A new computing model, called the active element machine (AEM), is presented that demonstrates Tu... more A new computing model, called the active element machine (AEM), is presented that demonstrates Turing incomputable computation using quantum random input. The AEM deterministically executes a universal Turing machine (UTM) program η with random active element firing patterns. These firing patterns are Turing incomputable when the AEM executes a UTM having an unbounded number of computable steps. For an unbounded number of computable steps, if zero information is revealed to an adversary about the AEM’s representation of the UTM’s state and tape and the quantum random bits that help determine η’s computation and zero information is revealed about the dynamic connections between the active elements, then there does not exist a “reverse engineer” Turing machine that can map the random firing patterns back to the sequence of UTM instructions. This casts a new light on Turing’s notion of a computational procedure. In practical terms, these methods present an opportunity to build a new cl...
Among the fundamental questions in computer science, at least two have a deep impact on mathemati... more Among the fundamental questions in computer science, at least two have a deep impact on mathematics. What can computation compute? How many steps does a computation require to solve an instance of the 3-SAT problem? Our work addresses the first question, by introducing a new model called the ex-machine. The ex-machine executes Turing machine instructions and two special types of instructions. Quantum random instructions are physically realizable with a quantum random number generator. Meta instructions can add new states and add new instructions to the ex-machine. A countable set of ex-machines is constructed, each with a finite number of states and instructions; each ex-machine can compute a Turing incomputable language, whenever the quantum randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan Turing posed the halting problem for Turing machines and proved that this problem is unsolvable for Turing machines. Consider an enumeration E_a(i) = (M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist an ex-machine X that has at least one evolutionary path X → X1 → X2 → ... → Xm, so at the mth stage ex-machine Xm can correctly determine for 0 ≤ i ≤ m whether M_i’s execution on tape T_i eventually halts? We demonstrate an ex-machine Q(x) that has one such evolutionary path. The existence of this evolutionary path suggests that David Hilbert was not misguided to propose in 1900 that mathematicians search for finite processes to help construct mathematical proofs. Our refinement is that we cannot use a fixed computer program that behaves according to a fixed set of mechanical rules. We must pursue methods that exploit randomness and self-modification so that the complexity of the program can increase as it computes.
A new computing model, called the active element machine (AEM), is presented that demonstrates Tu... more A new computing model, called the active element machine (AEM), is presented that demonstrates Turing incomputable computation using quantum random input. The AEM deterministically executes a universal Turing machine (UTM) program η with random active element firing patterns. These firing patterns are Turing incomputable when the AEM executes a UTM having an unbounded number of computable steps. For an unbounded number of computable steps, if zero information is revealed to an adversary about the AEM's representation of the UTM's state and tape and the quantum random bits that help determine η's computation and zero information is revealed about the dynamic connections between the active elements, then there does not exist a " reverse engineer " Turing machine that can map the random firing patterns back to the sequence of UTM instructions. This casts a new light on Turing's notion of a computational procedure. In practical terms, these methods present an opportunity to build a new class of computing machines where the program's computational steps are hidden. This non-Turing computing behavior may be useful in cybersecurity and in other areas such as machine learning where multiple, dynamic interpretations of firing patterns may be applicable.
A new computing model, called the active element machine (AEM), is presented that demonstrates Tu... more A new computing model, called the active element machine (AEM), is presented that demonstrates Turing incomputable computation using quantum random input. The AEM deterministically executes a universal Turing machine (UTM) program η with random active element firing patterns. These firing patterns are Turing incomputable when the AEM executes a UTM having an unbounded number of computable steps. For an unbounded number of computable steps, if zero information is revealed to an adversary about the AEM’s representation of the UTM’s state and tape and the quantum random bits that help determine η’s computation and zero information is revealed about the dynamic connections between the active elements, then there does not exist a “reverse engineer” Turing machine that can map the random firing patterns back to the sequence of UTM instructions. This casts a new light on Turing’s notion of a computational procedure. In practical terms, these methods present an opportunity to build a new cl...
Among the fundamental questions in computer science, at least two have a deep impact on mathemati... more Among the fundamental questions in computer science, at least two have a deep impact on mathematics. What can computation compute? How many steps does a computation require to solve an instance of the 3-SAT problem? Our work addresses the first question, by introducing a new model called the ex-machine. The ex-machine executes Turing machine instructions and two special types of instructions. Quantum random instructions are physically realizable with a quantum random number generator. Meta instructions can add new states and add new instructions to the ex-machine. A countable set of ex-machines is constructed, each with a finite number of states and instructions; each ex-machine can compute a Turing incomputable language, whenever the quantum randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan Turing posed the halting problem for Turing machines and proved that this problem is unsolvable for Turing machines. Consider an enumeration E_a(i) = (M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist an ex-machine X that has at least one evolutionary path X → X1 → X2 → ... → Xm, so at the mth stage ex-machine Xm can correctly determine for 0 ≤ i ≤ m whether M_i’s execution on tape T_i eventually halts? We demonstrate an ex-machine Q(x) that has one such evolutionary path. The existence of this evolutionary path suggests that David Hilbert was not misguided to propose in 1900 that mathematicians search for finite processes to help construct mathematical proofs. Our refinement is that we cannot use a fixed computer program that behaves according to a fixed set of mechanical rules. We must pursue methods that exploit randomness and self-modification so that the complexity of the program can increase as it computes.
A new computing model, called the active element machine (AEM), is presented that demonstrates Tu... more A new computing model, called the active element machine (AEM), is presented that demonstrates Turing incomputable computation using quantum random input. The AEM deterministically executes a universal Turing machine (UTM) program η with random active element firing patterns. These firing patterns are Turing incomputable when the AEM executes a UTM having an unbounded number of computable steps. For an unbounded number of computable steps, if zero information is revealed to an adversary about the AEM's representation of the UTM's state and tape and the quantum random bits that help determine η's computation and zero information is revealed about the dynamic connections between the active elements, then there does not exist a " reverse engineer " Turing machine that can map the random firing patterns back to the sequence of UTM instructions. This casts a new light on Turing's notion of a computational procedure. In practical terms, these methods present an opportunity to build a new class of computing machines where the program's computational steps are hidden. This non-Turing computing behavior may be useful in cybersecurity and in other areas such as machine learning where multiple, dynamic interpretations of firing patterns may be applicable.
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