Operational semantics

Examples

In this paper, we have given an equational characterization of programs. Categorically, we have:

To give a treatment of the dynamics of progams, we would like to give an operational semantics, for example:
In this example, the reductions are given as a type-preserving relation between terms. Following Milner's operational semantics for action calculi, we would expect to have the categorical picture: In the above case, the 2-cells are just a preorder, but in general we may be interested in labelled transition systems, which can be viewed as 2-categories where the 2-cells are labelled with an appropriate monoid, for example strings of actions. For example, we can give an lts semantics for programs containing print statements as a string-labelled 2-category:
'hello'
'world'

These 2-cells are generated from atomic reductions, in these examples:

(where x + y = z)
and:
s

Pre-2-categories

Unfortunately, we cannot just replay Milner's presentation in the premonoidal setting, since one of the axioms of a 2-category is functoriality of composition, which implies:

If
and
then ;
For example this would give us:
'world'
'hello'
This is a similar problem to that solved by premonoidal categories, and the solution is to lift the premonoidal structure into the 2-cells. In the same way as we divided the 1-cells into central and process 1-cells, we divide the 2-cells into central and process 2-cells. Central reductions can take place anywhere in a computation, but process reductions have to happen in left-to-right order:
If
and
then ;
and:
If
and
then ;
and:
If
and
then ;
This is just the usual definition of big-step reduction to normal form, replayed in categorical language.

Formally, define C P to be a pre-2-category whenever:

A pre-2-functor is a commuting square of functors respecting the pre-2-categorical structure.

Note that any 2-category is automatically a pre-2-category, since we can take C and P to be the same 2-category, and identify all of the hom-categories and composition functors.

We can then replay the definition of a premonoidal category at the 2-level to get a categorical presentation of a language with its operational semantics.

A premonoidal pre-2-category is a pre-2-category C P with:

Following Power's presentation of premonoidal categories as Subset-enriched categories, we can present pre-2-categories using enriched categories. Define a category to be centralized if some of its objects are tagged as central, some of its morphisms are tagged as central, and satisfying the property that if f : X  X and X is central, then f and X are central. Let Central be the category of centralized categories with functors respecting the central structure. Central has an asymmetric monoidal structure where C * D is the subcategory of C  D where for any pair of morphisms (f,g) : (X,Y)  (X,Y) either X is central or g is central. Then the above definition of a pre-2-category is equivalent to a Central-category with composition given as a functor P[X,Y] * P[Y,Z]  P[X,Z]. We can then replay Power's definition of a premonoidal category replacing Subset-Cat with Central-Cat to find a definition of a premonoidal pre-2-category equivalent to the above.

For the closed structure, the isomorphisms:

V[X (Y Z)] V[(X Y) Z]
V[X (Y Z)] C[(X Y) Z]
V[X (Y Z)] P[(X Y) Z]

allow value and central reductions to take place under function bodies, but not process reductions. For example we allow:

but not:
'hello'
As two extremes, we have: Thus the division of 2-cells into central and non-central categories gives a natural description of both the call-by-value lambda calculus with reduction allowed anywhere, and the canonical left-to-right reduction strategy.

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