Enrico giunchiglia, armando tacchella theory and tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về...
[...]... ones implied by the results of Erdös and Rado, and Berlekamp We proceed as follows For each triple of positive integers we define a propositional CNF theory and then show that is satisfiable if and only if With such encodings, one can use SAT solvers (at least in principle) to determine the satisfiability of and, consequently, 4 Michael R Dransfield, Victor W Marek, and find Since without loss of generality... proof of Szemerédi theorem Geometric and Functional Analysis, 11:465-588, 2001 10 R.L Graham, B.L Rotschild, and J.H Spencer Ramsey Theory, Wiley, 1990 11 A Hales and R.I Jewett Regularity and positional games, Transactions of American Mathematical Society, 106:222–229, 1963 12 R.E Jeroslaw and J Wang solving propositional satisfiability problems, Annals of Mathematics and Artificial Intelligence, 1:167–187,... by cardinality atoms and apply to them solvers capable of handling such atoms directly To describe we will use a standard first-order language, without function symbols, but containing a predicate symbol in_block and constants An intuitive reading of a ground atom is that an integer is in block We now define the theory by including in it the following clauses: for every vdW1: and every such that vdW2:... denote integers and blocks) such theory, after grounding, coincides with In fact, we have defined an appropriate syntax that allows us to specify both data and schemata and implemented a grounding program psgrnd [4] that generates their equivalent ground (propositional) representation This grounder accepts arithmetic expressions as well as simple regular expressions, and evaluates and eliminates them... least such number This E Giunchiglia and A Tacchella (Eds.): SAT 2003, LNCS 2919, pp 1–13, 2004 © Springer-Verlag Berlin Heidelberg 2004 2 Michael R Dransfield, Victor W Marek, and number is called the van der Waerden number Exact values of are known only for five pairs For other combinations of and there are some general lower and upper bounds but they are very coarse and do not give any good idea about... lemma; both it and its proof are from [17] Lemma 1 Let M, be independent binomial random variables If then where is monotonically increasing Proof For any the probability is bounded above by Choosing this sum can be bounded above by routine computations using the Chernoff bounds and We now present the notation and definitions used throughout the paper Let denote a 3-SAT formula, and let A and denote true/false... formula, and that almost all satisfiable formulas are good – hence, the algorithm can handle almost all satisfiable formulas 3 Algorithm In this section, we present our algorithm, and show that it finds a satisfying assignment on all good formulas The algorithm, which is parameterized by and takes a 3-SAT formula as input, is as follows: Pick a random assignment A While no assignment in Randomly pick... that, given and computes the van der Waerden number for consecutive integers we test whether the theory is satisfiable If so, we continue If not, we return and terminate the algorithm By the van der Waerden theorem, this algorithm terminates It is also clear that there are simple symmetries involved in the van der Waerden problem If a set M of atoms of the form is a model of the theory and is a permutation... clear that all clauses (vdW1) and (vdW2) from can be represented in a more concise way by the following collection of c-clauses: for every Indeed, c-clauses enforce that their models, for every contain exactly one atom of the form — precisely the same effect as that of clauses (vdW1) and (vdW2) Let be a PS+ theory consisting of clauses and (vdW3) It follows that Proposition 1 and Corollary 1 can be reformulated... kcnfs: an Efficient Solver for Random Gilles Dequen, Olivier Dubois 486 Formulae An Extensible SAT-solver Niklas Eén, Niklas Sörensson 502 Survey and Belief Propagation on Random K-SAT Alfredo Braunstein, Riccardo Zecchina 519 Author Index 529 This page intentionally left blank Satisfiability and Computing van der Waerden Numbers Michael R Dransfield1, Victor W Marek2, and 2 1 National Security Agency,