Notation for Conceptual Blending

Formal Notation for Conceptual Blending

This note concerns formal notation for conceptual spaces, conceptual morphisms, and conceptual blending, in the sense developed in An Introduction to Algebraic Semiotics, with Applications to User Interface Design (hereafter, we may refer to this as "the paper"). But instead of the notation developed there, we will use the notation of the OBJ algebraic specification language; this notation is actually executable, though we do not use that capability except for checking the syntax of our theories and morphisms, i.e., we use the OBJ3 implementation as a type checker for algebraic semiotics. For much more detail on OBJ3, see its manual, which is called Introducing OBJ. If you want to try OBJ3 yourself, it is available for UCSD Unix machines at /net/cat/disk1/goguen/obj; alternatively, you could try a newer version of OBJ, called BOBJ, which is available on UCSD Unix machines at /net/cat/disk1/bobj/bobj; the code given here runs under both these versions of OBJ.

Readers of this document should refer to the paper, especially the large diagram on page 20, and they will also need some of the definitions given in the paper, especially those on pages 18 to 22. We will refer to a diagram having the shape of the one on page 19 as a diamond diagram, and in such a diagram, refer to the composition of the two morphisms on its left as its left morphism, to the composition of the two morphism on its right as its right morphism, to the middle upward morphism as its center morphism, to the triangle on its left as its left triangle, and to the triangle on its right as its right triangle. The extent to which these two triangles commute will be an important part of our analysis.

Before we go further, please note that the names used in our formal descriptions are arbitrary, in the sense that changing them does not in any way change the formal meaning of the descriptions. This is because these descriptions only concern formal structure, and do not even try to capture meaning in any deep human sense. However, we do try to choose names that will help the reader's intuition.

It always helps to consider a concrete example, so let's look at how to blend the two words "boat" and "house", as in the paper. We first specify the builtin data types that are used in the example. These are the Booleans (which are imported via the phrase pr BOOL) and a so-called "enumerated type" for what we shall call "media," which in this example can be either land or water:

obj DATA is pr BOOL . sort Medium . ops land water : -> Medium . endo An important convention operating here is that constants are considered operations with no arguments. The keyword obj indicates that a theory defines data, i.e., things with a fixed interpretation, whereas the theories given below use the keyword th, which indicates a "loose" interpretation for its sorts and its operations.

Next, we give theories for the three given conceptual spaces of our example, beginning with the generic space; all of these import the DATA theory, as indicated by "pr DATA". Note that this generic space not only has sorts for the two kinds of thing in the example other than media, but it also has a "generic element" for each sort, where the generic element has the same name as its sort, but without the initial capitalization. (Of course, this is an arbitrary convention, chosen to be helpful to human readers, as are the names chosen for the sorts, elements, relations, and the space itself.)

th GENERIC is sorts Object Person . pr DATA . op object : -> Object . op person : -> Person . op medium : -> Medium . op use : Person Object -> Bool . op on : Object Medium -> Bool . eq use(person, object) = true . eq on(object, medium) = true . endth Other important conventions operating here are that relations are represented by Boolean valued functions, and that a relation holds of two arguments if the value of the function is true for those arguments; this can be expressed by equations, as in the above OBJ code. A different kind of convention relates OBJ notation to algebraic semiotics by assuming that the "top" sort of a theory is the first sort mentioned in it; however, OBJ notation is incapable of expressing which sorts, if any, are secondary, tertiary, etc. It is also incapable of expressing priorities on constructors, but that is less relevant here because there are no constructors other than constants. Next we give theories for the two input spaces: th HOUSE is sorts Object Person . pr DATA . op house : -> Object . op resident : -> Person . op live-in : Person Object -> Bool . op on : Object Medium -> Bool . eq live-in(resident, house) = true . eq on(house, land) = true . endth th BOAT is sorts Object Person . pr DATA . op boat : -> Object . op psngr : -> Person . op ride : Person Object -> Bool . op on : Object Medium -> Bool . eq ride(psngr, boat) = true . eq on(boat, water) = true . endth For historical reasons, what are called "morphisms" in algebraic semiotics are called "views" in OBJ. Here are the two morphisms that are given as part of this setup for blending, again using OBJ notation: view M1 from GENERIC to HOUSE is op object to house . op person to resident . op use to live-in . op medium to land . endv view M2 from GENERIC to BOAT is op object to boat . op person to psngr . op use to ride . op medium to water . endv Another OBJ convention is that if a morphism preserves something with exactly the same name, then that need not be explicitly mentioned. In particular, everything in DATA is automatically preseved, because it is a shared subtheory. For example, each of the above morphisms is assumed by this convention to map the sort Object in the GENERIC theory to the sort Object in its target theory. The same holds for the operation on, which is not mentioned in either M1 or M2. This convention makes it tricky to represent cases where there is some thing with the same name in each the source and the target space of a morphism, but we do not want this thing to be preserved. However, we can get around this by defining a subtheory of the source theory that does not include the unpreserved material, and then defining the view from that subtheory. In fact, the generic element medium cannot be preserved by the central morphism of any blend, because of the inconsistency of the way it is mapped by M1 and M2; for this reason, we will actually use the following subtheory of the generic space given above: th GENERIC is sorts Object Person . pr DATA . op object : -> Object . op person : -> Person . op use : Person Object -> Bool . op on : Object Medium -> Bool . eq use(person, object) = true . endth There is an obvious inclusion view from this to the previous generic space, i.e., the new generic space is a subtheory of the old one. Because of another OBJ convention (which of course is also part of the implementation), the name "GENERIC" will from now on refer to the new space, rather than to the earlier one of the same name. We also need new morphisms M1 and M2 from the new generic space to HOUSE and BOAT, respectively; these are the same as the old ones, except that they omit the generic element medium. view M1 from GENERIC to HOUSE is op object to house . op person to resident . op use to live-in . endv view M2 from GENERIC to BOAT is op object to boat . op person to psngr . op use to ride . endv

We now give the theory and three morphisms (three because the center morphism must be included, since we consider the generic theory to be auxiliary) for our first blend of our given two spaces and three morphisms:

th HOUSEBOAT is sorts Object Person . pr DATA . op houseboat : -> Object . op resident : -> Person . op live&ride : Person Object -> Bool . op on : Object Medium -> Bool . eq live&ride(resident, houseboat) = true . eq on(houseboat, water) = true . endth view M3 from HOUSE to HOUSEBOAT is op house to houseboat . op live-in to live&ride . endv view M4 from BOAT to HOUSEBOAT is op boat to houseboat . op psngr to resident . op ride to live&ride . endv view C from GENERIC to HOUSEBOAT is op object to houseboat . op person to resident . op use to live&ride . endv A careful discussion of blends needs to distinguish between a space and three morphisms for which the diamond diagram weakly commutes, and a space and three morphisms which really constitute a blend. Therefore let us call the former case a blendoid. For example, in almost no cases do we want to consider the completely empty conceptual space to be a blend; in this case, everything commutes but nothing is preserved. Using this concept, we can define a blend to be a maximal blendoid, not just as a theory, but in terms of the commutativity and preservation properties of the theory and three morphisms together. (A precise, but difficult, mathematical definition is given in Appendix B of the paper.)

In the above HOUSEBOAT blend, the two triangles commute for all three sorts in the generic space (noting that the data sort Medium is required to be preserved in any case), and similarly for the two generic constants object and person (but not for medium, which is why we have left it out of the second GENERIC space - if it were still there, then commutativity would be impossible - see the discussion in Section 3 of the Appendix below), because each one of them maps to the same element of the blend space via the three different paths. Similarly, each of the two relations (which appear as Boolean valued operations) in the generic space map to the same relation in the blend via the three different paths. Moreover, for each pair of elements in the generic space for which a relation holds, the corresponding elements in the blend space satisfy the corresponding relation, which means that all three paths preserve these two axioms in the same way. Thus we have commutativity in the strong sense for this blend. But note that because the GENERIC theory is auxiliary in this example, strong commutativity is not required; in fact, although strong commutativity holds for this blend with the new generic space, it does not hold with the original generic space. Hence this blend has commutativity for everything except the constant medium, but weak commutativity altogether.

Now let's consider a second blend. Here are its space and three morphisms:

th BOATHOUSE is sorts Object Person . pr DATA . subsort Object < Person . op boathouse : -> Object . op boat : -> Object . op live-in : Person Object -> Bool . op on : Object Medium -> Bool . eq live-in(boat, boathouse) = true . eq on(boathouse, land) = true . eq on(boat, water) = true . op psngr : -> Person . op ride : Person Object -> Bool . eq ride(psngr, boat) = true . endth view M3 from HOUSE to BOATHOUSE is op house to boathouse . op resident to boat . endv view M4 from BOAT to BOATHOUSE is endv view C from GENERIC to BOATHOUSE is op object to boathouse . op person to boat . op use to live-in . endv Note that in the morphism M4 above, all of its mappings are defined by default: this works because everything that appears in the BOAT theory also appears in the BOATHOUSE theory, with the same name, so that M4 is really an inclusion morphism.

In this case, even with the second (smaller) generic space, neither triangle commutes in the strong sense, because the boat ends up in the house, as reflected by the generic element object being mapped to boat in the blend by the right morphism (which is the composition of morphisms M2 and M4), but being mapped to boathouse by the left morphism (the composition of M1 with M3); as a result, the central morphism cannot preserve the element object, because there is no consistent place to map it. Similarly, since M4 preserves psngr, the central morphism cannot preserve the generic element person, and the some goes for the generic use operation. On the other hand, the generic relation on goes to the same place under all three maps, although its generic axiom is not preserved. It is interesting to notice that in order for the relation live-in to type check in the BOATHOUSE theory, the subsort declaration that appears there is needed; the same holds even more dramatically for the map of resident to boat under the morphism M3. Thus we should define a different, even more restricted subtheory of the original GENERIC space, on which everything commutes in the strong sense, and therefore weakly commutes on the older generic space. (This discussion differs from that in the paper, which gave a space that did not include all possible material from BOAT, e.g., the constant psngr and the relation ride which holds of the passenger; but otherwise this blend is the same.)

There is a third blend which is similar to (in fact, symetrical with) the above BOATHOUSE blend, in which the house ends up riding in the boat. (As noted in the paper, there are real world examples of this blend, where a boat is used to transport prefabricated houses across some body of water, e.g., for a new housing development on a nearby island; some people have told me that they have actually seen this, though I have not seen it myself.) This blend has similar (in fact, symmetrical) commutativity properties to those of the BOATHOUSE blend discussed above. It is left as an interesting exercise for the reader to write OBJ code for this blend.

There is also a fourth blend, the meaning of which is less familiar than the first three, but the preservation and commutativity properties of which are very good, corresponding to the fact that this concept is a very pure blend of its input spaces, even though its physical existence is dubious. This concept is that of an amphibious RV (recreational vehicle): a vehicle that you can live in, and that you can ride in on land and on water. Here all aspects of HOUSE and BOAT are realized, without any surprising transpositions, like a house riding a boat. It is left as an interesting exercise for the reader to write OBJ code for this blend. (The3 paper says that this blend has worse preservation and commutativity properties than the previous three, but this is an error.)

All of the OBJ code on this webpage has been successfully run in OBJ3, though it did not always run the first time, because OBJ3 caught some bugs in the initial versions; it has also been run in BOBJ, after fixing some bugs in that system. It is interesting to notice that, depending on how we choose to handle truth values in OBJ, we can implement either the assumption that relations not declared true are false by default, or the assumption that it is not known whether they are true or false. (Of course, we can in any case write an equation that explicitly declares a relation to be false on some arguments.)

It seems exciting that our intuitive sense of the relative purity of these four blends corresponds precisely to the extent to which their triangles commute, since this is the criterion given for measuring the quality of blends that is suggested in the paper.


Four `Tricks of the Trade'

In order to analyze "real" examples like those described above (though of course they are not really real, but are only idealized models of some key aspects that are especially relevant to the blends in questions), within the rigorous mathematical framework of algebraic semiotics (and OBJ), it is necessary to apply some `tricks' in writing the conceptual spaces and their morphisms. The purpose of this appendix is to explain in some detail some of the tricks that were used in the preceeding discussion.

1. Functions vs. Relations

A first challenge in writing formal definitions of conceptual spaces is how to represent relations and facts that a certain relation holds, or does not hold, on certain arguments. If one is using OBJ, as we are here, relations are not directly available, and so one must somehow represent them with functions. The most obvious way to do this is by represent a relation as a Boolean values function. It is then natural to use an equation to assert that a relation is true of certain arguments, and also to assert that it is false certain arguments; by using variables, such assertions can be made for large (even infinite) classes of arguments. Moreover, it is easy to treat unknown cases as either true of false, using OBJ's builtin function ==. In the first case, one uses tests of the form

not t == false . where t is a relation with some arguments, and in the second case, one uses tests of the form t == true . (These somewhat sophisticated representations are not needed in the examples discussed above because only the simplest reasoning is required to check that our axioms are preserved.)

2. Partial Preservation

A second challenge is to use a notation that was intended for total preservation to represent partial preservation; we showed above how to do this using an OBJ view from subtheory of the source, instead of the original theory. The idea here is that every partial morphism from the original theory corresponds to a total morphism from a subtheory of the source theory.

3. Strong and Weak Commutativity

We can apply the trick of Section 2 above to reduce any instance of weak commutativity to a corresponding instance of strong commutativity over a subtheory of the source theory, and in fact, this is what we did in the main discussion above, by replacing the generic space by a subtheory.

However, because we are trying to model "real" phenomena of blending, we should give some further justification of our choice to make M1, M2 and C partial. Let us consider the second blend. Here one can argue that C really should map the generic constant object to the main object of BOATHOUSE, which is boathouse. Actually, this argument is right for our conventional boathouse concept, and C can perfectly well do this, but its restriction to the subtheory will still be the morphism that we have given above, so there is no problem here. (But it should be noted that there is an alternative conceptualization where the focus of attention is on the boat instead of the house, so this choice is somewhat arbitrary.) Of course, the situation here is that the composition of M1 with M3 maps object to boathouse, whereas the composition of M2 with M4 maps object to boat, so that commutativity would fail for the right triangle if all the morphisms were defined on object, even though the left triangle would commute (or vice versa if we took the other choice for C).

4. Conjunctive Elements

Our final trick appears in our first blend, the HOUSEBOAT blend. Here we combined the two relations live-in and ride into a single relation, live&ride, which plays the role of the generic use relation. Although this example does not require it, it would be possible to provide a "fine structure," showing that the live&ride relation implies each of the live-in and ride relations. It would then also be possible to map live-in and ride in the input theories to the corresponding relations in the blend theory, although this would result in less preservation and commutativity. For this reason, the way we have set things up seems preferable.


I thank Lisa Tolentino, with whom discussing these ideas has helped me to clarify some of them (though they may still not be in their final form). Thanks also to Kai Lin for fixing BOBJ so that it could run this code.


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Last modified: Sun Jul 22 13:56:30 PDT 2001