The Holy Trinity

March 27, 2011

The Christian doctrine of trinitarianism states that there is one God that is manifest in three persons, the Father, the Son, and the Holy Spirit, who together form the Holy Trinity.   The doctrine of computational trinitarianism holds that computation manifests itself in three forms: proofs of propositions, programs of a type, and mappings between structures.  These three aspects give rise to three sects of worship: Logic, which gives primacy to proofs and propositions; Languages, which gives primacy to programs and types; Categories, which gives primacy to mappings and structures.  The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation.  There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.

Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two.  If you arrive at an insight that has importance for logic, languages, and categories, then you may feel sure that you have elucidated an essential concept of computation—you have made an enduring scientific discovery.  Advances in our understanding of computation may arise from insights gained in many ways (any data is useful and relevant), but their essential truth does not depend on their popularity.

Logic tells us what propositions exist (what sorts of thoughts we wish to express) and what constitutes a proof (how we can communicate our thoughts to others).  Languages (in the sense of programming) tells us what types exist (what computational phenomena we wish to express) and what constitutes a program (how we can give rise to that phenomenon).  Categories tell us what structures exist (what mathematical models we have to work with) and what constitutes a mapping between them (how they relate to one another).  In this sense all three have ontological force; they codify what is, not how to describe what is already given to us.  In this sense they are foundational; if we suppose that they are merely descriptive, we would be left with the question of where these previously given concepts arise, leading us back again to foundations.  It is the foundations that I wish to describe here, because I believe it will help to clarify some common misunderstandings about the notions of proposition, type, and structure.  Of particular interest here is that a “type system” is not, under this conception, an arbitrary collection of conditions imposed on a previously given notion of program (whether written with horizontal lines, or not).  It is, rather, a way to say what the programs are in the first place, and what they mean as proofs and as mappings.

Here I will outline the basic correspondences between logic, languages, and categories by examining their structural properties (and, for now, nothing more).

The fundamental notion in logic is that of entailment, written P_1,\dots,P_n\vdash P, expressing derivability of P from P_1,\dots, P_n.  This means that P is derivable from the rules of logic, given the P_i as axioms.  In contrast to admissibility (which I will not discuss further here) this form of entailment does not express implication!  In particular, an entailment is never vacuously true.  Entailment enjoys at least two crucial structural properties, making it a pre-order:

\displaystyle{\strut\over{P\vdash P}}

\displaystyle{{P\vdash Q\quad Q\vdash R}\over{P\vdash R}}.

In addition we often have the following additional structural properties:

\displaystyle{{P_1,\dots,P_n\vdash Q}\over{P_1,\dots,P_n,P_{n+1}\vdash Q}}

\displaystyle{{P_1,\dots,P_i,P_{i+1},\dots,P_n\vdash Q}\over{P_1,\dots,P_{i+1},P_{i},\dots,P_n\vdash Q}}

\displaystyle{{P_1,\dots,P_i,P_i,\dots,P_n\vdash Q}\over{P_1,\dots,P_i,\dots,P_n\vdash Q}}.

These state that “extra” axioms do not affect deduction; the “order” of axioms does not matter; “duplication” of axioms does not matter.  (These may seem inevitable, but in substructural logics any or all of these may be denied.)

In languages we have the fundamental concept of a typing judgement, written x_1{:}A_1,\dots,x_n{:} A_n\vdash M{:}A, stating that M is an expression of type $A$ involving variables x_i of type $A_i$.  A typing judgement must satisfy the following basic structural properties:

\displaystyle{\strut\over{x:A\vdash x:A}}

\displaystyle{{y:B\vdash N:C \quad x:A\vdash M:B}\over{x:A\vdash [M/y]N:C}}

We may think of the variables as names for “libraries”, in which case the first states that we may use any library we wish, and the second states closure under “linking” (as in the Unix tool ld or its relatives), with [M/x]N being the result of linking x in N to the library M.  Typically we expect analogues of the “extra”, “reordering”, and “duplication” axioms to hold as well, though this ain’t necessarily so.  I will leave their formulation as an exercise for the reader.

In categories we have the fundamental concept of a mapping f:X\longrightarrow Y between structures X and Y.  The most elementary structures, perhaps, are sets, and mappings are functions, but it is more common to consider, say, that X and Y are topological spaces, and $f$ is a continuous function between them.  Mappings satisfy analogous structural properties:

\displaystyle{\strut\over{\textit{id}_X : X \longrightarrow X}}

\displaystyle{{f:X\longrightarrow Y \quad g : Y\longrightarrow Z}\over{g\circ f:X\longrightarrow Z}}

These express, respectively, the existence of the identity map, and the closure of maps under composition.  They correspond to reflexivity and transitivity of entailment, and to the library and linking rule of languages.  As with types, one may expect additional closure conditions corresponding to the “extra”, “reordering”, and “duplication” axioms by giving suitable meaning to multiple assumptions.  I will not go into this here, but numerous standard sources treat these conditions in detail.

What I find captivating about computational trinitarianism is that it is beautiful!  Imagine a world in which logic, programming, and mathematics are unified, in which every proof corresponds to a program, every program to a mapping, every mapping to a proof!  Imagine a world in which the code is the math, in which there is no separation between the reasoning and the execution, no difference between the language of mathematics and the language of computing.  Trinitarianism is the central organizing principle of a theory of computation that integrates, unifies, and enriches the language of logic, programming, and mathematics.  It provides a framework for discovery, as well as analysis, of computational phenomena.  An innovation in one aspect must have implications for the other; a good idea is a good idea, in whatever form it may arise.  If an idea does not make good sense logically, categorially, and typically (sorry for the neologism), then it cannot be a manifestation of the divine.