scholarly journals Polynomially Bounded Sequences and Polynomial Sequences

2015 ◽  
Vol 23 (3) ◽  
pp. 205-213
Author(s):  
Hiroyuki Okazaki ◽  
Yuichi Futa
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Turing Machine ◽  
Polynomial Time ◽  
Computation Time ◽  
Decision Problems ◽  
Computational Complexity Theory ◽  
Polynomial Sequences ◽  
Deterministic Turing Machine ◽  
Bounded Sequences

Abstract In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].

scholarly journals The universality of polynomial time Turing equivalence

2016 ◽  
Vol 28 (3) ◽  
pp. 448-456 ◽  
Author(s):  
ANDREW MARKS
Keyword(s):  
Computational Complexity ◽  
Large Class ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Equivalence Relations ◽  
Turing Degrees ◽  
Computational Complexity Theory ◽  
Borel Sets ◽  
Borel Equivalence Relations

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

scholarly journals The trouble with the second quantifier

4OR ◽  
2021 ◽  
Author(s):  
Gerhard J. Woeginger
Keyword(s):  
Computational Complexity ◽  
Robust Optimization ◽  
Complexity Theory ◽  
Operational Research ◽  
Optimization Problems ◽  
Bilevel Optimization ◽  
Theoretical Background ◽  
Research Community ◽  
Universal Quantifier ◽  
Computational Complexity Theory

AbstractWe survey optimization problems that allow natural simple formulations with one existential and one universal quantifier. We summarize the theoretical background from computational complexity theory, and we present a multitude of illustrating examples. We discuss the connections to robust optimization and to bilevel optimization, and we explain the reasons why the operational research community should be interested in the theoretical aspects of this area.

The future of computational complexity theory: part II

1996 ◽  
Vol 27 (4) ◽  
pp. 3-7
Author(s):  
E. Allender ◽  
J. Feigenbaum ◽  
J. Goldsmith ◽  
T. Pitassi ◽  
S. Rudich
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Computational Complexity Theory ◽  
The Future

scholarly journals Computational Complexity Theory and the Philosophy of Mathematics†

2019 ◽  
Vol 27 (3) ◽  
pp. 381-439
Author(s):  
Walter Dean
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Complexity Theory ◽  
Philosophy Of Mathematics ◽  
Computability Theory ◽  
Computational Complexity Theory ◽  
Mathematical Problems ◽  
Np Problem

Abstract Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

scholarly journals Incremental FPT Delay

Algorithms ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 122
Author(s):  
Arne Meier
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Complexity Classes ◽  
Computational Complexity Theory ◽  
Function Classes ◽  
Classical Setting ◽  
Classical Function ◽  
Multivalued Functions ◽  
Relationship Of ◽  
The Relationship

In this paper, we study the relationship of parameterized enumeration complexity classes defined by Creignou et al. (MFCS 2013). Specifically, we introduce two hierarchies (IncFPTa and CapIncFPTa) of enumeration complexity classes for incremental fpt-time in terms of exponent slices and show how they interleave. Furthermore, we define several parameterized function classes and, in particular, introduce the parameterized counterpart of the class of nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist, TFNP, known from Megiddo and Papadimitriou (TCS 1991). We show that this class TF(para-NP), the restriction of the function variant of NP to total functions, collapsing to F(FPT), the function variant of FPT, is equivalent to the result that OutputFPT coincides with IncFPT. In addition, these collapses are shown to be equivalent to TFNP = FP, and also equivalent to P equals NP intersected with coNP. Finally, we show that these two collapses are equivalent to the collapse of IncP and OutputP in the classical setting. These results are the first direct connections of collapses in parameterized enumeration complexity to collapses in classical enumeration complexity, parameterized function complexity, classical function complexity, and computational complexity theory.

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