1,258 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
The logic of interactive Turing reduction
The paper gives a soundness and completeness proof for the implicative
fragment of intuitionistic calculus with respect to the semantics of
computability logic, which understands intuitionistic implication as
interactive algorithmic reduction. This concept -- more precisely, the
associated concept of reducibility -- is a generalization of Turing
reducibility from the traditional, input/output sorts of problems to
computational tasks of arbitrary degrees of interactivity. See
http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on
computability logic
The intuitionistic fragment of computability logic at the propositional level
This paper presents a soundness and completeness proof for propositional
intuitionistic calculus with respect to the semantics of computability logic.
The latter interprets formulas as interactive computational problems,
formalized as games between a machine and its environment. Intuitionistic
implication is understood as algorithmic reduction in the weakest possible --
and hence most natural -- sense, disjunction and conjunction as
deterministic-choice combinations of problems (disjunction = machine's choice,
conjunction = environment's choice), and "absurd" as a computational problem of
universal strength. See http://www.cis.upenn.edu/~giorgi/cl.html for a
comprehensive online source on computability logic
On algorithm and robustness in a non-standard sense
In this paper, we investigate the invariance properties, i.e. robust- ness, of phenomena related to the notions of algorithm, finite procedure and explicit construction. First of all, we provide two examples of objects for which small changes completely change their (non)computational behavior. We then isolate robust phenomena in two disciplines related to computability
Propositional computability logic I
In the same sense as classical logic is a formal theory of truth, the
recently initiated approach called computability logic is a formal theory of
computability. It understands (interactive) computational problems as games
played by a machine against the environment, their computability as existence
of a machine that always wins the game, logical operators as operations on
computational problems, and validity of a logical formula as being a scheme of
"always computable" problems. The present contribution gives a detailed
exposition of a soundness and completeness proof for an axiomatization of one
of the most basic fragments of computability logic. The logical vocabulary of
this fragment contains operators for the so called parallel and choice
operations, and its atoms represent elementary problems, i.e. predicates in the
standard sense. This article is self-contained as it explains all relevant
concepts. While not technically necessary, however, familiarity with the
foundational paper "Introduction to computability logic" [Annals of Pure and
Applied Logic 123 (2003), pp.1-99] would greatly help the reader in
understanding the philosophy, underlying motivations, potential and utility of
computability logic, -- the context that determines the value of the present
results. Online introduction to the subject is available at
http://www.cis.upenn.edu/~giorgi/cl.html and
http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi
Weihrauch goes Brouwerian
We prove that the Weihrauch lattice can be transformed into a Brouwer algebra
by the consecutive application of two closure operators in the appropriate
order: first completion and then parallelization. The closure operator of
completion is a new closure operator that we introduce. It transforms any
problem into a total problem on the completion of the respective types, where
we allow any value outside of the original domain of the problem. This closure
operator is of interest by itself, as it generates a total version of Weihrauch
reducibility that is defined like the usual version of Weihrauch reducibility,
but in terms of total realizers. From a logical perspective completion can be
seen as a way to make problems independent of their premises. Alongside with
the completion operator and total Weihrauch reducibility we need to study
precomplete representations that are required to describe these concepts. In
order to show that the parallelized total Weihrauch lattice forms a Brouwer
algebra, we introduce a new multiplicative version of an implication. While the
parallelized total Weihrauch lattice forms a Brouwer algebra with this
implication, the total Weihrauch lattice fails to be a model of intuitionistic
linear logic in two different ways. In order to pinpoint the algebraic reasons
for this failure, we introduce the concept of a Weihrauch algebra that allows
us to formulate the failure in precise and neat terms. Finally, we show that
the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which
also implies that the theory of our Brouwer algebra is Jankov logic.Comment: 36 page
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