941 research outputs found
On the Weak Computability of Continuous Real Functions
In computable analysis, sequences of rational numbers which effectively
converge to a real number x are used as the (rho-) names of x. A real number x
is computable if it has a computable name, and a real function f is computable
if there is a Turing machine M which computes f in the sense that, M accepts
any rho-name of x as input and outputs a rho-name of f(x) for any x in the
domain of f. By weakening the effectiveness requirement of the convergence and
classifying the converging speeds of rational sequences, several interesting
classes of real numbers of weak computability have been introduced in
literature, e.g., in addition to the class of computable real numbers (EC), we
have the classes of semi-computable (SC), weakly computable (WC), divergence
bounded computable (DBC) and computably approximable real numbers (CA). In this
paper, we are interested in the weak computability of continuous real functions
and try to introduce an analogous classification of weakly computable real
functions. We present definitions of these functions by Turing machines as well
as by sequences of rational polygons and prove these two definitions are not
equivalent. Furthermore, we explore the properties of these functions, and
among others, show their closure properties under arithmetic operations and
composition
Further Results on Size and Power of Heteroskedasticity and Autocorrelation Robust Tests, with an Application to Trend Testing
We complement the theory developed in Preinerstorfer and P\"otscher (2016)
with further finite sample results on size and power of heteroskedasticity and
autocorrelation robust tests. These allows us, in particular, to show that the
sufficient conditions for the existence of size-controlling critical values
recently obtained in P\"otscher and Preinerstorfer (2018) are often also
necessary. We furthermore apply the results obtained to tests for hypotheses on
deterministic trends in stationary time series regressions, and find that many
tests currently used are strongly size-distorted.Comment: Revised version. Some restructuring, some errors corrected, new
results adde
Beta-Conversion, Efficiently
Type-checking in dependent type theories relies on conversion, i.e. testing given lambda-terms for equality up to beta-evaluation and alpha-renaming.
Computer tools based on the lambda-calculus currently implement conversion by means of algorithms whose complexity has not been identified, and in some cases even subject to an exponential time overhead with respect to the natural cost models (number of evaluation steps and size of input lambda-terms).
This dissertation shows that in the pure lambda-calculus it is possible to obtain conversion algorithms with bilinear time complexity when evaluation is carried following evaluation strategies that generalize Call-by-Value to the stronger case required by conversion
A Dependently Typed Language with Nontermination
We propose a full-spectrum dependently typed programming language, Zombie, which supports general recursion natively. The Zombie implementation is an elaborating typechecker. We prove type saftey for a large subset of the Zombie core language, including features such as computational irrelevance, CBV-reduction, and propositional equality with a heterogeneous, completely erased elimination form. Zombie does not automatically beta-reduce expressions, but instead uses congruence closure for proof and type inference. We give a specification of a subset of the surface language via a bidirectional type system, which works up-to-congruence, and an algorithm for elaborating expressions in this language to an explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system. Zombie also features an optional termination-checker, allowing nonterminating programs returning proofs as well as external proofs about programs
In Search of Effectful Dependent Types
Real world programming languages crucially depend on the availability of
computational effects to achieve programming convenience and expressive power
as well as program efficiency. Logical frameworks rely on predicates, or
dependent types, to express detailed logical properties about entities.
According to the Curry-Howard correspondence, programming languages and logical
frameworks should be very closely related. However, a language that has both
good support for real programming and serious proving is still missing from the
programming languages zoo. We believe this is due to a fundamental lack of
understanding of how dependent types should interact with computational
effects. In this thesis, we make a contribution towards such an understanding,
with a focus on semantic methods.Comment: PhD thesis, Version submitted to Exam School
On Quasi-Interpretations, Blind Abstractions and Implicit Complexity
Quasi-interpretations are a technique to guarantee complexity bounds on
first-order functional programs: with termination orderings they give in
particular a sufficient condition for a program to be executable in polynomial
time, called here the P-criterion. We study properties of the programs
satisfying the P-criterion, in order to better understand its intensional
expressive power. Given a program on binary lists, its blind abstraction is the
nondeterministic program obtained by replacing lists by their lengths (natural
numbers). A program is blindly polynomial if its blind abstraction terminates
in polynomial time. We show that all programs satisfying a variant of the
P-criterion are in fact blindly polynomial. Then we give two extensions of the
P-criterion: one by relaxing the termination ordering condition, and the other
one (the bounded value property) giving a necessary and sufficient condition
for a program to be polynomial time executable, with memoisation.Comment: 18 page
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