Abstract. The Infinite Time Turing Machine model  of Hamkins and Kidder is, in an essential sense, a “Σ2-machine ” in that it uses a Σ2 Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σn rule. Such machines either halt or enter an infinite loop by stage ζ(n) =df µζ(n)[∃Σ(n)> ζ(n) Lζ(n) ≺Σn LΣ(n)], again generalising precisely the ITTM case. The collection of such machines taken together, computes precisely those reals of the least model of analysis. §1. Introduction. The Infinite Time Turing Machine (ITTM) model described in  is an attractive model of transfinite time computation based on the standard Turing machine with an infinite one way tape, and a finite transition table or instruction set. The latter specifies how the machine behaves at successor steps as is usual, and one needs really only to specify precisely how such a machine behaves at limit steps in time t
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