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Frank Vega - One of the best experts on this subject based on the ideXlab platform.
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A Solution of the P versus NP Problem
2019Co-Authors: Frank VegaAbstract:P versus NP is a major unsolved problem in computer science. It is considered by many to be the most important open problem in the field. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute. Another major Complexity Class is coNP. We show a problem that is in coNP. However, we prove this one cannot be solved in polynomial time. Hence, we demonstrate the separation from the Classes P and coNP. In addition, this also shows the Complexity Class P is not equal to NP.
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The Complexity of Class L
2019Co-Authors: Frank VegaAbstract:A major Complexity Classes are $L$ and $POLYLOGTIME$. A logarithmic Turing machine has a read-only input tape, a write-only output tape, and some read/write work tapes. The work tapes may contain at most $O(\log n)$ symbols. $L$ is the Complexity Class containing those decision problems that can be decided by a deterministic logarithmic Turing machine. We define the Complexity Class $POLYLOGTIME$ as the problems which can be decided by a random access machine in poly-logarithmic time. We prove there is problem in the Complexity Class $L$ which cannot be solved by a random access machine in poly-logarithmic time. Hence, we show $L \nsubseteq POLYLOGTIME$.
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P versus NP
2018Co-Authors: Frank VegaAbstract:P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major Complexity Class is FP. We show a problem that is not in FP. Under the assumption of P = NP, we prove the membership of this problem in FP is also hold. In this way, we demonstrate the Complexity Class P is not equal to NP by the reduction ad absurdum rule.
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UP versus NP
2018Co-Authors: Frank VegaAbstract:P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major Complexity Class is UP. Whether UP = NP is another fundamental question that it is as important as it is unresolved. To attack the UP = NP question the concept of NP-completeness is very useful. If any single NP-complete problem is in UP, then UP = NP. Quadratic Congruences is a well-known NP-complete problem. We prove Quadratic Congruences is also in UP. In this way, we demonstrate that UP = NP.
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On P versus NP
2017Co-Authors: Frank VegaAbstract:P versus NP is considered one of the great open problems of science. This consists in knowing the answer of the following question: Is P equal to NP? This incognita was first mentioned in a letter written by John Nash to the National Security Agency in 1955. However, a precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this huge problem have failed. Another major Complexity Class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. We prove there exists a problem in coNP that is not in P. In this way, we show that P is not equal to coNP. Since P = NP implies P = coNP, then we also demonstrate that P is not equal to NP.
Milind Sohoni - One of the best experts on this subject based on the ideXlab platform.
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geometric Complexity theory ii towards explicit obstructions for embeddings among Class varieties
SIAM Journal on Computing, 2008Co-Authors: Ketan Mulmuley, Milind SohoniAbstract:In [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496-526], henceforth referred to as Part I, we suggested an approach to the $P$ vs. $NP$ and related lower bound problems in Complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the $NP \not \subseteq P$ conjecture to the problem of showing that a variety associated with the Complexity Class $NP$ cannot be embedded in a variety associated with the Complexity Class $P$. We shall call these Class varieties associated with the Complexity Classes $P$ and $NP$. This paper develops this approach further, reducing these lower bound problems—which are all nonexistence problems—to some existence problems: specifically to proving the existence of obstructions to such embeddings among Class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel-Weil theorem to a Class of orbit closures, which include Class varieties. The second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the Class variety associated with the Complexity Class $NC$. These results indicate that the fundamental lower bound problems in Complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory. The results here were announced in [K. D. Mulmuley and M. Sohoni, in Advances in Algebra and Geometry (Hyderabad, $2001$), Hindustan Book Agency, New Delhi, India, 2003, pp. 239-261].
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geometric Complexity theory ii towards explicit obstructions for embeddings among Class varieties
arXiv: Computational Complexity, 2006Co-Authors: Ketan Mulmuley, Milind SohoniAbstract:In part I we reduced the arithmetic (characteristic zero) version of the P \not \subseteq NP conjecture to the problem of showing that a variety associated with the Complexity Class NP cannot be embedded in the variety associated the Complexity Class P. We call these Class varieties. In this paper, this approach is developed further, reducing the nonexistence problems, such as the P vs. NP and related lower bound problems, to existence problems: specifically to proving existence of obstructions to such embeddings among Class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel-Weil theorem to a Class of orbit closures, which include Class varieties. The recond result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the Class variety associated with the Complexity Class NC. These results indicate that the fundamental lower bound problems in Complexity theory are intimately linked with explicit construction problems in algebraic geometry and representation theory.
Jonathan P Dowling - One of the best experts on this subject based on the ideXlab platform.
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sampling arbitrary photon added or photon subtracted squeezed states is in the same Complexity Class as boson sampling
Physical Review A, 2015Co-Authors: Jonathan P Olson, Kaushik P Seshadreesan, Keith R Motes, Peter P Rohde, Jonathan P DowlingAbstract:Boson sampling is a simple model for non-universal linear optics quantum computing using far fewer physical resources than universal schemes. An input state comprising vacuum and single photon states is fed through a Haar-random linear optics network and sampled at the output using coincidence photodetection. This problem is strongly believed to be Classically hard to simulate. We show that an analogous procedure implements the same problem, using photon-added or -subtracted squeezed vacuum states (with arbitrary squeezing), where sampling at the output is performed via parity measurements. The equivalence is exact and independent of the squeezing parameter, and hence provides an entire Class of new quantum states of light in the same Complexity Class as boson sampling.
Jack H. Lutz - One of the best experts on this subject based on the ideXlab platform.
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Dimension Characterizations of Complexity Classes
computational complexity, 2008Co-Authors: Jack H. LutzAbstract:We use derandomization to show that sequences of positive pspace-dimension − in fact, even positive $$\Delta ^{p}_{k}$$ -dimension for suitable k − have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose $$\Delta ^{p}_{3}$$ -dimension is positive, then $${\rm BPP} \subseteq {\rm P} ^S$$ and, moreover, every BPP promise problem is P^ S -separable. We prove analogous results at higher levels of the polynomial-time hierarchy. The dimension-almost-Class of a Complexity Class $$\mathcal C$$ , denoted by dimalmost- $$\mathcal C$$ , is the Class consisting of all problems A such that $$A \in \mathcal C^{S}$$ for all but a Hausdorff dimension 0 set of oracles S . Our results yield several characterizations of Complexity Classes, such as BPP = dimalmost-P, Promise-BPP = dimalmost-P-Sep, and AM = dimalmost-NP, that refine previously known results on almost-Classes.
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STACS - Equivalence of Measures of Complexity Classes
Lecture Notes in Computer Science, 1997Co-Authors: Josef M. Breutzmann, Jack H. LutzAbstract:The resource-bounded measures of Complexity Classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number δ > 0, any uniformly polynomial-time computable sequence β=(β0,β1,β2), ... of real numbers (biases) β i e [δ, 1−δ], and any Complexity Class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomial-time, truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent.
Peter P Rohde - One of the best experts on this subject based on the ideXlab platform.
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sampling arbitrary photon added or photon subtracted squeezed states is in the same Complexity Class as boson sampling
Physical Review A, 2015Co-Authors: Jonathan P Olson, Kaushik P Seshadreesan, Keith R Motes, Peter P Rohde, Jonathan P DowlingAbstract:Boson sampling is a simple model for non-universal linear optics quantum computing using far fewer physical resources than universal schemes. An input state comprising vacuum and single photon states is fed through a Haar-random linear optics network and sampled at the output using coincidence photodetection. This problem is strongly believed to be Classically hard to simulate. We show that an analogous procedure implements the same problem, using photon-added or -subtracted squeezed vacuum states (with arbitrary squeezing), where sampling at the output is performed via parity measurements. The equivalence is exact and independent of the squeezing parameter, and hence provides an entire Class of new quantum states of light in the same Complexity Class as boson sampling.