Representing Discrete Structures in a Hopfield-Style Network
We have developed a variant (essentially a special case) of the discrete Hopfield network, which we call Hopfield-Style Network (HSN). The stable states of HSN are the maximal cliques of an underlying graph. We exploit this graph-theoretic characterization to represent — as associative memories — several discrete structures in HSN. All representable structures are stored in HSN via its associative memory storage rule. We describe representations of sets (with PDP schemata Uas example), relations (with PDP "Jets and Sharks" as example), multi-relations (with word-dictionaries as example), graphs (with PDP schemata and binary relations as examples), Boolean formulae, and *-free regular expressions (with restaurant script as example). We also discuss robustness of these representations. Our main result is that several different kinds of discrete structures are representable — in distributed fashion — in HSN — a simple Hopfield-type energy-minimizing (constraint-satisfaction) parallel-distributed network. For knowledge representation and retrieval, we have extended the scope of representations possible in Hopfield networks, while retaining (and improving upon) the good features of such networks: (1) spontaneous constraint-satisfaction and (2) retrieval of stored schemata from noisy and incomplete information.