Implementing Chyp into OrbitMines' Rays - (Cospans of HYPergraphs - Aleks Kissinger)

[FadiShawki]

Discord equivalent: https://discord.com/channels/1055502602365845534/1185009740103811122

[FadiShawki]

Some notes from last year < 2023-12-20

People

• [[Filippo Bonchi]]

• [[Fabio Gadducci]]

• [[Aleks Kissinger]]

• [[Paweł Sobocinski]]

• [[Fabio Zanasi]]

• [[2012.01847.pdf|String Diagram Rewrite Theory I: Rewriting with Frobenius Structure]]

  • "Two diagrams that can be toepologically deformed into each other without cutting or joining wires must necessarily describe the same map"
    • An example of homotopy if that frame is identified as some complicated hyperedge deemed homotopically as "the same" - "ignorantly, as the same"
  • "(i.e. the set of endpoints of aconnected component) matters"
    • Ignorant at the vertices. (which may as well be described as the same thing) - this is generally probably not the case.
  • Splitting wires/merging:
    • "These generators and rules are known in the literature as a [[Special Commutative Frobenius Algebra (SCFA)]]. A category where every object is equipped with an [[Special Commutative Frobenius Algebra (SCFA)]] is called a [[Hypergraph Category]]."
      • Non-branching (so no equivalencies at the edges) are discussed in the 2nd paper. (or: [Generic Symmetric Monoidal Category])
  • "This rewriting-modulo step can be seen as the formal, syntactic underpinning to the intuitive notion of rewriting defined directly on string diagrams"
    • This misses the point of that similarly one might say that for the syntactic one - it's just the historically used one. So if the diagrammatic way was the usual way of thinking it would be the other way around. Better phrased as merely a translation to the other way of thinking.
  • "two Frobenius algebras interacting as a bialgebra"
    • "ZX-Calculus" / involve two interacting Frobenius algebras at their heart
  • [[Peter Selinger]]
  • 2.2 : Entire thing can merely be descriped as moving connections to different vertix'es frames (/rays). As with ZX-Calculus or any rewriting.
  • 3.4 : Merely additional access to information one doesn't ignore which alters whether it's a match. - All intuitive, though not necessarily obvious whether the writer here or generally wants to self-referentially construct it, similar to my thing with Rays.
  • redex ; Something to be reduced according to the rules of a formal system.
  • [[Monoidal PROduct and Permutation Category (PROP)]]

• [[2104.14686.pdf|String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure]]

  • References
    • [[John Power]]
    • [[Gordon Plotkin]]
    • [[Steve Lack]]
    • [[Burroni]]
    • [[Samuel Mimram]]
    • [[Obradovic]]
    • [[Hadzihasanovic]]
  • "Pushout complements always exist in HypΣ , but they are not necessarily unique. They are so if the rule is left-linear, that is, if K → L is monic" /
  • Identities and symmetries in CspD (HypΣ ) are monogamous
  • What about just restructure the whole thing in a separate cage, then it's dissallowed by internal cyclicity right? -0 if that's not what this paper is after.
  • Termination/confluence
    • A rewriting relation is terminating if it admits no infinite sequence of rewrites
    • it is confluent if any pair of hypergraphs (or terms, etc.) arising from G by a sequence of rewriting steps can eventually be rewritten to the same hypergraph
    • Taken together, these properties imply the existence of unique normal forms.
  • Frobenius semi-algebras are Frobenius algebras lacking the unit and counit equations (No termination, no instantiation?, still dup/merge though as a form of termination/instantiation)
    • [[Quantum Physics]] ; relevant to quantum theory, such as H*-algebras

• [[string-diagram-rewrite-theory-iii-confluence-with-and-without-frobenius.pdf|String diagram rewrite theory III: Confluence with and without Frobenius]]