Syntactic string diagrams

Posted in Category theory, Math, Programming by Mike Stay on 2009 January 24

I hit on the idea of making lambda a node in a string diagram, where its inputs are an antivariable and a term in which the variable is free, and its output is the same term, but in which the variable is bound.  This allows a string diagram notation for lambda calculus that is much closer to the syntactical description than the stuff in our Rosetta Stone paper. Doing it this way makes it easy to also do pi calculus and blue calculus.

There are two types, V (for variable) and T (for term). I’ve done untyped lambda calculus, but it’s straightforward to add subscripts to the types V and T to do typed lambda calculus.

There are six function symbols:

  • \lambda:V^* \times T \to T. Lambda takes an antivariable and a term that may use the corresponding variable.
  • \cap:1 \to V^* \times T. This turns an antivariable “x” introduced by lambda into the term “x”.
  • A:T \times T \to T. (Application) This takes f and x and produces f(x).
  • {\rm swap}:T \times T \to T \times T
  • !:T \to 1
  • \Delta:T \to T \times T. These two mean we can duplicate and delete terms.

The \beta-\eta rule is the real meat of the computation. The “P” is an arbitrary subdiagram. The effect is replacing the “A” application node with the “P” subdiagram, modulo some wiring.

I label the upwards arrows out of lambdas with a variable name in parentheses; this is just to assist in matching up the syntactical representation with the string diagram.

In the example, I surround part of the diagram with a dashed line; this is the part to which the \beta-\eta rule applies. Within that, I surround part with a dash-dot line; this is the subdiagram P in the rule.

When I do blue calculus this way, there are a few more function symbols and the relations aren’t confluent, but the flavor is very much the same.

String diagrams for untyped lambda calculus

String diagrams for untyped lambda calculus

An example calculation

An example calculation


One Response

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  1. reperiendi said, on 2009 January 27 at 12:44 pm

    Peter Selinger was kind enough to send this response:

    Hi Mike,

    your string diagrams are known as “sharing graphs” in the literature,
    and are extremely well-studied. See e.g.

    [1] Stefano Guerrini, A general theory of sharing graphs (1997)

    There is a reduction strategy on sharing graphs that is provably the
    most efficient possible reduction strategy for lambda calculus; this
    goes back to the work of Levy and Lamping (both cited in the above

    Guerrini also gives an algebraic semantics of these sharing graphs,
    see section 7. I don’t know whether it is related to your categorical

    I also like the book by Peyton-Jones:
    [2] Simon Peyton Jones, The Implementation of Functional Programming
    Languages (1987)

    It shows how to use sharing graphs as the basis for a practical
    implementation of lazy programming languages. As far as I know, this
    is still state-of-the-art and is used in the implementation of Haskell.
    In keeping with the practical nature of the book, the sharing graphs
    are represented in slightly different form (with syntactic variables
    rather than backpointers), but this is of course equivalent.

    As for the categorical semantics, what you have in mind is a kind of
    abstract syntax with variable binding. To put this into perspective,
    the semantics of ordinary abstract syntax (i.e., without variable
    binding), is given by an object A in a cartesian category, together
    with interpretations for each n-ary function symbol f : A^n -> A. One
    can then inductively define the interpretation of terms, speak of the
    free such object, etc.

    Things get slightly more complicated if one adds variable binding to
    this picture. This has also been studied, though perhaps not in the
    same form as you are proposing. Perhaps the closest to your approach

    [3] Martin Hofmann, Semantical analysis of higher-order abstract
    syntax (1999)

    Here one has var : V -> T, app : T x T -> T, and lam : (V => T) -> T
    (see top of p.9, with the obvious changes in notation). You will note
    the use of a function space (V => T) in place of your cartesian
    product V^* x T.

    Another very similar paper is

    [4] M.P.Fiore, G.D.Plotkin and D.Turi. Abstract syntax and variable binding (1999)

    Here, one has var : V -> T, app : TxT -> T, and lam : delta T -> T (as
    contained e.g. in the commutative diagram on p.6). Again, delta T is
    something akin to the function space V => T, but is also isomorphic,
    in a suitable sense, to T => T, as far as I remember (this is
    important for substitution, see below).

    A third, technically slightly different (but conceptually similar)
    approach to abstract syntax with variable binders is:

    [5] Murdoch J. Gabbay, Andrew M. Pitts: A new approach to abstract syntax
    with variable binding (1999)

    [6] Murdoch J. Gabbay, Andrew M. Pitts: A new approach to abstract
    syntax with variable binding (2002)

    See especially the second-to-last line of [6, p.356], and you
    immediately see the similarity. Their [A]T operation is akin to
    V => T in [3] and delta T in [4].

    It is fun to note that [3-5] all appeared (independently) in the same
    conference in 1999. Semantics of variable binders was clearly a
    pressing problem that year.

    I can comment briefly on the main difference between these works and
    what you are proposing. One thing that is important in syntax is
    substitution (of a term for a variable). In fact, in the usual
    abstract syntax (without binders), a term with n variables is
    represented as a morphism A^n -> A. This is useful for substitution:
    namely, if f : A x A^n -> A represents the term t(x,y_1,…,y_n) and
    g : A^n -> A represents s(y_1,…,y_n), then f o represents

    In the presence of variables, a term with n variables is represented
    as 1 -> V*^n x T (in your notation). However, for reasons of
    substitution, one would still like this hom-set to be in 1-1
    correspondence with T^n -> T. Somehow this is what the papers [3-6]
    manage to do, each in their own way.

    I hope these references will be useful. It would be great if you had a
    more abstract monoidal framework of which the particular constructions
    in [3-6] are concrete examples. I have always wondered about the
    precise connection between [3-6], and whether there is a bigger

    Good luck, and let me know how it goes! — Peter

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