## The complexity of propositional proofs

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Venue: | Bulletin of Symbolic Logic |

Citations: | 20 - 0 self |

### BibTeX

@ARTICLE{Segerlind_thecomplexity,

author = {Nathan Segerlind},

title = {The complexity of propositional proofs},

journal = {Bulletin of Symbolic Logic},

year = {},

volume = {13},

pages = {2007}

}

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### Abstract

Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents

### Citations

2920 | Graph-Based Algorithms for Boolean Function Manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...is unsatisfiable, we would need only check that its canonical form is the constant false. Ordered binary decision diagrams (OBDDs) are data structures for canonically representing Boolean functions 4 =-=[46, 47, 117]-=-. The catch is that the canonical OBDD can sometimes be exponentially large. However, OBDDs often have reasonable sizes for Boolean functions encountered in engineering practice, and they are widely u... |

2400 | Compositional model checking
- Clarke, Long, et al.
- 1989
(Show Context)
Citation Context ...mes be exponentially large. However, OBDDs often have reasonable sizes for Boolean functions encountered in engineering practice, and they are widely used in circuit synthesis and model checking, cf. =-=[46, 47, 114, 65]-=-. Presently, there are two kinds of satisfiability algorithms based upon OBDDs in the satisfiability literature. The first kind builds the OBDD for the given CNF and tests if it is the constant false ... |

1293 |
Symbolic Model Checking
- McMillan
- 1993
(Show Context)
Citation Context ...mes be exponentially large. However, OBDDs often have reasonable sizes for Boolean functions encountered in engineering practice, and they are widely used in circuit synthesis and model checking, cf. =-=[46, 47, 114, 65]-=-. Presently, there are two kinds of satisfiability algorithms based upon OBDDs in the satisfiability literature. The first kind builds the OBDD for the given CNF and tests if it is the constant false ... |

1112 | Chaff: Engineering an Efficient SAT Solver
- Moskewicz, Madigan, et al.
- 2001
(Show Context)
Citation Context ...ecent years there has been progress with other methods that generate DAG-like resolution proofs in a more efficient manner than the Davis-Putnam approach. Algorithms based on DLL with clause learning =-=[157, 24, 111, 119, 83, 77]-=- perform a DLL backtracking search augmented with the ability to create new (“learned”) clauses and remove these new clauses when unneeded. This process constructs DAG-like resolution refutations [109... |

1073 |
A computing procedure for quantification theory
- Davis, Putnam
- 1960
(Show Context)
Citation Context ...hm deems a CNF F to be unsatisfiable within s steps, then there is a resolution refutation of F of size at most s. The Davis-Putnam procedure is another satisfiability algorithm based upon resolution =-=[71]-=-. Below we present pseudocode for a simple DP-based satisfiability algorithm. Again, the input F is a CNF represented as a set of clauses. The procedure returns 0 if F is unsatisfiable and 1 if F is s... |

739 |
A Machine Program for Theorem Proving
- Davis, Logemann, et al.
- 1962
(Show Context)
Citation Context ... of the clauses C1,...Cm, or follows from the preceding clauses Dj, j < i by application of one of the inference rules. A basic satisfiability algorithm is the Davis-Logemann-Loveland (DLL) procedure =-=[112]-=-. Below we present pseudocode for a simple DLL-based satisfiability algorithm 2 . The input F is a CNF represented as a set of clauses and the input π is a partial assignment to the variables, represe... |

459 |
An extensible SAT-solver
- Een
(Show Context)
Citation Context ...ecent years there has been progress with other methods that generate DAG-like resolution proofs in a more efficient manner than the Davis-Putnam approach. Algorithms based on DLL with clause learning =-=[157, 24, 111, 119, 83, 77]-=- perform a DLL backtracking search augmented with the ability to create new (“learned”) clauses and remove these new clauses when unneeded. This process constructs DAG-like resolution refutations [109... |

358 | GRASP – a New Search Algorithm for Satisfiability
- Marques-Silva, Sakallah
(Show Context)
Citation Context ...ecent years there has been progress with other methods that generate DAG-like resolution proofs in a more efficient manner than the Davis-Putnam approach. Algorithms based on DLL with clause learning =-=[157, 24, 111, 119, 83, 77]-=- perform a DLL backtracking search augmented with the ability to create new (“learned”) clauses and remove these new clauses when unneeded. This process constructs DAG-like resolution refutations [109... |

326 | The relative efficiency of propositional proof systems
- Cook, Reckhow
- 1977
(Show Context)
Citation Context ...ifying tautologies: Every tautology has a proof, only tautologies have proofs, and valid proofs are computationally easy to verify.sCOMPLEXITY OF PROPOSITIONAL PROOFS 3 Definition 1.2. (modified from =-=[69]-=-) Let F denote the set of propositional formulas over the connectives ∧, ∨, → and ¬, with a countably infinite supply of propositional variables. An abstract propositional proof system is a polynomial... |

281 |
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
- Smolensky
- 1987
(Show Context)
Citation Context ..., however, no unconditional proof of this is known. Interestingly, superpolynomial size lower bounds are known constant alternation depth formulas built from ∧, ∨, ¬, and modular counting connectives =-=[138, 155, 45]-=-, but it not known how to extend the techniques from formulas to proof systems. Polynomial calculus: Clauses correspond naturally to polynomials over a field, for example the clause x ∨ ¬y ∨ z can be ... |

260 | Cones of matrices and set-functions and 0–1 optimization
- Lovász, Schrijver
- 1991
(Show Context)
Citation Context ...n run on unsatisfiable CNFs [72, 17, 73]. Lovász-Schrijver Refutations: The Lovász-Schrijver lift-and-project method is a way to convert zero-one programming problems into linear programming problems =-=[108]-=-. The first observation is that if one knows that a linear inequality f(�x) ≥ t holds and that all variables xi take values in [0,1], then for any variable xi, xif(�x) ≥ xit and (1−xi)f(x) ≥ (1−xi)t. ... |

235 | Almost optimal lower bounds for small depth circuits
- H˚astad
- 1986
(Show Context)
Citation Context ...riables to 0 or 1 in order to collapse a k-DNF into a CNF of narrow clauses. Consider the standard formulation for distributions that set bits independently: Theorem 9.1. (“H˚astad’s switching lemma” =-=[88]-=-, cf. [45, 25]) Let positive integers k and w be given. Setting φ = (1 + √ 5)/2 and γ = 2/ln φ (note that γ > 4), we have that for any k-DNF F, if we construct an assignment ρ by independently setting... |

218 | Hard and easy distribution of SAT problems
- Mitchell, Selman, et al.
- 1992
(Show Context)
Citation Context ...uggests that there is a threshold value for ∆ (it seems to be approximately 4.2), above which a random 3-CNF is almost surely unsatisfiable and below which a random 3-CNF is almost surely satisfiable =-=[118]-=-. Rigorously, it is known that there is some threshold but its value has not been been rigorously determined [81]. This value is called the satisfiability threshold. EmpiricalsCOMPLEXITY OF PROPOSITIO... |

211 |
Bounded Arithmetic, Propositional Logic, and Complexity Theory
- Krajíček
- 1994
(Show Context)
Citation Context ...ction 2.1. Resolution also arises from translations of very weak theories of arithmetic into propositional logic, for example, the fragment of I∆0(R) that allows induction only on Σ b 1 formulas, cf. =-=[100, 102]-=-. Theorem 2.4 shows that resolution is not polynomially bounded. Res (k): The Res (k) systems generalize resolution by using formulas that are k-DNFs instead of only clauses [102, 20]. The inference r... |

210 |
Many hard examples for resolution
- Chvátal, Szemerédi
- 1988
(Show Context)
Citation Context ...st refutation of F is no smaller than the smallest refutation of PHP m n . Some striking results obtaineds20 NATHAN SEGERLIND System Lower Bound � Resolution 2 Res O( � � log n/log log n) n ∆4/k−2 +ɛ =-=[64, 29, 38]-=- 2 n/2O(k2 ) [20, 152, 8] Constant Depth Frege Ω(n) Polynomial calculus 2 Ω(n) [93, 36, 13] Cutting planes Ω(n) Lovász-Schrijver Ω(n) OBDD Refutations Ω(n) Figure 7. Best known lower bounds for refuti... |

209 |
An algorithm for integer solutions to linear programs
- Gomory
- 1963
(Show Context)
Citation Context ...problems into linear programming problems by repeatedly applying the following “cutting planes inference rule”: From �n i=1 caixi ≥ a, where c ∈ N, c > 0, and each ai ∈ Z, infer � n i=1 aixi ≥ ⌈a c ⌉ =-=[85, 63]-=-. Cutting planes derivations can be viewed as a Frege-like refutation system that manipulates linear inequalities: There are axioms 0 ≤ x and x ≤ 1 for each variable x, and in addition to the cutting ... |

183 | A.: Expander graphs and their applications
- Linial, Wigderson
- 2002
(Show Context)
Citation Context ...el constraints) to guarantee that the CNF requires large width to refute in resolution. For a thorough introduction to expansion and its applications in discrete mathematics and computer science, see =-=[90]-=-. The following definition is more often phrased in the language of bipartite graphs, but matrix notation better suits our perspective. Definition 7.1. Let A be a Boolean matrix with m rows and n colu... |

180 | Short proofs are narrow—resolution made simple
- Ben-Sasson, Wigderson
- 1999
(Show Context)
Citation Context ...lity to reuse previously derived formulas, rather than repeatedly rederiving them, can make general resolution exponentially more efficient than tree-like resolution. Theorem 2.3. ([37] building upon =-=[67, 159, 44, 38]-=-) There exists a family of unsatisfiable CNFs, {Fn} ∞ n=1 , with |Fn| = O(n), so that treelike resolution refutations of Fn are all of size 2 Ω(n/ log n) but Fn possesses DAG-like resolution refutatio... |

163 | Sharp thresholds of graph properties, and the k-sat problem
- Friedgut
- 1999
(Show Context)
Citation Context ...most surely unsatisfiable and below which a random 3-CNF is almost surely satisfiable [118]. Rigorously, it is known that there is some threshold but its value has not been been rigorously determined =-=[81]-=-. This value is called the satisfiability threshold. EmpiricalsCOMPLEXITY OF PROPOSITIONAL PROOFS 21 ¢£¤¥¦¥§¨§©� �¦©§��§¦¥¨�s¡ ����� ������������ Figure ���������� 8. Overlay of graphs depicting the p... |

155 |
Concentration, in: Probabilistic Methods for Algorithmic Discrete Mathematics
- McDiarmid
- 1998
(Show Context)
Citation Context ...ry expander.sCOMPLEXITY OF PROPOSITIONAL PROOFS 29 In some of the lower bound arguments, we make use of the following form of the Chernoff-Hoeffding bounds: Lemma 7.3. (Chernoff-Hoeffding bounds, cf. =-=[113]-=-) Let X1,... ,Xn be independent random indicator variables. Let µ = E [ � n i=1 Xi]. For every ɛ > 0: Pr[ �n i=1 Xi < (1 − ɛ)µ] ≤ e−ɛ2 � µ/2 n and Pr[ i=1 Xi > (1 + ɛ)µ] ≤ e − ɛ2 µ 2(1+ɛ/3). Corollary... |

142 | Edmonds polytopes and a hierarchy of combinatorial problems
- Chvátal
- 1973
(Show Context)
Citation Context ...problems into linear programming problems by repeatedly applying the following “cutting planes inference rule”: From �n i=1 caixi ≥ a, where c ∈ N, c > 0, and each ai ∈ Z, infer � n i=1 aixi ≥ ⌈a c ⌉ =-=[85, 63]-=-. Cutting planes derivations can be viewed as a Frege-like refutation system that manipulates linear inequalities: There are axioms 0 ≤ x and x ≤ 1 for each variable x, and in addition to the cutting ... |

134 | Lower bounds for resolution and cutting plane proofs and monotone computations
- Pudlák
- 1997
(Show Context)
Citation Context ...atz F-polynomial calculus resolution [48] F-Nullstellensatz Res � Θ(log 2 n) � F-PCR [141, 110] F polynomial calculus, resolution Res � Θ(log 2 n) � [141, 110] Cutting planes resolution Frege systems =-=[132]-=- Lovász-Schrijver resolution OBDD refutations resolution, Gaussian elimination, cutting planes with unary coefficients [22] Frege systems [103] Figure 4. Some known p-simulations and non-psimulations ... |

122 |
Exponential lower bounds for the pigeonhole principle
- Pitassi, Beame, et al.
- 1993
(Show Context)
Citation Context ...depth O(d) and size (maxi ni) O(1) . A break-through result of Miklós Ajtai showed that there are no polynomialsize, constant-depth Frege proofs of the n+1 to n pigeonhole principle [6]. Theorem 2.9. =-=[6, 106, 130]-=-. All depth d Frege proofs of PHP n+1 � n require size Ω 2n1/6d�. By Theorem 2.8, if I∆0(R) could prove php n+1 n (R), then that proof would translate into a family of polynomial-size, constant altern... |

115 |
Feasibly constructive proofs and the propositional calculus
- Cook
- 1975
(Show Context)
Citation Context ...roof systems P and Q that can p-simulate Frege systems, if P does not p-simulate a proof system Q, then P requires superpolynomial size to prove the partial consistency statements for Q. Theorem 7.1. =-=[68, 104, 52]-=- Let P be a propositional proof system that p-simulates Frege systems, and let Q be any propositional proof system. P + ConQ p-simulates Q .sCOMPLEXITY OF PROPOSITIONAL PROOFS 27 Separations based on ... |

107 |
Metamathematics of First-Order Arithmetic
- Hájek, Pudlák
- 1998
(Show Context)
Citation Context ...rom a plausible hypothesis or to entail an implausible consequence. Metamathematical apsects could be worth investigating as well. Further reading. For more on theories of bounded arithmetic, consult =-=[50, 134, 100]-=-. A survey by Alexander Razborov [144] provides furthersCOMPLEXITY OF PROPOSITIONAL PROOFS 57 material on the proof complexity of the propositional pigeonhole principle, and gives proofs for a connect... |

103 | The complexity of propositional proofs
- Urquhart
- 1995
(Show Context)
Citation Context ...lity to reuse previously derived formulas, rather than repeatedly rederiving them, can make general resolution exponentially more efficient than tree-like resolution. Theorem 2.3. ([37] building upon =-=[67, 159, 44, 38]-=-) There exists a family of unsatisfiable CNFs, {Fn} ∞ n=1 , with |Fn| = O(n), so that treelike resolution refutations of Fn are all of size 2 Ω(n/ log n) but Fn possesses DAG-like resolution refutatio... |

97 | Using the Groebner basis algorithm to find proofs of unsatisfiability
- Clegg, Edmonds, et al.
- 1996
(Show Context)
Citation Context ...tisfiability problem is to translate the given CNF into a system of polynomials over a field, and then use Groebner’s basis algorithm to decide if the system of polynomials has a common zero-one root =-=[66]-=-. The steps of the Groebner basis algorithm over a field F can be simulated by the following refutation system: Treat as axioms the clauses of the input CNF (translated into polynomials), as well as x... |

87 | Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithemtic - Krajíček - 1997 |

86 | Relations between average case complexity and approximation complexity
- Feige
(Show Context)
Citation Context ...equires superpolynomial size refutations in all abstract proof systems, then several approximation problems (that resist analysis via current PCP-based techniques) cannot be solved in polynomial time =-=[80]-=-. 6 Recently Galesi and Lauria announced an exponential lower bound for refuting random 3-CNFs of constant clause density in the “polynomial calculus plus Res(k)” over finite fields of characteristic ... |

84 |
Propositional proof systems, the consistency of first order theories and the complexity of computations
- Krajicek, Pudl'ak
- 1989
(Show Context)
Citation Context ...on has been guaranteed for non-trivial proof systems only under cryptographic assumptions. Among these results are: “If one-way functions exist, then Frege systems do not have feasible interpolation” =-=[104]-=-, and “if factoring Blum integers is hard, then constant-depth Frege systems do not have feasible interpolation” [42, 43]. It is not known, even under cryptographic assumptions, whether or not Res (k)... |

81 |
Existence and feasibility in arithmetic
- Parikh
- 1971
(Show Context)
Citation Context ...he definable functions have restricted growth rates. A well known example of such a system is Parikh’s theory I∆0, which formalizes strongly finitist arguments that disallow the use of exponentiation =-=[126, 55]-=-. Definition 2.2. The bounded formulas over the language +, ·, ≤, 0, 1 are those meeting the following recursive definition: 1. All quantifier free formulas are bounded. 2. If φ(y) is a bounded formul... |

81 |
Counting problems in bounded arithmetic
- Paris, Wilkie
- 1985
(Show Context)
Citation Context ...nting gates: A natural extension to bounded arithmetic is the introduction of a bounded modular counting quantifier Qmx < t ψ(x), meaning that the number of x < t with ψ(x) satisfied is zero modulo m =-=[127]-=-. Consider the system that extends I∆0(R) with counting quantifiers modulo m. The analog of Theorem 2.8 for this system is that its proofs translate into propositional proofs in a constant-depth Frege... |

75 | Lower bounds for cutting planes proofs with small coefficients
- Bonet, Pitassi, et al.
- 1997
(Show Context)
Citation Context ...mial in the size of the proof of φ(�x,�y) → ψ(�x,�z). This phenomenon is called feasible interpolation. Feasible interpolation has been used to prove size lower bounds for propositional proof systems =-=[92, 41, 132, 101, 22, 151, 103]-=-, and it has found applications in formal verification and theorem proving [115, 116]. Systems known to have feasible interpolation include resolution [132], cutting planes [132], Lovász-Schrijver ref... |

74 | Some consequences of cryptographical conjectures for S1 2 and EF - Krajíček, Pudlák - 1998 |

73 |
Propositional proof complexity: past, present, future
- Beame, Pitassi
- 2001
(Show Context)
Citation Context ...xity, and the others offer a different emphasis. Results prior to 1995 are more thoroughly covered in [159] and [100]. Parallels between circuit and proof complexity are dealt with more thoroughly in =-=[31]-=-. Feasible interpolation, automatizability, and lower bounds for constant-depth systems via H˚astad’s switching lemma are covered more thoroughly in [26] than in this article. Acknowledgments. The aut... |

72 |
Provability of the pigeonhole principle and the existence of infinitely many primes, thisJournal
- Paris, Wilkie, et al.
- 1988
(Show Context)
Citation Context .... In particular, it is not known whether or not I∆0 can prove the infinitude of the primes. It is known that if I∆0 can prove the pigeonhole principle, then I∆0 can prove the infinitude of the primes =-=[160, 128]-=-. However, the relationship between I∆0 and the pigeonhole principle is sticky. Definition 2.3. Let I∆0(R) denote I∆0 with its language expanded to include the relation symbol R. Let php(R) denote the... |

69 | Towards Understanding and Harnessing the Potential of Clause Learning - Beame, Kautz, et al. |

68 | Bounded arithmetic
- Buss
- 1986
(Show Context)
Citation Context ...rom a plausible hypothesis or to entail an implausible consequence. Metamathematical apsects could be worth investigating as well. Further reading. For more on theories of bounded arithmetic, consult =-=[50, 134, 100]-=-. A survey by Alexander Razborov [144] provides furthersCOMPLEXITY OF PROPOSITIONAL PROOFS 57 material on the proof complexity of the propositional pigeonhole principle, and gives proofs for a connect... |

67 | On the weak pigeonhole principle
- Krajíček
(Show Context)
Citation Context ...ction 2.1. Resolution also arises from translations of very weak theories of arithmetic into propositional logic, for example, the fragment of I∆0(R) that allows induction only on Σ b 1 formulas, cf. =-=[100, 102]-=-. Theorem 2.4 shows that resolution is not polynomially bounded. Res (k): The Res (k) systems generalize resolution by using formulas that are k-DNFs instead of only clauses [102, 20]. The inference r... |

66 | Exponential lower bounds to the size of bounded depth frege proofs of the pigeonhole principle. Random Structure and Algorithms
- Kraj´ı˘cek, ák, et al.
- 1995
(Show Context)
Citation Context ...depth O(d) and size (maxi ni) O(1) . A break-through result of Miklós Ajtai showed that there are no polynomialsize, constant-depth Frege proofs of the n+1 to n pigeonhole principle [6]. Theorem 2.9. =-=[6, 106, 130]-=-. All depth d Frege proofs of PHP n+1 � n require size Ω 2n1/6d�. By Theorem 2.8, if I∆0(R) could prove php n+1 n (R), then that proof would translate into a family of polynomial-size, constant altern... |

63 |
BerkMin: A fast and robust SAT solver
- Goldberg, Novikov
- 2002
(Show Context)
Citation Context |

61 | Lower bounds on Hilbert’s Nullstellensatz and propositional proofs
- Beame, Krajíček, et al.
- 1996
(Show Context)
Citation Context ...m, and more generally, arguments based on Hilbert’s Nullstellensatz over Zm [95]. It is known that for every m, constant-depth Frege systems with counting axioms modulo m are not polynomially bounded =-=[5, 7, 27, 57]-=-. Furthermore, when p and q are coprime, there is no subexponential size derivation of the counting principles modulo q from the counting principles modulo p [7, 27, 57]. Constant-depth Frege with cou... |

54 | Bounds to the Size of Constant-depth Propositional Proofs
- Krajíček
- 1994
(Show Context)
Citation Context ...esolution Res(2) [19, 152], cutting Res (k) resolution, Res (k − 1) planes [6], Nullstellensatz Cutting planes, Res (k + 1) [152, 150] d-Frege Res(k), (d − 1)-Frege Cutting planes [6], (d + 1)- Frege =-=[99]-=- d-Frege + CAp Zp-Nullstellensatz [95], d-Frege polynomial calculus mod p, constant-depth Frege + CGp [94] d-Frege + CGp d-Frege + CAp F-Nullstellensatz F-polynomial calculus resolution [48] F-Nullste... |

53 | The efficiency of resolution and Davis-Putnam procedures
- Beame, Karp, et al.
- 2002
(Show Context)
Citation Context ...k”. (This CNF is so important that we give it a name, PHP n+1 n .) A famous result of Armin Haken shows that the pigeonhole principle requires exponentially large resolution refutations. Theorem 2.4. =-=[87, 60, 29, 38]-=- Resolution refutations of PHP n+1 n quire size 2 Ω(n) . Corollary 2.5. All DLL, Davis-Putnam or DLL with clause learning algorithms run for 2 Ω(n) many steps when processing PHP n+1 n . resCOMPLEXITY... |

51 | Resolution proofs of generalized pigeonhole principle
- Buss, Turán
- 1988
(Show Context)
Citation Context ...k”. (This CNF is so important that we give it a name, PHP n+1 n .) A famous result of Armin Haken shows that the pigeonhole principle requires exponentially large resolution refutations. Theorem 2.4. =-=[87, 60, 29, 38]-=- Resolution refutations of PHP n+1 n quire size 2 Ω(n) . Corollary 2.5. All DLL, Davis-Putnam or DLL with clause learning algorithms run for 2 Ω(n) many steps when processing PHP n+1 n . resCOMPLEXITY... |

48 | Resolution is not automatizable unless w[p] is tractable
- Alekhnovich, Razborov
(Show Context)
Citation Context ...s in computational complexity and cryptography. It is known that neither resolution no tree-resolution is polynomial-time automatizable unless the W[P] hierarchy in parameterized complexity collapses =-=[12]-=-. Moreover, there is no automatizability for Frege systems if one-way functions exist [104], and under the assumption that “factoring Blum integers is hard”, there is no automatizability for any syste... |

48 | Space bounds for resolution
- Esteban, Torán
(Show Context)
Citation Context ...e of such algorithms, so it is natural to ask “How large must the clause database be to refute a given CNF?” This leads to the notion of space complexity for resolution refutations. Definition 6.1. ( =-=[79]-=- and [10]) Let F be a CNF. A resolution refutation presented in configuration form is a sequence of sets of clauses S1,...Sm satisfying the following properties: 1. S1 = ∅ 2. The empty clause belongs ... |

47 |
A switching lemma primer
- Beame
- 1994
(Show Context)
Citation Context ... 0 or 1 in order to collapse a k-DNF into a CNF of narrow clauses. Consider the standard formulation for distributions that set bits independently: Theorem 9.1. (“H˚astad’s switching lemma” [88], cf. =-=[45, 25]-=-) Let positive integers k and w be given. Setting φ = (1 + √ 5)/2 and γ = 2/ln φ (note that γ > 4), we have that for any k-DNF F, if we construct an assignment ρ by independently setting each bit to 0... |

47 | Near-optimal separation of treelike and general resolution
- Ben-Sasson, Impagliazzo, et al.
- 2000
(Show Context)
Citation Context ...used twice. The ability to reuse previously derived formulas, rather than repeatedly rederiving them, can make general resolution exponentially more efficient than tree-like resolution. Theorem 2.3. (=-=[37]-=- building upon [67, 159, 44, 38]) There exists a family of unsatisfiable CNFs, {Fn} ∞ n=1 , with |Fn| = O(n), so that treelike resolution refutations of Fn are all of size 2 Ω(n/ log n) but Fn possess... |

47 | Resolution lower bounds for the weak pigeonhole principle
- Raz
(Show Context)
Citation Context ... as n → ∞, a 3-CNF on ∆n clauses requires size S to be refuted in that system. through such techniques show that resolution-based methods cannot prove superpolynomial circuit size lower bounds for NP =-=[137, 144]-=-. In the study of bounded arithmetic, it is known that I∆0 can prove the infinitude of primes from the 2n to n weak pigeonhole principle [128]. By Theorem 2.8, a necessary condition for I∆0(R) to be a... |