scholarly journals A New Method for 3-Satisfiability Problem Solving Space Structure on Structural Entropy

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2005
Author(s):  
Chen Liang ◽  
Xiaofeng Wang ◽  
Lei Lu ◽  
Pengfei Niu
Keyword(s):  
Phase Transition ◽  
Belief Propagation ◽  
Solution Space ◽  
Space Structure ◽  
Label Propagation ◽  
Frustration Theory ◽  
Problem Solution ◽  
Analysis Model ◽  
Sat Problem ◽  
Structure Information

Analyzing the solution space structure and evolution of 3satisfiability (3-SAT) problem is an important way to study the difficulty of the solving satisfiability (SAT) problem. However, there is no unified analysis model for the spatial structure and evolution of solutions under different constraint densities. The analysis of different phase transition points and solution regions is based on different metric analysis models. The solution space of 3-SAT problem is obtained by planting strategy and belief propagation. According to the distribution of the influence of frozen variables on the solution, a label propagation algorithm based on planting strategy is proposed, is used to find the solution cluster, and then the structure entropy is used to measure its structure information. The structure entropy analysis model of 3-SAT problem solution space is established, and the unified analysis framework of solution space evolution and satisfiability phase transition is given. The experimental results show that the model is effective and can accurately analyze the evolution process of solution space and satisfiability phase transition, and verify the accuracy of interference phase transition point threshold predicted by long-range frustration theory.

scholarly journals The Phase Transition Analysis for the Random Regular Exact 2-(d,k)-SAT Problem

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1231
Author(s):  
Guoxia Nie ◽  
Daoyun Xu ◽  
Xiaofeng Wang ◽  
Xi Wang
Keyword(s):  
Phase Transition ◽  
High Probability ◽  
Solution Space ◽  
Regular Structure ◽  
Generation Model ◽  
Transition Analysis ◽  
Sat Problem ◽  
Second Moments ◽  
Instance Generation ◽  
Theoretical Results

In a regular (d,k)-CNF formula, each clause has length k and each variable appears d times. A regular structure such as this is symmetric, and the satisfiability problem of this symmetric structure is called the (d,k)-SAT problem for short. The regular exact 2-(d,k)-SAT problem is that for a (d,k)-CNF formula F, if there is a truth assignment T, then exactly two literals of each clause in F are true. If the formula F contains only positive or negative literals, then there is a satisfiable assignment T with a size of 2n/k such that F is 2-exactly satisfiable. This paper introduces the (d,k)-SAT instance generation model, constructs the solution space, and employs the method of the first and second moments to present the phase transition point d* of the 2-(d,k)-SAT instance with only positive literals. When d<d*, the 2-(d,k)-SAT instance can be satisfied with high probability. When d>d*, the 2-(d,k)-SAT instance can not be satisfied with high probability. Finally, the verification results demonstrate that the theoretical results are consistent with the experimental results.

scholarly journals Constraint Satisfaction by Survey Propagation

2005 ◽  
Author(s):  
Alfredo Braunstein ◽  
Marc Mézard
Keyword(s):  
Constraint Satisfaction ◽  
Message Passing ◽  
Statistical Physics ◽  
Belief Propagation ◽  
Solution Space ◽  
Error Correcting Codes ◽  
Constraint Satisfaction Problems ◽  
Probabilistic Methods ◽  
Survey Propagation ◽  
Algorithmic Strategy

Methods and analyses from statistical physics are of use not only in studying the performance of algorithms, but also in developing efficient algorithms. Here, we consider survey propagation (SP), a new approach for solving typical instances of random constraint satisfaction problems. SP has proven successful in solving random k-satisfiability (k -SAT) and random graph q-coloring (q-COL) in the “hard SAT” region of parameter space [79, 395, 397, 412], relatively close to the SAT/UNSAT phase transition discussed in the previous chapter. In this chapter we discuss the SP equations, and suggest a theoretical framework for the method [429] that applies to a wide class of discrete constraint satisfaction problems. We propose a way of deriving the equations that sheds light on the capabilities of the algorithm, and illustrates the differences with other well-known iterative probabilistic methods. Our approach takes into account the clustered structure of the solution space described in chapter 3, and involves adding an additional “joker” value that variables can be assigned. Within clusters, a variable can be frozen to some value, meaning that the variable always takes the same value for all solutions (satisfying assignments) within the cluster. Alternatively, it can be unfrozen, meaning that it fluctuates from solution to solution within the cluster. As we will discuss, the SP equations manage to describe the fluctuations by assigning joker values to unfrozen variables. The overall algorithmic strategy is iterative and decomposable in two elementary steps. The first step is to evaluate the marginal probabilities of frozen variables using the SP message-passing procedure. The second step, or decimation step, is to use this information to fix the values of some variables and simplify the problem. The notion of message passing will be illustrated throughout the chapter by comparing it with a simpler procedure known as belief propagation (mentioned in ch. 3 in the context of error correcting codes) in which no assumptions are made about the structure of the solution space. The chapter is organized as follows. In section 2 we provide the general formalism, defining constraint satisfaction problems as well as the key concepts of factor graphs and cavities, using the concrete examples of satisfiability and graph coloring.

scholarly journals VORTEX OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM

2018 ◽  
Vol 6 (1) ◽  
pp. 266-276
Author(s):  
K. Lenin
Keyword(s):  
Reactive Power ◽  
Solution Space ◽  
Stability Margin ◽  
Solution Procedure ◽  
Problem Solution ◽  
Static Voltage Stability ◽  
Optimal Reactive Power Dispatch ◽  
Voltage Stability Margin ◽  
Vortex Nature ◽  
Power Problem

In this paper, a new Vortex Optimization (VO) algorithm is proposed to solve the reactive power problem. The idea is generally focused on a typical Vortex flow in nature and enthused from some dynamics that are occurred in the sense of Vortex nature. In a few words, the algorithm is also a swarm-oriented evolutional problem solution methodology; since it comprises numerous techniques related to removal of feeble swarm members and trying to progress the solution procedure by supporting the solution space through fresh swarm members. In order to evaluate the performance of the proposed Vortex Optimization (VO) algorithm, it has been tested in Standard IEEE 30 bus systems and compared to other standard algorithms. Simulation results reveal about the best performance of the proposed algorithm in reducing the real power loss and static voltage stability margin index has been enhanced.

scholarly journals A Computational Study of the Effect of Experience on Problem/Solution Space Exploration in Teams

2019 ◽  
Vol 1 (1) ◽  
pp. 11-20
Author(s):  
Marija Majda Perisic ◽  
Tomislav Martinec ◽  
Mario Storga ◽  
John S Gero
Keyword(s):  
Computational Study ◽  
Space Exploration ◽  
Solution Space ◽  
Problem Solution ◽  
Design Team ◽  
Design Teams ◽  
Agent Based Model ◽  
Problem Space ◽  
Agent Based ◽  
The Impact

AbstractThis paper presents the results of computational experiments aimed at studying the effect of experience on design teams’ exploration of problem-solution space. An agent-based model of a design team was developed and its capability to match theoretically-based predictions is tested. Hypotheses that (1) experienced teams need less time to find a solution and that (2) in comparison to the inexperienced teams, experienced teams spend more time exploring the solution-space than the problem-space, were tested. The results provided support for both of the hypotheses, demonstrating the impact of learning and experience on the exploration patterns in problem and solution space, and verifying the system's capability to produce the reliable results.

scholarly journals A framework for studying design thinking through measuring designers’ minds, bodies and brains

2020 ◽  
Vol 6 ◽  
Author(s):  
John S. Gero ◽  
Julie Milovanovic
Keyword(s):  
Cognitive Processes ◽  
Design Thinking ◽  
Near Infrared ◽  
Electrodermal Activity ◽  
Solution Space ◽  
Integrated Approach ◽  
Problem Solution ◽  
Design Cognition ◽  
Functional Near Infrared Spectroscopy ◽  
Design Reasoning

This paper presents a framework for studying design thinking. Three paradigmatic approaches are described to measure design cognitive processes: design cognition, design physiology and design neurocognition. Specific tools and methods serve each paradigmatic approach. Design cognition is explored through protocol analysis, black-box experiments, surveys and interviews. Design physiology is measured with eye tracking, electrodermal activity, heart rate and emotion tracking. Design neurocognition is measured using electroencephalography, functional near infrared spectroscopy and functional magnetic resonance imaging. Illustrative examples are presented to describe the types of results each method provides about the characteristics of design thinking, such as design patterns, design reasoning, design creativity, design collaboration, the co-evolution of the problem solution space, or design analysis and evaluation. The triangulation of results from the three paradigmatic approaches to studying design thinking provides a synergistic foundation for the understanding of design cognitive processes. Results from such studies generate a source of feedback to designers, design educators and researchers in design science. New models, new tools and new research questions emerge from the integrated approach proposed and lay down future challenges in studying design thinking.

scholarly journals Revisiting Problem-Solution Co-Evolution in the Context of Team Conceptual Design Activity

2020 ◽  
Vol 10 (18) ◽  
pp. 6303 ◽  
Author(s):  
Tomislav Martinec ◽  
Stanko Škec ◽  
Marija Majda Perišić ◽  
Mario Štorga
Keyword(s):  
Conceptual Design ◽  
Design Education ◽  
Design Research ◽  
Solution Space ◽  
Design Activity ◽  
Problem Solution ◽  
Conceptual Design Phase ◽  
Design Activities ◽  
Conventional Models ◽  
Concept Review

The conventional prescriptive and descriptive models of design typically decompose the overall design process into elementary processes, such as analysis, synthesis, and evaluation. This study revisits some of the assumptions established by these models and investigates whether they can also be applied for modelling of problem-solution co-evolution patterns that appear during team conceptual design activities. The first set of assumptions concerns the relationship between performing analysis, synthesis, and evaluation and exploring the problem and solution space. The second set concerns the dominant sequences of analysis, synthesis, and evaluation, whereas the third set concerns the nature of transitions between the problem and solution space. The assumptions were empirically tested as part of a protocol analysis study of team ideation and concept review activities. Besides revealing inconsistencies in how analysis, synthesis, and evaluation are defined and interpreted across the literature, the study demonstrates co-evolution patterns, which cannot be described by the conventional models. It highlights the important role of analysis-synthesis cycles during both divergent and convergent activities, which is co-evolution and refinement, respectively. The findings are summarised in the form of a model of the increase in the number of new problem and solution entities as the conceptual design phase progresses, with implications for both design research and design education.

Design of a Massively Reconfigurable Origami Space Structure Incorporating Shape Memory Alloys

2012 ◽  
Author(s):  
Darren Hartl ◽  
Kathryn Lane ◽  
Richard Malak
Keyword(s):  
Shape Memory ◽  
Three Dimensional ◽  
Passive Layer ◽  
Space Structure ◽  
Integration Scheme ◽  
Cylindrical Shape ◽  
Material System ◽  
Analysis Model ◽  
Element Analysis ◽  
Design Assessment

The subject of origami design has recently garnered increasing attention from the science, mathematics, and engineering communities. Mathematically rigorous frameworks have been developed that allow the identification of folding patterns needed to obtain a final three-dimensional goal shape. However, relatively little research exists on the problem of understanding the behavioral aspects of the material system undergoing the folding operations. This work considers the design and analysis of a novel concept for a self-folding material system. The system consists of an active, self-morphing laminate structure that includes thermally actuated shape memory alloy (SMA) layers and a compliant passive layer. Multiple layers allow folds in both directions (e.g., cross-folds). The layers are configured to allow continuously variable folding operations based only on which regions are heated. For the purposes of demonstration, an example problem is considered whereby an autonomous planetary landing craft is designed that can be stored in a flat sheet configuration, morph using a set of folds into a stable shape for safe descent through a gaseous atmosphere, and then, once landed, morph again toward a cylindrical shape for the purpose of rolling locomotion. We examine the effects of fold width, layer thicknesses, and activation parameters on the geometric configurations that can be obtained. The design efforts are supported by realistic morphing structural analysis tools. These include a comprehensive and accurate three-dimensional constitutive model for SMAs implemented into a finite element analysis (FEA) framework (the Abaqus Unified FEA suite) using a robust and efficient numerical integration scheme. Shell elements and laminate theory are used to increase the computational efficiency of the analysis. Model pre-processing, submission, and post-processing scripting methods are used to automate the design assessment tasks.

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