# Functions

## Syntax

Define the language RecFun(V,C,P) as extending RecExp(V,C,P) with new types:

T | ::= | ...as before...
| | | C T : T

and new expressions:

M | ::= | ...as before...
| | | fn C P {M} | (Anonymous function)
| | | M M | (Function application)

The typing for anonymous functions is:

|  | (P : T) :
, |  | M : T in C
|  | fn C P {M} : (C T : T) in val

and for function application is:

|  | M : (C T : T) in val
|  | M : T in C
|  | M M : T in C

In particular, note that anonymous functions are always values, even if the function body is central or process. This agrees with the usual definition of normal forms for operational semantics.

## Graphical semantics

### Cartesian closed

To give a graphical semantics for the cartesian closed structure, we extend the category of cyclic flow graphs CGraph(V) to that of closed cyclic flow graphs CCGraph(V) by allowing edges to be labelled by value function types:

A,B,C ::= X | A1,...,Am B1,...,Bn

and allowing nodes of the form:

with labellings:
: (B1,...,Bn C1,...,Co), B1,...,Bn C1,...,Co in CCGraph(V)
: A1,...,Am (B1,...,Bn C1,...,Co) in CCGraph(V)
where:
: A1,...,Am, B1,...,Bn C1,...,Co in CCGraph(V)
These graphs are factored up to the equivalence required for a cartesian closed category:

CCGraph(V)[A, B C] CCGraph(V)[A B, C]

Graphically these axioms are:

Beta: |  | = |
Eta: |  | = |
Naturality: |  | = |

The graphical semantics is extended with:

[[fn val P {M} in val]] | = |
[[M N in val]] | = |

### Symmetric monoidal closed

To give a semantics for the symmetric monoidal closed structure:

- we extend the category of cyclic flow graphs CGraph(V,C) to that of closed cyclic flow graphs CCGraph(V,C), and

- we extend the category CCGraph(V) to CCGraph(V,C).

by allowing edges to be labelled by central function types:

A,B,C ::= ... | A1,...,Am B1,...,Bn

We extend flow graphs to allow nodes of the form:

with labellings:
: (B1,...,Bn C1,...,Co), B1,...,Bn C1,...,Co in CCGraph(V,C)
: A1,...,Am (B1,...,Bn C1,...,Co) in CCGraph(V,C)
where:
: A1,...,Am, B1,...,Bn C1,...,Co in CCGraph(V,C)
These graphs are factored up to the equivalence required for the adjunction:

CCGraph(V,C)[A, B C] CCGraph(V,C)[A B, C]

Graphically:

Beta: |  | = |
Eta: |  | = |
Naturality: |  | = |

The graphical semantics is extended with:

[[fn central P {M} in val]] | = |
[[M N in central]] | = |

### Symmetric premonoidal closed

To give a semantics for the symmetric monoidal closed structure:

- we extend the category of cyclic flow graphs CGraph(V,C,P) to that of closed cyclic flow graphs CCGraph(V,C,P),

- we extend the category CCGraph(V,C) to CCGraph(V,C,P), and

- we extend the category CCGraph(V,C) to CCGraph(V,C,P).

by allowing edges to be labelled with process function types:

A,B,C ::= ... | A1,...,Am B1,...,Bn

and allowing nodes of the form:

with labellings:
: (B1,...,Bn C1,...,Co), B1,...,Bn C1,...,Co in CCGraph(V,C,P)
: A1,...,Am (B1,...,Bn C1,...,Co) in CCGraph(V,C,P)
where:
: A1,...,Am, B1,...,Bn C1,...,Co in CCGraph(V,C,P)
These graphs are factored up to the equivalence required for the adjunction:

CCGraph(V,C,P)[A, B C] CCGraph(V,C,P)[A B, C]

Graphically:

Beta: |  | = |
Eta: |  | = |
Naturality: |  | = |

The graphical semantics is extended with:

[[fn proc P {M} in val]] | = |
[[M N in proc]] | = |

## Recursive functions

To allow recursive functions, we just add a new judgement:

|  | proc T : T traceable

For example, with appropriate string and I/O primitives, we can write a simple `hello' program:

No change is required to the graphical semantics, except to make edges labelled with types of the form A1,...,Am B1,...,Bn traceable.

## Initiality

Since we have taken the initial cartesian/monoidal/premonoidal categories and factored them by equations for the closed structure, the categories:

CCGraph(V,C,P) CCGraph(V,C,P) CCGraph(V,C,P)

form the initial triple of categories:

V C P

with:

- V a partially traced cartesian category over V.

- C a strict symmetric monoidal category over C.

- P a strict symmetric premonoidal category over P.

- The inclusions are identity on objects symmetric premonoidal functors.

- Adjunctions:

  V[X, Y Z] V[X Y, Z]
  V[X, Y Z] C[X Y, Z]
  V[X, Y Z] P[X Y, Z]

- All objects of the form X Y are traceable.

Thus the graphical semantics defined above uniquely determines a semantics in any such triples of categories.

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