The emergence of electronic computers in the last thirty years has given rise to many interesting questions. Many of these questions are technical, relating to a machine’s ability to perform complex operations in a variety of circumstances. While some of these questions are not without philosophical interest, the one question which above all others has stimulated philosophical interest is explicitly non-technical and it can be expressed crudely as follows: Can a machine be said to think and, if so, in what sense? The issue has received much attention in the scholarly journals with articles and arguments appearing in great profusion, some resolutely answering this question in the affirmative, some, equally resolutely, answering this question in the negative, and others manifesting modified rapture. While the ramifications of the question are enormous I believe that the issue at the heart of the matter has gradually emerged from the forest of complications. It is easy to answer the question “Can machines/computers think?” Easy, that is, once we know with some degree of precision what thinking is, what machine/computers are, and, last but not least, how to understand the word “can.” The possibility of answering the question, “Can a computer think?” is rendered either question-beggingly trivial or mind-bogglingly impossible unless one is able to give some independent signification to its key terms. To define a computer as a sort of machine, and then to define machine in such a narrow way that it becomes meaningless to ask if it thinks viciously begs the question unless the definitions can be given independent justification. Of course, a similar consideration applies to arguments in favour of artificial intelligence. I assume that the majority of those who would put this question intend by “think” something like

https://researchrepository.ucd.ie/bitstream/10197/5469/1/Minds_and_Machines.pdf

Minds and machines

The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics

The Church-Turing Thesis is widely regarded as true, because of evidence that there is only one genuine notion of computation. By contrast, there are nowadays many different formal logics, and different corresponding foundational frameworks. Which ones can deliver a theory of computability? This question sets up a difficult challenge: the meanings of basic mathematical terms (like "set", "function", and "number") are not stable across frameworks. While it is easy to compare what different frameworks say, it is not so easy to compare what they mean. We argue for some minimal conditions that must be met if two frameworks are to be compared; if frameworks are radical enough, comparison becomes hopeless. Our aim is to clarify the dialectical situation in this bourgeoning area of research, shedding light on the nature of non-classical logic and the notion of computation alike.

Computation in Non-Classical Foundations?

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Computability and Unsolvability.

Computability. Computable functions, logic, and the foundations of mathematics

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra-Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.

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Polynomial ring calculus for modal logics: A new semantics and proof method for modalities

We present the new model of paraconsistent Turing machines and revise the models of quantum Turing machines and quantum circuits, stressing the concepts of entangled states and quantum parallelism which are important features for efficient quantum algorithms. We revise a quantum circuit that solves the so-called Deutsch’s problem, and then provide another solution to Deutsch’s problem by means of paraconsistent Turing machines. This raises some stimulating questions, and we close the paper by discussing some relationships between paraconsistent Turing machines and quantum Turing machines.

Quantum Algorithms, Paraconsistent Computation and Deutsch's Problem.

We describe a method to axiomatize computations in deterministic Turing machines (TMs). When applied to computations in non-deterministic TMs, this method may produce contradictory (and therefore trivial) theories, considering classical logic as the underlying logic. By substituting in such theories the underlying logic by a paraconsistent logic we define a new computation model, the paraconsistent Turing machine. This model allows a partial simulation of superposed states of quantum computing. Such a feature allows the definition of paraconsistent algorithms which solve (with some restrictions) the well-known Deutsch's and Deutsch-Jozsa problems. This first model of computation, however, does not adequately represent the notions of entangled states and relative phase, which are key features in quantum computing. In this way, a more sharpened model of paraconsistent TMs is defined, which better approaches quantum computing features. Finally, we define complexity classes for such models, and establish some relationships with classical complexity classes.

Paraconsistent Machines and their Relation to Quantum Computing

Resumen es: Se define un metodo para axiomatizar las maquinas de Turing, mediante el cual, dada una maquina M y una entrada n, se construye una teoria en la logica c...

Máquinas de Turing paraconsistentes : una posible definición

We present an original model of paraconsistent Turing machines (PTMs), a generalization of the classical Turing machines model of computation using a paraconsistent logic. Next, we briefl y describe the standard models of quantum computation: quantum Turing machines and quantum circuits, and revise quantum algorithms to solve the so-called Deutsch's problem and Deutsch-Jozsa problem. Then, we show the potentialities of the PTMs model of computation simulating the presented quantum algorithms via paraconsistent algorithms. This way, we show that PTMs can resolve some problems in exponentially less time than any classical deterministic Turing machine. Finally, We show that it is not possible to simulate all characteristics (in particular entangled states) of quantum computation by the particular model of PTMs here presented, therefore we open the possibility of constructing a new model of PTMs by which it is feasible to simulate such states.

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Juan C. Agudelo

Papers

Polynomial ring calculus for modal logics: A new semantics and proof method for modalities

Quantum Algorithms, Paraconsistent Computation and Deutsch's Problem.

Paraconsistent Machines and their Relation to Quantum Computing

Máquinas de Turing paraconsistentes : una posible definición

Quantum Computation via Paraconsistent Computation