Dayanand S. Rajan

Lattice computers for approximating Euclidean space

In the context of mesh-like, parallel processing computers for (i) approximating continuous space and (ii) <italic>analog</italic> simulation of the motion of objects and waves in continuous space, the present paper is concerned with <italic>which</italic> mesh-like interconnection of processors might be particularly suitable for the task and why. Processor interconnection schemes based on nearest neighbor connections in geometric lattices are presented along with motivation. Then two major threads are exploded regarding which lattices would be good: the <italic>regular lattices</italic>, for their symmetry and other properties in common with continuous space, and the well-known <italic>root lattices</italic>, for being, in a sense, the lattices required for physically natural basic algorithms for motion. The main theorem of the present paper implies that <italic>the</italic >well-known lattice A<italic><subscrpt>n</subscrpt></italic> is the regular lattice having the maximum number of nearest neighbors among the <italic>n</italic>-dimensional regular lattices. It is noted that the only <italic>n</italic>-dimensional lattices that are both regular and root are A<italic><subscrpt>n</subscrpt></italic> and Z<italic><supscrpt>n</supscrpt></italic> (Z<italic><supscrpt>n</supscrpt></italic> is the lattice of <italic>n</italic>-cubes. The remainder of the paper specifies other desirable properties of A<italic><subscrpt>n</subscprt</italic> including other ways it is superior to Z<italic><supscrpt>n</supscrpt></italic> for our purposes.

A characterization of root lattices

Abstract In this note we show that root lattices are all and only those lattices in which the set of minimal length vectors equals the set of relevant vectors.

The adjoints to the derivative functor on species

Abstract As a direct consequence of the Kan Extension Theorem, the derivative functor, D, on (combinatorial) species, has both a left adjoint, MX, and a right adjoint, II. The functor MX can be described as “tensoring by X,” whereas the functor II is new. We show that there is an injective homomorphism, Card, from the rig (informally, a “ring without negatives”) of isomorphism classes of finitary species Speciesfx into the rig of functions NMol under pointwise addition and multiplication, which “preserves” D and its adjoints. This, in effect, allows one to translate some of the calculations involving these functors to solving linear equations over the natural numbers.

Representing the Spatial/Kinematic Domain and Lattice Computers

An approach to analogical representation for objects and their motions in space is proposed.

Spatial/kinematic domain and lattice computers

Abstract An approach to analogical representation for objects and their motions in space is proposed. This approach involves lattice computer architectures and associated algorithms and is shown to be abstracted from the behaviour of human beings mentally solving spatial/kinematic puzzles. There is also discussion of where in this approach the modelling of human cognition leaves off and the engineering begins. The possible relevance of the approach to a number of issues in artificial intelligence is discussed. These issues include efficiency of sentential versus analogical representations, common sense reasoning, update propagation, learning performance tasks, diagrammatic representations, spatial reasoning, metaphor, human categorization, and pattern recognition. Lastly there is a discussion of the somewhat related approach involving cellular automata applied to computational physics.