Synthesising Graphical Theories

In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category. In particular, categorical quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of quantum theory into abstract structural properties, expressed in the form of diagrammatic identities. One way we search for these properties is to start with a concrete model (e.g. a set of linear maps or finite relations) and start composing generators into diagrams and looking for graphical identities. Naively, we could automate this procedure by enumerating all diagrams up to a given size and check for equalities, but this is intractable in practice because it produces far too many equations. Luckily, many of these identities are not primitive, but rather derivable from simpler ones. In 2010, Johansson, Dixon, and Bundy developed a technique called conjecture synthesis for automatically generating conjectured term equations to feed into an inductive theorem prover. In this extended abstract, we adapt this technique to diagrammatic theories, expressed as graph rewrite systems, and demonstrate its application by synthesising a graphical theory for studying entangled quantum states.Comment: 10 pages, 22 figures. Shortened and one theorem adde

Review of Noterapion Kissinger from Chile and Argentina (Coleoptera: Apionidae)

Descriptions and a key are provided for 7 South American species of Note rap ion Kissinger (2002) (type species Apion meorrhynchum Philippi and Philippi) including N. bruchi (Beguin-Billecocq), N. meorrhynchum (Philippi and Philippi), N. philippianum (Alonso-Zarazaga) and four new species described from Chile: N. chilense Kissinger, N. lwscheli Kissinger, N. nothofagi Kissinger, and N. saperion Kissinger. A lectotype designation is published for Apion meorrhynchum Philippi and Philippi and Apion uestitum Philippi and Philippi. Apion fuegianum Enderlein and A. pingue Beguin-Billecocq are synonymized with N. meorrhynchum (Philippi and Philippi), new synonymy. Noterapionini (new tribe) is erected for Noterapion Kissinger (type genus) within Apioninae. Extension of a phylogenetic analysis of Brentidae s. lato by Wanat (2001) places Noterapion near the base of Apioninae and shows the genus sharing various symplesiomorphies with primitive apionid subfamilies from Africa and not found otherwise in the New World apionids. The weevils are associated with the southern beech, Nothofagus Blume (in Nothofagaceae, see Manos, 1997), also known from the Australasian Region. Noterapion meorrhynchum develops in abandoned cynipid wasp leaf galls. The combination of a plant host with biogeographic significance and the possession of very primitive characters suggests that Noterapion may represent an ancient lineage dating back to the time of the Cretaceous and the breakup of Gondwana

Apionidae from North and Central America : 5. Description of genus Apionion and 4 new species (Coleoptera)

Apionion (type species Apion crassum Fall) is described for 14 species formerly assigned to the Apion annulatum species group of Coelocephalapion Wagner, namely, championi Sharp, crassum Fall, derasum Sharp, dilatatum Smith, fenyesi Kissinger, howdeni Kissinger, inflatipenne Sharp, latipenne Sharp, latipes Sharp, len tum Sharp, neolentum Kissinger, samson Sharp, and subauratum Sharp from North and Central America, and annulatum Gerstaecker from South America, all originally included in Apion Herbst. Four new species are described: delion (panama), eranion (Costa Rica, Panama), humongum (Mexico, El Salvador, Honduras), and sapphirum (Mexico, Costa Rica). New records and/or supplemental descriptions are given for championi, derasum, dilatatum, fenyesi, howdeni, inflatipenne, latipenne, latipes, and neolentum

Description of a new genus, Sayapion, from North and Central America (Coleoptera: Apionidae)

A new genus, Sayapion Kissinger (type-species: Apion segnipes Say) is proposed for the 10 members of the Apion segnipes species group of Coelocephalapion Wagner orphaned when Coelocephalapion Wagner was raised to generic level (Kissinger, 1992). The species transferred from Apion to Sayapion as new combinations are as follows: Sayapion aponipes (Kissinger), S. arizonae (Fall), S. basale (Sharp), S. cinereum (Gerstaecker), S. laterale (Sharp), S. paranipes (Kissinger), S. pronipes (Kissinger), S. segnipes (Say), S. sublaterale (Kissinger), and S. terale (Kissinger)

Apionidae from North and Central America : 6. Description of new species of Apionion Kissinger, Coelocephalapion Wagner and Trichapion Wagner (Coleoptera)

Two new species of Trichapion Wagner, T. baranowskii and T. santaritae, are described from Madera Canyon, near Tucson, Arizona. Six new species of Coelocephalapion Wagner are described: C. dilox (Mexico), C. goldilox (Costa Rica, Panama), C.johnsoni (Panama) with host Vatairea erythrocarpa Ducke (Fabaceae), C. nirostrum (Mexico), C. tellum (Texas, Mexico), and C. turnbowi (Mexico). Apionion opetion is described from Mexico. A closely similar species, Apionion bettyae (Kissinger), new combination, with probable host plant Lonchocarpus sp. (Fabaceae), is transferred from Trichapion

Tensors, !-graphs, and non-commutative quantum structures

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entire families of diagrams. One candidate is the language of !-graphs, which consist of graphs with certain subgraphs marked with boxes (called !-boxes) that can be repeated any number of times. New !-graph equations can then be proved using a powerful technique called !-box induction. However, previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.Comment: In Proceedings QPL 2014, arXiv:1412.810

A first-order logic for string diagrams

Equational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of equations using a simple graphical syntax called !-box notation. While this does greatly increase the proving power of string diagrams, previous attempts to go beyond equational reasoning have been largely ad hoc, owing to the lack of a suitable logical framework for diagrammatic proofs involving !-boxes. In this paper, we extend equational reasoning with !-boxes to a fully-fledged first order logic called with conjunction, implication, and universal quantification over !-boxes. This logic, called !L, is then rich enough to properly formalise an induction principle for !-boxes. We then build a standard model for !L and give an example proof of a theorem for non-commutative bialgebras using !L, which is unobtainable by equational reasoning alone.Comment: 15 pages + appendi

ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.Comment: In Proceedings QPL 2018, arXiv:1901.0947