## Linear logic for generalized quantum mechanics (1993)

Venue: | In Proc. Workshop on Physics and Computation (PhysComp'92 |

Citations: | 16 - 2 self |

### BibTeX

@INPROCEEDINGS{Pratt93linearlogic,

author = {Vaughan Pratt},

title = {Linear logic for generalized quantum mechanics},

booktitle = {In Proc. Workshop on Physics and Computation (PhysComp'92},

year = {1993},

pages = {166--180},

publisher = {IEEE}

}

### Years of Citing Articles

### OpenURL

### Abstract

Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the “structure veil. ” When VLSI shrinks to where quantum effects are felt, their computer-aided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1

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Citation Context ...tems? That is, will it be enough just to change those nonlogical axioms defining how electronics works, or will it prove necessary to dig deeper and tamper with logic itself? Birkhoff and von Neumann =-=[BvN36]-=- incorporate quantum uncertainty into Boolean logic by interpreting conjunction A ∧ B standardly but modifying negation A ⊥ to mean certainly not A, in the sense that A ⊥ holds in just those states ha... |

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Citation Context ...be a set of states ordered by information content, while an event space will be a set of events ordered by time. State spaces generalize bc-domains (Scott domains or bounded-complete algebraic cpo's) =-=[Gun92]-=-, while their complementary event spaces generalize Winskel's event structures [Win86]. This particular duality is one small fragment of Birkhoff-Stone duality [Bir33, Sto36, Sto37, Pri70], with the p... |

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Citation Context ... of time and Heisenberg uncertainty. Time is standardly added to quantum logic in terms of the group of automorphisms of the underlying ortholattice, and from this one recovers Heisenberg uncertainty =-=[Var68]-=-. In this paper we raise the problem of formulating this extension as a language extension. That is, what propositional logic of quantum mechanics incorporates not only QM's uncertainty but also its c... |

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Citation Context ... is necessarily of this static kind since it lacks a separate dynamic intersection distinct from static intersection, cf. Stone's topological treatment of intuitionistic logic at the end of his paper =-=[Sto37]-=-.) We define the dual notion of closure, ?A, as (!(A ? )) ? , and static implication A)B as !A\GammaffiB. We have seen that the language of linear logic is that of quantum logic expanded with dynamic ... |

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Citation Context ...es Heisenberg uncertainty appear, even implicitly; all that is represented is whether or not any two given states are mistakable for each other. Second, QL has no reasonable implication. Dalla Chiara =-=[Dal86]-=- surveys a number of candidates for a quantum logic implication, all of which are lacking in various ways. There does exist the well-known Sasaki hook A!B = A ?s(AsB), and Finch [Fin70] has given a co... |

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Citation Context ...tems? That is, will it be enough just to change those nonlogical axioms defining how electronics works, or will it prove necessary to dig deeper and tamper with logic itself? Birkhoff and von Neumann =-=[BvN36]-=- incorporate quantum uncertainty into Boolean logic by interpreting conjunction AB standardly but modifying negationsA ? to mean certainly not A, in the sense that A ? holds in just those states havin... |

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Citation Context ...d class of linear automata and their dual linear schedules, state spaces and their dual event spaces. (Elsewhere we describe in detail the yet more restricted class of chains as rigid local behaviors =-=[Pra92b]-=-.) In the section on measurement we anticipate a broader class of linear automata, partial distributive lattices, but defer its detailed treatment to another paper. However the notion of time contempl... |

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Citation Context .... Dalla Chiara [Dal86] surveys a number of candidates for a quantum logic implication, all of which are lacking in various ways. There does exist the well-known Sasaki hook A!B = A ?s(AsB), and Finch =-=[Fin70]-=- has given a conjunction A:B = As(A ?sB) to go with it satisfying A:1 = A = 1:A, A:0 = 0 = 0:A (i.e. A:B has the same unit and annihilator as AB), A:A = A (idempotence), and A:B ` C iff B ` A!C. Howev... |

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Citation Context ...llowing section we then give the informal behavioral intuition behind this interpretation. This language of sums and products of spaces is a close cousin of Birkhoff's generalize arithmetic of posets =-=[Bir42]-=-, elaborated on elsewhere [Pra92a]. The dual A ? of a state space A is obtained as the order dual of A \Gamma fq 0 g with q 0 then added back in at the bottom. That is, holding q 0 fixed, turn the res... |

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Citation Context ...uct space, made an orthoframe by interpreting orthogonality standardly for Euclidean space. Although completeness does not have a first-order definition for either inner product spaces or orthoframes =-=[Gol84]-=-, the condition of orthomodularity [BvN36] on the associated orthoframe does express completeness exactly [AA66]. (Orthomodularity can be defined as either A ` B and A ? B = 0 implies A = B, or A(A ? ... |

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Citation Context ...ive model of linear logic consists of a *-autonomous category D with all finite products and a comonad ! with natural isomorphisms !(A \Theta B) �� =!A\Omega !B and !1 �� = ?. A *-autonomous c=-=ategory [Bar79] is -=-a representably self-dual closed symmetric monoidal category D. Monoidal means that there exists a binary operation (functor)\Omega : D 2 ! D and an object ? of D having A\Omega (B\Omega C) �� = (... |

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Citation Context ...ve closure, a central concern for both database managers and program verifiers. Instead both first-order approximations (induction in Peano arithmetic) and exact second-order solutions (dynamic logic =-=[Pra80]-=-, relation algebras with transitive closure [NT77, Ng84]) have been proposed and used. The same considerations should apply to orthomodularity, which has a straightforward secondorder definition which... |

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Citation Context ...informal behavioral intuition behind this interpretation. This language of sums and products of spaces is a close cousin of Birkhoff's generalize arithmetic of posets [Bir42], elaborated on elsewhere =-=[Pra92a]-=-. The dual A ? of a state space A is obtained as the order dual of A \Gamma fq 0 g with q 0 then added back in at the bottom. That is, holding q 0 fixed, turn the rest of A upside down. This operation... |

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Citation Context ...ction. We understand classical propositional logic in terms of the class of its models, namely Boolean algebras. These can be succinctly defined as complemented distributive lattices. Following Seely =-=[See89]-=- we may define linear logic analogously, with constructivity calling for the substitution of categories for algebras. A constructive model of linear logic consists of a *-autonomous category D with al... |

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Citation Context ...ic state about that plane [BvN36]. 2.4 Limitations of Quantum Logic Quantum logic has been intensively studied in the intervening 56 years. A very recent bibliography of the subject lists 1851 papers =-=[Pav92]-=-. Evidently this appealingly simple system has touched a responsive chord with many philosophers of physics. But quantum logic may be too abstract to be useful. First, it is a very minimal logic of QM... |

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1 |
A remark on Piron's paper. Pubs. of the Res
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Citation Context ... does not have a first-order definition for either inner product spaces or orthoframes [Gol84], the condition of orthomodularity [BvN36] on the associated orthoframe does express completeness exactly =-=[AA66]. (Orthomodularity c-=-an be defined as either A ` B and A ? B = 0 implies A = B, or A(A ? (AB)) ` B. The second "ortho" in "orthomodular ortholattice" can be dropped.) Quantum logic is defined as the lo... |