CDMTCS Research Report Series Logical Equivalence Between Generalized Urn Models and Finite Automata

To every generalized urn model there exists a finite (Mealy) automaton with identical propositional calculus. The converse is true as well. Introduction of concepts In what follows we shall explicitly and constructively demonstrate the equivalence of the empirical logics (i.e., the propositional calculi) associated with the generalized urn models (GUM) suggested by Ron Wright [Wright(1978), Wright(1990)], and automaton partition logics (APL) [Svozil(1993), Schaller and Svozil(1996), Dvurečenskij et al.(1995)Dvurečenskij, Pulmannová, and Svozil, Calude et al.(1997)Calude, Calude, Svozil, and Yu, Svozil(1998)]. This result has been already mentioned in [Svozil(1998), p.145]. The restriction to Mealy automata is for convenience only. The considerations are robust with respect to variations of different types of finite input/output automata. Although the automaton models are dynamic in nature, their logics are equivalent to the static generalized urn models. This equivalence suggests that these logics are more general and “robust” with respect to changes of the particular model than can be expected from the particular instances of their first appearance. Generalized urn models A generalized urn model U = 〈U,C,L,Λ〉 is characterized as follows. Consider an ensemble of balls with black background color. Printed on these balls are some symbols from a symbolic alphabet L. These symbols are colored. The colors are elements of a set of colors C. A particular ball type is associated with a unique combination of mono-spectrally (no mixture of wavelength) colored symbols printed on the black ball background. U is the set of all ball types. To make life easier we shall assume that every ball contains just one single symbol per color. (Not all types of balls; i.e., not all color/symbol combinations, may be present in the ensemble, though.) Let |U | be the number of different types of balls, |C| be the number of different monospectral colors, |L| be the number of different output symbols. Consider the deterministic “output” or “lookup” function Λ(u,c) = v, u ∈U , c ∈C, v ∈ L, which returns one symbol per ball type and color. One interpretation of this lookup function Λ is as follows. Consider a set of |C| eyeglasses build from filters for the |C| different colors. Let us assume that these mono-spectral filters are “perfect” in that they totally absorb all other colors but a particular single one. In that way, every color is associated with a particular eyeglass and vice versa. When a spectator looks at a particular ball through such an eyeglass, the only operationally recognizable symbol will be the one in the particular color which is transmitted through the eyeglass. All other colors get absorbed, and the symbols printed in them will appear black and cannot therefore be different from the black background. Hence the ball appears to carry a different “message” or symbol, depending on the color at which it is viewed. An empirical logic can be constructed as follows. Consider the set of all ball types. With respect to a particular colored eyeglass, this set disjointly “decays” or gets partitioned into those ball types which can be separated by the particular color of the eyeglass. Every such state partition can then be identified with a Boolean algebra whose atoms are the elements of the partition. A pasting of all of these Boolean algebras yields the empirical logic associated with the particular urn model.