Constraint-satisfaction problems are among the computationally hardest tasks: solutions are efficiently checkable, but no efficient algorithms are known to compute those solutions. Fresh insight might come from physics. A study mapping optimization hardness onto the phenomena of turbulence and chaos suggests that constraint-satisfaction problems can be tackled using analog devices. Boolean satisfiability1 (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for k≥3) implies efficient solutions to a large number of hard optimization problems2,3. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic4,5,6,7, and the boundaries between the basins of attraction8 of the solution clusters become fractal7,8,9, signalling the appearance of optimization hardness10. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT (ref. 11) and of locked occupation problems12 (considered among the hardest algorithmic benchmarks), a property partly due to the system’s hyperbolic4,13 character. The system finds solutions in polynomial continuous time, however, at the expense of exponential fluctuations in its energy function.

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Optimization hardness as transient chaos in an analog approach to constraint satisfaction

In recent years, different optimization methods have been developed for optimization problem. Many of these methods are inspired by swarm behaviors in nature. In this paper, a new algorithm for optimization inspired by the gases brownian motion and turbulent rotational motion is introduced, which is called Gases Brownian Motion Optimization (GBMO). The proposed algorithm is created using the features of gas molecules. The proposed algorithm is an efficient approach to search and find an optimum solution in search space. The efficiency of the proposed method has been compared with some well-known heuristic search methods. The obtained results confirm the high performance of the proposed method in solving various functions.

Gases Brownian Motion Optimization: an Algorithm for Optimization (GBMO)

Constraint satisfaction problems are ubiquitous in many domains They are typically solved using conventional digital computing architectures that do not reflect the distributed nature of many of these problems, and are thus ill-suited for solving them Here we present a parallel analogue/digital hardware architecture specifically designed to solve such problems We cast constraint satisfaction problems as networks of stereotyped nodes that communicate using digital pulses, or events Each node contains an oscillator implemented using analogue circuits The non-repeating phase relations among the oscillators drive the exploration of the solution space We show that this hardware architecture can yield state-of-the-art performance on random SAT problems under reasonable assumptions on the implementation We present measurements from a prototype electronic chip to demonstrate that a physical implementation of the proposed architecture is robust to practical non-idealities and to validate the theory proposed

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An event-based architecture for solving constraint satisfaction problems

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Review: J. W. S. Cassels, An introduction to the geometry of numbers

We present a survey on analog models of computation. Analog can be understood both as computing by analogy, or as working on the continuum. We consider both approaches, often intertwined, with a point of view mostly oriented by computation theory.

A Survey on Analog Models of Computation.

On the stability of Lagrange programming neural networks for satisfiability problems of prepositional calculus

Parallel State Space Search for SAT with Lagrange Programming Neural Network

Solving SAT by Lagrange Programming Neural Network with Long and Short Term Memories.

We proposed a neural network called LPPH-CSP (Lagrange Programming neural network with Polarized High-order connections for Constraint Satisfaction Problem) to solve the CSP. The CSP is a problem to find a variable assignment which satisfies all given constraints. Because the CSP has a well defined representation ability, it can represent many problems in AI compactly. From experimental results of LPPH-CSP and GENET which is a famous CSP solver, we confirmed that our method is as efficient as the GENET. In addition, unlike the other conventional CSP solvers which are discrete-valued methods, our method is a continuous-valued method and it can update all variables simultaneously, while the conventional csp solvers cannot find a solution by updating all variables simultaneously Because of the oscilation of the states. Therefore, we can expect the speed-up of LPPH-CSP if it is implemented by the hardware such as FPGA. In this paper, we extend LPPH-CSP to deal with the linear inequality constraints. By using this type of constraint, we can represent various practical problems more briefly. In this paper, we also define the CSP which has an objective function, and we extend LPPH-CSP to solve this problem. In experiment, we apply our method and OPBDP to the warehouse location problem and compare the effectiveness.

Lagrange neural network for solving CSP which includes linear inequality constraints

The satisfiability problem (SAT) of the propositional calculus is a well-known NP-complete problem It requires exponential computation time as the problem size increases We proposed a neural network, called LPPH, for the SAT The equilibrium point of the dynamics of the LPPH exactly corresponds to the solution of the SAT, and the dynamics does not stop at any point that is not the solution of the SAT Experimental results show the effectiveness of the LPPH for solving the SAT In this paper we extend the dynamics of the LPPH to solve several variations of the SAT, such as, the SAT with an objective function, the SAT with a preliminary solution, and the MAX-SAT The effectiveness of the extensions is shown by the experiments

Masahiro Nagamatu

Papers

On the stability of Lagrange programming neural networks for satisfiability problems of prepositional calculus

Parallel State Space Search for SAT with Lagrange Programming Neural Network

Solving SAT by Lagrange Programming Neural Network with Long and Short Term Memories.

Lagrange neural network for solving CSP which includes linear inequality constraints

Extensions of Lagrange programming neural network for satisfiability problem and its several variations