A. Haken, "The intractability of resolution," Theoretical Computer Science, vol. 39, pp. 297--308, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the solver from effectively processing the search space. For example, the pigeonhole problem states that n 1 pigeons cannot be placed in n holes without sharing. The length of the shortest resolution proof of unsatisfiability of the corresponding CNF problem is exponential in the number of holes [6]. Therefore, every DavisPutnam Logemann Loveland style (DPLL) solver [7, 8] will exercise an exponential runtime. In contrast, a description based on cardinality constraints suits this problem naturally and the length of the shortest cutting plane proof [9, 10] of unsatisfiability is only ....

A. Haken, "The intractability of resolution," Theoretical Computer Science, vol. 39, pp. 297--308, 1985.

....hole principle. There are trivial resolution proofs (and regular resolution proofs) of length 2 poly(n) for the pigeon hole principle and for the weak pigeon hole principle. In a seminal paper, Haken proved that for the pigeon hole principle, the trivial proof is (almost) the best possible [Hak]. More specifically, Haken proved that any resolution proof (or regular resolution proof) for the tautology PHP n is of length n) Haken s argument was further developed in several other papers (e.g. Urq, BeP, BSW] It was shown that a similar argument gives lower bounds also for the weak ....

Haken, A. "The intractability of Resolution," Theoretical Computer Science, 39, 1985, pp. 297-308.

....truth assignment with probability at least = r) 0: If there exist unit clauses, pick one randomly and satisfy it; else pick a random unset variable and set it to 0. A seminal result in the area was established a few years later by Chvatal and Szemeredi [8] Extending the work of Haken [18] and Urquhardt [27] they proved the following: for all k 3, if r 2 ln 2, then w.h.p. F k (n; rn) is unsatisfiable and every resolution proof of its unsatisfiability must contain at least (1 ) clauses, for some = k; r) 0. Random k SAT owes a lot of its popularity to the ....

A. Haken. The intractability of resolution. Theoret. Comput. Sci., 39(2-3):297--308, 1985.

....most once. There are trivial Resolution proofs (and Regular Resolution proofs) of length 2 Delta poly(n) for the pigeon hole principle and for the weak pigeon hole principle. In a seminal paper, Haken proved that for the pigeon hole principle, the trivial proof is (almost) the best possible [Hak]. More specifically, Haken proved that any Resolution proof for the tautology PHP n is of length 2 Omega Gamma n) Haken s argument was further developed in several other papers (e.g. Urq, BeP, BSW] In particular, it was shown that a similar argument gives lower bounds also for the weak ....

Haken, A. "The intractability of resolution," Theoretical Computer Science, vol. 39, 1985, pp. 297-308.

....it can be formalized in propositional logic, it is natural to ask in which propositional proof systems such a principle can be proved in polynomial size, with respect to the size of the encoding. A fair amount of information is known about sizes of proofs of PHP n in various proof systems. Haken [12] proved that this principle requires exponentialsize proofs in Resolution. His proof techniques were later extended and simplified [4, 5] Also Beame et al. 2] proved that PHP n requires exponential size proofs in bounded depth Frege systems. Regarding upper bounds, Buss [8] gave polynomial size ....

A. Haken. The intractability of resolution. TCS, 39(2-3):297--308, Aug. 1985.

....Pigeon hole CNF formulas, by means of propositional logic, describe the fact that n objects (pigeons) cannot be placed in m holes so that no two objects occupy the same hole if n m. Pigeon hole formulas was the first class of CNF formulas for which resolution was proven to be exponential [6]. Definition 13. Denote by ph(i,k) the Boolean variable whose value indicates if i th pigeon is in k th hole (ph(i,k) 1 means that the pigeon is in the hole) Pigeon hole CNF formula (written PH(n,m) consists of the following two sets of clauses (denote them by H 1 (n,m) and H 2 (n,m) Set H 1 ....

A.Haken. The intractability of resolution. Theor. Comput. Sci. 39 (1985),297-308.

....1 Introduction Many real world problems have interesting symmetries. Encoding these problems into CNF formulas generally results in hard SAT problems. Examples includes the pigeon hole problem, which, although very simple, is nonpolynomial for any resolution based method when encoded into SAT [6], and the n queen problem. In order to speed up search algorithms for these problems, intrinsic symmetries should be exploited to avoid repeated search of equivalent portions of search space. The general strategy to exploit symmetries is to divide the objects of the search space into equivalence ....

A. Haken. The intractability of resolution. In Theoretical Computer Science 39, pages 297--308, 1985.

....such must also be exponential (in the number of arguments defining H ) It is worth noting, at this point, that there is a rich corpus of research concerning the length of proofs in various proof systems. Results on the complexity of General Resolution date back to the seminal paper of Haken [20] in which this approach was shown to require exponential length proofs for tautologies corresponding to the combinatorial Pigeon Hole Principle, with important subsequent work in, e.g [1, 3, 4, 27, etc. Excellent introductory surveys discussing progress involving proof complexity may be found in ....

.... has chromatic number greater than 3, 7, 25] It is the case, however, that these analyses are effectively only dealing with Classical (Propositional) Logic, and such results as extend to non classical Logics do so only by virtue of propositional logic being treatable as a sub case, e.g. Haken[20] trivially applies to the csd rep gentzen.tex; 17 10 2001; 12:47; p.26 27 Resolution Calculus for Temporal Logic of [17] simply by expressing the relevant tautology without the use of any temporal operators, i.e. exactly as its propositional form. We conclude by reviewing some directions for ....

Haken, A.: 1985, `The intractability of resolution'. Theoretical Computer Science 39(2- 3), 297--308.

....Logic of algorithms 5.1 A big achievement of proof complexity is the result that every algorithm for 3 SAT based on Davis Putnam procedure has worst case complexity at least 2 c:n for a positive constant c. This follows from a result of Urquhart [18] which uses ideas of Tseitin [17] and Haken [5]. It is worthwhile to explain this result in more details. The propositional resolution calculus is the system based on the resolution rule described above. Successive applications of the rule produce new clauses from a given set of clauses. The system is complete in the sense that for any clause ....

A. Haken, The intractability of resolution, Theor. Computer Science, 39, 1985, 297-308.

.... Phi 2 is necessarily false. In spite of the clear intuition, this example is a challenge for mechanical proof systems. In 1985 Haken showed, that there is an exponential lower bound in n for the number of steps any proof of unsatisfiability needs, when using resolution as decision procedure (cf. Hak85] 4.4.2 Expressing Pigeon Hole in the Bit Vector Theory For each boolean variable m ij , a corresponding bit vector m ij [1] is introduced. Then the formulae Phi 1 ; Phi 2 and Phi 3 can be built as OBDDs by means of applying the boolean connectives on bit vector terms of width one. In the ....

Armin Haken. The intractability of resolution. In TCS-39-1 [TCS84], pages 297--308. Journal.

....while keeping the ratio of clauses to variables fixed. Unfortunately, this result does not directly tell us much about the expected time to find a single assignment. A recent result by Chvatal and Szemeredi (1988) can be used to obtain some further insights. Extending a ground breaking result byHaken (1985),they showed that any resolution strategy requires exponential time with probability approaching 1 on formulas where the ratio of clauses to variables is a constant greater than 5.2. They also show that with probability approaching 1 such formulas are unsatisfiable. Given that DP corresponds to ....

Haken, A. (1985). The intractability of resolution. Theoretical Computer Science, 39, 1985, 297--308.

....the existence of hard tautologies for Frege systems, and for extended Frege systems. So far, however, such lower bounds have been given only for restricted versions of Frege systems. The first general lower bound was given for Resolution proofs of the propositional pigeonhole principle by Haken [H]. Resolution proofs can be viewed as depth 1 Frege proofs. Later, in a remarkable paper by Ajtai [Ajt] it was shown that no bounded depth Frege proof can prove the pigeonhole principle in polynomial size. Then, Kraj icek [K2] proved exponential lower bounds for constant depth Frege proofs of a ....

....Resolution. Secondly, the propositional pigeonhole principle (PHP ) has a very simple polynomial size CP proof [CCT] This is interesting because PHP is the canonical hard tautology that has been previously used to prove lower bounds for Resolution as well as for bounded depth Frege systems (e.g. [H], BIKPPW] Since PHP has a short CP proof, CP is strictly stronger than Resolution (with respect to what can be proven by polynomial size proofs) It was shown in [G] that any Frege system can polynomially simulate CP , and therefore CP lies between Resolution and Frege. Thus, understanding the ....

Haken, A. "The intractability of Resolution," Theoretical Computer Science, 39, 1985. pp. 297-308.

....powerful are Cutting Planes and tree like Cutting Planes proof systems It is straightforward to verify that CP is a generalization of Resolution i.e. any Resolution proof can be polynomially simulated by CP. Moreover, there are polynomial sized CP proofs of the propositional PHP, but Haken [H] has shown that Resolution proofs of PHP require exponential size. Thus, Resolution cannot p simulate CP. It was shown in [G1] that any Cutting Planes proof can be p simulated by a Frege proof. This result is not surprising because each formula in a CP refutation is just a depth 1 threshold ....

Haken, A. "The intractability of Resolution," Theoretical Computer Science, 39, 1985. pp. 297-308.

.... Simon Dept of Computer Science University of Chicago Chicago, IL 60637, USA simon cs.uchicago.edu Shi Chun Tsai Information Management Department National Chi Nan University Pu Li, Nan Tou 545, TAIWAN tsai csie.ncnu.edu.tw January 26, 1999 Abstract Both the bottleneck counting argument [7, 8] and Razborov s approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance ....

....introduces only a small number of new errors and the number of errors for the whole circuit is large, M must have many gates. Many more lower bounds were proven using the approximation method, for example Yao [15] Goldmann and Hastad [6] The bottleneck counting argument, introduced by Haken [7], defines a mapping from a subset of the inputs to the gates in the circuit. The number of inputs that are mapped to the gates divided by the maximum number of inputs that can be mapped to a gate in the circuit is the lower bound of the circuit size. Thus by finding a proper mapping, we can show ....

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A. Haken. The intractability of resolution. Theoret. Comput. Sci., 39:297--308, 1985.

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A. Haken, "The intractability of resolution," Theoretical Computer Science, vol. 39, pp. 297--308, 1985.

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A. Haken, "The intractability of resolution," Theoretical Comput. Sci., vol. 39, pp. 297--308, 1985.

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A. Haken, The intractability of resolution, Theor. Comp. Sci., 39 (1985), 297--308.

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A. Haken, The intractability of resolution. Theor. Comp. Sci., 39 (1985), 297--308.

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A. Haken. The intractability of Resolution. Theoret. Comp. Sci. 39, pp. 297--308, 1985.

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Armin Haken, The intractability of resolution, Theoret. Comput. Sci. 39 (1985), no. 2-3, 297--308.

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A. Haken, "The Intractability of Resolution," Theoretical Computer Science, vol. 39, pp. 297-- 308, 1985.

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A. Haken. "The Intractability of Resolution". Theoretical Computer Science, 39:297--308, 1985.

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A. Haken. The Intractability of Resolution. In Theoretical Computer Science, 39 (1985), pp. 297-308.

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A. Haken. The Intractability of Resolution. In Theoretical Computer Science, 39 (1985), pp. 297-308.

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Haken, A. (1985), `The intractability of resolution', Theoretical Computer Science 39, 297 - 308.

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