Hard Formulas Research Articles

Hardness Measurements via an Ellipsoid-Shaped Indenter

In this theoretical study, we have chosen to use a body of an ellipsoidal geometric form as an indenter, where we determined the mathematical expression of the static hardness as function of the depth and the radii of the area of projected imprint. We used the general formula of the static hardness expressed by the ratio of a force applied perpendicular on the indenter to the resulting area of the imprint; also, we have established the real imprint (cap) of an indenter of revolution ellipsoid form. Finally, geometrical and mathematical approaches have been used to derive the formula of the static hardness expression.

Local Search for Hard SAT Formulas: The Strength of the Polynomial Law

Random k-CNF formulas at the anticipated k-SAT phase-transition point are prototypical hard k-SAT instances. We develop a stochastic local search algorithm and study it both theoretically and through a large-scale experimental study. The algorithm comes as a result of a systematic study that contrasts rates at which a certain measure concentration phenomenon occurs. This study yields a new stochastic rule for local search. A strong point of our contribution is the conceptual simplicity of our algorithm. More importantly, the empirical results overwhelmingly indicate that our algorithm outperforms the state-of-the-art. This includes a number of winners and medalist solvers from the recent SAT Competitions.

Hiding multiple solutions in a hard 3-SAT formula

Hiding solutions in 3-SAT formulas can be used in privacy protection and data security. Although the typical q-hidden algorithm could cancel the attraction to the unique predefined solution, and generate deceptive 3-SAT formulas with unique predefined solution, few works have mentioned that with multiple predefined solutions. Therefore, the objective of this paper is to propose algorithms which could cancel the attraction to the multiple predefined solutions simultaneously. The core element of these proposed algorithms is misguiding the SAT solvers with local search strategy to the reverse direction of the centre solution of the multiple predefined solutions, so that the attraction to the multiple predefined solutions can be cancelled simultaneously. Experimental results verify the behaviour of the two classical SAT solvers: the SAT solvers with local search strategy (such as WalkSAT) and that with DPLL strategy (such as zChaff). And a real-world application is introduced based on the proposed algorithm.

On the computational complexity of finding hard tautologies

It is well known (cf. Krajíček and Pudlák [‘Propositional proof systems, the consistency of first order theories and the complexity of computations’, J. Symbolic Logic 54 (1989) 1063–1079]) that a polynomial time algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1(k) and outputs a tautology τk of size at least k such that P is not p-bounded on the set of all formulas τk. We consider two more general search problems involving finding a hard formula, Cert and Find, motivated by two hypothetical situations: that one can prove that NP≠coNP and that no optimal proof system exists. In Cert one is asked to find a witness that a given non-deterministic circuit with k inputs does not define TAUT∩{0, 1}k. In Find, given 1(k) and a tautology α of size at most k c 0 , one should output a size k tautology β that has no size k c 1 P-proof from substitution instances of α. We will prove, assuming the existence of an exponentially hard one-way permutation, that Cert cannot be solved by a time 2O(k) algorithm. Using a stronger hypothesis about the proof complexity of the Nisan–Wigderson generator, we show that both problems Cert and Find are actually only partially defined for infinitely many k (that is, there are inputs corresponding to k for which the problem has no solution). The results are based on interpreting the Nisan–Wigderson generator as a proof system.

A note on SAT algorithms and proof complexity

We apply classical proof complexity ideas to transfer lengths-of-proofs lower bounds for a propositional proof system P into examples of hard unsatisfiable formulas for a class Alg(P) of SAT algorithms determined by P. The class Alg(P) contains those algorithms M for which P proves in polynomial size tautologies expressing the soundness of M. For example, the class Alg(Fd) determined by a depth d Frege system contains the commonly considered enhancements of DPLL (even for small d). Exponential lower bounds are known for all Fd. Such results can be interpreted as a form of consistency of P≠NP.Further we show how the soundness statements can be used to find hard satisfiable instances, if they exist.

Information loss in knowledge compilation: A comparison of Boolean envelopes

Since Selman and Kautz's seminal work on the use of Horn approximation to speed up the querying of knowledge bases, there has been great interest in Boolean approximation for AI applications. There are several Boolean classes with desirable computational properties similar to those of the Horn class. The class of affine Boolean functions, for example, has been proposed as an interesting alternative to Horn for knowledge compilation. To investigate the trade-offs between precision and efficiency in knowledge compilation, we compare, analytically and empirically, four well-known Boolean classes, and their combinations, for ability to preserve information. We note that traditional evaluation which explores unit-clause consequences of random hard 3-CNF formulas does not tell the full story, and we complement that evaluation with experiments based on a variety of assumptions about queries and the underlying knowledge base.

First-principle calculation on structures and properties of diamond-like B 3C 10N 3 compound

Diamond-like B 3C 10N 3 model was built by stacking monolayer B 3C 10N 3 which has the lowest energy. The first-principle calculations were performed by CASTEP code based on density functional theory (DFT) and basal physical properties of the model were investigated. The diamond-like B 3C 10N 3 is semiconducting with direct energy gap of 1.65 eV. The results of elastic constants ( C ij ) satisfy the Born criteria, showing the mechanical stability. Bulk modulus calculated is about 398 GPa, with hardness about 60.6 GPa calculated by hardness formula of the covalent crystal, which might be one of superhard materials. The transformation pressure from layered B 3C 10N 3 to diamond-like was proposed at about 25.3 GPa.

Whose side are you on?1

We introduce a new jump strategy for look-ahead based satisfiability (Sat) solvers that aims to boost their performance on satisfiable formulae, while maintaining their behavior on unsatisfiable instances. Direction heuristics select which Boolean value to assign to a decision variable. They are used in various state-of-the-art Sat solvers and may bias the distribution of solutions in the search-tree. This bias can clearly be observed on hard random k-Sat formulae. While alternative jump / restart strategies are based on exploring a random new part of the search-tree, the proposed one examines a part that is more likely to contain a solution based on these observations. Experiments with - so-called distribution jumping - in the march ks Sat solver, show that this new technique boosts the number of satisfiable formulae it can solve. Moreover, it proved essential for winning the satisfiable crafted category during the Sat 2007 competition.

Hardness and elastic properties of covalent/ionic solid solutions from first-principles theory

Most of the engineering materials are alloys (solid solutions) and inevitably contain some impurities or defects such as vacancies. However, theoretical predictions of the hardness of this kind of materials have rarely been addressed in literature. In this paper, a hardness formula for multicomponent covalent solid solution is proposed based on the work of Šimůnek and Vackář [Phys. Rev. Lett. 96, 085501 (2006)]. With this formula, the composition dependence of the hardness is investigated for titanium nitrogencarbide (TiN1−xCx), off-stoichiometric transition-metal nitrides (TiN1−x and VN1−x), and B-doped semiconductors. The predicted hardness is in good agreement with experiments. To investigate the most frequently quoted correlation between hardness and elastic modulus, the elastic moduli of the systems involved in this paper have also been calculated. The results show that the elastic moduli cannot be used for rigorous predictions of the hardness of the solid solutions.

An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams

AbstractWe prove an exponential lower bound on the size of proofs in the proof system operating with ordered binary decision diagrams introduced by Atserias, Kolaitis and Vardi [2]. In fact, the lower bound applies to semantic derivations operating with sets defined by OBDDs. We do not assume any particular format of proofs or ordering of variables, the hard formulas are in CNF. We utilize (somewhat indirectly) feasible interpolation.We define a proof system combining resolution and the OBDD proof system.

Predicting hardness of covalent∕ionic solid solution from first-principles theory

We introduce a hardness formula for the multicomponent covalent and ionic solid solutions. This expression is tested on nitride spinel materials A3N4 (A=C,Si,Ge) and applied to titanium nitrogen carbide (TiN1−xCx with 0⩽x⩽1), off-stoichiometric transition-metal nitride (TiN1−x and VN1−x with x⩽0.25), and B-doped semiconductors (C1−xBx, Si1−xBx, and Ge1−xBx with x⩽0.1). In all cases, the theoretical hardness is in good agreement with experiments.

Generating Hard Satisfiable Formulas by Hiding Solutions Deceptively

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the "hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to "deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT.

Generating hard satisfiable formulas by hiding solutions deceptively

To test incomplete search algorithms for constraint satisfaction problems such as 3- SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a rand...

Theoretical model of intrinsic hardness

Intense theoretical and experimental interest has been focused on the possibility of new materials with hardness exceeding that of diamond. However, building the link between the information that first-principles calculations can produce and the hardness of materials remains one of the challenges of computational materials science. In this paper, a calculated method of hardness based on the Mulliken overlap population analysis in first principles has been presented. In particular, the effects of stress strain on intrinsic hardness were studied, and a formula of hardness under stress is established. It can be employed to explain the hardening phenomenon resulting from the stress of film and grain boundary and nanoeffects. The theoretical results revealed that nanodiamond films obtained by careful experiments can be harder than bulk single-crystal diamonds.

Evaluation of Bending Strength of Carburized Gears Based on a Quantification of Defect Size in the Surface Layer

Generally, fatigue strength of steel correlates with hardness up to a certain point. However, the relationship of hardness and fatigue strength is more complicated for high strength materials, mainly because strength is sensitive to material defects. This paper presents a determination of defect size in carburized gear teeth and a strength evaluation method based on that size. A micro notch is artificially processed at the fillet of carburized gear tooth by means of a Focused Ion Beam, and a bending fatigue test is performed. Threshold notch depth anth, the limit beyond which fatigue strength declines, is obtained by the test. Threshold notch depth is used as a measure to determine latent defect size in the surface layer. The anth of gears from which a nonmartensitic layer is removed by electro-polishing is shorter than the anth of gears with a nonmartensitic layer. The surface layer is observed by scanning electron microscope and electron probe micro analyzer to clarify latent defect characteristics. Many micro cracks, in addition to long cracks in the nonmartensitic layer, Cr oxides and Mn oxides near the surface, and Si segregated in grain boundary are observed. This demonstrates that they trigger fatigue crack initiation; and therefore defect size limits strength. Consequently, strength is expressed by an experimental formula of Vickers hardness and the square root of projected area of the defect obtained above.

Order- α radiative correction calculations for oriented allowed nuclear β decays

A general method for the order- α radiative correction calculation in oriented nuclear β decays is presented. This methodis based upon the Monte Carlo event generation and integration procedure of F. Glück. The calculations are described in detail in the case of the zeroth order oriented β decays. The virtual, soft and hard photon bremsstrahlung formulae are then given. The method is useful for radiative correction calculations and for weighted or unweighted event generations in experimental off-line data analyses.

A new method for hardness determination from depth sensing indentation tests

A new semiempirical formula is developed for the hardness determination of the materials from depth sensing indentation tests. The indentation works measured both during loading and unloading periods are used in the evaluation. The values of the Meyer hardness calculated in this way agree well with those obtained by conventional optical observation, where this latter is possible. While the new hardness formula characterizes well the behavior of the conventional hardness number even for the ideally elastic material, the mean contact pressure generally used in hardness determination differs significantly from the conventional hardness number when the ideally elastic limiting case is being approached.

A model for the evolution of the genome: the effect of stochasticity on genetic loads.

In this paper a general model is given for the evolution of the genome incorporating stochastic factors. The model is applied to the substitutional genetic load problem. All of the major hard selection load formulae in the literature are extended and, where necessary, corrected (for stochasticity). Turning to rank selection, formulae for stochastic factors are also corrected and harmful mutations included. A simple formula for the selection coefficient as a function of the nonneutral substitution rate and the mutation profile is obtained. Further, it is noted that the formulae derived also apply (for different parameter values) to the mutation load, thus unifying the two loads under a single theory. A general formula for the mutation load under hard selection is given, extending previous results. Finally, the author derives a formula showing how many harmful mutations can be effectively eliminated by rank selection and discusses its relevance to the question of the possible buildup of harmful mutations in the human gene pool due to long-term exposure to low-level radiation.

Tensile properties of heat affected zone of medium strength low carbon, C–Mn, and 2·25Cr–1Mo steels

AbstractThe grain coarsened tensile properties in the heat affected zone (HAZ) of medium strength (Rp0·2⋍330–470 MN m−2) low carbon microalloyed, C–Mn, and 2·25Cr–1Mo steels have been examined using the weld simulation technique. Very high values of yield and tensile strength (Rp0·2>900 MN m−2; Rm>1100 MN m−2) can be obtained for low heat input welding (E 2 MJ m−1; ∆t8/5> 10 s), however, a substantial reduction in the values of Rp0·2 and Rm occurs, with the exception of the 2·25Cr–IMo steel. Based on the empirical equations developed, and the Suzuki BL70 hardness formula, it is shown that the grain coarsened HAZ yield and tensile strength can be calculated to an accuracy of ±50 MN m−2, while elongation at fracture and reduction in area can be predicted to an accuracy of ±3% and ±7%, respectively.MST/I077

A New Myxosporidan in Red and Golden Shiners

Myxosoma cyprini sp. n. is described in the gills of the red shiner, Notropis lutrensis, and golden shiner, Notemigonus crysoleucas, from Oklahoma. Cysts measuring 0.3 ua or less occur adjacent to capillary endothelial cells of secondary gill lamellae. Infected fish were collected in October 1971 and in June 1972. M. cyprini is disporoblastic and development is asynchronous. Mature spores are pyriform in front view and lenticular when viewed from the side. Fresh spores average 14.4 A in length by 8.0 uL in width by 7.5 u in thickness. Polar capsules are pyriform and 7.1 u long by 3.0 u wide. Approximately 9 windings of the polar filaments are seen inside each capsule. Extruded polar filaments are 100 to 110 A long. Mucous envelopes are present. All of the 35 red shiners and 9 of 10 golden shiners contained M. cyprini. Host reactive cells do not appear at the locus of the parasite. A new species of Myxosoma was found in the branchial tissues of two cyprinoid fishes, the red shiner, Notropis lutrensis, and the golden shiner, Notemigonus crysoleucas. Immature and mature stages of the parasite are described herein. The specific name M. cyprini, referring to the hosts' family name, is proposed. MATERIALS AND METHODS Twenty-two red shiners were collected by seining Little Stillwater Creek, Payne County, Oklahoma, on 2 October 1971. Thirteen more red shiners and 10 golden shiners were collected from the same site on 8 June 1972. Portions of infected gills were macerated in water on glass slides or were fixed for 4 to 16 hr in a cold 2% glutaraldehyde solution containing 0.5% acrolein, 4.0 mM calcium chloride, and buffered by 0.2 M sodium cacodylate to pH 7.4. Fixed tissues containing the parasite were washed in cold 0.22 M sucrose in the cacodylate buffer and either examined as squashed preparations with phase contrast (Zeiss system) or embedded for sectioning. The embedding steps included postfixation for 30 min in 2% Os04 in the cacodylate buffer, dehydration in a graded ethanol series, and infiltration in Spurr's low-viscosity medium (hard formula; Polysciences, Industrial Park, Warren, Pennsylvania). The embedment was polymerized at 37 to 40 C for 4 hr and 64 to 70 C for 6 to 7 hr. Sections, 1 /t thick, were cut with glass knives on a Porter-Blum MT-2 ultramicrotome, mounted on glass slides, and stained on a hot plate with 0.5% toluidine blue in 1% sodium borate. Stained sections were rinsed in running water, air-dried, flooded with immersion oil, and cover-slipped. Observations of stained sections were made by both bright-field and phasecontrast optics using 125 to 2,500X magnification. Mucous envelopes were demonstrated in fresh spores by the method of Lom and Vavra (1963). Received for publication 18 July 1973. Description of the sporogenic stages and spores is based on composite findings in both fresh and fixed unstained smears. Description of the plasmodium and of its spatial relationship to host tissues is based on observations of sectioned and