A CONTINUUM OF DISCRETE-SYSTEMS

We show how to regard covered logic programs as cellular automata. Cov ered logic programs are ones for which every variable occurring in the body of a given clause also occurs in the head of the same clause. We generalize the class of register machine programs to permit negative literals and characterize the members of this class of programs as n-s tate 2-dimensional cellular automata. We show how monadic covered prog rams, the class of which is computationally universal, can be regarded as I-dimensional cellular automata. We show how to continuously (and differentiably) deform 1-dimensional cellular automata from one to ano ther and understand the arrangement of these cellular automata in a se parable Hilbert space over the real numbers. The embedding of the cell ular automata of fixed radius r is a linear mapping into R22r+1 in whi ch a cellular automaton's transition function is the attractor of a st ate-governed iterated function system of affine contraction mappings. The class of covered monadic programs having a particular fixed point has a uniform arrangement in an affine subspace of the Hilbert space l (2). Furthermore, these programs are construable as almost everywhere continuous functions from the unit interval {x \ 0 less than or equal to x less than or equal to 1} to the real numbers R. As one consequenc e, in particular, we can define a variety of natural metrics on the cl ass of these programs. Moreover, for each program in this class, the s et of initial segments of the program's fixed points, with respect to an ordering induced by the program's dependency relation, is a regular set.