The ZX& calculus: A complete graphical calculus for classical circuits using spiders

@article{Comfort2021TheZC,
  title={The ZX\& calculus: A complete graphical calculus for classical circuits using spiders},
  author={Cole Comfort},
  journal={ArXiv},
  year={2021},
  volume={abs/2004.05287}
}
We give a complete presentation for the fragment, ZX&, of the ZX-calculus generated by the Z and X spiders (corresponding to copying and addition) along with the not gate and the and gate. To prove completeness, we freely add units and counits to the category TOF generated by the Toffoli gate and ancillary bits, showing that this yields the strictification of spans of powers of the two element set; and then perform a two way translation between this category and ZX&. A translation to some… 

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