89,565 research outputs found
Extending Nunchaku to Dependent Type Theory
Nunchaku is a new higher-order counterexample generator based on a sequence
of transformations from polymorphic higher-order logic to first-order logic.
Unlike its predecessor Nitpick for Isabelle, it is designed as a stand-alone
tool, with frontends for various proof assistants. In this short paper, we
present some ideas to extend Nunchaku with partial support for dependent types
and type classes, to make frontends for Coq and other systems based on
dependent type theory more useful.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
A Purely Functional Computer Algebra System Embedded in Haskell
We demonstrate how methods in Functional Programming can be used to implement
a computer algebra system. As a proof-of-concept, we present the
computational-algebra package. It is a computer algebra system implemented as
an embedded domain-specific language in Haskell, a purely functional
programming language. Utilising methods in functional programming and prominent
features of Haskell, this library achieves safety, composability, and
correctness at the same time. To demonstrate the advantages of our approach, we
have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's
and , in a composable way.Comment: 16 pages, Accepted to CASC 201
Limit theorems for nondegenerate U-statistics of continuous semimartingales
This paper presents the asymptotic theory for nondegenerate -statistics of
high frequency observations of continuous It\^{o} semimartingales. We prove
uniform convergence in probability and show a functional stable central limit
theorem for the standardized version of the -statistic. The limiting process
in the central limit theorem turns out to be conditionally Gaussian with mean
zero. Finally, we indicate potential statistical applications of our
probabilistic results.Comment: Published in at http://dx.doi.org/10.1214/13-AAP983 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Discriminating quantum states: the multiple Chernoff distance
We consider the problem of testing multiple quantum hypotheses
, where an arbitrary prior
distribution is given and each of the hypotheses is copies of a quantum
state. It is known that the average error probability decays
exponentially to zero, that is, . However, this error
exponent is generally unknown, except for the case that .
In this paper, we solve the long-standing open problem of identifying the
above error exponent, by proving Nussbaum and Szko\l a's conjecture that
. The right-hand side of this equality is
called the multiple quantum Chernoff distance, and
has been previously
identified as the optimal error exponent for testing two hypotheses,
versus .
The main ingredient of our proof is a new upper bound for the average error
probability, for testing an ensemble of finite-dimensional, but otherwise
general, quantum states. This upper bound, up to a states-dependent factor,
matches the multiple-state generalization of Nussbaum and Szko\l a's lower
bound. Specialized to the case , we give an alternative proof to the
achievability of the binary-hypothesis Chernoff distance, which was originally
proved by Audenaert et al.Comment: v2: minor change
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
An Exercise in Invariant-based Programming with Interactive and Automatic Theorem Prover Support
Invariant-Based Programming (IBP) is a diagram-based correct-by-construction
programming methodology in which the program is structured around the
invariants, which are additionally formulated before the actual code. Socos is
a program construction and verification environment built specifically to
support IBP. The front-end to Socos is a graphical diagram editor, allowing the
programmer to construct invariant-based programs and check their correctness.
The back-end component of Socos, the program checker, computes the verification
conditions of the program and tries to prove them automatically. It uses the
theorem prover PVS and the SMT solver Yices to discharge as many of the
verification conditions as possible without user interaction. In this paper, we
first describe the Socos environment from a user and systems level perspective;
we then exemplify the IBP workflow by building a verified implementation of
heapsort in Socos. The case study highlights the role of both automatic and
interactive theorem proving in three sequential stages of the IBP workflow:
developing the background theory, formulating the program specification and
invariants, and proving the correctness of the final implementation.Comment: In Proceedings THedu'11, arXiv:1202.453
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