrency (Part I).
- Caires, Cardelli
- 2003
(Show Context)
Citation Context ...tions preserve the normal form, so that this translation only need be applied once in the beginning. We also note that composition and quotient operators are used in other logics such as e.g. spatial =-=[14]-=- or separation logics [29], [33]. However, in these logics they are treated as first-class operators, i.e. as part of the formal syntax. In our approach, on the other hand, they are defined as operati... |

87 |
Modal specifications
- Larsen
- 1989
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Citation Context ...ld wish specifications to have. As an alternative to logical specification formalisms and with an eye to compositionality and decomposition, automatabased behavioral specifications were introduced in =-=[22]-=-. Here the specification formalism is a generalization of the modeling AG(req ⇒ AX(work AW grant)) νX. ( [grant, idle,work]X∧ [req]νY.(〈work〉Y ∨ 〈grant〉X) ∧ [idle, req]ff ) X ν = [grant, idle,work]X ∧... |

77 |
Equation solving using modal transition systems, in:
- Larsen, Xinxin
- 1990
(Show Context)
Citation Context ...le(ing) is allowed, but only work, until grant is executed” using the logical formalisms of CTL [15] and the modal µ-calculus [21] and the behavioral formalism of disjunctive modal transition systems =-=[26]-=-. The precise relationship between logical and behavioral specification formalisms has been subject to some investigation. In [22], Larsen shows that any modal transition system can be translated to a... |

71 |
Papers on Time and Tense
- Prior
- 1968
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Citation Context ... logic We introduce a hybrid modal logic which can serve as compact representation of AA. This logic is closely related to the Boolean modal transition systems of [7] and hybrid in the sense of [10], =-=[30]-=-: it contains nominals, and the semantics of a nominal is given as all sets which contain the nominal. For a finite set X of nominals, let L(X) be the set of formulae generated by the abstract syntax ... |

64 |
Acceptance Trees”, in
- Hennessy
- 1985
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Citation Context ...ence also AA implementations are precisely LTS. Acceptance automata were first introduced in [31] (see also [32], where a language-based approach is taken), based on the notion of acceptance trees in =-=[20]-=-; however, there they are restricted to be deterministic. We employ no such restriction here. The following notion of modal refinement for AA was also introduced in [31]. Let A1 = (S1, S01 ,Tran1) and... |

62 |
Verification of Modal Properties Using Boolean Equation Systems
- Mader
- 1997
(Show Context)
Citation Context ... “deterministic context systems”), but we know of no extension (other than ours) to more general logics. We also note that our hybrid modal logic appears related to the Boolean equation systems [25], =-=[27]-=- which are used in some µ-calculus model checking algorithms. The precise relation between the modal ν-calculus, our L-expressions and Boolean equation systems should be worked out. Similarly, accepta... |

61 | Discounting the future in systems theory. - Alfaro, Henzinger, et al. - 2003 |

57 | Modal mu-calculi
- Bradfield, Stirling
- 2007
(Show Context)
Citation Context ...s. The precise relation between the modal ν-calculus, our L-expressions and Boolean equation systems should be worked out. Similarly, acceptance automata bear some similarity to the modal automata of =-=[12]-=-. Lastly, we should note that we have in [3], [4] introduced quantitative specification theories for weighted modal transition systems. These are well-suited for specification and analysis of systems ... |

42 |
Proof Systems for Satisfiability in HennessyMilner Logic with Recursion,
- Larsen
- 1990
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Citation Context ...he modal µ-calculus [21], [34] with only maximal fixed points. Instead of an explicit maximal fixed point operator, we use the representation by equation systems in Hennessy-Milner logic developed in =-=[24]-=-. For a finite set X of variables, let H(X) be the set of Hennessy-Milner formulae, generated by the abstract syntax H(X) ∋ φ ::= tt | ff | x | 〈a〉φ | [a]φ | φ ∧ φ | φ ∨ φ, for a ∈ Σ and x ∈ X . A dec... |

41 |
Efficient local correctness checking
- Larsen
- 1992
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Citation Context ...ted to “deterministic context systems”), but we know of no extension (other than ours) to more general logics. We also note that our hybrid modal logic appears related to the Boolean equation systems =-=[25]-=-, [27] which are used in some µ-calculus model checking algorithms. The precise relation between the modal ν-calculus, our L-expressions and Boolean equation systems should be worked out. Similarly, a... |

32 |
Graphical versus logical specifications
- Boudol, Larsen
- 1990
(Show Context)
Citation Context ...sition system can be translated to a formula in Hennessy-Milner logic (its characteristic formula) which is equivalent in the sense of admitting the same models. Conversely, Boudol and Larsen show in =-=[11]-=- that any formula in Hennessy-Milner logic is equivalent to a finite disjunction of modal transition systems. We have picked up this work in [6], where we show that any disjunctive modal transition sy... |

31 |
Residual for component specifications.
- Raclet
- 2008
(Show Context)
Citation Context ...∅. An AA is an implementation if it holds for all s ∈ S that Tran(s) = {M} is a singleton; hence also AA implementations are precisely LTS. Acceptance automata were first introduced in [31] (see also =-=[32]-=-, where a language-based approach is taken), based on the notion of acceptance trees in [20]; however, there they are restricted to be deterministic. We employ no such restriction here. The following ... |

27 | Representation, reasoning, and relational structures: a hybrid logic manifesto.
- Blackburn
- 2000
(Show Context)
Citation Context ... modal logic We introduce a hybrid modal logic which can serve as compact representation of AA. This logic is closely related to the Boolean modal transition systems of [7] and hybrid in the sense of =-=[10]-=-, [30]: it contains nominals, and the semantics of a nominal is given as all sets which contain the nominal. For a finite set X of nominals, let L(X) be the set of formulae generated by the abstract s... |

21 | On determinism in modal transition systems.
- Benes, Kretınsky, et al.
- 2009
(Show Context)
Citation Context ...ontext of specification theories, this is what one would expect, and we only include it for completeness of presentation. It can be shown that modal refinement is also complete for deterministic DMTS =-=[8]-=-, but we will not need this here. Proposition 1: For all DMTS D1, D2, D1 ≤m D2 implies D1 ≤th D2. Ds t1 t3 u1 u2 u3 v1 v3 a a a a a d a b, c b mc(D) s′ t′ 1 t′ 3 u′ 1 u′ 2 u′ 3 v′ 1 v′ 3 a a a a a a b... |

20 | Quantitative verification and control via the mu-calculus. - Alfaro - 2003 |

20 |
The structure of commutative residuated lattices
- Hart, Rafter, et al.
(Show Context)
Citation Context ... ν-calculus expressions forms a commutative residuated lattice up to ≡m. The unit of ‖ (up to ≡m) is the ν-calculus expression corresponding to the LTS U = ({u}, {u}, {(u, a, u) | a ∈ Σ}). Proof: See =-=[19]-=- for a good reference on commutative residuated lattices. We have already seen that the class of νcalculus expressions forms a lattice, up to ≡m, under ∧ and ∨, and by Theorem 21, / is the residual, u... |

12 | Checking thorough refinement on modal transition systems is EXPTIME-complete
- Beneš, Křet́ınský, et al.
- 2009
(Show Context)
Citation Context ...ote that to compute the may-completion of a DMTS, one has to decide thorough refinements, hence this computation (or, more precisely, deciding whether a given DMTS is maycomplete) is EXPTIME-complete =-=[9]-=-. We show an example of a may-completion in Fig. 2. Proposition 2: For any DMTS D, D ≤m mc(D) and D ≡th mc(D). Proof: It is always the case that D ≤m D, and adding may transitions on the right side pr... |

10 |
Modal transition systems: Composition and LTL model checking
- Beneš, Černá, et al.
- 2011
(Show Context)
Citation Context ...tion if −→ = {(s, {(a, t)}) | s a 99K t}; DMTS implementations are precisely LTS. DMTS were introduced in [26] in the context of equation solving, or quotient, for specifications and are used e.g. in =-=[5]-=- for LTL model checking. They are a natural closure of modal transition systems (MTS) [22] in which all disjunctive musttransitions s −→ N lead to singletons N = {(a, t)}. Let D1 = (S1, S01 , 99K1,−→1... |

9 |
A.: Moving from specifications to contracts in component-based design
- Bauer, David, et al.
- 2012
(Show Context)
Citation Context ...und abstraction of logical implication), we show that one can freely translate between the modal ν-calculus and DMTS, while preserving structural refinement. DMTS form a complete specification theory =-=[2]-=- in that they both admit logical operations of conjunction and disjunction and structural operations of composition and quotient [6]. Hence they support full compositionality and decomposition in the ... |

6 | Quantitative refinement for weighted modal transition systems
- Bauer, Fahrenberg, et al.
- 2011
(Show Context)
Citation Context ...alculus, our L-expressions and Boolean equation systems should be worked out. Similarly, acceptance automata bear some similarity to the modal automata of [12]. Lastly, we should note that we have in =-=[3]-=-, [4] introduced quantitative specification theories for weighted modal transition systems. These are well-suited for specification and analysis of systems with quantitative information, in that they ... |

5 | General quantitative specification theories with modalities
- Bauer, Fahrenberg, et al.
- 2012
(Show Context)
Citation Context ...us, our L-expressions and Boolean equation systems should be worked out. Similarly, acceptance automata bear some similarity to the modal automata of [12]. Lastly, we should note that we have in [3], =-=[4]-=- introduced quantitative specification theories for weighted modal transition systems. These are well-suited for specification and analysis of systems with quantitative information, in that they repla... |

5 |
Ideal specification formalism = expressivity + compositionality + decidability + testability
- Larsen
- 1990
(Show Context)
Citation Context ...lus expressions in normal form, and it is an interesting question whether they can be defined for general ν-calculus expressions. (For disjunction and conjunction this is of course trivial.) Larsen’s =-=[23]-=- has composition and quotient operators for Hennessy-Milner logic (restricted to “deterministic context systems”), but we know of no extension (other than ours) to more general logics. We also note th... |

3 |
de Bakker. A theory of programs. Unpublished manuscript
- Scott, W
- 1969
(Show Context)
Citation Context ...then proceed to propose a new translation which fixes the mismatch. A. Syntax and semantics We first recall the syntax and semantics of the modal νcalculus, the fragment of the modal µ-calculus [21], =-=[34]-=- with only maximal fixed points. Instead of an explicit maximal fixed point operator, we use the representation by equation systems in Hennessy-Milner logic developed in [24]. For a finite set X of va... |

2 |
Hennessy-Milner logic with greatest fixed points as a complete behavioural specification theory
- Beneš, Delahaye, et al.
- 2013
(Show Context)
Citation Context ...the same models. Conversely, Boudol and Larsen show in [11] that any formula in Hennessy-Milner logic is equivalent to a finite disjunction of modal transition systems. We have picked up this work in =-=[6]-=-, where we show that any disjunctive modal transition system (DMTS) is equivalent to a formula in the modal ν-calculus, the safety fragment of the modal µ-calculus which uses only maximal fixed points... |

1 |
Reactive Systems. Cambridge Univ
- Aceto, Ingólfsdóttir, et al.
- 2007
(Show Context)
Citation Context ...o ν-calculus. Our translation from DMTS to ν-calculus is new, but similar to the translation from AA to ν-calculus given in [6]. This in turn is based on the characteristic formulae of [22] (see also =-=[1]-=-). For a DMTS D = (S, S0, 99K,−→) and all s ∈ S, we define ♦(s) = {N | s −→ N} and, for each a ∈ Σ, a(s) = {t | s a 99K t}. Then, let ∆(s) = ∧ N∈♦(s) ( ∨ (a,t)∈N 〈a〉t ) ∧ ∧ a∈Σ [a] ( ∨ t∈a(s) t ) an... |

1 | editors. Automated Technology for Verification and Analysis - Bultan, Hsiung - 2011 |

1 | Compositionality of probabilistic HennessyMilner logic through structural operational semantics - Gebler, Fokkink - 2012 |

1 |
Residual for component specifications. Publication interne 1843
- Raclet
- 2007
(Show Context)
Citation Context ...0, Tran(s0) 6= ∅. An AA is an implementation if it holds for all s ∈ S that Tran(s) = {M} is a singleton; hence also AA implementations are precisely LTS. Acceptance automata were first introduced in =-=[31]-=- (see also [32], where a language-based approach is taken), based on the notion of acceptance trees in [20]; however, there they are restricted to be deterministic. We employ no such restriction here.... |