7,000 research outputs found
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
Comments on Unitarity in the Antifield Formalism
It is shown that the local completeness condition introduced in the analysis
of the locality of the gauge fixed action in the antifield formalism plays also
a key role in the proof of unitarity.Comment: 15 pages, Latex error corrected, otherwise unchange
Completion of Choice
We systematically study the completion of choice problems in the Weihrauch
lattice. Choice problems play a pivotal role in Weihrauch complexity. For one,
they can be used as landmarks that characterize important equivalences classes
in the Weihrauch lattice. On the other hand, choice problems also characterize
several natural classes of computable problems, such as finite mind change
computable problems, non-deterministically computable problems, Las Vegas
computable problems and effectively Borel measurable functions. The closure
operator of completion generates the concept of total Weihrauch reducibility,
which is a variant of Weihrauch reducibility with total realizers. Logically
speaking, the completion of a problem is a version of the problem that is
independent of its premise. Hence, studying the completion of choice problems
allows us to study simultaneously choice problems in the total Weihrauch
lattice, as well as the question which choice problems can be made independent
of their premises in the usual Weihrauch lattice. The outcome shows that many
important choice problems that are related to compact spaces are complete,
whereas choice problems for unbounded spaces or closed sets of positive measure
are typically not complete.Comment: 30 page
Complexity of equivalence relations and preorders from computability theory
We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .Comment: To appear in J. Symb. Logi
Consistency and Completeness of Rewriting in the Calculus of Constructions
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism,
such as Coq, may greatly improve usability of the tool. Unfortunately adding an
arbitrary set of rewrite rules may render the underlying formal system
undecidable and inconsistent. While ways to ensure termination and confluence,
and hence decidability of type-checking, have already been studied to some
extent, logical consistency has got little attention so far. In this paper we
show that consistency is a consequence of canonicity, which in turn follows
from the assumption that all functions defined by rewrite rules are complete.
We provide a sound and terminating, but necessarily incomplete algorithm to
verify this property. The algorithm accepts all definitions that follow
dependent pattern matching schemes presented by Coquand and studied by McBride
in his PhD thesis. It also accepts many definitions by rewriting, containing
rules which depart from standard pattern matching.Comment: 20 page
The World of Combinatorial Fuzzy Problems and the Efficiency of Fuzzy Approximation Algorithms
We re-examine a practical aspect of combinatorial fuzzy problems of various
types, including search, counting, optimization, and decision problems. We are
focused only on those fuzzy problems that take series of fuzzy input objects
and produce fuzzy values. To solve such problems efficiently, we design fast
fuzzy algorithms, which are modeled by polynomial-time deterministic fuzzy
Turing machines equipped with read-only auxiliary tapes and write-only output
tapes and also modeled by polynomial-size fuzzy circuits composed of fuzzy
gates. We also introduce fuzzy proof verification systems to model the
fuzzification of nondeterminism. Those models help us identify four complexity
classes: Fuzzy-FPA of fuzzy functions, Fuzzy-PA and Fuzzy-NPA of fuzzy decision
problems, and Fuzzy-NPAO of fuzzy optimization problems. Based on a relative
approximation scheme targeting fuzzy membership degree, we formulate two
notions of "reducibility" in order to compare the computational complexity of
two fuzzy problems. These reducibility notions make it possible to locate the
most difficult fuzzy problems in Fuzzy-NPA and in Fuzzy-NPAO.Comment: A4, 10pt, 10 pages. This extended abstract already appeared in the
Proceedings of the Joint 7th International Conference on Soft Computing and
Intelligent Systems (SCIS 2014) and 15th International Symposium on Advanced
Intelligent Systems (ISIS 2014), December 3-6, 2014, Institute of Electrical
and Electronics Engineers (IEEE), pp. 29-35, 201
Finitary reducibility on equivalence relations
We introduce the notion of finitary computable reducibility on equivalence
relations on the natural numbers. This is a weakening of the usual notion of
computable reducibility, and we show it to be distinct in several ways. In
particular, whereas no equivalence relation can be -complete under
computable reducibility, we show that, for every , there does exist a
natural equivalence relation which is -complete under finitary
reducibility. We also show that our hierarchy of finitary reducibilities does
not collapse, and illustrate how it sharpens certain known results. Along the
way, we present several new results which use computable reducibility to
establish the complexity of various naturally defined equivalence relations in
the arithmetical hierarchy
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