So far, we have shown how the machinery of situation theory models the fact that one situation can carry information about another. It is easy to find examples of
this flow beyond those already discussed. The manner in which thediaphragm of a microphone is vibrating carries informationabout what the announcer is saying. The modulation of the electromagnetic signal arriving at some antenna carries information about the way in which that diaphragm isvibrating. And finally, the modulation of the electromagneticsignal arriving at the antenna carries information about whatthe announcer is saying, for instance, ``Hillary Clinton is irritated''. Situation theory provides tools for answering to the question: how does the modulation of the electromagnetic signal at the antennacarry information about the words that the announcer spoke? The theory relies on some fundamental principles about the nature of information as it flows across distributed systems, that is, systems that can be analyzed in sub-parts:
(1) The availability of information depends on the existence of regularities between connected parts of a distributed system.
(2) These regularities are relative to how the system is analyzed into parts.
(3) Flow of information crucially involves types and their concrete instances. It is in virtue of constraints (relations on types) that concrete situations, being of certain type, can act as carriers of (concrete or general) information.
We will now present a second, more mathematical way of formalizing the idea of one situation carrying information about another. This is done via the theory of
classificationsand channels (Barwise & Seligman 1997). This paradigm, based on category theory, is an elegant formal distillation of the ideas presented so far. 5.4 Classifications and channel theory
The notions of information flow presented in the previous section involve facts of the form s |= T, where s is a situation and T a type. These facts are about how situations are classified. The notion of classification, independently discovered around 1990 as ‘Chu Spaces’ (Gupta 1994) and ‘Formal Contexts’ (Ganter & Wille 1997), is the basic notion on which the theory of classifications and channels is built. Classifications are triples, often depicted as follows:
SA
|= A
WA
Here WA is a set of tokens (for example, situations), SA is a set of types (conceived of as anything that can classify tokens), and |= is a relation between tokens and types. If s is a token and T a type, then s |= A T reads as 's is of type T '.
The natural 'part-of' relationships that exist between parts of a system are called
infomorphisms. An infomorphism f: AÆ C between two classifications is a pair
< fŸ, f⁄ > of functions
SA SC
|= A |= C
WA WC
such that for all tokens b Œ WC and all types T Œ SA (*) f ⁄(c) |=
AT if and only if c |=C f Ÿ(T )
Infomorphisms are of independent interest as an abstract invariance behind translation between theories, and that of general category-theoretic adjunctions. But here we look at their concrete uses as an intuitive model for information flow. A concrete scenario The Smoke-on-the-Mountain scenario of Indian logic, mentioned in Section 1 of this chapter, involves at least two classifications A and C. Tokens of A, WA might be situations where somebody is facing a mountain, while the relevant typesSA might include SEESSMOKE, LOOKINGUP, LOOKINGDOWN,
BLIND etc. On the other hand, the classification C might correspond to the overall setting including the observer and the mountain. Its tokens are situations extending those of WA, and types in SC might include OBSERVERSEESSMOKE,
THEREISAFIREONTOP, etc. The map f⁄ maps each large situation to the sub-situation
capturing just the point of view of the observer. The map fŸsends S
MOKEOBSERVED to OBSERVERSEESSMOKE, LOOKINGDOWN to OBSERVERLOOKINGDOWN, etc. Thus type
T of A is mapped to a type of C intended to mean that 'the observing situation' is of type T. Condition (*) ensures that things work out just this way.
As before, it is the existence of constraints, in the form of regularities between types that makes information flow within a distributed system. In this more general abstract setting, a constraint S1 |- S2 of classification A consists of two sets of types such that for all a in WA, if a |=AŸS1, then a |=A ⁄S2. If A were the classification of observers facing a mountain, SEESSMOKE, BLIND |- ∅ would be a constraint of A, saying that no blind observer sees smoke on top of the mountain. Channels and information flow Let us now add to our observer and his mountain a third classification B for what is happening at the mountain top. Its tokens are situations located on mountain tops, and its types include things such
f Ÿ
as THEREISFIRE, THEREISFOG, etc. B is also a 'part' of the big component C – say, via the infomorphism g depicted here:
C
A B
A collection of infomorphisms sharing codomain C is called a channel with core C. Tokens of the core are called connections, because they connect subparts into a whole. Tokens a from WA and b from WB are connected in channel C if there is a
token cŒWC such that f
⁄(c) = a and g⁄(c) = b. In the example, an observing
situation and a mountain top are connected if they belong to the same overall situation. We can now formulate a notion of (incremental!) information flow:
a |=AT carries the information thatb |=BT ' (relative to C) if a, b are connected in C and fŸ (T) |- gŸ (T') is a constraint of C.
This notion of information flow is relative to a channel –and hence, to an analysis of a whole distributed system into its parts. Again we see that 'carrying information' is not an absolute property: the mere fact that a token or situation is of a certain type does not completely determine what information it carries. 26 Here is our observer-mountain example in these terms. Let sŒA be a situation of type SEESSMOKE: the observer in it sees smoke on top of the mountain. Let s'Œ B be the top-of-the mountain observed from the base of the mountain. Our choice of C makes s and s' be linked by some connection in C. In addition,
fŸ (S
EESSMOKE) = OBSERVERSEESSMOKE
gŸ(T