TY - JOUR

T1 - A continuum of discrete systems

AU - Blair, Howard A.

AU - Chidella, Jagan

AU - Dushin, Fred

AU - Ferry, Audrey

AU - Humenn, Polar

PY - 1997

Y1 - 1997

N2 - We show how to regard covered logic programs as cellular automata. Covered 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-state 2-dimensional cellular automata. We show how monadic covered programs, the class of which is computationally universal, can be regarded as 1-dimensional cellular automata. We show how to continuously (and differentiably) deform 1-dimensional cellular automata from one to another and understand the arrangement of these cellular automata in a separable Hilbert space over the real numbers. The embedding of the cellular automata of fixed radius r is a linear mapping into R22r+1 in which a cellular automaton's transition function is the attractor of a state-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 ℓ2. Furthermore, these programs are construable as almost everywhere continuous functions from the unit interval {x | 0 ≤ x ≤ 1} to the real numbers R. As one consequence, in particular, we can define a variety of natural metrics on the class of these programs. Moreover, for each program in this class, the set 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.

AB - We show how to regard covered logic programs as cellular automata. Covered 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-state 2-dimensional cellular automata. We show how monadic covered programs, the class of which is computationally universal, can be regarded as 1-dimensional cellular automata. We show how to continuously (and differentiably) deform 1-dimensional cellular automata from one to another and understand the arrangement of these cellular automata in a separable Hilbert space over the real numbers. The embedding of the cellular automata of fixed radius r is a linear mapping into R22r+1 in which a cellular automaton's transition function is the attractor of a state-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 ℓ2. Furthermore, these programs are construable as almost everywhere continuous functions from the unit interval {x | 0 ≤ x ≤ 1} to the real numbers R. As one consequence, in particular, we can define a variety of natural metrics on the class of these programs. Moreover, for each program in this class, the set 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.

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U2 - 10.1023/a:1018913302060

DO - 10.1023/a:1018913302060

M3 - Article

AN - SCOPUS:21944445206

SN - 1012-2443

VL - 21

SP - 153

EP - 186

JO - Annals of Mathematics and Artificial Intelligence

JF - Annals of Mathematics and Artificial Intelligence

IS - 2-4

ER -