Automating algebraic proof systems is NP-hard

2021 ◽  
Author(s):  
Susanna F. de Rezende ◽  
Mika Göös ◽  
Jakob Nordström ◽  
Toniann Pitassi ◽  
Robert Robere ◽  
...  
Keyword(s):  
Np Hard ◽  
Algebraic Proof ◽  
Proof Systems

scholarly journals Algebraic proof systems over formulas

2003 ◽  
Vol 303 (1) ◽  
pp. 83-102 ◽  
Author(s):  
Dima Grigoriev ◽  
Edward A. Hirsch
Keyword(s):  
Algebraic Proof ◽  
Proof Systems

Minimum propositional proof length is NP-hard to linearly approximate

2001 ◽  
Vol 66 (1) ◽  
pp. 171-191 ◽  
Author(s):  
Michael Alekhnovich ◽  
Sam Buss ◽  
Shlomo Moran ◽  
Toniann Pitassi
Keyword(s):  
Assignment Problem ◽  
Sequent Calculus ◽  
Hardness Of Approximation ◽  
Np Hard ◽  
Satisfying Assignment ◽  
Proof Systems ◽  
Frege Systems ◽  
Propositional Proof ◽  
Almost All ◽  
Polynomial Calculus

AbstractWe prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within a factor of . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution. Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs.

A Framework for Space Complexity in Algebraic Proof Systems

2015 ◽  
Vol 62 (3) ◽  
pp. 1-20 ◽  
Author(s):  
Ilario Bonacina ◽  
Nicola Galesi
Keyword(s):  
Space Complexity ◽  
Algebraic Proof ◽  
Proof Systems

scholarly journals Uniformly Generated Submodules of Permutation Modules

1998 ◽  
Vol 5 (20) ◽  
Author(s):  
Søren Riis ◽  
Meera Sitharam
Keyword(s):  
Representation Theory ◽  
Vector Space ◽  
Complexity Theory ◽  
Singular Values ◽  
Singular Value ◽  
Proof Complexity ◽  
Algebraic Proof ◽  
Proof Systems ◽  
Special Cases ◽  
Np Problem

This paper is motivated by a link between algebraic proof<br />complexity and the representation theory of the finite symmetric<br />groups. Our perspective leads to a series of non-traditional<br />problems in the representation theory of Sn.<br />Most of our technical results concern the structure of "uniformly"<br />generated submodules of permutation modules. We consider<br />(for example) sequences Wn of submodules of the permutation<br />modules M(n−k;1k) and prove that if the modules Wn are<br />given in a uniform way - which we make precise - the dimension<br />p(n) of Wn (as a vector space) is a single polynomial with rational<br />coefficients, for all but finitely many "singular" values of n. Furthermore, we show that dim(Wn) < p(n) for each singular value of n >= 4k. The results have a non-traditional flavor arising from the study of the irreducible structure of the submodules Wn beyond isomorphism types. We sketch the link between our structure theorems and proof complexity questions, which can be viewed as special cases of the famous NP vs. co-NP problem in complexity theory. In particular, we focus on the efficiency of proof systems for showing membership in polynomial ideals, for example, based on Hilbert's Nullstellensatz.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

Secure Cloud Quantum Computing with Verification Based on Quantum Interactive Proof

Impact ◽  
2019 ◽  
Vol 2019 (10) ◽  
pp. 30-32
Author(s):  
Tomoyuki Morimae
Keyword(s):  
Quantum Computing ◽  
Quantum Information ◽  
Theoretical Physics ◽  
View Point ◽  
Research Subjects ◽  
Interactive Proof ◽  
Open Problems ◽  
Main Research ◽  
Proof Systems ◽  
Information Group

In cloud quantum computing, a classical client delegate quantum computing to a remote quantum server. An important property of cloud quantum computing is the verifiability: the client can check the integrity of the server. Whether such a classical verification of quantum computing is possible or not is one of the most important open problems in quantum computing. We tackle this problem from the view point of quantum interactive proof systems. Dr Tomoyuki Morimae is part of the Quantum Information Group at the Yukawa Institute for Theoretical Physics at Kyoto University, Japan. He leads a team which is concerned with two main research subjects: quantum supremacy and the verification of quantum computing.

scholarly journals Optimizing Transmit Sequence and Instrumental Variables Receiver for Dual-Function Complexity System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Yu Yao ◽  
Junhui Zhao ◽  
Lenan Wu
Keyword(s):  
Approximate Solution ◽  
Instrumental Variables ◽  
Performance Index ◽  
Dual Function ◽  
Np Hard ◽  
Match Filter ◽  
Filter Output ◽  
Phase Code ◽  
Discrete Phase ◽  
Simulation Results

This correspondence deals with the joint cognitive design of transmit coded sequences and instrumental variables (IV) receive filter to enhance the performance of a dual-function radar-communication (DFRC) system in the presence of clutter disturbance. The IV receiver can reject clutter more efficiently than the match filter. The signal-to-clutter-and-noise ratio (SCNR) of the IV filter output is viewed as the performance index of the complexity system. We focus on phase only sequences, sharing both a continuous and a discrete phase code and develop optimization algorithms to achieve reasonable pairs of transmit coded sequences and IV receiver that fine approximate the behavior of the optimum SCNR. All iterations involve the solution of NP-hard quadratic fractional problems. The relaxation plus randomization technique is used to find an approximate solution. The complexity, corresponding to the operation of the proposed algorithms, depends on the number of acceptable iterations along with on and the complexity involved in all iterations. Simulation results are offered to evaluate the performance generated by the proposed scheme.

On the Complexity of Deadlock Recovery

1986 ◽  
Vol 9 (3) ◽  
pp. 323-342
Author(s):  
Joseph Y.-T. Leung ◽  
Burkhard Monien
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Arbitrary Number ◽  
The Other ◽  
Np Hard ◽  
Total Cost ◽  
Other Hand ◽  
Input Parameters ◽  
Resource Type

We consider the computational complexity of finding an optimal deadlock recovery. It is known that for an arbitrary number of resource types the problem is NP-hard even when the total cost of deadlocked jobs and the total number of resource units are “small” relative to the number of deadlocked jobs. It is also known that for one resource type the problem is NP-hard when the total cost of deadlocked jobs and the total number of resource units are “large” relative to the number of deadlocked jobs. In this paper we show that for one resource type the problem is solvable in polynomial time when the total cost of deadlocked jobs or the total number of resource units is “small” relative to the number of deadlocked jobs. For fixed m ⩾ 2 resource types, we show that the problem is solvable in polynomial time when the total number of resource units is “small” relative to the number of deadlocked jobs. On the other hand, when the total number of resource units is “large”, the problem becomes NP-hard even when the total cost of deadlocked jobs is “small” relative to the number of deadlocked jobs. The results in the paper, together with previous known ones, give a complete delineation of the complexity of this problem under various assumptions of the input parameters.

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