scholarly journals The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines

2020 ◽  
Author(s):  
Matheus Santana Lima
Keyword(s):  
Communication Systems ◽  
Decision Problem ◽  
Communication Channel ◽  
Solution Space ◽  
Search Space ◽  
Decision Problems ◽  
Turing Machines ◽  
Bernoulli Process ◽  
General Process ◽  
New Interpretation

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.

scholarly journals The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines

2020 ◽  
Author(s):  
Matheus Santana Lima
Keyword(s):  
Communication Systems ◽  
Decision Problem ◽  
Communication Channel ◽  
Solution Space ◽  
Search Space ◽  
Decision Problems ◽  
Turing Machines ◽  
Bernoulli Process ◽  
General Process ◽  
New Interpretation

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.

scholarly journals The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines

2020 ◽  
Author(s):  
Matheus Santana Lima
Keyword(s):  
Communication Systems ◽  
Decision Problem ◽  
Communication Channel ◽  
Solution Space ◽  
Search Space ◽  
Decision Problems ◽  
Turing Machines ◽  
Bernoulli Process ◽  
General Process ◽  
New Interpretation

We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials problems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines.

scholarly journals Algorithmic-Information Theory interpretation to the Traveling Salesman Problem

2019 ◽  
Author(s):  
Matheus Santana Lima
Keyword(s):  
Information Theory ◽  
Traveling Salesman Problem ◽  
Time Complexity ◽  
Communication Systems ◽  
Communication Channel ◽  
Transmission Rate ◽  
Solution Space ◽  
Traveling Salesman ◽  
Computational Time ◽  
The Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a important optimization problem in computer science, mathematics and logistics. It belongs to the class of NP-Hard problems and can be very time consuming to find solutions to large instances with guarantee optimality. As number of city-nodes in the graph increases, the amount of valid route tours also growths rapidly and thus requiring considerable time to evaluate and classify each permutation. The objective of the heuristic process is to search the solution space for the optimal solution while maximizing the attached utility-cost function (i.e. finding the shortest euclidean distance tour) and minimizing the computational time complexity of the algorithm.Many complex real world scenarios can be reduced to a simulation of a salesman trying to find the shortest (length) Hamiltonian (cycle) route in a euclidean super-graph G*. If each city-node is modeled as a input symbol in a communication channel represented by an output pair with consistent probabilities distribution thus an polynomial-time probabilistic algorithm can use this information to improve the solution quality at the same rate of transmission of information over the channel.In this paper we explore an quantitative stochastic process based in Algorithm Information Theory and the Shannon-Kelly criterion to find valid near optimal solutions using a new growth- optimal strategy applied to the TSP problem that have statistically significant transmission rate even when no encoding scheme is available, regardless of time-complexity of the problem.Previous heuristics such as 2 opt, Genetic Algorithms (GA) and Simulated Annealing (SA) approach’s the TSP problem by relying on a priori knowledge about the data distribution in order to reduce the probability of error in finding the best candidate solution tour.In this work we propose a method that models the solution space boundaries of the TSP problem as a communication channel by means of Information Theory. We describe a search algorithm that check for patterns (i.e information content) in the elements of a constrained solution space modeled as messages transmitted through communication systems. The boundaries of the search space are defined by the Kolmogorov complexity of the candidate solutions sequences. We conclude with an discussion about the quality of the results and implications for general decision problem in Turing machines.

Some undecidability results in strong algebraic languages

1984 ◽  
Vol 49 (3) ◽  
pp. 951-954
Author(s):  
Cornelia Kalfa
Keyword(s):  
Decision Problem ◽  
Decision Problems ◽  
Turing Machines ◽  
Constant Symbol ◽  
Finite Sets ◽  
Operation Symbol ◽  
Halting Problem ◽  
First Order ◽  
Order Language ◽  
Image Position

The recursively unsolvable halting problem for Turing machines is reduced to the problem of the existence or not of an algorithm for deciding whether a field is finite. The latter problem is further reduced to the decision problem of each of propertiesfor recursive sets Σ of equations of strong algebraic languages with infinitely many operation symbols.Decision problems concerning properties of sets of equations were first raised by Tarski [9] and subsequently examined by Perkins [6], McKenzie [4], McNulty [5] and Pigozzi [7]. Perkins is the only one who studied recursive sets; the others investigated finite sets. Since the undecidability of properties Pi for recursive sets of equations does not imply any answer to the corresponding decision problems for finite sets, the latter problems remain open.The work presented here is part of my Ph.D. thesis [2]. I thank Wilfrid Hodges, who supervised it.An algebraic language is a first-order language with equality but without relation symbols. It is here denoted by , where Qi is an operation symbol and cj, is a constant symbol.

Logical Decision Problems and Complexity of Logic Programs

1987 ◽  
Vol 10 (1) ◽  
pp. 1-33
Author(s):  
Egon Börger ◽  
Ulrich Löwen
Keyword(s):  
Fixed Point ◽  
Computational Complexity ◽  
Decision Problem ◽  
Decision Problems ◽  
Complexity Classes ◽  
Logic Programs ◽  
Finite Structures ◽  
Syntactic Restrictions

We survey and give new results on logical characterizations of complexity classes in terms of the computational complexity of decision problems of various classes of logical formulas. There are two main approaches to obtain such results: The first approach yields logical descriptions of complexity classes by semantic restrictions (to e.g. finite structures) together with syntactic enrichment of logic by new expressive means (like e.g. fixed point operators). The second approach characterizes complexity classes by (the decision problem of) classes of formulas determined by purely syntactic restrictions on the formation of formulas.

scholarly journals COVID-19 Vaccination Priority Evaluation

2021 ◽  
Author(s):  
Jozo J Dujmovic ◽  
Daniel Tomasevich
Keyword(s):  
Decision Problem ◽  
Evaluation Method ◽  
Social Conditions ◽  
Decision Problems ◽  
Organ Transplants ◽  
Quantitative Criterion

Computing the COVID-19 vaccination priority is an urgent and ubiquitous decision problem. In this paper we propose a solution of this problem using the LSP evaluation method. Our goal is to develop a justifiable and explainable quantitative criterion for computing a vaccination priority degree for each individual in a population. Performing vaccination in the order of the decreasing vaccination priority produces maximum positive medical, social, and ethical effects for the whole population. The presented method can be expanded and refined using additional medical and social conditions. In addition, the same methodology is suitable for solving other similar medical priority decision problems, such as priorities for organ transplants.

scholarly journals Flow complexity in open systems: interlacing complexity index based on mutual information

2017 ◽  
Vol 825 ◽  
pp. 704-742 ◽  
Author(s):  
Jose M. Pozo ◽  
Arjan J. Geers ◽  
Maria-Cruz Villa-Uriol ◽  
Alejandro F. Frangi
Keyword(s):  
Mutual Information ◽  
Communication Systems ◽  
Communication Channel ◽  
Communication Theory ◽  
Open Systems ◽  
Straight Tube ◽  
Natural Distribution ◽  
Ergodic Processes ◽  
Complexity Index ◽  
Crucial Component

Flow complexity is related to a number of phenomena in science and engineering and has been approached from the perspective of chaotic dynamical systems, ergodic processes or mixing of fluids, just to name a few. To the best of our knowledge, all existing methods to quantify flow complexity are only valid for infinite time evolution, for closed systems or for mixing of two substances. We introduce an index of flow complexity coined interlacing complexity index (ICI), valid for a single-phase flow in an open system with inlet and outlet regions, involving finite times. ICI is based on Shannon’s mutual information (MI), and inspired by an analogy between inlet–outlet open flow systems and communication systems in communication theory. The roles of transmitter, receiver and communication channel are played, respectively, by the inlet, the outlet and the flow transport between them. A perfectly laminar flow in a straight tube can be compared to an ideal communication channel where the transmitted and received messages are identical and hence the MI between input and output is maximal. For more complex flows, generated by more intricate conditions or geometries, the ability to discriminate the outlet position by knowing the inlet position is decreased, reducing the corresponding MI. The behaviour of the ICI has been tested with numerical experiments on diverse flows cases. The results indicate that the ICI provides a sensitive complexity measure with intuitive interpretation in a diversity of conditions and in agreement with other observations, such as Dean vortices and subjective visual assessments. As a crucial component of the ICI formulation, we also introduce the natural distribution of streamlines and the natural distribution of world-lines, with invariance properties with respect to the cross-section used to parameterize them, valid for any type of mass-preserving flow.

Space Contraction and Partition Algorithm for Combinatorial Optimization

2011 ◽  
Vol 421 ◽  
pp. 559-563
Author(s):  
Yong Chao Gao ◽  
Li Mei Liu ◽  
Heng Qian ◽  
Ding Wang
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Solution Space ◽  
Search Space ◽  
Small Scale ◽  
Optimal Solutions ◽  
Solution Quality ◽  
Intelligent Algorithms ◽  
Combinatorial Optimization Problems

The scale and complexity of search space are important factors deciding the solving difficulty of an optimization problem. The information of solution space may lead searching to optimal solutions. Based on this, an algorithm for combinatorial optimization is proposed. This algorithm makes use of the good solutions found by intelligent algorithms, contracts the search space and partitions it into one or several optimal regions by backbones of combinatorial optimization solutions. And optimization of small-scale problems is carried out in optimal regions. Statistical analysis is not necessary before or through the solving process in this algorithm, and solution information is used to estimate the landscape of search space, which enhances the speed of solving and solution quality. The algorithm breaks a new path for solving combinatorial optimization problems, and the results of experiments also testify its efficiency.

Chapter 13. Symmetry and Satisfiability

2021 ◽  
Author(s):  
Karem A. Sakallah
Keyword(s):  
Group Theory ◽  
Normal Form ◽  
Boolean Functions ◽  
Search Algorithm ◽  
Conjunctive Normal Form ◽  
Search Space ◽  
Decision Problems ◽  
Mathematical Language ◽  
Sat Solvers ◽  
Speed Up

Symmetry is at once a familiar concept (we recognize it when we see it!) and a profoundly deep mathematical subject. At its most basic, a symmetry is some transformation of an object that leaves the object (or some aspect of the object) unchanged. For example, a square can be transformed in eight different ways that leave it looking exactly the same: the identity “do-nothing” transformation, 3 rotations, and 4 mirror images (or reflections). In the context of decision problems, the presence of symmetries in a problem’s search space can frustrate the hunt for a solution by forcing a search algorithm to fruitlessly explore symmetric subspaces that do not contain solutions. Recognizing that such symmetries exist, we can direct a search algorithm to look for solutions only in non-symmetric parts of the search space. In many cases, this can lead to significant pruning of the search space and yield solutions to problems which are otherwise intractable. This chapter explores the symmetries of Boolean functions, particularly the symmetries of their conjunctive normal form (CNF) representations. Specifically, it examines what those symmetries are, how to model them using the mathematical language of group theory, how to derive them from a CNF formula, and how to utilize them to speed up CNF SAT solvers.

scholarly journals Research of the Channel Estimation in Wireless Communication Systems

2019 ◽  
Vol 25 ◽  
pp. 01002 ◽  
Author(s):  
Lili Zhao ◽  
Peng Zhang ◽  
Qicai Dong ◽  
Xiangyang Huang ◽  
Jianhua Zhao ◽  
...  
Keyword(s):  
Wireless Communication ◽  
Channel Estimation ◽  
Communication Technology ◽  
Communication Systems ◽  
Communication Channel ◽  
Data Transfer ◽  
Estimation Methods ◽  
Wireless Communication Systems ◽  
Data Transfer Rate

Wireless communication technology has been developed rapidly after entering the 21st century. Data transfer rate increased significantly as well as the bandwidth became wider and wider from 2G to 4G in wireless communication systems. Channel estimation is an import part of any communication systems; its accuracy determines the quality of the whole communication. Channel estimation methods of typical wireless communication systems such as UWB, 2G and 3G have been researched.

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