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A Generalised Dynamical System, Infinite Time Register Machines, and Π 1 1CA0.
"... Abstract. We identify a number of theories of strength that of Π 1 1CA0. In particular: (a) the theory that the set of points attracted to the origin in a generalised transfinite dynamical system of any ndimensional integer torus exists; (b) the theory asserting that for any Z ⊆ ω and n, the halti ..."
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Abstract. We identify a number of theories of strength that of Π 1 1CA0. In particular: (a) the theory that the set of points attracted to the origin in a generalised transfinite dynamical system of any ndimensional integer torus exists; (b) the theory asserting that for any Z ⊆ ω and n, the halting set H Z n of infinite time nregister machine with oracle Z exists. Suppose f: N n − → N n. We are going to consider transfinite iterations of such f: N n − → N n as a generalised dynamical system. If one wishes, one may think of f acting on the points of an ndimensional lattice torus where we identify ∞ with 0. We set this up as follows. Given a point r = (r1,..., rn) ∈ N n set: r 0 = (r 0 1,..., r 0 n) = (r1,..., rn); r α+1 = (r α+1 1,..., r α+1 n) = f((r α 1,..., r α n)); r λ = (r λ 1,..., r λ n) = (Liminf ∗ α→λ r α 1, Liminf ∗ α→λ r α 2,..., Liminf ∗ α→λ r α n) where we define Liminf ∗ α→λ r α 1 = Liminf α→λ r α 1 if the latter is < ω, and set it to 0 otherwise, thus: r λ i = Liminfα→λ r α i if the latter is < ω = 0 otherwise. We may ask after the behaviour of points under this dynamic. For example which points ultimately end up at the origin O? As a more amusing example let p = (p0, p1, p2) ∈ (N n) 3 be a triple of three points on the ndimensional lattice. In general they thus form a proper triangle. Then define: Tf = {(p0, p1, p2) ∈ N n3  ∃α p α 0 = p α 1 = p α 2}. Tf is thus the set of possible starting triangles, which at some point collapse and become coincident after iteration of their vertices (and remain collapsed of course from some point α0 onwards).
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
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In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original