dynamics.md

The Road to Dynamical Systems

Basic steps

• Tuple types

• [-] Symbolic functions Data type:

  struct AlgebraicFunction
      theory::GAT
      args::TypeScope
      ret::AlgType
      body::AlgTerm
  end

  Affordances:

  • DSL for writing down functions, composing, etc.
  • A function tcompose(t::Dtry{AlgebraicFunction})::AlgebraicFunction, implementing the Dtry-algebra structure on morphisms
  • Interpret/compile a symbolic function into a real function
  • Serialize symbolic functions

• Compilation

• Serialization

Lens-based dynamical systems

• Arenas Sketch:

  struct Arena
      in::AlgType
      out::AlgType
  end

  Affordances:

  • A function tcompose(arena::Dtry{Arena})::Arena, implementing the Dtry-algebra structure on objects

• Multilenses Sketch:

  struct MultiLens
      inner_boxes::Dtry{Arena}
      outer_box::Arena
      # used for namespacing `params` in composition, must not overlap with `inner_boxes`
      name::Symbol 
      params::AlgType
      # (params, tcompose(inner_boxes[...].out)) -> outer_box.out
      output::AlgebraicFunction
      # (params, tcompose(inner_boxes[...].out), outer_box.in) -> tcompose(inner_boxes[...].in)
      update::AlgebraicFunction
  end

  Affordances:

  • A function ocompose(l::MultiLens, args::Dtry{MultiLens})::MultiLens implementing the Dtry-multicategory structure

• Systems Sketch:

  struct System
      interface::Arena
      state::AlgType
      params::AlgType
      # (params, state) -> interface.out
      output::AlgebraicFunction
      # (params, state, interface.in) -> state
      input::AlgebraicFunction
  end

  Affordances:

  • A function oapply(l::MultiLens, args::Dtry{System})::System implementing the action of the Dtry-multicategory of multilenses on systems.

Resource sharers

• Interfaces

• Rhizomes (epi-mono uwds)

  struct VariableType
      type::AlgType
      exposed::Bool
  end
  
  struct Rhizome
      boxes::Dtry{Interface}
      junctions::Dtry{VariableType}
      mapping::Dict{DtryVar, DtryVar}
  end

  Affordances:

  • ocompose(r::Rhizome, rs::Dtry{Rhizome})::Rhizome

  In ocompose, the names of the junctions in the top-level rhizome dominate.

• Systems

  struct ResourceSharer
      variables::Dtry{VariableType}
      params::AlgType
      output::AlgType
      # (params, state) -> state
      update::AlgebraicFunction
      # (params, state) -> output
      readout::AlgebraicFunction
  end

  Affordances:

  • oapply(r::Rhizome, sharers::Dtry{ResourceSharer})::ResourceSharer

  In oapply, variables get renamed to the junctions that they are attached to.