Stephen A. Cook, The complexity of theorem-proving procedures, 3rd Annual ACM Symposium on Theory of Computing (Shaker Heights, OH,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the results of [Sh 89, LFKN 89] do not relativize, there is an oracle construction in [FRS 88] that shows that the characterization of NEXP in terms of two prover interactive proof systems [BFL 90] does not relativize. the proofs in [BF 90] The standard reduction given by Cook s theorem [Co 71] shows that any computation tree on inputs of length n can be described with a polynomialsized instance of 3SAT. Replace each of the disjunctions in this 3SAT instance by an equivalent disjunction of 7 conjunctions on the same variables. Now treat this resulting formula as an arithmetic polynomial ....

S. Cook, The complexity of theorem proving procedures, Proc. 3rd Annual ACM Symposium on Theory of Computing, pp. 151--158.

....CNF, and v be a variable. Then OptVal(F ) max( OptVal(F [v] OptVal(F [v] We also note that a polynomial time algorithm for 2 SAT is known. In our context, a formula F is satis able if OptVal(F ) is equal to the sum of the weights of all clauses occurring in F . Lemma 4 (see, e.g. [2]) There is a polynomial time algorithm for 2 SAT. Yannakakis presented in [23] an algorithm which transforms a formula in 2 CNF into a formula in 2E CNF which has the same optimal value. This algorithm consists of two stages. The rst stage is a removal of a maximum symmetric ow from a graph ....

....exactly once positively in a clause C and exactly once negatively in a clause D, then F : F fC; Dg) R(C;D) 4) If F has been changed at steps (2) 3) then go to step (1) 5) If F is satis able 5 , return the sum of the weights of all its clauses. 5 We can check it in a polynomial time [2], see Lemma 4. 5 (6) If there is a variable v occurring in the clauses of F of the total weight at least 4, execute Algorithm 1 for the formulas F [v] and F [v] and return the maximum of its answers. 7) Find 6 in F a clause ( fl 1 ; l 2 g) such that l 1 and l 2 are (2;1) literals, and the ....

S. A. Cook, The complexity of theorem-proving procedure, Proc. 3rd Annual ACM Symposium on the Theory of Computing, 1971, pp. 151-159.

....which are subsets of V [ V and are such that no two literals are complementary. A random formula contains m clauses, each chosen uniformly and with replacement from S. We use the notation F k (n; m) to denote an instance of a xed width distribution. It is well known that SAT is NP complete [11] and k SAT is NP complete even if k 3. However, there are many subclasses of SAT that are known to be solvable in polynomial time: the two most notable being 2 SAT and Horn formulas (every clause has at most one positive literal) 5 3 A History of Threshold Related Results Logic has always ....

....Clause Algorithm for CNF formulas. whether the threshold exists and, if so, where is it. Chv atal and Reed [10] Goerdt [21] and Fernandez de la Vega [14] independently answered these questions: they determined r 2 = 1. It is important to observe that 2 SAT being solvable in polynomial time [11] means that there is a simple characterization of unsatis able 2 SAT formulas. Indeed, both [10] and [21] make full use of this characterization as they proceed by focusing on the emergence of the most likely unsatis able formulas in F 2 (n; rn) Also using this characterization, Bollob as et ....

Stephen A. Cook, The complexity of theorem-proving procedures, 3rd Annual ACM Symposium on Theory of Computing (Shaker Heights, OH, 1971), ACM, New York, 1971, pp. 151-158.

....CNF, and v be a variable. Then OptVal(F ) max( OptVal(F [v] OptVal(F [v] We also note that a polynomial time algorithm for 2 SAT is known. In our context, a formula F is satis able if OptVal(F ) is equal to the sum of the weights of all clauses occurring in F . Lemma 4 (see, e.g. [1]) There is a polynomial time algorithm for 2 SAT. Yannakakis presented in [21] an algorithm which transforms a formula in 2 CNF into a formula in 2E CNF which has the same optimal value. This algorithm consists of two stages. The rst stage is a removal of a maximum symmetric ow from a graph ....

....l 2 are (2;1) literals, and the two other clauses C and D containing the literals l 1 ; l 1 do not contain the literals l 2 ; l 2 . Execute Algorithm 1 for the formulas (F [l 1 ] fC; Dg) R(C;D) and F [l 1 ; l 2 ] and return the maximum of its answers. ut 3 We can check it in a polynomial time [1], see Lemma 4. 4 Theorem 1 proves that it is possible to nd a clause satisfying the conditions of this step. 5 Theorem 1 Given a formula F in 2 CNF, Algorithm 1 always correctly nds OptVal(F ) in time O(2 K2 (F ) 4 ) Proof. Correctness. If Algorithm 1 outputs an answer, then its ....

S. A. Cook, The complexity of theorem-proving procedure, Proc. 3rd Annual ACM Symposium on the Theory of Computing, 1971, pp. 151-159.

....we will omit oors and ceilings when this does not cause confusion. 2.1 Random 2 SAT For k = 2, Chv atal and Reed [6] Goerdt [19] and Fernandez de la Vega [14] independently proved the conjecture, in fact determining r 2 = 1. It is important to note that 2 SAT being solvable in polynomial time [7] means that we have a simple characterization of unsatis able 2 SAT formulas. Indeed, both [6] and [19] make full use of this characterization as they proceed by focusing on the 2 emergence of the most likely unsatis able subformulas in F 2 (n; rn) Also using this characterization, Bollob as ....

Stephen A. Cook, The complexity of theorem-proving procedures, 3rd Annual ACM Symposium on Theory of Computing (Shaker Heights, OH, 1971), ACM, New York, 1971, pp. 151-158.

....Typically, fomulae are considered to be in Conjunctive Normal Form (CNF) i.e. a conjunction of disjunctions (clauses) and one needs to determine if there exists an assignment of truth values to the formula s variables so that at least one literal is satis ed from each clause. Cook s Theorem [12] asserts that satis ability is NP complete and thus at least as hard as any y Address: Microsoft Research, One Microsoft Way, Redmond WA 98052, U.S.A. Email: optas microsoft.com. Work done while at the Department of Computer Science, University of Toronto. z Address: Department of Computer ....

....Science, Carleton University, Ottawa, Ontario, Canada K1S 5B6. Email: fkranakis,krizancg scs.carleton.ca. 1 problem whose solutions can be veri ed in polynomial time. A canonical version of the satis ability problem is k SAT, where each clause of the input formula has precisely k literals. Cook [12] proved that for k 3, k SAT is NP complete, while for k = 2 it can be solved in polynomial time. Given that satis ability is NP complete, practitioners seek heuristic solutions to the problem of deciding the satis ability of large fomulae. The most common approach is to employ some variation of ....

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Stephen A. Cook, The complexity of theorem-proving procedures, 3rd Annual ACM Symposium on Theory of Computing (Shaker Heights, OH, 1971), ACM, New York, 1971, pp. 151-158.

....contents of the memory location, usually logarithmically. This prevents arti cial coding of large amounts of data in a few memory locations, to be manipulated at unit cost. Polynomial time transformations between SAM and RAM computations are one of the cornerstones of the NP completeness theory [1]. In the next section, we show some examples, where SAM (and respectively RAM) models well the computation, and where it doesn t. In many occasions, neither of the two models captures a right abstraction of the computation by a modern computer. We suggest a minor but signi cant extension for the ....

Cook, S.: The complexity of theorem proving procedures. Proc. 3rd Annual ACM Symposium on Theory of Computing, 151-158.

....the following three cases. First case: A has a maximum a. Then co NP is p m reducible to Absolute Counting (A) Let a language L in co NP be given and let M be a machine for L in the sense that x 2 L iff no path of M(x) is accepting. Let m(x) denote the number of accepting paths of M(x) Cook [7] established a method to construct in polynomial time a circuit Cook M x with inputs y 1 ; y n (n depends on x and is bounded by a polynomial in the length of x) such that accepting computation paths of M(x) and satisfying assignments of Cook M x (y 1 ; y n ) correspond to each ....

S. A. Cook. The complexity of theorem proving procedures, Proc. 3rd Annual ACM Symposium on the Theory of Computing (STOC), 1971, pp. 151--158.

....NP complete, unapproximable, randomized reduction, clique, counting problems, permanent, 2SAT AMS subject classifications. 68Q15, 68Q25, 68Q99 1. Introduction. Previous Work. The theory of NP completeness was developed in order to explain why certain computational problems appeared intractable [10, 14, 12]. Yet certain optimization problems, such as MAX KNAPSACK, while being NP complete to compute exactly, can be approximated very accurately. It is therefore vital to ascertain how difficult various optimization problems are to approximate. One problem that eluded attempts at accurate approximation ....

S.A. Cook, The Complexity of Theorem-Proving Procedures, 3rd Annual ACM Symposium on Theory of Computing, 1971, pp. 151-158.

....where each clause C i is a disjunction of literals and each literal is either a variable x i or its negation :x i (1 i n) The SAT(isfiability) problem consists in determining whether such an expression OE is true for some assignment of boolean values to the variables x 1 ; xn . Cook [2] has shown that SAT is NP complete. It is now the reference NP complete problem [5] and, for this reason, it is one of the key problems of automatic deduction. Recently, several authors [3, 1, 6, 4] have exhibited a threshold phenomena on randomly generated K SAT instances: when the number of ....

S. A. Cook. The complexity of theorem proving procedures. 3rd Annual ACM Symposium on the theory of computing, (8):151--158, 1971.

....logic of type theory, we can now code a binary relation ffi D n 3 Theta D n 3 , where ffi(ID; ID 0 ) means ID 0 is reachable from ID in one machine transition. The logical specification of ffi is straightforward, more or less on the level of the detailed coding in Cook s Theorem [Coo71, GJ79]. Let ffi : Delta n 3 Delta n 3 Bool be the calculus interpretation of ffi; instantiating Bool j oe oe oe in this type as Delta n 3 Delta n 3 Delta n 3 , we can define the transition function ffi : Delta n 3 Delta n 3 as: ffi j ID : Delta n 3 :D n 3 (ID 0 : Delta n 3 ....

S. A. Cook. The complexity of theorem-proving procedures. 3rd Annual ACM Symposium on the Theory of Computing, pp. 151--158.

....The proof distinguishes the following three cases. First case: has a maximum . Then co NP is reducible to Absolute Counting( Let a language in co NP be given and let be a machine for in the sense that iff no path of ( is accepting. Let ( denote the number of accepting paths of ( Cook [7] established a method to construct in polynomial time a circuit with inputs . depends on and is bounded by a polynomial in the length of ) such that accepting computation paths of ( and satisfying assignments of ( correspond to each other. Therefore, evaluates to 1 for ....

S. A. Cook. , Proc. 3rd Annual ACM Symposium on the Theory of Computing (STOC), 1971, pp. 151--158.

....versions of these problems where jc i j (the number of literals in clause i) is limited. When the size of each clause is restricted to k literals, the above problems are referred to as k CNF SAT and MAX k CNF SAT respectively. It is known that for k 3 the k CNF SAT problem is NP Complete [1] but the 2 CNF SAT problem is in P [2] Interestingly, the MAX k CNF SAT problem is NP Complete for k 2 [3] So, the 2 CNF SAT problem is solvable in polynomial time but the MAX 2 CNF SAT problem is NP Complete. Recently it was shown that the 2 CNF SAT problem can be solved in O(logn) parallel ....

S. Cook, The complexity of theorem-proving procedures, Proc. 3rd Annual ACM Symposium on Theory of Computing, 1971, 151--158.

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Stephen A. Cook, The complexity of theorem-proving procedures, 3rd Annual ACM Symposium on Theory of Computing (Shaker Heights, OH,

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Stephen A. Cook. The complexity of theorem-proving procedures. 3rd Annual ACM Symposium on the Theory of Computing, pp. 151-158.

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S. A. Cook, The complexity of theorem-proving procedure, Proc. 3rd Annual ACM Symposium on the Theory of Computing, 1971, pp. 151-159.

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Stephen A. Cook. The complexity of theorem-proving procedures. 3rd Annual ACM Symposium on the Theory of Computing, pp. 151--158.

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