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### The probability of pure literals

J. Rosenthal
1999 Journal of Logic and Computation
Speci cally, we show with probability approaching 1 as m goes to 1 one can apply the pure literal rule repeatedly to a random instance of 2-SAT until the number of clauses is \small" provided n=m > 1.  ...  We describe an error in earlier probabilistic analyses of the pure literal heuristic as a procedure for solving k-SAT.  ...  probability the sum of the lengths of all cycles of C is at least t and no cycles of C are connected is phenomenal success of the pure literal heuristic under the hypotheses of Theorem 3.3 it is reasonable  ...

### Average time for the full pure literal rule

Paul Walton Purdom
1994 Information Sciences
The random problems are characterized by t h e n umber of variables v, the number of clauses t, and the probability that a literal is in a clause p.  ...  The simpli ed pure literal algorithm solves satis ability problems by c hoosing variables in a xed order and then generating subproblems for various values of the chosen variable.  ...  Extensive assistance of Professor Khaled Bugrara with this research is greatly appreciated. I also wish to thank Northeastern university for the use of computer resources for some of the  ...

### A sharp threshold in proof complexity

Dimitris Achlioptas, Paul Beame, Michael Molloy
2001 Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01
We give the first example of a sharp threshold in proof complexity.  ...  A consequence of our result also yields the first proof that typical random 3-CNF formulas at ratios below the generally accepted range of the satisfiability threshold (and thus expected to be satisfiable  ...  Define the set È È´Àµ of literals based on À as follows: È consists of the pure literals of À¾, the smallest numbered literal in each pure cycle of À¾, and every literal on the variables of Î´Àµ Î´À¾µ.  ...

### Linear Upper Bounds for Random Walk on Small Density Random 3‐CNFs

Mikhail Alekhnovich, Eli Ben‐Sasson
2007 SIAM journal on computing (Print)
We use the analysis of the pure literal heuristic presented by Broder, Frieze, and Upfal [show that for small clause densities good terminators exist.  ...  Thus we show that the pure literal threshold (≈1.63) is a lower bound on the terminator threshold. (We conjecture the terminator threshold to be in fact higher.)  ...  We thank Bart Selman and Andrew Parkes for valuable information on the empirical results regarding RWalkSAT and Balint Virag for his help with the analysis of martingales.  ...

### Hard satisfiable formulas for DPLL-type algorithms [article]

Sergey I. Nikolenko
2003 arXiv   pre-print
Namely, we consider two DPLL-type algorithms, enhanced with the unit clause and pure literal heuristics.  ...  We address lower bounds on the time complexity of algorithms solving the propositional satisfiability problem.  ...  Hirsch, who suggested the problem and supervised my work.  ...

### Structure of random r-SAT below the pure literal threshold [article]

Alexander D. Scott, Gregory B. Sorkin
2010 arXiv   pre-print
It remains open what happens between the pure literal threshold and the satisfiability threshold.  ...  By contrast, we show that for the (rare) unsatisfiable formulae below the pure literal threshold, the unsatisfiability is whp due to a unique MUF with smallest possible "excess", failing this whp due to  ...  ), and a similar asymptotic expansion for the probability that the pure literal rule fails.  ...

### A sharp threshold in proof complexity yields lower bounds for satisfiability search

Dimitris Achlioptas, Paul Beame, Michael Molloy
2004 Journal of computer and system sciences (Print)
Our result also yields the first proof that random 3-CNF formulas with densities well below the generally accepted range of the satisfiability threshold (and thus believed to be satisfiable) cause natural  ...  Let F ðrn; DnÞ denote a random CNF formula consisting of rn randomly chosen 2-clauses and Dn randomly chosen 3-clauses, for some arbitrary constants r; DX0: It is well-known that, with probability 1 À  ...  probability that F Ã has a small subformula H with few pure items by bounding the probability for each set of literals T that there is a subformula H of F Ã with P Ã ðHÞ ¼ T and with at most jTj=10 pure  ...

### Page 539 of None Vol. 83, Issue [page]

1927 None
The usual result probably contains something of both: the original representative (or realistic; or impression- istic!) element being present, but subjected to a purely musical destiny.  ...  Beethoven’s sixth and seventh symphonies are clearly on the edge, if not of the literal, then of the metaphorical.  ...

### Results related to threshold phenomena research in satisfiability: lower bounds

John Franco
2001 Theoretical Computer Science
., this volume) the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal, intuitive manner.  ...  We present a history of results related to the threshold phenomena which arise in the study of random conjunctive normal form (CNF) formulas. In a companion paper (D. Achlioptas, Theoret. Comput.  ...  Formula (1) is said to be the result of an application of the unit clause rule in the former case and an application of the pure literal rule in the latter case.  ...

### Page 101 of Journal of Chemical Physics Vol. 35, Issue 1 [page]

1961 Journal of Chemical Physics
This may probably be due to the greater concentration quenching that results from the much lower viscosity of pure anisole as compared to anisole in paraffin oil.  ...  (e) Variation of viscosity with temperature of solutions of 0.5 g/liter anthracene in mixtures of anisole and paraffin oil. Curve (1): pure anisole.  ...

### On smoothedk-CNF formulas and the Walksat algorithm [chapter]

Amin Coja-Oghlan, Uriel Feige, Alan Frieze, Michael Krivelevich, Dan Vilenchik
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Starting from an arbitrary instance F with n variables and m = dn clauses, apply the ε-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability  ...  In this paper we study the model of ε-smoothed k-CNF formulas.  ...  Hence, the probability that there is a set S of size s with µ clauses and without a pure literal is at most s ≤ e s n k−1 km n µ .  ...

### Page 331 of The Antijacobin Review and Protestant Advocate, Or, Monthly Political, and Literary Censor Vol. 41, Issue [page]

1812 The Antijacobin Review and Protestant Advocate, Or, Monthly Political, and Literary Censor
if it be not generally the land of delights, is at least within the Jand of delights, a part of the whole, and probably the capital, and probably the Mount Sion, that joy of the whole earth, within that  ...  As these enemies are thus, however, gathered toa war, there must be a hostile force on the pure side: there must be, at this time, some pure power, of such growing consequence as to require these extraordinary  ...

### Page 210 of Rubber Chemistry and Technology Vol. 1, Issue 2 [page]

1928 Rubber Chemistry and Technology
Eight gallons (36 liters) of the railway hydro- carbon gave a little over a liter of these hydrocarbons, from which about 200 grams of the crude solid tetrabromide and 145 grams of the liquid tetrabromides  ...  Preparation of Pure Butadiene Pure butadiene can be readily and conveniently prepared from the solid tetrabromide by the method described by Thiele.2 For most purposes the once-crystallized material is  ...

### Attached growth of Sphaerotilus and mixed populations in a continuous-flow apparatus

F F Dias, N A Dondero, M S Finstein
1968 Applied microbiology
The effects of NH(4)Cl concentration, organic nitrogen compounds, glucose concentration, dissolved oxygen concentration, and flow rate on the attached growth of pure cultures of Sphaerotilus natans and  ...  The conditions that allowed maximal attached growth in pure culture did not correspond to the conditions that promoted attached growth of Sphaerotilus in a mixed population.  ...  However, the open channels used by most investigators probably are more difficult to control and are difficult, if not impossible, to use for pure-culture studies.  ...

### THE QUANTITATIVE DETERMINATION OF CHLOROPHYLL

F. M. Schertz
1928 Plant Physiology
PROBABLE ERROR OF A SINGLE DETERMINATION CONCENTRATION IN GRAMS PER LITER 0.20 0.15 0.10 0.05 mm.  ...  Calculating, using the values for grams of From the curve in fig. 1 , these readings are interpreted in terms of grams of chlorophyll per liter respectively as follows: The probable error of a single  ...  The spectrophotometric method offers great promise of being a very accurate method for determining chlorophyll, though the data for the method are yet to be obtained. LITERATURE CITED  ...
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