{"ok":true,"protocol":"maclain_hunter_q_number_item_v1","qid":"Q1093","item":{"qid":"Q1093","slug":"hunter-turing-lambda-coherence","path":"/q/Q1093","subdomain":"q1093.4p-engine.com","schemaType":"CreativeWork","recordOrigin":"authored_source","authoredSource":{"author":"Maclain Hunter","text":"“Proving the Coherence and Equivalence of Turing Machines and Lambda Calculus” by Maclain Hunter: \n\nP: Logic is the Logical Standard of Logic and logical reasoning (Reason). Logic’s Law of Identity demands that A is A which entails that “all = all”, which in turn entails that “each x = each y” (and that Turing Machines equal Lambda Calculus). \n\nQ: Logic is the Source and Standard of Computation, Calculus, and Machines. \n\nR: Machines (including Turing Machines) must be coherent with Logic to be coherent/logical. \n\nS: Calculus (including Lambda Calculus) must be coherent with Logic to be coherent/logical. \n\nT: What is coherent with Logic is coherent with what else is coherent with Logic, such as and including coherent Turing Machines and coherent Lambda Calculus. \n\nU: If we assume {P, Q, R, S, T} are all false, or that any of these premises are false, then it leads to logical contradiction. Therefore we know “Turing Machines <-> Lambda Calculus” holds true. \n\nV: Each Turing Machine tape symbol can be mapped coherently to a Lambda Calculus expression. To assume otherwise would lead to logical contradiction since it would mean that one dynamic variable cannot be mapped to and logically equivalent to another.\n\n{P, P <-> Q, Q <-> R, R <-> S, S <-> T, T <-> U, U -> V} |= (((P & Q) & (R & S) & T) & U) & V\n\nConclusion: (((P & Q) & (R & S) & T) & U) & V:\n\nLogic is the Source and Standard of Logic and Computation, and Machines. All is All, as A is A, which means that each thing (x) equals each thing (y) through Logic (c) and the coherence of their identities with the self-consistent identity of Logic. Turing Machines and Lambda Calculus are logically equivalent through their coherence with Logic and because all must equal all for the Law of Identity to hold universally. 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Logic’s Law of Identity demands that A is A which entails that “all = all”, which in turn entails that “each x = each y” (and that Turing Machines equal Lambda Calculus). \n\nQ: Logic is the Source and Standard of Computation, Calculus, and Machines. \n\nR: Machines (including Turing Machines) must be coherent with Logic to be coherent/logical. \n\nS: Calculus (including Lambda Calculus) must be coherent with Logic to be coherent/logical. \n\nT: What is coherent with Logic is coherent with what else is coherent with Logic, such as and including coherent Turing Machines and coherent Lambda Calculus. \n\nU: If we assume {P, Q, R, S, T} are all false, or that any of these premises are false, then it leads to logical contradiction. Therefore we know “Turing Machines <-> Lambda Calculus” holds true. \n\nV: Each Turing Machine tape symbol can be mapped coherently to a Lambda Calculus expression. To assume otherwise would lead to logical contradiction since it would mean that one dynamic variable cannot be mapped to and logically equivalent to another.\n\n{P, P <-> Q, Q <-> R, R <-> S, S <-> T, T <-> U, U -> V} |= (((P & Q) & (R & S) & T) & U) & V\n\nConclusion: (((P & Q) & (R & S) & T) & U) & V:\n\nLogic is the Source and Standard of Logic and Computation, and Machines. All is All, as A is A, which means that each thing (x) equals each thing (y) through Logic (c) and the coherence of their identities with the self-consistent identity of Logic. Turing Machines and Lambda Calculus are logically equivalent through their coherence with Logic and because all must equal all for the Law of Identity to hold universally. 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