## Automaton Logic (1996)

Venue: | International Journal of Theoretical Physics |

Citations: | 79 - 47 self |

### BibTeX

@ARTICLE{Schaller96automatonlogic,

author = {M. Schaller and K. Svozil},

title = {Automaton Logic},

journal = {International Journal of Theoretical Physics},

year = {1996},

volume = {35},

number = {35},

pages = {911--940}

}

### Years of Citing Articles

### OpenURL

### Abstract

The experimental logic of Moore and Mealy type automata is investigated. key words: automaton logic; partition logic; comparison to quantum logic; intrinsic measurements 1

### Citations

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Neumann The Logic of Quantum Mechanics
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Citation Context ...ir subsets which do not admit ‘large’ set of two-valued states. We show connections with other results of this type. 1 Introduction Quantum logic, as it has been pioneered by Birkhoff and von Neumann =-=[2]-=-, is usually derived from Hilbert space. There, the logical primitives, such as propositions and the logical operators “and”, “or” and “not” are defined by Hilbert space entities. For instance, consid... |

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Citation Context ... connection between the nonexistence of a two-valued state and density in R 3 . This might give better insight into the nature of subortholattices of H3 and the connection with famous Gleason theorem =-=[6, 20]-=-, which (among other things) states that there is no two-valued state on H3. It should be noted that Greechie diagrams of subortholattices of H3 are relatively ‘complex’—the distance of every pair of ... |

136 | The Philosophy of Quantum Mechanics - Jammer - 1974 |

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76 |
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Citation Context ...ads of mutually orthogonal rays. Such a measurement can be performed with the operator discussed by Peres, or with an arrangement of beam splitters discussed by Reck, Zeilinger, Bernstein and Bertani =-=[21]-=-. For instance, after c is found to be “true” (corresponding to s(c) = 1), then measurement of the original values of a or b is no longer possible. However, suppose one would be willing to believe in ... |

74 |
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Citation Context ...tangled subsystems. On each one of the 16 different entangled subsystems one could measure one of the 16 different orthogonal triads. This is similar to a proposal by Greenberger, Horne and Zeilinger =-=[7]-=-, which 25use three particles and eight-dimensional Hilbert space. Indeed, only in such a way—namely by (counterfactually) inferring non-comeasurable propositions— one would encounter a complete Koch... |

69 |
Orthomodular structures as quantum logics
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Citation Context ...e is such a state that the first is true while the second is not. These properties are largely studied. An orthomodular poset with a full set of two-valued states is called a concrete logic (see e.g. =-=[20]-=-), an orthomodular poset with a separating set of twovalued states is called a partition logic—this notion is within orthomodular posets equivalent to the notion of automaton logic (see e.g. [22, 23, ... |

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34 |
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Citation Context ...ct that propositions are either “true” (≡ measure value 1) or “false” (≡ measure value 0). Kochen and Specker’s original construction used 117 lines. The number of lines has been subsequently reduced =-=[17, 18, 14, 4]-=-. These constructions are examples of propositional structures without any two-valued measures. This paper deals with the following questions: which orthomodular structure— finite or infinite—underlie... |

29 |
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Citation Context ...“true” (corresponding to s(c) = 1), then measurement of the original values of a or b is no longer possible. However, suppose one would be willing to believe in the existence of “elements of reality” =-=[8, 16]-=-, which could merely be counterfactually inferred. Then one could for instance—at least in principle—“measure” all 16 orthogonal triads by the production of a state with 16 entangled subsystems. On ea... |

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Citation Context ...t in several important aspects. They are non-Boolean, i.e., non-distributive, algebraic structures. Furthermore, as has first been pointed out by Kochen and Specker in the context of partial algebras =-=[12]-=-, there exist certain finite sets of lines, such that the associated propositional structure cannot be classically embedded. That is, there does not exist any classical, i.e., two-valued, measure whic... |

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Citation Context ...and blocks which preserve incidence relations. A Greechie diagram does not contain any loop of order 4 if and only if the corresponding orthomodular poset is a lattice. The proof can be found e.g. in =-=[15]-=-. Let us reserve the notion Greechie logic for an orthomodular poset which can be represented by a Greechie diagram with only finite edges. It is easy to see that such an orthomodular poset does not c... |

15 |
Automaton partition logic versus quantum logic
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Citation Context ...e.g. [20]), an orthomodular poset with a separating set of twovalued states is called a partition logic—this notion is within orthomodular posets equivalent to the notion of automaton logic (see e.g. =-=[22, 23, 24]-=-). It is easy to see that a full set of states is separating and that a separating set of two-valued states is unital. Before we give examples demonstrating differences in the above defined notions le... |

13 | Introduction to lattice theory - Szasz - 1963 |

12 | Automata simulating quantum logics - Grib - 1990 |

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Quantum Theory
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Citation Context ...ct that propositions are either “true” (≡ measure value 1) or “false” (≡ measure value 0). Kochen and Specker’s original construction used 117 lines. The number of lines has been subsequently reduced =-=[17, 18, 14, 4]-=-. These constructions are examples of propositional structures without any two-valued measures. This paper deals with the following questions: which orthomodular structure— finite or infinite—underlie... |

10 |
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Citation Context ... ′ ; a,b∈Pn such that a⊥b}. 3 Two-valued states and Greechie diagrams Let us present the main definition. 3.1. Definition. Let P be an orthomodular poset and let G⊂P. A state s on G is a mapping s: P→=-=[0,1]-=- such that: (1) s(0)=0, (2) s(a)≤s(b) whenever a,b∈G with a≤b, (3) ∑a∈O s(a)≤1 for every orthogonal set O⊂G, (4) ∑a∈O s(a)=1 for every orthogonal set O⊂G with ∨ O=1. A two-valued state is a state with... |

8 | Measures and Hilbert Lattices - Kalmbach - 1986 |

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other theorem and its connection with non–locality. Part 1
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Citation Context ... has given rise to a number of interpretations, by Kochen and Specker and others. A detailed overview of the history of the subject can, for instance, be found in the reviews by Mermin [14] and Brown =-=[3]-=-. What does it physically mean that three nonorthogonal rays in threedimensional Hilbert space are sufficient to generate a finite system of rays which have no two-valued state? To state the associate... |

5 |
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Citation Context ...which are not crossed, a set of orthogenerators is e.g. the set of vertices of both pentagons with ai’s and with the middle point, a set of generators is marked by circles. It should be noted that in =-=[4, 27]-=- there is an example of an 11-element set of lines orthogenerating a 25-element set of lines and a 76-element (37-atomic) suborthoposet of H3 without a unital set of two-valued states. This suborthopo... |

4 | Lattice Theory, Second Edition - Birkhoff - 1948 |

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letter to Professor E. P. Specker, dated April 22nd, 1965; published in E. Clavadetscher-Seeberger: Eine partielle Prädikatenlogik. (Dissertation
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Citation Context ...ct that propositions are either “true” (≡ measure value 1) or “false” (≡ measure value 0). Kochen and Specker’s original construction used 117 lines. The number of lines has been subsequently reduced =-=[17, 18, 14, 4]-=-. These constructions are examples of propositional structures without any two-valued measures. This paper deals with the following questions: which orthomodular structure— finite or infinite—underlie... |

1 |
Validité dans les algébres de Boole partielles
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Citation Context ...“not” ( ′ ) to form new expressions. Thus, it would be incorrect to state that there exists a classical tautology in the three variables a, b and c which is no quantum tautology. Indeed, Coray proved =-=[5]-=- that all classical tautologies in three variables are tautologies in all partial algebras, in particular in the one associated with the logic of quantum observables. However, also Schütte’s example i... |

1 |
Specker: The problem of hidden varibles in quantum mechanics
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Citation Context ...on of an orthomodular lattice given in Fig. 2.2. Then ̸ (a,b)∈〈arccos 1 3 , π 2 ). On the other hand, for every α∈〈arccos 1 3 , π 2 ) there is a realization of L such that ̸ (a,b)=α. Proof. (See also =-=[13]-=-). Let us choose a coordinate system such that c=Sp(1,0,0), d = Sp(0,1,0). Hence e=Sp(0,0,1). Since ca ⊥ c and db ⊥ d, there are x,y∈ R\{0} such that ca = Sp(0,y,1), db = Sp(x,0,1). Since cb ⊥ c,ca an... |

1 |
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Citation Context |

1 | Specker: Die Logik nicht gleichzeitig entscheidbarer Aussagen. Dialectica 14 - P - 1960 |