gB:Infomorphism

Collection	GlossariumBITri
Author	Francisco Salto Alemany
Editor	Francisco Salto Alemany
Year	2010
Volume	1
Number	1
ID	50
Object type	Concept
Domain	Logics Semantics Situation Theory Transdisciplinary
es	Infomorfismo
fr	Infomosphisme
de	Infomorphismus

The mathematical concept of morphism tries to produce an image of a set that captures its structure. The notion of infomorphism generalizes and extends this idea by means of defining certain homomorphism among structures supporting infons. The concept emerged originally in situation semantics and it has been applied in distinct contexts.

Any set $A$ includes all elements or tokens defining a family $R$ of relations on $A$. Let us call relational structure ${\boldsymbol {A}}$ the set $A$ with these relations. Let ${\boldsymbol {A}}$, ${\boldsymbol {B}}$ respectively be relational structures $\langle A,R\rangle$ y $\langle B,S\rangle$. Taken with benevolence, an homomorphism from ${\boldsymbol {A}}$ to ${\boldsymbol {B}}$ is defined as any function f from ${\boldsymbol {A}}$ into ${\boldsymbol {B}}$ such that: If $R(a_{1},...a_{n})$, then $S(f(a_{1},...f(a_{n}))$. ${\boldsymbol {B}}$ is then a homomorphic image of ${\boldsymbol {A}}$.

Consider now the specific relational structure which we may call classificatory relational structure ${\boldsymbol {A}}$, taken as the result of classifying the elements or tokens of $A$ by means of a set $\Sigma _{A}$ of types. For example, the set of tokens: $\{a,a,a\}$ corresponds to a unique type $a$. We write

$y\vDash _{A}x$(1)

true

to say that the token x instantiates the type y. Barwise and Seligman (1997) called classifications such classificatory structures

${\boldsymbol {A}}=\langle A,\Sigma _{A},\vDash _{A}\rangle$(2)

true

where $A$ is the grounding token set, $\Sigma _{A}$ the set of individual types and $\vDash _{A}$ the relation of being an instance of.

Let ${\boldsymbol {A}}$ and ${\boldsymbol {B}}$ both be classificatory structures:

An infomorphism i relating ${\boldsymbol {A}}$ and ${\boldsymbol {B}}$ consists in a pair of functions $f^{+}$ (from $\Sigma _{A}$ to $\Sigma _{B}$ ) and f^- (from $B$ to $A$) such that, for every type $\alpha$ of ${\boldsymbol {A}}$ and every token $\beta$ of ${\boldsymbol {B}}$: $f^{+}(b)\vDash _{A}\alpha \Leftrightarrow \beta \vDash _{B}f^{-}(\alpha )$.

Schematically:

${\begin{array}{ccc}\Sigma _{A}&\xrightarrow {} &\Sigma _{B}\\\downarrow &&\downarrow \\{\boldsymbol {A}}&\xleftarrow {} &{\boldsymbol {B}}\end{array}}$(3)

true

As an homomorphism preserves structure, so an infomorphism preserves the instantiation relation, among sets that can be quite distinct, but informationally analogous.

In the references (Devlin, Gunji) you may find relevant examples of infomorphisms.

References

• BARWISE, J. & SELIGMAN, J. (1997). Information Flow. The Logic of Distributed Systems. Cambridge: C.U.P.
• BREMER, M. & COHNITZ, D. (2004). Information and Information Flow. Frankfurt: Ontos Verlag.
• DEVLIN, K. (2001). The Mathematics of Information. Lecture 4: Introduction to Channel Theory. ESSLLI 2001, Helsinki, Finland
• GUNJI, Y.P., TAKAHASHI, T. & AONO, M. (2004) “Dynamical infomorphism: form of endo-perspective”. Chaos, Solitrons & Fractals, 22, 1077-1101.