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 ....

[Article contains additional citation context not shown here]

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

....all ( cylinders of) solutions, or that no solutions exist. y Research supported in part by an NSERC Postdoctoral Fellowship. Address: Microsoft Research, One Microsoft Way, Redmond WA 98052, U.S.A. Email: optas microsoft.com 1 In a seminal paper, extending the ground breaking result of Haken [21] on the worst case complexity of resolution, Chv atal and Szemer edi [5] used F k (n; rn) to provide examples of formulas that are hard to prove unsatis able for any resolution type strategy (such as the DP algorithm) In particular, they showed that for all k 3, if r2 k 0:7 then there ....

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

....proof system. The theories of interest here are the bounded arithmetic system I1 0 (R) and its extensions by 5 0 2 axioms, cf. 17] The corresponding propositional proof systems are constant depth Frege systems. The first strong lower bound for a natural proof system was obtained by Haken [11] who proved an exponential lower bound to the resolution proofs of the pigeonhole principle (PHP ) Then Ajtai [1] showed that constant depth Frege systems (which are stronger than resolution) do not admit polynomial size proofs of PHP . The first exponential lower bound for such systems was ....

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

....How 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 polynomiallysimulated by CP. Moreover, there are polynomialsized 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.

.... m n H =q m n N H PHP m n O tautology H J k PHP n 1 n O pps , super G J 3 H r ( 9 a Ntautology N8uJd H 7 F [CR] K h C FDs ( 5 l [CR] O eF , PHP n 1 n Npoly size N ZL r; D 3 H r [Bu1] OF ,PHP n 1 n Npoly size N ZL r; D 3 H r ( 7 F k = l KBP 7 [H] Oresolution NPHP n 1 n N ZL ,exponential size H J k 3 H r ( 9 3 H Gresolution ,super G J 3 H r ZL 7 5 i K [Aj] ObdF NPHP n 1 n N ZL ,exponential size H J k 3 H r ( 9 3 H GbdF ,super G J 3 H r ZL 7 F k [Bu1] H[Aj] ibdF HF N4V K:9 , k 3 H ,J, k [BT] ....

.... Oresolution NPHP n 1 n N ZL ,exponential size H J k 3 H r ( 9 3 H Gresolution ,super G J 3 H r ZL 7 5 i K [Aj] ObdF NPHP n 1 n N ZL ,exponential size H J k 3 H r ( 9 3 H GbdF ,super G J 3 H r ZL 7 F k [Bu1] H[Aj] ibdF HF N4V K:9 , k 3 H ,J, k [BT] O[H] N7k2L ,PHP 2n n N l9g K3HD G k 3 H r ( 7 F k , PW] H[PWW] ibdF ,PHP 2n n Nn log n size N ZL r; D 3 H ,J, k N G 3 N [BT] N7k2L i resolution H bdF N4V K:9 , k 3 H , 5 l k F deF ,super I F HeF N4V K:9 , k I O8= G bL 2r7h N=EMW ....

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

....lower bounds for existing proof systems. There are still a number of well known proof systems for which no exponential lower bounds have been found, such as Frege systems [3] Resolution is one of the most popular and simplest proof systems. Even so, it took more than two decades before Haken [6] finally obtained an exponential lower bound for the pigeonhole principle. This settlement of the major open question, however, has stimulated continued research on the topic [1, 4, 8] The reason is that Haken s lower bound is quite far from being tight and his proof, although based on an ....

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

....is bounded by O(n 2k ) Hence the size of the whole proof is bounded by a polynomial of n. 2 Theorem 1 and theorem 2 witness the fact that none of resolution, bounded depth Frege, polynomial calculus, Hilbert s Nullstellensatz do not p simulate the system of GCNF permutation. Proposition 3 [9] There exists a constant c , c 1 such that, for sufficiently large n, every resolution refutation of PHPn contains at least c n different cedents. Proposition 4 [1] There exists a constant c , c 1 so that, for sufficiently large n, every constant depth Frege proof of : k Eq(n) contains ....

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

....is exponential in the number of atoms. It appears however, that this is actually the most efficient algorithm known: there is no proof system for which a subexponential upper bound would be established. For a few particular systems some lower bounds to the size of proofs are known. Haken [10] established exponential lower bound for the resolution system, preceeded by Cejtin [7] who showed a similar lower bound for regular resolution. Ajtai [1] then proved that there is no universal polynomial upper bound for proofs in a Frege system when the logical depth of all formulas in proofs is ....

....ik ; x jk ) one for each i 6= j 2 D and k 2 R, and (x i1 ; x in ) one for each i 2 D. Thinking about x ik as saying f(i) k, the cedents from :PHP (D; R) assert that f is an injective function from D into R. Hence for jDj n the set :PHP (D; R) is refutable. The argument from Haken [10] was generalized by Buss and Tur an to yield the following lower bound which we restate for our system (Haken had m = n 1) Proposition 2.1 (S. Buss G. Tur an [6] Let jDj = m n. Then every depth 0 refutation of :PHP (D; R) must have length at least exp i Omega i n 2 m jj . Note that ....

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

.... 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.

....bound was proved for this model [1, 2] based on a combinatorial condition of [4] One direction of study of propositional proof systems is to prove lower bounds on the length of proofs in certain restricted proof systems. Exponential lower bounds were obtained for such systems as resolution [12], bounded depth Frege systems [15, 17] cutting planes [6, 18] and Nullstellensatz refutations [3, 8] In this paper we are interested in systems that use uses polynomials instead of boolean formulas. From the previous list this includes the Nullstellensatz refutations. Recently a stronger system ....

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

....Classical resolution was developed by Quine (1952,1955) for propositional logic and extended by Robinson (1965) to predicate logic. Various forms of it are widely used in logic programming and theorem proving systems. Its worstcase complexity was first investigated by Tseitin (1968) and shown by Haken (1985) to be exponential. In practice, however, one would use resolution to generate only a few cuts rather than carry it to completion. Resolution for propositional logic is defined as follows. Suppose two clauses are given for which exactly one atomic proposition occurs positively in one and ....

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

....system. The theories of interest here are the bounded arithmetic system I Delta 0 (R) and its extensions by Pi 0 2 axioms, cf. 17] The corresponding propositional proof systems are constant depth Frege systems. The first strong lower bound for a natural proof system was obtained by Haken [11] who proved an exponential lower bound to the resolution proofs of the pigeonhole principle (PHP ) Then Ajtai [1] showed that constant depth Frege systems (which are stronger than resolution) do not admit polynomial size proofs of PHP . The first exponential lower bound for such systems was ....

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

....Boolean circuits over the complete basis f; g. Moreover, in the case of unbounded fanin gates previous lower bounds were known only for AND OR gates under additional restriction that circuits have constant depth (cf. 21] Our proof combines two ideas: the bottlenecks counting idea of Haken [11, 12, 13] and Sipser s idea of finite limits [28, 29] The resulting argument becomes extremely simple and is different from Razborov s method of approximations [23, 24, 26] although the general idea remains the same: we try to map a large set of input vectors to gates in the circuit so that not too many ....

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

....axioms, see [28] or [26, Chp.14] Obviously then, one expects that no proof system admits polynomial size proofs for all tautologies (as NP 6= coNP is a generally accepted conjecture) This was, however, demonstrated only for some particular subsystems of F . Most notable examples are resolution ([20]) constant depth systems ( 2, 24, 33, 44] and constant depth systems augmented by additional axiom schemes expressing various combinatorial principles ( 3, 50, 8, 4, 7, 51] see [26, Chpts.4 and 12] Nothing is known for the constant depth systems if the language contains also the equivalence ....

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

....such a proof is non trivial because of the huge number of irrelevant clauses that can be derived. Although many strategies have been proposed for choosing the order in which clauses are resolved to help control the complexity in certain situations, resolution in general is known to be NP complete (Haken 1985). Hybrid Reasoning Algorithm Suppose we are given a domain theory in the hybrid language and a query whose entailment we are to prove. As with standard resolution refutation theorem provers, the first steps are to add the negation of the queried sentence to the domain theory, and then convert it ....

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

....steps [4] 5] Hence these problems admit only exponentially large closed clause trees. However, such problems can sometimes be solved in polynomial time [4] 7] 39] One method that allows the pigeonhole problem to be solved polynomially is to allow a single variable to replace a clause [9]. The negation of a clause is a conjunction, so that conjunctions can be dealt with. How such substitutions can be represented as clause trees, and when such substitutions should be performed are two more open questions. 9.2 Summary and Conclusions Clause trees offer new insights for ....

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

....hard, it seems reasonable to prove superpolynomial lower bounds gradually for stronger and stronger proof systems. For quite a long time there were no nontrivial lower bounds for any proof system, except for a very special system, regular resolution, see [20] The real progress started when Haken [13] proved an exponential lower bound for the (unrestricted) resolution system. This is a rather weak system, but it is very important for practical applications. In [11] Cook and Reckhow defined certain important proof systems, the most important is the class of the so called Frege systems. This ....

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

....tree in memory at any one time) Most work on systematic search concentrates on heuristics for variable ordering and value selection, all in order to the reduce size of the tree. However, there are known fundamental limitations on the size of the shortest resolution proofs for certain problems (Haken 1985; Chvatal and Szemeredi 1988) For example, pigeon hole problems (showing that n pigeons cannot fit in n Gamma 1 holes) are intuitively easy, but shortest resolution refutation proofs are of exponential length. Shorter proofs do exist in more powerful proof systems. Examples of proof systems ....

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

.... e.g. or the formulas with the implication as the only operator and with every variable occurring twice, 11] Classes of instances of the SAT problem have been studied in order to show proof systems like resolution to be exponential time provers for these classes, as the pigeonhole formulas, [10], or Tseitin s graph formulas, 18] The satisfiability test of these instances is hard for certain proof systems only, but not for a human solver, who knows in advance due to an understanding of the idea behind the construction principle of the formulas whether they are satisfiable or ....

Haken, A.: The Intractability of Resolution, Theor. Comput. Sci., 39, 1985, 297--308

....(k; b; d) design with k = n, b = m, 0 and d = 1. Since jF j = m n = k(d 1) 2, we can apply (3) which yields the lower bound 2 Omega (n 2 = ml) Recall that 2 Omega Gamma n 2 =m) is the best known lower bound for the minimal length of a Resolution refutation proof of PHP m n [13, 23, 9, 11]. So, the reason why PHP m n is hard for Resolution, seems to lie not in the weakness of the resolution rule itself, but rather in the impossibility to keep enough information about possible outcomes, using small (up to l) sets of clauses. Example 2 (Affine planes) Take an affine plane AG(2; q) ....

....complexity (cf. the famous Switching Lemma for depth 2 AND OR circuits) The forcing (large clauses) idea was used by Chv atal and Szemer edi [10] to generate hard examples for resolution. In a recent work [2] Beame and Pitassi accumulated both ideas into a direct and elegant proof of Haken s [13] lower bound for the pigeonhole principle PHP n 1 n . Our work is motivated by the exposition in [2] All the combinatorics we need is accumulated in two lemmas: the killing large clauses lemma (Lemma 1) and the forcing large clause lemma (Lemma 2) 3.1. Combinatorics Lemma 1. Killing Lemma) ....

[Article contains additional citation context not shown here]

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

....interpolation theorems for the following proof systems: a) resolution. b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff) c) linear equational calculus. d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) a) for resolution ([15]) b) for the cutting planes proof system with coefficients written in unary ( 4] 3. An alternative proof of the independence result of [43] concerning the provability of circuit size lower bounds in the bounded arithmetic theory S 2 2 (ff) 1991 Mathematics Subject Classification. Primary ....

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

....1 as the number of variables approaches infinity. 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 insight. Extending a ground breaking result by Haken (1985), they showed that any resolution strategy requires exponential time with probability 1 on formulas where the ratio of clauses to variables is a constant greater than 5.6. They also show that with probability approaching 1 such formulas are unsatisfiable. Given that DP corresponds to a ....

Haken, A. (1985). The intractability of resolution.

.... [2] e.g. or the formulas with the implication as the only operator and with every variable occurring twice, 7] Classes of instances of the SAT problem have been studied in order to show proof systems like resolution to be exponential time provers for these classes, as the pigeonhole formulas, [6], or Tseitin s graph formulas, 11] The satisfiability test of these instances is hard for certain proof systems only, but not for a human solver, who knows in advance due to an understanding of the idea behind the construction principle of the formulas whether they are satisfiable or ....

Haken A.: The Intractability of Resolution, Theor. Comput. Sci., 39, 1985, 297--308

....demonstrate the unprovability of a principle in I Delta 0 (R) or S 2 (R) it suffices to prove that it has no constant depth Frege proofs of a certain size. In a series of results, it has been shown that any constant depth Frege proof of the pigeonhole principle (PHP ) requires exponential size [12, 1, 15, 6, 17, 19]. These results were extended to show that even with the pigeonhole principle as an additional axiom constant depth Frege systems require exponential size to prove the Count 2 tautologies [2, 7, 20] We use the notation Count q for the generic version of Count m q where m 6j 0 (mod q) Ajtai ....

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

....gate in the circuit at which a certain amount of progress in classifying the input is made. The measure of progress is the length of a fence which intuitively keeps track of how many bits of the input are actually used by the computation at that point. This method is similar to that used by Haken [Hak85] (and subsequently others [Urq87] CS88] to prove an exponential lower bound on the length of resolution proofs in the propositional calculus. In that case a large set of critical truth assignment vectors was mapped to clauses in the proof. The result in the present paper has an application to ....

A. Haken. The intractability of resolution. Theor. Comp. Sci. 39, pages 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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Haken, A. (1985) The intractability of resolution, Theoretical Computer Science, 39:297--308.

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