25,695 research outputs found
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
Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes
Diagonalization in the spirit of Cantor's diagonal arguments is a widely used
tool in theoretical computer sciences to obtain structural results about
computational problems and complexity classes by indirect proofs. The Uniform
Diagonalization Theorem allows the construction of problems outside complexity
classes while still being reducible to a specific decision problem. This paper
provides a generalization of the Uniform Diagonalization Theorem by extending
it to promise problems and the complexity classes they form, e.g. randomized
and quantum complexity classes. The theorem requires from the underlying
computing model not only the decidability of its acceptance and rejection
behaviour but also of its promise-contradicting indifferent behaviour - a
property that we will introduce as "total decidability" of promise problems.
Implications of the Uniform Diagonalization Theorem are mainly of two kinds:
1. Existence of intermediate problems (e.g. between BQP and QMA) - also known
as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class
is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the
original Uniform Diagonalization Theorem the extension applies besides BQP and
QMA to a large variety of complexity class pairs, including combinations from
deterministic, randomized and quantum classes.Comment: 15 page
Complex Algebras of Arithmetic
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a
sequence of arithmetic and logical operations to be performed on sets of
natural numbers. Arithmetic circuits can also be viewed as the elements of the
smallest subalgebra of the complex algebra of the semiring of natural numbers.
In the present paper, we investigate the algebraic structure of complex
algebras of natural numbers, and make some observations regarding the
complexity of various theories of such algebras
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
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