1,104 research outputs found
With a Few Square Roots, Quantum Computing Is as Easy as Pi
Rig groupoids provide a semantic model of Π, a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on 1+1. As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of Π, called √Π, that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to ≤2 qubits, and the computationally universal Gaussian Clifford+T gate set
Southern Adventist University Undergraduate Catalog 2023-2024
Southern Adventist University\u27s undergraduate catalog for the academic year 2023-2024.https://knowledge.e.southern.edu/undergrad_catalog/1123/thumbnail.jp
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
From G\"odel's Incompleteness Theorem to the completeness of bot beliefs (Extended abstract)
Hilbert and Ackermann asked for a method to consistently extend incomplete
theories to complete theories. G\"odel essentially proved that any theory
capable of encoding its own statements and their proofs contains statements
that are true but not provable. Hilbert did not accept that G\"odel's
construction answered his question, and in his late writings and lectures,
G\"odel agreed that it did not, since theories can be completed incrementally,
by adding axioms to prove ever more true statements, as science normally does,
with completeness as the vanishing point. This pragmatic view of validity is
familiar not only to scientists who conjecture test hypotheses but also to real
estate agents and other dealers, who conjure claims, albeit invalid, as
necessary to close a deal, confident that they will be able to conjure other
claims, albeit invalid, sufficient to make the first claims valid. We study the
underlying logical process and describe the trajectories leading to testable
but unfalsifiable theories to which bots and other automated learners are
likely to converge.Comment: 19 pages, 13 figures; version updates: changed one word in the title,
expanded Introduction, improved presentation, tidied up some diagram
Southern Adventist University Undergraduate Catalog 2022-2023
Southern Adventist University\u27s undergraduate catalog for the academic year 2022-2023.https://knowledge.e.southern.edu/undergrad_catalog/1121/thumbnail.jp
Kolmogorov's Calculus of Problems and Its Legacy
Kolmogorov's Calculus of Problems is an interpretation of Heyting's
intuitionistic propositional calculus published by A.N. Kolmogorov in 1932.
Unlike Heyting's intended interpretation of this calculus, Kolmogorov's
interpretation does not comply with the philosophical principles of
Mathematical Intuitionism. This philosophical difference between Kolmogorov and
Heyting implies different treatments of problems and propositions: while in
Heyting's view the difference between problems and propositions is merely
linguistic, Kolmogorov keeps the two concepts apart and does not apply his
calculus to propositions. I stress differences between Kolmogorov's and
Heyting's interpretations and show how the two interpretations diverged during
their development. In this context I reconstruct Kolmogorov's philosophical
views on mathematics and analyse his original take on the Hilbert-Brouwer
controversy. Finally, I overview some later works motivated by Kolmogorov's
Calculus of Problems and propose a justification of Kolmogorov's distinction
between problems and propositions in terms of Univalent Mathematics.Comment: 66 pages including Appendi
With a Few Square Roots, Quantum Computing is as Easy as {\Pi}
Rig groupoids provide a semantic model of \PiLang, a universal classical
reversible programming language over finite types. We prove that extending rig
groupoids with just two maps and three equations about them results in a model
of quantum computing that is computationally universal and equationally sound
and complete for a variety of gate sets. The first map corresponds to an
root of the identity morphism on the unit . The second map
corresponds to a square root of the symmetry on . As square roots are
generally not unique and can sometimes even be trivial, the maps are
constrained to satisfy a nondegeneracy axiom, which we relate to the Euler
decomposition of the Hadamard gate. The semantic construction is turned into an
extension of \PiLang, called \SPiLang, that is a computationally universal
quantum programming language equipped with an equational theory that is sound
and complete with respect to the Clifford gate set, the standard gate set of
Clifford+T restricted to qubits, and the computationally universal
Gaussian Clifford+T gate set
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Undergraduate and Graduate Course Descriptions, 2023 Spring
Wright State University undergraduate and graduate course descriptions from Spring 2023
Morris Catalog 2023-2025
This document serves as an official historical record for a specific period in time. The information found is subject to change without notice. Colleges and departments make changes to their degree requirements and course descriptions frequently. More information is available at catalogs.umn.edu.https://digitalcommons.morris.umn.edu/catalog/1034/thumbnail.jp
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