476,802 research outputs found
A Proof of the S-m-n theorem in Coq
This report describes the implementation of a mechanisation of the theory of computation in the Coq proof assistant which leads to a proof of the Smn theorem. This mechanisation is based on a model of computation similar to the partial recursive function model and includes the definition of a computable function, proofs of the computability of a number of functions and the definition of an effective coding from the set of partial recursive functions to natural numbers. This work forms part of a comparative study of the HOL and Coq proof assistants
A universal adiabatic quantum query algorithm
Quantum query complexity is known to be characterized by the so-called
quantum adversary bound. While this result has been proved in the standard
discrete-time model of quantum computation, it also holds for continuous-time
(or Hamiltonian-based) quantum computation, due to a known equivalence between
these two query complexity models. In this work, we revisit this result by
providing a direct proof in the continuous-time model. One originality of our
proof is that it draws new connections between the adversary bound, a modern
technique of theoretical computer science, and early theorems of quantum
mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's
theorem, while the upper bound relies on the adiabatic theorem, as it goes by
constructing a universal adiabatic quantum query algorithm. Another originality
is that we use for the first time in the context of quantum computation a
version of the adiabatic theorem that does not require a spectral gap.Comment: 22 pages, compared to v1, includes a rigorous proof of the
correctness of the algorithm based on a version of the adiabatic theorem that
does not require a spectral ga
Bipolar Proof Nets for MALL
In this work we present a computation paradigm based on a concurrent and
incremental construction of proof nets (de-sequentialized or graphical proofs)
of the pure multiplicative and additive fragment of Linear Logic, a resources
conscious refinement of Classical Logic. Moreover, we set a correspon- dence
between this paradigm and those more pragmatic ones inspired to transactional
or distributed systems. In particular we show that the construction of additive
proof nets can be interpreted as a model for super-ACID (or co-operative)
transactions over distributed transactional systems (typi- cally,
multi-databases).Comment: Proceedings of the "Proof, Computation, Complexity" International
Workshop, 17-18 August 2012, University of Copenhagen, Denmar
Energy-recycling Blockchain with Proof-of-Deep-Learning
An enormous amount of energy is wasted in Proofof-Work (PoW) mechanisms
adopted by popular blockchain applications (e.g., PoW-based cryptocurrencies),
because miners must conduct a large amount of computation. Owing to this, one
serious rising concern is that the energy waste not only dilutes the value of
the blockchain but also hinders its further application. In this paper, we
propose a novel blockchain design that fully recycles the energy required for
facilitating and maintaining it, which is re-invested to the computation of
deep learning. We realize this by proposing Proof-of-Deep-Learning (PoDL) such
that a valid proof for a new block can be generated if and only if a proper
deep learning model is produced. We present a proof-of-concept design of PoDL
that is compatible with the majority of the cryptocurrencies that are based on
hash-based PoW mechanisms. Our benchmark and simulation results show that the
proposed design is feasible for various popular cryptocurrencies such as
Bitcoin, Bitcoin Cash, and Litecoin.Comment: 5 page
An Algebraic Weak Factorisation System on 01-Substitution Sets: A Constructive Proof
We will construct an algebraic weak factorisation system on the category of
01 substitution sets such that the R-algebras are precisely the Kan fibrations
together with a choice of Kan filling operation. The proof is based on Garner's
small object argument for algebraic weak factorization systems. In order to
ensure the proof is valid constructively, rather than applying the general
small object argument, we give a direct proof based on the same ideas. We use
this us to give an explanation why the J computation rule is absent from the
original cubical set model and suggest a way to fix this
Focusing and Polarization in Intuitionistic Logic
A focused proof system provides a normal form to cut-free proofs that
structures the application of invertible and non-invertible inference rules.
The focused proof system of Andreoli for linear logic has been applied to both
the proof search and the proof normalization approaches to computation. Various
proof systems in literature exhibit characteristics of focusing to one degree
or another. We present a new, focused proof system for intuitionistic logic,
called LJF, and show how other proof systems can be mapped into the new system
by inserting logical connectives that prematurely stop focusing. We also use
LJF to design a focused proof system for classical logic. Our approach to the
design and analysis of these systems is based on the completeness of focusing
in linear logic and on the notion of polarity that appears in Girard's LC and
LU proof systems
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
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