## Quantum institutions (2006)

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Venue: | Algebra, Meaning, and Computation – Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday, volume 4060 of Lecture Notes in Computer Science |

Citations: | 5 - 4 self |

### BibTeX

@INPROCEEDINGS{Caleiro06quantuminstitutions,

author = {C. Caleiro and P. Mateus and A. Sernadas and C. Sernadas},

title = {Quantum institutions},

booktitle = {Algebra, Meaning, and Computation – Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday, volume 4060 of Lecture Notes in Computer Science},

year = {2006},

pages = {50--64},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

The exogenous approach to enriching any given base logic for probabilistic and quantum reasoning is brought into the realm of institutions. The theory of institutions helps in capturing the precise relationships between the logics that are obtained, and, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract. 1

### Citations

1529 |
Chuang “Quantum Computation and Quantum Information
- Nielsen, Isaac
- 2000
(Show Context)
Citation Context ...ore abstract. 1 Introduction A new logic was proposed in [16, 18] for modeling and reasoning about quantum states, embodying the relevant postulates of quantum physics (as presented, for instance, in =-=[22]-=-). The logic was designed from the semantics upwards, starting with the key idea of adopting superpositions of classical models as the models of the quantum logic. In [19], other instances of the exog... |

520 |
Institutions: Abstract model theory for specification and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...s-style construction of a quantum logic from any given base logic in order to assess how general the construction is. To this end, a logic is identified with an institution, as originally proposed in =-=[12, 13]-=- with great success in both theoretical issues and applications. The construction is carried out in three main steps. Given an arbitrary institution we first build its global extension where each mode... |

436 |
Probabilistic logic
- Nilsson
- 1986
(Show Context)
Citation Context ...kin to the society semantics introduced in [6] for many-valued logic and to the possible translations semantics proposed in [5] for paraconsistent logic. The possible worlds approach was also used in =-=[23, 24, 2, 1, 10]-=- for probabilistic logic. Our semantics to quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics (as f... |

303 |
1936]. The Logic of Quantum Mechanics
- BIRKHOFF, NEUMANN
- 1936
(Show Context)
Citation Context ... sets of such structures. This novel approach to quantum logic semantics is completely different from the traditional approach [6, 7] to the problem, as initially proposed by Birkhoff and von Neumann =-=[8]-=-, that focuses on the lattice of closed subspaces of a Hilbert space. The main drawback of Birkhoff and von Neumann’s approach is that it does not yield an extension of classical logic. Our semantics ... |

261 |
Representing and Reasoning with Probabilistic Knowledge
- Bacchus
- 1990
(Show Context)
Citation Context ...kin to the society semantics introduced in [6] for many-valued logic and to the possible translations semantics proposed in [5] for paraconsistent logic. The possible worlds approach was also used in =-=[23, 24, 2, 1, 10]-=- for probabilistic logic. Our semantics to quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics (as f... |

232 | A logic for reasoning about probabilities
- Fagin, Halpern, et al.
- 1990
(Show Context)
Citation Context ...kin to the society semantics introduced in [6] for many-valued logic and to the possible translations semantics proposed in [5] for paraconsistent logic. The possible worlds approach was also used in =-=[23, 24, 2, 1, 10]-=- for probabilistic logic. Our semantics to quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics (as f... |

180 |
General logics
- Meseguer
- 1987
(Show Context)
Citation Context ...ach other. A recent systematization of the field can be found in [14]. The notion of arrow that we will be using in this paper can be classified as a comorphism (or a plain map as originally named in =-=[20]-=-, or also a representation as rebaptized in [25]). It is however a modified comorphism that maps models to sets of models, which can be explained as an instance of the general monad construction of [2... |

158 |
A semantical analysis of modal logic I: normal modal propositional calculi. Zeitschrift f ur Mathematische Logik und Grundlagen der Mathematik, 9:67--96
- Kripke
- 1963
(Show Context)
Citation Context ...ing the design of the language around the underlying concepts of quantum physics while keeping the classical connectives and was inspired by the possible worlds approach originally proposed by Kripke =-=[15]-=- for modal logic. It is also akin to the society semantics introduced in [6] for many-valued logic and to the possible translations semantics proposed in [5] for paraconsistent logic. The possible wor... |

66 | Institution morphisms
- Goguen, Roşu
- 2002
(Show Context)
Citation Context ...r, as a first step towards the full understanding of the proposed approach to enriching logics, we shall adopt an original variant of the notion of institution, without morphisms between models (c.f. =-=[14]-=-). For simplicity we shall just call it an institution, without any further qualifiers. Definition 2.1. Institution An institution is a tuple I = 〈Sig, Sen, Mod, �〉 where: • Sig is a category (of sign... |

56 |
The Logic of Quantum Mechanics
- Birkho¤, Neumann
- 1936
(Show Context)
Citation Context ...s of the original logic. This novel approach to quantum logic semantics is completely different from the traditional approach [11, 8] to the problem, as initially proposed by Birkhoff and von Neumann =-=[3]-=-, that focuses on the lattice of closed subspaces of a Hilbert space. Our semantics has the advantage of closely guiding the design of the language around the underlying concepts of quantum physics wh... |

55 | Moving between logical systems
- Tarlecki
- 1996
(Show Context)
Citation Context ... can be found in [14]. The notion of arrow that we will be using in this paper can be classified as a comorphism (or a plain map as originally named in [20], or also a representation as rebaptized in =-=[25]-=-). It is however a modified comorphism that maps models to sets of models, which can be explained as an instance of the general monad construction of [21]. The definition will take advantage of the us... |

48 |
I borrow your logic? (Transporting logical structure along maps
- Cerioli, Meseguer
(Show Context)
Citation Context ...fact that comorphisms preserve entailment. A further simple condition on the surjectivity of the translation of models can also guarantee the reflection of entailment. Such properties were studied in =-=[7]-=-. These 3sresults can easily be lifted to the level of power-model comorphisms, as stated below. (Power-model) comorphisms compose in the usual way. Proposition 2.3. Let I = 〈Sig, Sen, Mod, �〉 and I ′... |

47 |
I.L.: Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ..., more abstract. 1 Introduction A new logic was proposed in [1–3] for modeling and reasoning about quantum states, embodying the relevant postulates of quantum physics (as presented, for instance, in =-=[4]-=-) and adopting the exogenous approach (the original models are kept). The logic was designed from the semantics upwards, starting with the key idea of adopting superpositions of classical models as th... |

45 |
A study in the foundations of programming methodology: Speci institutions, charters and parchments
- Goguen, Burstall
- 1986
(Show Context)
Citation Context ...s-style construction of a quantum logic from any given base logic in order to assess how general the construction is. To this end, a logic is identified with an institution, as originally proposed in =-=[12, 13]-=- with great success in both theoretical issues and applications. The construction is carried out in three main steps. Given an arbitrary institution we first build its global extension where each mode... |

37 |
Possible-translations semantics for paraconsistent logics
- Carnielli
- 2000
(Show Context)
Citation Context ...ds approach originally proposed by Kripke [15] for modal logic. It is also akin to the society semantics introduced in [6] for many-valued logic and to the possible translations semantics proposed in =-=[5]-=- for paraconsistent logic. The possible worlds approach was also used in [23, 24, 2, 1, 10] for probabilistic logic. Our semantics to quantum logic, although inspired by modal logic, is also completel... |

36 |
Institution morphisms. Formal aspects of computing 13
- Goguen, Roşu
- 2002
(Show Context)
Citation Context ...er, as a first step towards the full understanding of the proposed approach to enriching logics, we shall adopt a variant of the original notion of institution, without morphisms between models (c.f. =-=[17]-=-). For simplicity we shall just call it an institution, without any further qualifiers. We denote by Cls the category with classes as objects and maps between classes as morphisms. An institution is a... |

32 | Weakly complete axiomatization of exogenous quantum propositional logic
- Mateus, Sernadas
- 2006
(Show Context)
Citation Context ...ate of the systems. The logic resulting from the quantization of classical propositional logic was introduced and studied in [16, 18]. A sound and weak complete calculus for the logic was obtained in =-=[17]-=- using an iterated Henkin construction inspired by the technique in [10]. The qubits of interest in this case were the propositional symbols. Using the logic it is possible, for instance, to model and... |

30 |
Society semantics and multiple-valued logics
- Carnielli, Lima-Marques
- 2005
(Show Context)
Citation Context ...ics while keeping the classical connectives and was inspired by the possible worlds approach originally proposed by Kripke [15] for modal logic. It is also akin to the society semantics introduced in =-=[6]-=- for many-valued logic and to the possible translations semantics proposed in [5] for paraconsistent logic. The possible worlds approach was also used in [23, 24, 2, 1, 10] for probabilistic logic. Ou... |

26 |
Probabilistic logic revisited
- Nilsson
- 1993
(Show Context)
Citation Context |

20 | Exogenous semantics approach to enriching logics
- Mateus, Sernadas, et al.
- 2005
(Show Context)
Citation Context ... presented, for instance, in [22]). The logic was designed from the semantics upwards, starting with the key idea of adopting superpositions of classical models as the models of the quantum logic. In =-=[19]-=-, other instances of the exogenous approach to enriching logics were presented in detail. In short, the exogenous approach is based on adopting as models of the new envisaged logic (enriched) sets of ... |

19 | Reasoning about Quantum Systems
- Mateus, Sernadas
- 2004
(Show Context)
Citation Context ...d, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract. 1 Introduction A new logic was proposed in =-=[16, 18]-=- for modeling and reasoning about quantum states, embodying the relevant postulates of quantum physics (as presented, for instance, in [22]). The logic was designed from the semantics upwards, startin... |

17 | On probability distributions over possible worlds
- BACCHUS
- 1988
(Show Context)
Citation Context |

17 | Two’s company: “The humbug of many logical values
- Caleiro, Carnielli, et al.
- 2005
(Show Context)
Citation Context ...ically equivalent models. 18sThe case of classical propositional logic is paradigmatic, once we take as qubits all the propositional symbols. But similar choices are possible in many other logics. In =-=[4]-=- it is shown how to do this choice in any suitable finitely-valued logic. For instance, in Lukasiewicz’s three-valued logic it suffices to consider as qubits all propositional symbols and negations of... |

14 |
A Half Century of Quantum Logic − What Have we
- Foulis
- 1999
(Show Context)
Citation Context ...d) sets of models of the given base logic without tempering with the models of the original logic. This novel approach to quantum logic semantics is completely different from the traditional approach =-=[11, 8]-=- to the problem, as initially proposed by Birkhoff and von Neumann [3], that focuses on the lattice of closed subspaces of a Hilbert space. Our semantics has the advantage of closely guiding the desig... |

12 |
Reasoning in quantum theory
- Chiara, Giuntini, et al.
- 2004
(Show Context)
Citation Context ...d) sets of models of the given base logic without tempering with the models of the original logic. This novel approach to quantum logic semantics is completely different from the traditional approach =-=[11, 8]-=- to the problem, as initially proposed by Birkhoff and von Neumann [3], that focuses on the lattice of closed subspaces of a Hilbert space. Our semantics has the advantage of closely guiding the desig... |

10 |
Moving between logical systems. Recent Trends in Data Type Specifications. Selected Papers
- Tarlecki
- 1995
(Show Context)
Citation Context ...eld can be found in [17]. The notion of arrow that we will be using in this paper can be classified as a comorphism (or a plain map as originally named in [18], or also a representation as renamed in =-=[19]-=-). It is however a modified comorphism that maps models to sets of models, which can be explained as an instance of the general monad construction of [20]. The definition will take advantage of the us... |

9 | Exogenous quantum logic
- Mateus, Sernadas
(Show Context)
Citation Context ...d, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract. 1 Introduction A new logic was proposed in =-=[16, 18]-=- for modeling and reasoning about quantum states, embodying the relevant postulates of quantum physics (as presented, for instance, in [22]). The logic was designed from the semantics upwards, startin... |

7 |
Semantics of the Minimal Logic of Quantum Mechanics
- Dishkant
- 1972
(Show Context)
Citation Context ...c logic. Our semantics to quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics (as first proposed in =-=[9]-=-) still closely related to the lattice-oriented operations. The resulting quantum logic also incorporates probabilistic reasoning (in the style of Nilsson’s calculus [23, 24]) since the postulates of ... |

5 | Different types of arrow between logical frameworks
- Mossakowski
- 1996
(Show Context)
Citation Context ...0], or also a representation as rebaptized in [25]). It is however a modified comorphism that maps models to sets of models, which can be explained as an instance of the general monad construction of =-=[21]-=-. The definition will take advantage of the usual covariant powerset endofunctor Pw, in this case extended to classes, that is, Pw : Cls → Cls is such that Pw(X) = 2 X , and Pw(f : X → X ′ ) maps each... |

3 |
Semantics of the minimal logic of quantum mechanics. Studia Logica 30
- Dishkant
- 1972
(Show Context)
Citation Context ...c logic. Our semantics to quantum logic, although inspired by modal logic, is also completely different from the alternative Kripke semantics given to traditional quantum logics (as first proposed in =-=[14]-=-) still closely related to the lattice-based operations. The resulting quantum logic also incorporates probabilistic reasoning (in the style of Nilsson’s calculus [9, 10]) since the postulates of quan... |

1 |
Computability. Volume 146 of Graduate Texts in Mathematics
- Bridges
- 1994
(Show Context)
Citation Context ...the logic will be augmented to allow probabilistic assertions and reasoning. For that sake, we assume fixed a set X of variables. We shall also denote by R the set of all computable real numbers (see =-=[22]-=-). Definition 3. The probability institution I p = 〈Sig, Sen p , Mod p , � p 〉 based on I is defined as follows: – Sen p (Σ) is the least set containing Sen(Σ) such that: (⊟ δ), (δ1 ⊐ δ2) ∈ Sen p (Σ) ... |