2,130 research outputs found

    Strings from Logic

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    What are strings made of? The possibility is discussed that strings are purely mathematical objects, made of logical axioms. More precisely, proofs in simple logical calculi are represented by graphs that can be interpreted as the Feynman diagrams of certain large-N field theories. Each vertex represents an axiom. Strings arise, because these large-N theories are dual to string theories. These ``logical quantum field theories'' map theorems into the space of functions of two parameters: N and the coupling constant. Undecidable theorems might be related to nonperturbative field theory effects.Comment: Talk, 19 pp, 7 figure

    On product, generic and random generic quantum satisfiability

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    We report a cluster of results on k-QSAT, the problem of quantum satisfiability for k-qubit projectors which generalizes classical satisfiability with k-bit clauses to the quantum setting. First we define the NP-complete problem of product satisfiability and give a geometrical criterion for deciding when a QSAT interaction graph is product satisfiable with positive probability. We show that the same criterion suffices to establish quantum satisfiability for all projectors. Second, we apply these results to the random graph ensemble with generic projectors and obtain improved lower bounds on the location of the SAT--unSAT transition. Third, we present numerical results on random, generic satisfiability which provide estimates for the location of the transition for k=3 and k=4 and mild evidence for the existence of a phase which is satisfiable by entangled states alone.Comment: 9 pages, 5 figures, 1 table. Updated to more closely match published version. New proof in appendi

    Generic-case complexity, decision problems in group theory and random walks

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    We give a precise definition of ``generic-case complexity'' and show that for a very large class of finitely generated groups the classical decision problems of group theory - the word, conjugacy and membership problems - all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.Comment: Revised versio
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