Supercritical space-width trade-offs for resolution

Space Bounds for Resolution

For a d-regular (k,s)-CNF formula, a problem is to determine whether it has a (1,0)-super solution. If so, it is called (1,0)-d-regular (k,s)-SAT. A (1,0)-super solution is an assignment that satisfies at least two literals of each clause. When the value of any one of the variables is flipped, the (1,0)-super solution is still a solution. Super solutions have gained significant attention for their robustness. Here, a d-regular (k,s)-CNF formula is a special CNF formula with clauses of size exactly k, in which each variable appears exactly s-times, and the absolute frequency difference between positive and negative occurrences of each variable is at most a nonnegative integer d. Obviously, the structure of a d-regular (k,s)-CNF formula is much more regular than other formulas. In this paper, we certify that, for k≥5, there is a critical function φ(k,d) such that, if s≤φ(k,d), all d-regular (k,s)-CNF formulas have a (1,0)-super solution; otherwise (1,0)-d-regular (k,s)-SAT is NP-complete. By the Lopsided Local Lemma, we get an existence condition of (1,0)-super solutions and propose an algorithm to find the lower bound of φ(k,d).

Experiments on the Flow around two tandem circular cylinders from sub- up to transcritical Reynolds numbers

Fractures have significantly impacted the preservation, enrichment, and productivity of shale gas in the Longmaxi Formation in the complex structural area of the southern Sichuan Basin. Based on outcrop, core, imaging logging, inclusion analysis, acoustic emission experiments, and burial and thermal evolution history data, the formation stage, evolution and preservation conditions of structural fractures were comprehensively studied. Shear fractures are the main features of tectonic origin and are characterized by small widths, wide spacing, and high filling degrees. The structural fractures of the Longmaxi Formation can be divided into three groups in terms of their formation stage. The first-stage fractures were formed in the middle-late period of the Yanshanian tectonic movement (86.2–68.5 Ma), when the palaeotectonic stress was 68.33–71.82 MPa; and the homogenization temperature of the fluid inclusions in the corresponding fracture fillings is 150–178°C. The second-stage fractures were formed at the end of the Yanshanian -early Himalayan tectonic movement (68.5–35.2 Ma), when the palaeotectonic stress was 85.2–100.5 MPa; the homogenization temperature of the corresponding fluid inclusions is 123–148°C. The third-stage tectonic fractures formed during the middle-late period of the Himalayan tectonic movement (35.2–0 Ma), when the palaeotectonic stress was 55.6–63.8 MPa; fillings are scarce in these fractures, but the measured homogenization temperature of the corresponding fluid inclusions is 90–110°C. Shale gas preservation conditions were affected by the fracture formation stage, maximum horizontal crustal stress, fault scale, structural burial depth, formation pressure coefficient, etc. Three types of favourable preservation conditions were identified. The Type I target area is the low slope area in the middle of the syncline, which occurs more than 2.0 km from the fault. The research results have guiding significance for the evaluation of the preservation conditions of deep shale gas in the study area and similar areas.

We show that if a k-CNF requires width w to refute in resolution, then it requires space square root of √ω to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is the first analogue, in polynomial calculus, of Atserias and Dalmau's result lower-bounding clause space in resolution by resolution width. As a by-product of our new approach to space lower bounds we give a simple proof of Bonacina's recent result that total space in resolution (the total number of variable occurrences that must be kept in memory) is lower-bounded by the width squared. As corollaries of the main result we obtain some new lower bounds on the PCR space needed to refute specific formulas, as well as partial answers to some open problems about relations between space, size, and degree for polynomial calculus.

Following the lines of Little (1965), who investigated the effect of various simulated and/or imagined spaces on critical interpersonal distance, the present study determined the distance at which Ss would move away from an approaching experimenter in a large, a small, a long, and a wide space. The dimensions of the spaces were, respectively, 12 ft. x 29 ft., 4 ft. x 6 ft., and 7.5 ft. x 65 ft. In the latter space they were tested at both the wide and the narrow wall. Ss were 13 male and 24 female student volunteers from introductory psychology. Each S was tested in each space, with order counterbalanced. S stood about 3 ft. from one wall, but was told he could move around a little in order to appreciate the room's spatial qualities. E (P. W. S.) asked S questions about the space, including feelings of crowding. E, while moving about and writing, at one point approached S and estimated to the nearest .5 ft. the distance between E's foot nearest S and S's first foot that moved away.' After being tested in each space S filled out the California F scale, the dogmatism scale, and the Rotter I-E scale. The distances ranged from less than 0.5 ft. to 3.0 ft. The greatest mean distances were in the large (1.2 ft.) and the long (1.3 ft.) spaces, followed by the small (1.0 ft.) and the wide (0.5 ft.) spaces (F = 7.37, df = 3/108, p < .01). However, of the three per:onalicy scales employed, only one, dogmatism, was significantly correlated with distance (r = .39, p < .01), even when interaction vectors were generated by taking the two- and three-way products of the personality scores and generating prediction equations in a stepwise fashion (Draper & Smith, 1966). Except for dogmatism, the largest F for increase in R' was less than 1.0. Thus, at most only about 16% of the variance in distance is associated with the personality measures employed, and, cons~dering the small N, even this value must be considered as tentative. The small space yielded 18 reports of crowding, with 3, 6, and 9 reports from the large, long, and wide rooms. This finding, considered with the fact that both the largest and the smallest distances were found in the same room, suggests that neither area nor perceived crowding per se determines the distance, but the extent of a single dimension of the space. The shortest distance was obtained with the longest (65 ft.) dimension to S's side. This finding tends to confirm Little (1965), who found the shortest distance in a simulated open space. The small but significant relationship between distance and dogmatism suggests that further research along these lines would be in order.

To observe and quantify the changes in channels and sand bars, a computer-controlled video camera was deployed on Paku Hill at Tairua Beach, on the East Coast of Coromandel Peninsula, New Zealand. Over the last 11 months, 600 images were captured every hour by the camera for 15.66 minutes and then averaged. Rip characteristics (such as spacing, shape, migration direction and rates) were identified by the decreased wave breaking in the channel leading to reduced light intensity in the image. Seven out of twelve storm events over the 11 months were chosen for detailed study. During these storms, the spatial separation between the rips channels did not vary significantly over the length of the beach or during a storm event. Rip channel spacing altered between storm events so that the average spacing for the main channels was either approximately 130 m, 169 m or 205 m, with the exception of a single storm in which spacing was approximately 100 m. A and beach state occurred with the smaller spacing and a bar-trough beach state occurred with the wider spacing. Migration of the main channel was measured for each of the eight rips typically present by calculating the distance moved between two consecutive images. Migration rate, averaged over all storms and rips, was 0.6 m/hr. However migration direction was generally not consistent over time, so that over all the storms there was no net migration (the mean net migration over an hour over all storms was -0.04 m/hr +/- a standard deviation of 0.4 m/hr). At any one time, the migration rate and direction for all rips during a particular storm was similar, so that spacing was preserved during a storm. Sequences of current genesis were also observed in the averaged images. The most common type occurred when a shore-parallel segment of the longshore bar evolved into a crescentic bar, in which case the channel formed between two horns of the crescentic bar. Rip currents also evolved within a circulation cell by a transverse bar extending either outward from the shoreline or inward from the crescentic bar. The most unusual pattern that was observed was a rip where a new formed at the one end of the alongshore extension of a pre-existing rip.

Given a Boolean formula in conjunctive normal form with \(n\) variables and \(m=rn\) clauses, if there exists a truth assignment satisfying \((1-2^{-k}-q(1-2^{-k}))m\) clauses, call the formula \(q\)-satisfiable. The Minimum Satisfiability Problem (MinSAT) is a special case of \(q\)-satisfiable, which asks for an assignment to minimize the number of satisfied clauses. When each clause contains \(k\) literals, it is called Min\(k\)SAT. If each clause is independently and randomly selected from all possible clauses over the \(n\) variables, it is called random MinSAT. In this paper, we give upper and lower bounds of \(r\) (the ratio of clauses to variables) for random \(k\)-CNF formula with \(q\)-satisfiable. The upper bound is proved by the first moment argument, while the proof of lower bound is the second moment with weighted scheme. Interestingly, our experimental results about MinSAT demonstrate that the lower and upper bounds are very tight. Moreover, these results give a partial explanation for the excellent performance of MinSatz, the state-of-the-art MinSAT solver, from the perspective of pruning effects. Finally, we give a conjecture about the relationship between the minimum number and the maximum number of satisfied clauses on random SAT instances.KeywordsMinSATUpper boundsLower bounds

Semantical and computational aspects of Horn approximations

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Theta(√(n)) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

Branch decomposition is a prominent method for structurally decomposing a graph, hypergraph or CNF formula. The width of a branch decomposition provides a measure of how well the object is decomposed. For many applications it is crucial to compute a branch decomposition whose width is as small as possible. We propose a SAT approach to finding branch decompositions of small width. The core of our approach is an efficient SAT encoding which determines with a single SAT-call whether a given hypergraph admits a branch decomposition of certain width. For our encoding we develop a novel partition-based characterization of branch decompositions. The encoding size imposes a limit on the size of the given hypergraph. In order to break through this barrier and to scale the SAT approach to larger instances, we develop a new heuristic approach where the SAT encoding is used to locally improve a given candidate decomposition until a fixed-point is reached. This new method scales now to instances with several thousands of vertices and edges.

A vibrational resonant cantilever beam device is a type of tactile vision substitution device for the visually impaired that has the potential to achieve a high resolution in a small space. In order to realize a device of this type, it is necessary to accurately model the resonant behavior of the mesoscale (0.1–1.0 mm) metal cantilever beams of which the device is composed. Specifically, the natural frequencies and damping ratios of these beams must be analytically modeled in order to design beam dimensions for the device. In this paper, the resonant frequencies and damping ratios of a set of A2 tool-steel mesoscale cantilever beams are obtained using three different methods: calculation based on a mass–spring–damper model of the beams; a forced response experiment; and a free response experiment. The results are compared, and the size effect on stiffness and elastic modulus is investigated. Based on the modified couple stress theory, the length scale parameter is calculated. The damping ratio difference depending on the dimension of material in the mesoscale is also investigated.

We construct uniquely satisfiable k-CNF formulas that are hard for the PPSZ algorithm, the currently best known algorithm solving k-SAT. This algorithm tries to generate a satisfying assignment by picking a random variable at a time and attempting to derive its value using some inference heuristic and otherwise assigning a random value. The "weak PPSZ" checks all subformulas of a given size to derive a value and the "strong PPSZ" runs resolution with width bounded by some given function. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most (2 + epsilon)/k; the saving of an algorithm on an input formula with n variables is the largest gamma such that the algorithm succeeds (i.e. finds a satisfying assignment) with probability at least 2^{- (1 - gamma) n}. Since PPSZ (both weak and strong) is known to have savings of at least (pi^2 + o(1))/6k, this is optimal up to the constant factor. In particular, for k=3, our upper bound is 2^{0.333... n}, which is fairly close to the lower bound 2^{0.386... n} of Hertli [SIAM J. Comput.'14]. We also construct instances based on linear systems over F_2 for which strong PPSZ has savings of at most O(log(k)/k). This is only a log(k) factor away from the optimal bound. Our constructions improve previous savings upper bound of O((log^2(k))/k) due to Chen et al. [SODA'13].

To explore the effect of chromofungin (CHR), a chromogranin A (CGA) derived peptide CGA47-66, on hyper-permeability of blood brain barrier in septic mice. 120 healthy male C57BL/6 mice were randomly divided into groups, with 12 mice in each group. Seventy-two mice were used for dynamic observation of the contents of water and Evan blue (EB) in brain tissue after being treated with lipopolysaccharide (LPS). Another 48 mice were divided into normal saline control group (NS group), LPS induced sepsis model group (LPS group), low-dose CHR pretreatment group (CL+LPS group), and high-dose CHR pretreatment group (CH+LPS group). The septic model was reproduced by intraperitoneal injection of 10 mg/kg LPS 0.1 mL, and the mice in NS group was given equal volume of normal saline. The mice in CL+LPS group and CH+LPS group were intraperitoneally injected with 15.5 μg/kg and 77.5 μg/kg CHR 10 minutes before LPS injection. Six hours after LPS injection, 4 mL/kg of 2% EB was injected via caudal vein, the contents of water and EB in brain tissue were determined, and EB immune fluorescence in brain tissue was determined to assess the changes in permeability of blood brain barrier. Brain pathology was observed with hematoxylin and eosin (HE) staining. With the extension of time after LPS injection, the contents of water and EB in brain tissue were gradually increased, and the time of difference with statistical significance appeared earlier when compared with that of control group in the contents of water than that in EB contents (3 hours and 6 hours, respectively). The contents of water and EB in brain tissue in LPS group were significantly increased as compared with NS group [water content: (79.77±0.62)% vs. (78.28±0.44)%, P < 0.01; EB content (μg/g): 13.87±4.50 vs. 7.13±1.76, P < 0.05]. CHR pretreatment with either of two dosages could reverse the increase in water and EB contents in brain tissue induced by LPS, and the effect was more significant in CH+LPS group [water content: (78.15±0.73)% vs. (79.77±0.62)%, EB (μg/g): 7.09±2.59 vs. 13.87±4.50, both P < 0.05]. It was shown by EB fluorescence observation that the fluorescence signal displayed only in the meninges in NS group, and EB fluorescence was widely distributed in brain parenchyma in LPS group, indicating that the EB leakage in LPS group was more marked than that of NS group. In CHR pretreatment groups, EB fluorescence was decreased in brain parenchyma, indicating that EB leakage was significantly less marked, while it was more obvious in high dose CHR group. It was shown by HE staining that cerebral blood vessel structure was intact in NS group, and the gap around blood vessel was not significant increased. On the other hand, brain structure in LPS group appeared loose, with widening of small perivascular spaces and obvious edema. Brain edema in CHR pretreatment groups was improved as compared with that of the LPS group, and it was more apparent in high dose CHR group. LPS induced change in blood brain barrier permeability in mice in a time-dependent manner. Exogenous CGA derived peptides CHR can inhibit LPS induced hyper-permeability of blood brain barrier in septic mice, thus reduces brain edema, protects the brain tissue, and the effect is more obvious with a high dose of CHR (77.5 μg/kg).

Treewidth for graphs with small chordality