612 research outputs found
Guessing with a Bit of Help
What is the value of a single bit to a guesser? We study this problem in a
setup where Alice wishes to guess an i.i.d. random vector, and can procure one
bit of information from Bob, who observes this vector through a memoryless
channel. We are interested in the guessing efficiency, which we define as the
best possible multiplicative reduction in Alice's guessing-moments obtainable
by observing Bob's bit. For the case of a uniform binary vector observed
through a binary symmetric channel, we provide two lower bounds on the guessing
efficiency by analyzing the performance of the Dictator and Majority functions,
and two upper bounds via maximum entropy and Fourier-analytic /
hypercontractivity arguments. We then extend our maximum entropy argument to
give a lower bound on the guessing efficiency for a general channel with a
binary uniform input, via the strong data-processing inequality constant of the
reverse channel. We compute this bound for the binary erasure channel, and
conjecture that Greedy Dictator functions achieve the guessing efficiency
Secure Numerical and Logical Multi Party Operations
We derive algorithms for efficient secure numerical and logical operations
using a recently introduced scheme for secure multi-party
computation~\cite{sch15} in the semi-honest model ensuring statistical or
perfect security. To derive our algorithms for trigonometric functions, we use
basic mathematical laws in combination with properties of the additive
encryption scheme in a novel way. For division and logarithm we use a new
approach to compute a Taylor series at a fixed point for all numbers. All our
logical operations such as comparisons and large fan-in AND gates are perfectly
secure. Our empirical evaluation yields speed-ups of more than a factor of 100
for the evaluated operations compared to the state-of-the-art
Quantum-mechanical machinery for rational decision-making in classical guessing game
In quantum game theory, one of the most intriguing and important questions
is, "Is it possible to get quantum advantages without any modification of the
classical game?" The answer to this question so far has largely been negative.
So far, it has usually been thought that a change of the classical game setting
appears to be unavoidable for getting the quantum advantages. However, we give
an affirmative answer here, focusing on the decision-making process (we call
'reasoning') to generate the best strategy, which may occur internally, e.g.,
in the player's brain. To show this, we consider a classical guessing game. We
then define a one-player reasoning problem in the context of the
decision-making theory, where the machinery processes are designed to simulate
classical and quantum reasoning. In such settings, we present a scenario where
a rational player is able to make better use of his/her weak preferences due to
quantum reasoning, without any altering or resetting of the classically defined
game. We also argue in further analysis that the quantum reasoning may make the
player fail, and even make the situation worse, due to any inappropriate
preferences.Comment: 9 pages, 10 figures, The scenario is more improve
A Casual Tour Around a Circuit Complexity Bound
I will discuss the recent proof that the complexity class NEXP
(nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial
size. The proof will be described from the perspective of someone trying to
discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News,
September 201
Proof-Pattern Recognition and Lemma Discovery in ACL2
We present a novel technique for combining statistical machine learning for
proof-pattern recognition with symbolic methods for lemma discovery. The
resulting tool, ACL2(ml), gathers proof statistics and uses statistical
pattern-recognition to pre-processes data from libraries, and then suggests
auxiliary lemmas in new proofs by analogy with already seen examples. This
paper presents the implementation of ACL2(ml) alongside theoretical
descriptions of the proof-pattern recognition and lemma discovery methods
involved in it
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