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Exploring Complexity in Computing and Consciousness

Dive deep into the realms of Turing Machines, Quantum Computing, Logic, and Consciousness, exploring diverse topics from Church-Turing Thesis to Quantum Entanglement and beyond. This survey delves into complex concepts with relevance to computation and human consciousness.

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Exploring Complexity in Computing and Consciousness

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  1. Survey of COMP 006 Time to Remember …

  2. Turing Machine definition (why TM?) • Turmites • Conway’s Game of Life • Program Turing machines in class • Does input have “a” in it • Is input of form w_w for binary w • Others (including HW 1 - 3) • Extensions and Restrictions of TM • Decidability • Partial Decidability

  3. Church-Turing Thesis • Universal Turing Machines • Small Universal Turing Machines • Undecidability of halting problem • Blank tape halting problem • Busy beaver function (uncomputable) • Busy Beaver machine simulations • Chaitin’s omega (uncomputable)

  4. Paradox of consciousness • Loop testing version • Four possible solutions • Quotes from Lucas, Penrose, Ramanujan • Simulate student’s turmites from hw4 • Hypercomputation • Wave particle duality • Double slit experiment • Schroedinger’s Cat • Uncertainty Principle

  5. Tunneling and Solar Energy • Quantum Superposition • Kieu’s Algorithm • Decoherence • C60, C70 molecules as waves • Feynman: Quantum wierdness • Faster than light travel • Quantum entanglement • Quantum computation (brief)

  6. Consciousness and time • Left and right brains and consciousness • Poincare and Mozart’s inspiration • Einstein and nonverbal thought • Inspiration in animals • Is language necessary for consciousness? • Logic • Interpretations, models, truth • Validity in a logic

  7. Standard Interpretation • Theorem Prover for a logic • Aristotelian logic (syllogisms) • Deciding syllogisms with Venn diagrams • Geometry – two axiom systems • Propositional calculus • Truth Tables

  8. First-order logic • For all • There exists • Translations into English • Peano arithmetic and induction • Zermelo-Fraenkel set theory • Nonstandard interpretations • Nonstandard integers • Statement H(Tj,j) true but not provable in L, asserts its own unprovability

  9. New form of paradox in terms of logics • We can know a statement L cannot derive • Independent statements about program complexity • Large number of them • 2001 A Space Odyssey

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