289 research outputs found
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
The scope of Fefermanâs semi-intuitionistic set theories and his second conjecture
The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Fefermanâs that relates the concepts for which the law of excluded middle obtains to those that are absolute with regard to the relevant test structures, or more precisely of â1 complexity. The latter is then used to show that a plethora of statements is indeterminate with respect to various semi-intuitionistic set theories
Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function
For a quantum computer acting on d-dimensional systems, we analyze the
computational power of circuits wherein stabilizer operations are perfect and
we allow access to imperfect non-stabilizer states or operations. If the noise
rate affecting the non-stabilizer resource is sufficiently high, then these
states and operations can become simulable in the sense of the Gottesman-Knill
theorem, reducing the overall power of the circuit to no better than classical.
In this paper we find the depolarizing noise rate at which this happens, and
consequently the most robust non-stabilizer states and non-Clifford gates. In
doing so, we make use of the discrete Wigner function and derive facets of the
so-called qudit Clifford polytope i.e. the inequalities defining the convex
hull of all qudit Clifford gates. Our results for robust states are provably
optimal. For robust gates we find a critical noise rate that, as dimension
increases, rapidly approaches the the theoretical optimum of 100%. Some
connections with the question of qudit magic state distillation are discussed.Comment: 14 pages, 1 table; Minor changes vs. version
Optimal Decision Trees for Nonlinear Metrics
Nonlinear metrics, such as the F1-score, Matthews correlation coefficient,
and Fowlkes-Mallows index, are often used to evaluate the performance of
machine learning models, in particular, when facing imbalanced datasets that
contain more samples of one class than the other. Recent optimal decision tree
algorithms have shown remarkable progress in producing trees that are optimal
with respect to linear criteria, such as accuracy, but unfortunately nonlinear
metrics remain a challenge. To address this gap, we propose a novel algorithm
based on bi-objective optimisation, which treats misclassifications of each
binary class as a separate objective. We show that, for a large class of
metrics, the optimal tree lies on the Pareto frontier. Consequently, we obtain
the optimal tree by using our method to generate the set of all nondominated
trees. To the best of our knowledge, this is the first method to compute
provably optimal decision trees for nonlinear metrics. Our approach leads to a
trade-off when compared to optimising linear metrics: the resulting trees may
be more desirable according to the given nonlinear metric at the expense of
higher runtimes. Nevertheless, the experiments illustrate that runtimes are
reasonable for majority of the tested datasets
Composable computation in discrete chemical reaction networks
We study the composability of discrete chemical reaction networks (CRNs) that
stably compute (i.e., with probability 0 of error) integer-valued functions
. We consider output-oblivious CRNs in which the
output species is never a reactant (input) to any reaction. The class of
output-oblivious CRNs is fundamental, appearing in earlier studies of CRN
computation, because it is precisely the class of CRNs that can be composed by
simply renaming the output of the upstream CRN to match the input of the
downstream CRN.
Our main theorem precisely characterizes the functions stably computable
by output-oblivious CRNs with an initial leader. The key necessary condition is
that for sufficiently large inputs, is the minimum of a finite number of
nondecreasing quilt-affine functions. (An affine function is linear with a
constant offset; a quilt-affine function is linear with a periodic offset)
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