By G.O. Okokiolu
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1 = 1 + 1 = 1) one determines which pairs of nodes in the graph are connected: after t iterations all connections by some path of length ≤ 2t are found. Since cycles don’t contribute to connections and the number of the nodes is bounded by 2O(spacebound) a polynomial number of iterations is sufficient. At the end of the computation the answer is found in the desired matrix element; the rest of the matrix is discarded. The details of this simulation depend on the precise model considered. Generic tasks are the construction of some object representing a list of all integers in the range 0 .
He came up with four principles (or constraints, or axioms) satisfied, he claimed, by all discrete deterministic machines. Contrast this with Turing’s analysis. While Turing’s analysis was convincing, it is hard to isolate first principles that, in Turing’s opinion, are satisfied by all symbolic sequential computations. Wilfried Sieg adopted Gandy’s approach and reworked Gandy’s axioms; see [23] and references there. Critical Remarks In a 2002 article [20], Oron Shagrir suggests that “there is an ambiguity regarding the types of machines that Gandy was postulating”.
6 Conclusion - Is There a Dragon Out There? I hope that the examples in the preceding sections have convinced the reader that the use of the Intrinsic Representation of Turing Machine configurations has been the enabler for several fundamental results in Theoretical Computer Science. The question remains whether this observation should affect our behaviour as theoreticians. Stated otherwise: do we need to start a Crusade against the use of the Mathematical Representation? Is there a dragon out there which should be slayed?