## Proof Nets as Formal Feynman Diagrams

### BibTeX

@MISC{Blute_proofnets,

author = {Richard Blute and Prakash Panangaden},

title = {Proof Nets as Formal Feynman Diagrams},

year = {}

}

### OpenURL

### Abstract

Summary. The introduction of linear logic and its associated proof theory has revolutionized many semantical investigations, for example, the search for fullyabstract models of PCF and the analysis of optimal reduction strategies for lambda calculi. In the present paper we show how proof nets, a graph-theoretic syntax for linear logic proofs, can be interpreted as operators in a simple calculus. This calculus was inspired by Feynman diagrams in quantum field theory and is accordingly called the φ-calculus. The ingredients are formal integrals, formal power series, a derivative-like construct and analogues of the Dirac delta function. Many of the manipulations of proof nets can be understood as manipulations of formulas reminiscent of a beginning calculus course. In particular, the “box ” construct behaves like an exponential and the nesting of boxes phenomenon is the analogue of an exponentiated derivative formula. We show that the equations for the multiplicative-exponential fragment of linear logic hold. 1

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Citation Context ...ve to reintroduce the locations to indicate how things are connected. In process terms these willProof Nets as Formal Feynman Diagrams 7 correspond to ports or channels or names as in the π-calculus =-=[Mil89]-=-. In our formal calculus, locations will be introduced with roughly the same status. The set of locations L is ranged over by x, y, z, . . .. The next important idea is that of located sequents, of th... |

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Citation Context ...e φ-calculus, which closely mimics some of the ideas of quantum field theory. In particular we have imitated some of the techniques, called “functional methods” in the quantum field theory literature =-=[IZ80]-=-, and shown how to represent the exponential types in linear logic as an exponential power series. The manipulations of boxes in linear logic amounts to certain simple exponential identities. Thus we ... |

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Citation Context ... to include a differential combinator, and explore the syntactic consequences. The possible relationships to the present work are striking. Ehrhard and Regnier’s work was subsequently categorified in =-=[BCS06]-=-. Finally it is possible to construct a mathematical model for the φ-calculus. The manipulations that we have done with variational derivatives and exponentials are very close to the calculations that... |

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Citation Context .... Given the correspondence between proof nets and the λ-calculus, a significant shift has occurred. One now has a local, asynchronous algorithm for β-reduction [DR93, ADLR94]. Abramsky and Jagadeesan =-=[AJ94b]-=- presented geometry of interaction using dataflow nets using fixed point theory instead of the apparatus of Hilbert spaces and operators. However, the information flow paradigm is clear in both presen... |

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Citation Context ... Fock space – also known as the symmetric tensor algebra – can be viewed as the space of analytic functions on a Banach space and, in a formal sense can be viewed as an exponential. Our original work =-=[RBS93]-=- fell short of modelling linear logic. Girard [Gir95b] later succeeded in modelling linear logic using analytic functions on what he called coherent Banach spaces. A key idea in that work is that the ... |