GN

Structural Rules

Exchange

$$\begin \frac' \vdash D}' \vdash D} \text \end$$

(A.1)

Weakening

$$\begin \frac \text \end$$

(A.2)

Contraction

$$\begin \frac\text \end$$

(A.3)

The Identity Group

Axiom

$$\begin A\vdash A\quad \text \end$$

(A.4)

Cut

$$\begin \frac \textsc \end$$

(A.5)

Multiplicative Logical Rules

Conjunctive (Multiplicative) Connective

$$\begin \frac\text \otimes \end$$

(A.6)

$$\begin \frac\text\otimes \end$$

(A.7)

Of Course operator

!

$$\begin \frac\text \end$$

(A.8)

$$\begin \frac\text \end$$

(A.9)

Capital Greeks stand for finite sequences of formulas including possibly the empty one, and D stands for either a single formula or no formula, i.e. the empty sequence, and when it appears in the form \(\otimes D\), the \(\otimes\) symbol is presumed to be absent when D is empty. If \(\Gamma\) denotes the sequence \(A_1,A_2, \ldots , A_n\) then \(!\Gamma\) will denote the sequence \(!A_1,!A_2, \ldots , !A_n\). The Girard of course exponential operator ! is sometimes pronounced “bang.” The sign \(\otimes\) should strictly speaking just be regarded as an abstract symbol though there is no harm and large benefit if the reader just thinks of this and all the other symbols involved as pertaining directly to the category of finite dimensional real Hilbert spaces. If this is done, then commas are replaced by \(\otimes\), \(A \vdash B\) is replaced by a linear map \(A \rightarrow B\), !A is replaced by the exterior algebra over the Hilbert space A, and blank spaces by \(\mathbb \). In applications, one adds “non-logical” axioms to the GN rules, for instance to depict a brain as a family of linked networks.

An associative algebra B over the ring R may be defined as follows. B is a module over R with a map \(m: B \otimes B \rightarrow B\) called multiplication (or sometimes abusively product). With \(m(a\otimes b):= ab\) associativity means \(a(bc) = (ab)c\). This condition may be expressed in the form of a commutative diagram specifying that \(m(1_B \otimes m) = m (m\otimes 1_B)\). One virtue of expressing this condition diagramatically is that it may easily be dualized by reversing the arrows.

A coassociative colagebra A over the ring R is a module over R with a map \(\psi : A \rightarrow A \otimes A\) called comultiplication (or sometimes abusively coproduct) with a dual version of the diagram above commuting, namely \((\psi \otimes 1_A) \psi = (1_A \otimes \psi ) \psi\).

Let A be such a coalgebra and B such an algebra. Given two R-module maps \(L, M: A \rightarrow B\) we get another module map \(L*M:A \rightarrow B\) as follows.

$$\begin L*M := m (L \otimes M) \psi . \end$$

(B.1)

This *-operation is called convolution. It is easily seen to be an associative product, due to the associativity of B and the coassociativity of A.

The simplest way to see that ordinary convolution may be expressed in this form is to consider the group algebra R[G] of a finite group G written multiplicatively. This is just the space of functions on G into R, and here we ignore the algebra product on it. Each such function may be expressed in the form

$$\begin f = \sum _ a_g \chi _g \end$$

(B.2)

where \(a_g \in R\) and \(\chi _g\) is the characteristic function on G which is 1 on g and zero elsewhere. We have a coassociative comultiplication on R[G] induced from the associative group multiplication \(G \times G \rightarrow G\), namely the map \(\psi : R[G] \rightarrow R[G \times G] \cong R[G] \otimes R[G]\) given for \(f \in R[G]\) by \(\psi (f)(g_1,g_2) = f(g_1g_2)\).

We can express this precisely in tensorial form for \(f = \chi _\) as follows.

$$\begin \psi (\chi _)(g_1,g_2) = \chi _(g_1g_2) \end$$

(B.3)

$$\begin = 1&\text \quad g_0 =g_1g_2\\ 0&\text \end\right. } \end$$

(B.4)

$$\begin = 1&\text \quad g_0g_2^ =g_1\\ 0&\text \end\right. } \end$$

(B.5)

$$\begin = (\sum _g \chi _} \otimes \chi _g)(g_1,g_2). \end$$

(B.6)

Now take B to be R and \(m: R\otimes R\rightarrow R\) to be multiplication in R and note that any module map \(L: R[G] \rightarrow R\) acting on some \(f = \sum _ a_g \chi _g\) takes the form \(L(f) = \sum _ a_g L(\chi _g)\). Of course, any such L may be regarded as a function on G via the one-to-one map \(g \leftrightarrow \chi _g\) so that we may write \(L(g):= L(\chi _g)\) and concomitantly \(L(f) = \sum _g a_g L(g)\). Then we have

$$\begin \begin (L*M)(f)&= (m(L\otimes M) \psi )(\sum _h a_h \chi _h) \\&= m(L\otimes M)( \sum _h a_h (\sum _g \chi _} \otimes \chi _g)) \end \end$$

(B.7)

$$\begin \qquad \qquad \text \; (\text .6) \end$$

(B.8)

$$\begin =\sum _h a_h (\sum _g L(hg^)M(g)) \end$$

(B.9)

so that the function of \(h\in G\) corresponding to the functional \(L*M\) is

$$\begin (L*M)(h) =(L*M) (\chi _h) = \sum _g L(hg^)M(g) \end$$

(B.10)

This is just the usual convolution of the functions L and M. This construction has generalizations to measures and functions on locally compact groups and in other areas of harmonic analysis.