The Spatial Transformation

Chapter 4. The Spatial Transformation

GTR1: from the map to movement and time

[ INSERT FIGURE 4: the Gativus architecture (GTR0 → SLGW → GTR1) ]

The Gativus architecture: from SEED through GTR0 and the SLGW gateway to the informational cycle GTR1. This chapter unfolds the orange group — MP10–MP13 and the three transformations over them.

4. 1. From the Map to Behavior

The MP00 map closed the previous chapter: the habitat space was formulated as an informational entity. Into the resulting coordinate system, parameters affecting movement were introduced (solid separated from hollow, the memory of where obstacles lie). The organism orients itself in space — it knows where there is a wall, where a passage, where it itself is. But it does nothing yet with this knowledge. There is a map, but no behavior yet.

Behavior is not a reflex. The reflex was already there in the first chapter: a stimulus at the input, a movement at the output, with no map between them. Behavior is arranged differently. The organism knows where it is; determines where the goal is; lays a path from the first to the second and passes this path on for execution. Between perception and movement there now stands a constructed route.

This is precisely the content of GTR1 — the second full cycle of Gativus transformations, now not material but informational. If GTR0 built the body, GTR1 builds behavior in the habitat space: the path from “where I am” to “where the goal is.”

GTR1 is arranged isomorphically to GTR0: four spaces and three transformations between them — the same triad of convolution / splice / chain. But the material is different. GTR0 worked with substance — cells, tissues, organs. GTR1 works with information — maps, objects, movements. In Fig. 4 this is the orange group, built over the gray GTR0 through the SLGW gateway.

4. 2. Multimaps

Before examining the four spaces separately, a property running through all of them must be introduced. Each GTR1 space is not one map but many. In the MP1x domain there coexist many elementary maps and one unified map composed of them. The unification is performed by gateways — common elements visible at once in two adjacent maps.

The reason for this arrangement is not speculative but engineering-physiological. A map of fixed size can be laid down at the embryonic level — its structure is predetermined. A map of variable size, growing as the world is mastered, is an incomparably more complex task. Nature solves it by a roundabout path: it builds many small maps of fixed size and stitches them together with gateways into a single whole. The capacity for growth is achieved not by stretching one map but by adding new elementary maps to the unified one.

Therefore, whenever one says “the MP10 map” or “the MP11 map,” one must keep the duality in mind: there is the elementary map — the atom of the given level — and there is the unified map that all of them compose. Below, for each level, it is indicated what its elementary map is.

4. 3. MP10 — Space

The first space of GTR1 is the three-dimensional environment inherited from SLGW.

From the GTR0 side it was called MP00; included in the informational cycle, the same map is read as MP10. These are not two maps but one organ at the boundary of two levels, read from two sides — in Fig. 4 the double name MP00 / MP10 stands on the adjacent faces of the SLGW block.

The elementary MP10 map is a page of space: a three-dimensional fragment of the inhabited environment of fixed size. The pages are stitched together by spatial components — cells that belong to two pages at once and serve as the point of transition between them. The unified MP10 map is all the mastered space, assembled from pages through these transitions.

MP10 carries the habitat space and the organism’s position within it — the coordinate system in which every further movement is specified. This is “where I am,” the beginning of the second contour of life; everything subsequent is built upon it.

4. 4. MP11 — Recognition

MP11 solves the task of identification: extracting invariants from the sensory stream and identifying them with known ones. The voxel occupancy in MP10 changes with every angle, distance, and position of the observer; a single object generates a continuum of projections. Recognition convolves this continuum down to an invariant.

The instrument is a convolutional neural network (CNN). A fragment of MP10 is mapped into a feature vector in the network’s own space. Identification reduces to a nearest-neighbor search: each previously defined object is a node with a stored vector, produced by the same CNN when the object was created. A match identifies the observation with a known node; the absence of a close value can be used to create a new one. The network and the database are inseparable — vectors are interpretable only in the feature space that produced them, and this pairing is MP11.

The elementary MP11 map is one convolution together with the objects it generates. There are many convolutions, on the order of thousands, and as many elementary maps; the unified MP11 is all of them together. Each node (object) of MP11 has a position on the MP10 map: the objects are seated on space. Thereby MP10 acquires the property of an object map — its elements become not single voxels but groups generated by objects. The map of passability becomes a map of things.

The transition from MP10 to MP11 is convolution. As in GTR0 convolution changed the very nature of the object (a static description turned into a living cell), here convolution turns a continuum of projections into an invariant, the nameless voxel occupancy into a recognized thing.

Neurobiologically — the ventral stream in conjunction with the hippocampal system: invariant categorization plus the spatial anchoring of what has been identified.

4. 5. MP12 — Movement

A change in an object’s position on the map is a vector. Such a vector may be the movement required to reach food or to solve other tasks of the organism. For a movement to take place, three things must be determined: who moves, what moves, and where. A triplet arises:

Subject — Movement — Object.

If the movement concerns only the organism itself, the object is its own body. In more developed animals the object may differ from the subject: the subject moves not itself but something external — prey, a branch, a tool.

The formation of the triplet Gativus regards as a splice — the stitching of subject and object around the vector of movement (the b-vector). The nature of the elements does not change in this: both subject and object remain nodes on the map. Something else changes — between them a b-vector is stretched, turning two positions into one action. The vector itself may be computed by various algorithms, giving the organism one or another advantage.

The elementary MP12 map is one splice: a b-vector with subjects and objects stitched around it. The unified MP12 is all the organism’s splices. This is the space of possible actions.

It is precisely here, in the triplet, that the subject appears for the first time. The previous chapters managed without it: GTR0 builds the organism without needing any “I”; SLGW builds the map without needing an observer. The subject arises only together with movement, as one of its supports: a movement is never no one’s — it always has someone who performs it. The subject is not an additional entity on top of the system but a necessary pole of the triplet.

4. 6. MP13 — Events and Time

One splice is an elementary action. Real behavior rarely reduces to a single action: to reach food means to skirt an obstacle, approach, reach out, grasp. MP13 is the space where coherent schemas are assembled from separate splices.

The objects of the MP13 map are the splices themselves, created in MP12, plus a definite technique for connecting them. The elementary MP13 map is an event schema (b-local map): a connected graph of splices. Importantly, its topology is a network, not a tree: an action may have several predecessors and several continuations, branches converge and diverge. The unified MP13 is all the organism’s event schemas. The nearest human analogue of such a schema is a business process: a network of operations with branchings, mergings, and parallel branches.

This is a chain — the third transition in GTR1. As in GTR0 the chain built a sequence over ready-made organs, here the chain builds a schema over ready-made splices. Actions are linked so that the result of one becomes the starting position for the next. Even when the subject is one and the same, the schema does not reduce to repetition: each splice changes the object of movement — first the object was one’s own body (approach), then an external thing (grasp), then the body together with the thing (carry off).

And here something is revealed that was present at none of the previous levels. When splices are arranged into a schema, an order arises between them: one earlier, another later. This order is time.

Time in the Gativus architecture is not a separate entity and not an axis added to space from outside. Time is the sequence generated by the change of an object’s position. When an object changes position, and then changes again, a “before” and an “after” arise — an orderedness of states. It is not that movement takes place in time, but that movement generates time as its own sequence. Time is a product of MP13.

This overturns the customary view. Usually time is conceived as an empty stage on which events unfold. Gativus asserts the opposite: no events — no time; the stage is woven from the events themselves and their order. The MP13 event schema is not placed in time — it creates it.

4. 7. The Closed Cycle and Learning

As GTR0 was closed by the return from Organism to SEED, GTR1 too is closed. The reverse pass is the return of the performed movement into the map: the result of an action changes how objects are recognized, and the refined recognition changes how space is read.

This is not an abstract symmetry. An organism whose movement has run into a new obstacle corrects the map: a passage that was considered open is marked as closed. An organism that has burned itself on an object recognizes it differently — and the event schemas around it are rebuilt. The performed action returns into the map. The cycle is closed.

And just as in GTR0 the inexactness of the return generated evolution, in GTR1 the inexactness of the return generates learning. New experience does not overwrite the map from scratch — it refines, extends, sometimes overturns it. Learning in the Gativus architecture is not added to perception from outside as a separate mechanism: it is the built-in inexactness of the closing of GTR1.

4. 8. GTR1 as Foundation

GTR0 gave life a body. GTR1 gives it movement in space and time: a learned map, recognized objects, action as a splice, an event schema as a graph of splices, and time itself as the order of those events. The organism has ceased to be a bundle of reflexes — it acts in a world it knows.

Modern robotics reproduces exactly this scheme: autonomous machines and humanoid robots build a map of space, localize themselves on it, recognize objects, plan movement, and execute it.

This is an engineering confirmation of GTR1 — the same cycle, assembled from silicon instead of nervous tissue.

But this level too has a boundary. Everything GTR1 operates with is given in perception. The object must be on the map, the goal — in the field of view, the action — directed at what is present. The subject of GTR1 cannot deal with what is not now before it: with the absent, the hidden, the not-yet-arrived. It is bound to the immediately perceived.

The next transformation will step across exactly this boundary. A sign will appear — something that substitutes for an absent object and allows one to deal with it in its absence. With this begins the symbolic level, GTR2, and with it — the subject detached from what is directly before the eyes.