Quantum Algorithms, Paraconsistent Computation And Deutsch's Problem

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 Deutsch's problem, and then provide another solution to Deutsch's problem by means of paraconsistent Turing machines. This raises some interesting questions, and we close the paper by discussing some relationships between paraconsistent Turing machines and quantum Turing machines. Copyright © IICAI 2005.16091628Agudelo, J.C., Máquinas de turing paraconsistentes: Algunas posibles defini-ciones y consecuencias (2003) Graduation Project, Universidad EAFIT, , http://sigma.eafit.edu.co:90/~asicard/archivos/mtps.ps.gzAgudelo, J.C., Sicard, A., Máquinas de turing paraconsistentes: Una posible definición (2004) Matemáticas: Enseñanza Universitaria, 12 (2), pp. 37-51. , http://revistaerm.univalle.edu.co/Enlaces/volXII2.htmlBenioff, P., The computer as a physical system: A microscopic quantum mechanical hamiltonian model of computers as represented by Turing machines (1980) Journal of Statistical Physics, 22, pp. 563-591Coniglio, W., Carnielli, M., Marcos, J., Logics of formal inconsistency (2005) Handbook of Philosophical Logic, 14. , http://www.cle.rniicamp.br/e-prints/vol5,nl,2005.html, To be published as a chapter in the, second edition, Kluwer Acad. Pub, vol 5, n. 1, 2005Carnielli, W.A., Many-valued logics and plausible reasoning (1990) Proceedings of the Twentieth International Symposium on Multiple-Valued Logic, pp. 328-335. , Charlotte, NC, USA, IEEE Computer SocietyCarnielli, W.A., Possible-translations semantics for paraconsistent logics (2000) Frontiers of Paraconsistent Logic: Proceedings of the I World Congress on Paraconsistency, pp. 149-163. , D. Batens, C. Mortensen, G. Priest, and J. P. Van Bendegem, editors, Logic and Computation Series, Baldock: Research Studies Press, King's College PublicationsCattaneo, G., Chiara, M.L.D., Giuntini, R., Leporini, R., An unsharp logic from quantum computation (2004) International Journal of Theoretical Physics, 43, pp. 1803-1817Chuang, I.L., Nielsen, M.A., (2000) Quantum Computation and Quantum Information, , Cambridge: Cambridge University PressCleve, R., Ekert, A., Macchiavello, C., Mosca, M., Quantum algorithms revisited (1998) Proceedings of the Royal Society of London, 454, pp. 339-354. , Series ACopeland, J., Hipercomputation (2002) Minds and Machines, 12, pp. 461-502Deutsch, D., Quantum theory, the church-turing principle and the universal quantum computer (1985) Proceedings of the Royal Society of London, 400, pp. 97-117. , Series ADeutsch, D., Quantum computational networks (1989) Proceedings of the Royal Society of London, 425, pp. 73-90. , Series AEpstein, R.L., Carnielli, W.A., Computability: Computable functions (2000) Logic, and the Foundations of Mathematics, , Belmont, CA: Wads worth/Thomson Learning, 2a editionFeynman, R.P., Simulating physics with computers (1982) International Journal of Theoretical Physics, 21, pp. 467-488Gruska, J., (1999) Quantum Computing, , Cambridge: McGraw-Hill International (UK) LimitedLloyd, S., Braunstein, S.L., Quantum computation over continuous variables (1999) Physical Review Letters, 82 (8), pp. 1784-1787De Almeida, J.M., (1999) Semǎnticas de Traduções Possiveis, , Master's Thesis, Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias HumanasMateus, P., Sernadas, A., Exogenous quantum logic (2004) Manuscript, , http://wslc.math.ist.utl.pt/ftp/pub/SernadasA/04-MS-fiblog24s.pdf, CLC, Department of Mathematics, IST., Available atOdifreddi, P., Classical recursion theory (1989) The Theory of Functions and Sets of Natural Numbers, 125. , Studies in Logic and the Foundations of Mathematics, volume, Amsterdam North-HollandOzawa, M., Nishimura, H., (1999) Local Transition Function of Quantum Turing Machines, , http://eirXiv.org/abs/quant-ph/9811069Penrose, R., (1989) The Emperor's New Mind: concerning Computers, Minds and the Laws of Physics, , Oxford: Oxford University PressRieffel, E., Polak, W., An introduction to quantum computing for non-physicists (2000) ACM Computing Surveys, 32 (3), pp. 300-335Shor, P.W., Algorithms for quantum computation: Discrete log and factoring (1994) Proc. 35th Symposium on Foundations of Computer Science, pp. 124-134. , IEEE Computer Society PressShor, P.W., Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer (1997) SI AM Journal on Computing, 26 (5), pp. 1484-1509Sicard, A., Computación paraconsistente (2002) Technical Report, Universi- Dad EAFIT, , http://sigma.eafit.edu.co:90/~asicard/archivos/proyectoCP.ps.gzSylvan, R., Copeland, J., Computability is logic-relative (2000) Sociative Logics and their Applications: Essays by the Late Richard Sylvan, pp. 189-199. , Graham Priest and Dominic Hyde, editors, London: Ashgate Publishing Company, 2000Turing, A.M., On computable numbers, with an application to the Entscheidungsproblem (1936) Proceedings of the London Mathematical Society, 43, pp. 230-265. , A correction, ibid, 1936-1937, pags, 544-546Veléz, M., Sicard, A., (2000) Sobre Un Modelo de Computación Cuántica Sobre Variables Continuas, , http://sigma.eafit.edu.co:90/~asicard/archivos/ccc.ps.gzYao, A.C., Quantum circuit complexity (1993) Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, pp. 352-360. , IEEE Computer Society Press, Los Alamitos, C

Unconventional Models Of Computation Through Non-standard Logic Circuits

The classical (boolean) circuit model of computation is generalized via polynomial ring calculus, an algebraic proof method adequate to non-standard logics (namely, to all truth-functional propcsitional logics and to some non-truth-functional logics). Such generalization allows us to define models of computation based on non-standard logics in a natural way by using 'hidden variables' in the constitution of the model. Paraconsistent circuits for the paraconsistent logic mbC (and for some extensions) are defined as an example of such models. Some potentialities are explored with respect to computability and computational complexity. © Springer-Verlag Berlin Heidelberg 2007.4618 LNCS2940Agudelo, J.C., Carnielli, W., Quantum algorithms, paraconsistent computation and Deutsch's problem (2005) Proceedings of the 2nd Indian International Conference on Artificial Intelligence, pp. 1609-1628. , Prasad, B, ed, Pune, India, ppBoolos, G., Jeffrey, R., (1989) Computability and logic, , 3rd edn. Cambridge University Press, CambridgeRichard Büchi, J., Turing-machines and the Entscheidungsproblem (1962) Mathematische Annalen, 148, pp. 201-213Calabro, C., Turing Machine, RAM. , http://www-cse.ucsd.edu/classes/fa06/cse200/ln2.ps, vs, Machine vs. Circuits. Lecture notes, available atConiglio Carlos Caleiro, M.E., Carnielli, W., Marcos, J.: Two's company: the humbug of many logical values. In: Beziau, J.-Y. (ed.) Logica Universalis, pp. 169-189. Birkhäuser Verlag, Basel, Switzerland (preprint available at) http://wslc.math.ist.utl.pt/ftp/pub/CaleiroC/05-CCCM-dyadic.pdfCarnielli, W.A.: Polynomial ring calculus for many-valued logics. In: Werner, B. (ed.) Proceedings of the 35th International Symposium on Multiple-Valued Logic, pp. 20-25. IEEE Computer Society, Los Alamitos (2005), (preprint available at CLE e-Prints 5(3), 2005), http://www.cle. unicamp.br/e-prints/vol_5,n_3,2005. htmlCarnielli, W.A., Coniglio, M.E., Marcos, J.: Logics of Formal Inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn. 14, Kluwer Academic Publishers, Dordrecht (2005), (in print, preprint available at CLE e-Prints 5(1), 2005), http ://www.cle.unicamp.br/e-prints/vol_5,n_1,2005.htmlChuang, I.L., Nielsen, M.A., (2000) Quantum Computation and Quantum Information, , Cambridge University Press, CambridgeClote, P., Kranakis, E., (2002) Boolean Functions and Computation Models, , Springer, HeidelbergCook, S.A., The complexity of theorem proving procedures (1971) Proceedings of the Third Annual ACM Symposium on the Theory of Computing, pp. 151-158. , ACM Press, New YorkCopeland, J., Hypercomputation (2002) Minds and machines, 12, pp. 461-502Davis, M., The myth of hypercomputation (2004) Alan Turing: Life and Legacy of a Great Thinker, pp. 195-212. , Teuscher, C, ed, Springer, HeidelbergDavis, M., Why there is no such discipline as hypercomputation (2004) Applied Mathematics and Computation, 178, pp. 4-7Genovese, M., Research on hidden variable theories: A review of recent progresses (2005) Physics Reports, , 413, 319-396Halpern, J.Y., Harper, R., Immerman, N., Kolaitis, P.G., Vardi, M.Y., Vianu, V., On the unusual effectiveness of logic in computer science (2001) The Bulletin of Symbolic Logic, 7 (2), pp. 213-236Jacobson, N., (1985) Basic Algebra I, , 2nd edn. W. H. Freeman and Company, New YorkPapadimitriou, C., (1994) Computational Complexity, , Adisson-Wesley, London, UKTuring, A.M.: On computable numbers, with an application to the Entscheidungsproblem. In: Proceedings of the London Mathematical Society, pp. 230-265 (1936) (A correction, ibid, 43, pp. 544-546, 1936-1937

Polynomizing: Logic Inference In Polynomial Format And The Legacy Of Boole

Polynomizing is a term that intends to describe the uses of polynomiallike representations as a reasoning strategy and as a tool for scientific heuristics. I show how proof-theory and semantics for classical and several non-classical logics can be approached from this perspective, and discuss the assessment of this prospect, in particular to recover certain ideas of George Boole in unifying logic, algebra and the differential calculus. © 2007 Springer-Verlag Berlin Heidelberg.64349364Agudelo, J.C., Carnielli, W.A.: Quantum algorithms, paraconsistent computation and Deutsch's problem. In Bhanu Prasad et al., eds.: Proceedings of the 2nd Indian International Conference on Artificial Intelligence, Pune, India, December 20-22 (2005) IICAI 2005, 1609-1628. Pre-print available from CLE e-Prints 5(10) (2005) ftp://logica.cle.unicamp.br/pub/e-prints/ MTPs-CompQuant%28Ing%29.pdfAhmed, T.S., Algebraic logic, where does it stand today? (2005) Bull. Symbolic Logic, 11 (4), pp. 465-516Beame, P., Impagliazzo, R., Krajicek, J.T., Pitassi, T., Pudlak, P., Lower bounds on Hilbert's Nullstellensatz and propositional proofs (1996) Proceedings of the London Mathematical Society, 73, pp. 1-26Boole, G., The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning (1847) Macmillan, Barclay and Macmillan, , London , Reprinted by Basil Blackwell, Oxford, 1965Boole, G., The calculus of logic (1848) Cambridge and Dublin Math. Journal, 3, pp. 183-198Boole, G., (1854) An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities, , Walton and Maberley, London , Reprinted by Dover Books, New York, 1954Boole, G., Calculus of Finite Differences (1970) Chelsea Publishing (originally published in 1860), , 5th EditionBurris, S.: The laws of Boole's thought. Unpublished (2002), Preprint at http://www.thoralf.uwaterloo.ca/htdocs/MYWORKS/ PREPRINTS/aboole.pdfCaleiro, C., Carnielli, W.A., Coniglio, M.E., Marcos, J.: Two's company: The humbug of many logical values. In Béziau J.-Y., Birkhäuser, eds.: Logica Universalis. Verlag, Basel, Switzerland (2005) 169-189 Preprint available at http:// wslc.math.ist.utl.pt/ftp/pub/CaleiroC/05-CCCM-dyadic.pdfCaicedo, X., Martín, A., Completud de dos cálculos lógicos de Leibniz (2001) Theoria, 16 (3), pp. 539-558Carnielli, W.A., Systematization of the finite many-valued logics through the method of tableaux (1987) The Journal of Symbolic Logic, 52 (2), pp. 473-493Carnielli, W.A., A polynomial proof system for Lukasiewicz logics (2001) Second Principia International Symposium, , August 6-10, Florianpolis, SC, BrazilCarnielli, W.A.: Polynomial ring calculus for many-valued logics. Proceedings of the 35th International Symposium on Multiple-Valued Logic. IEEE Computer Society. Calgary, Canad. IEEE Computer Society, pp. 20-25, 2005. Available from CLE e-Prints 5(3) (2005) at http://www.cle.unicamp.br/e-prints/vol_5,n_3,2005.htmlCarnielli, W.A., Coniglio, M.E., Polynomial formulations of non-deterministic semantics for logics of formal inconsistency (2006), ManuscriptClegg, M., Edmonds, J., Impagliazzo, R., Using the Gröbner bases algorithm to find proofs of unsatisfiability (1996) Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 174-183. , Philadelphia, Pennsylvania, USACorcoran, J., Aristotle's Prior Analytics and Boole's Laws of Thought (2003) Hist. and Ph. of Logic, 24, pp. 261-288Dummett, M., Review of "Studies in Logic and Probability by George Boole". Rhees, R., Open Court, 1952 (1959) J. of Symb. Log, 24, pp. 203-209Eves, H., (1990) An Introduction to the History of Mathematics, , 6th ed, Saunders, New YorkEuler, L., Variae observations circa series infinitas (1737) Commentarii academiae scientiarum Petropolitanae, 9, pp. 160-188Reprinted in Opera Omnia, Series I 14, Birkhuser, 216-244. Available on line at www.EulerArchive.orgGiusti, E.B., (1980) Cavalieri and the Theory of Indivisibles, , Cremonese, RomaGottwald, S., (2001) A Treatise on Many-Valued Logics, Studies in Logic and Computation, , Research Studies Press Ltd. Hertfordshire, EnglandHailperin, T., (1986) Boole's Logic and Probability: A Critical Exposititon from the Standpoint of Contemporary Algebra, Logic, and Probability Theory, , North-Holland Studies In Logic and the Foundations of MathematicsLloyd, G., Finite and infinite in Greece and China (1996) Chinese Science, 13, pp. 11-34MacHale, D., (1985) George Boole: His Life and Work, , Boole PressMates, B., (1953) Stoic Logic, , University of California Press, Berkeley, CAMartzloff, J.-C., (1997) A History of Chinese Mathematics, , Springer-Verlag, BerlinRoy, R., The discovery of the series formula for π by Leibniz, Gregory and Nilakantha (1990) Mathematics Magazine, 63 (5), pp. 291-306Schroeder, M., A brief history of the notation of Boole's algebra (1997) Nordic Journal of Philosophical Logic, 2 (1), pp. 41-62Schröder, E., (1891) Vorlesungen über die Algebra der Logik (exakte Logik), 2, p. 1. , B.G. Teubner, Leipzig, reprinted by Chelsea, New York, 1966Stone, M.H., The theory of representations for boolean algebras (1936) Trans. of the Amer. Math. Soc, 40, pp. 37-111Styazhkin, M.I., (1969) History of Mathematical Logic from Leibniz to Peano, , The M.I.T. Press, CambridgeWu, J.-Z., Tan, H.-Y., Li, Y., An algebraic method to decide the deduction problem in many-valued logics (1998) Journal of Applied Non-Classical Logics, 8 (4), pp. 353-360Weyl, H., (1932) God and the Universe: The Open World, , Yale University PressReprinted as, (1989) The Open World, , by Ox Bow PressZhegalkin, I.I., O tekhnyke vychyslenyi predlozhenyi v symbolytscheskoi logykye (On a technique of evaluation of propositions in symbolic logic) (1927) Matematicheskii Sbornik, 34 (1), pp. 9-28. , in Russia