We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant Problems. Compared to counting Constrained Satisfaction Problems (#CSP), it is a refinement with a more explicit role for the function constraints. Both graph homomorphism and #CSP can be viewed as special cases of Holant Problems. We prove complexity dichotomy theorems in this framework. Because the framework is more stringent, previous dichotomy theorems for #CSP problems no longer apply. Indeed, we discover surprising tractable subclasses of counting problems, which could not have been easily specified in the #CSP framework. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations. The study of Holant Problems led us to discover and prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the complex number field C.
We prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the complex number field C. This generalizes a theorem by Dyer, Goldberg and Jerrum [11] where each constraint function takes non-negative values. We first give a non-trivial tractable class of Boolean #CSP which was inspired by holographic reductions. The tractability crucially depends on algebraic cancelations which are absent for non-negative numbers. We then completely characterize all the tractable Boolean #CSP with complex valued constraints and show that we have found all the tractable ones, and every remaining problem is #P-hard. We also improve our result by proving the same dichotomy theorem holds for Boolean #CSP with max degree 3 (every variable appears at most three times). The concept of Congruity and Semi-congruity provides a key insight and plays a decisive role in both the tractability and hardness proofs. We also introduce local holographic reductions as a technique in hardness proofs.
The generated light can be tuned to cover almost the entire spectral range from deep‐ultraviolet to terahertz wavelengths by utilizing the nonlinear optical crystals with a simple frequency doubling process. Among them, the discovery of novel candidates for the production of deep‐ultraviolet light is by extension a great challenge toward realizing the vast potential. Actually, the availability for this process mainly depends on whether the critical performance can be well coexisted in one practical crystal. Herein, the first magnesium fluorooxoborate MgB5O7F3 was synthesized as a new competitive candidate for deep‐ultraviolet nonlinear optical application. It has a sufficiently large nonlinearity and a deep‐ultraviolet phase matching wavelength, indicating that it holds great potential for the production of coherent light below 200 nm. The critical performance enhancement of MgB5O7F3 when compared with its isomorphic phases was demonstrated and discussed. More importantly, we proposed that fluorooxoborate system with the general formula of MB5O7F3 (M=divalent metal) possesses stable fluorine terminated framework, which makes them tend to retain their crystallized space groups unchanged.
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