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26
The relative efficiency of propositional proof systems
 Journal of Symbolic Logic
, 1979
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Cited by 330 (5 self)
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On the Proof Complexity of Deep Inference
, 2000
"... We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential ..."
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Cited by 31 (13 self)
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We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential speedup over analytic Gentzen proof systems that they polynomially simulate.
Are there Hard Examples for Frege Systems?
"... It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. S ..."
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Cited by 20 (2 self)
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It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speedup of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl. It is
On a generalization of Extended Resolution
 Discrete Applied Mathematics
, 1997
"... Motivated by improved SAT algorithms ([13, 14, 15]; yielding new worst case upper bounds) a natural parameterized generalization GER of Extended Resolution (ER) is introduced. ER can simulate polynomially GER, but GER allows special cases for which exponential lower bounds can be proven. 1 Introduct ..."
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Cited by 17 (7 self)
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Motivated by improved SAT algorithms ([13, 14, 15]; yielding new worst case upper bounds) a natural parameterized generalization GER of Extended Resolution (ER) is introduced. ER can simulate polynomially GER, but GER allows special cases for which exponential lower bounds can be proven. 1 Introduction Extended Resolution G. Tseitin introduced in [21] the Extension Rule for the Resolution Calculus: F \Gamma! F [ n fv; a; bg; fv; ag; fv; bg o for arbitrary variables a; b and a new variable v (new relative to the set F of premises and to a; b). Thereby the clauseset \Phi fv; a; bg; fv; ag; fv; bg \Psi is the Conjunctive Normal Form of the formula v $ (a b). An Extended Resolution Proof (for short: ER proof) of the empty clause ? from the clauseset F is an ordinary resolution proof of ? from F , where F ' F is obtained by repeated applications of the Extension Rule. The length of an ER proof is the (total) number of (different) clauses in it. We denote by Comp ER (F...
How to Lie Without Being (easily) Convicted and the Lengths of Proofs in Propositional Calculus
"... We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the sec ..."
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Cited by 15 (1 self)
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We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.
Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
SAT Local Search Algorithms: WorstCase Study
 Journal of Automated Reasoning
, 2000
"... Recent experiments demonstrated that local search algorithms (e.g. GSAT) are able to nd satisfying assignments for many \hard" Boolean formulas. A wide experimental study of these algorithms demonstrated their good performance on some important classes of formulas as well as poor performance on some ..."
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Cited by 11 (6 self)
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Recent experiments demonstrated that local search algorithms (e.g. GSAT) are able to nd satisfying assignments for many \hard" Boolean formulas. A wide experimental study of these algorithms demonstrated their good performance on some important classes of formulas as well as poor performance on some other ones. In contrast, theoretical knowledge of their worstcase behaviour is very limited. However, many worstcase upper and lower bounds of the form 2 n ( < 1 is a constant) are known for other SAT algorithms, e.g. resolutionlike algorithms. In the present paper we prove both upper and lower bounds of this form for local search algorithms. The class of linearsize formulas we consider for the upper bound covers most of the DIMACS benchmarks, the satisability problem for this class of formulas is NPcomplete. 1 Introduction Recently there has been an increased interest to local search algorithms for the Boolean satisability problem. Though this problem is NPcomplete (see e.g. ...
Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
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Cited by 10 (0 self)
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This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
Logical omniscience via proof complexity
 In Computer Science Logic 2006, Lecture Notes in Computer Science, Vol 4207
, 2006
"... Abstract. The Hintikkastyle modal logic approach to knowledge has a wellknown defect of logical omniscience, i.e., an unrealistic feature that an agent knows all logical consequences of her assumptions. In this paper we suggest the following Logical Omniscience Test (LOT): an epistemic system E is ..."
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Cited by 9 (6 self)
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Abstract. The Hintikkastyle modal logic approach to knowledge has a wellknown defect of logical omniscience, i.e., an unrealistic feature that an agent knows all logical consequences of her assumptions. In this paper we suggest the following Logical Omniscience Test (LOT): an epistemic system E is not logically omniscient if for any valid in E knowledge assertion A of type ‘F is known ’ there is a proof of F in E, the complexity of which is bounded by some polynomial in the length of A. We show that the usual epistemic modal logics are logically omniscient (modulo some common complexity assumptions). We also apply LOT to Justification Logic, which along with the usual knowledge operator Ki(F) (‘agent i knows F ’) contain evidence assertions t:F (‘t is a justification for F ’). In Justification Logic, the evidence part is an appropriate extension of the Logic of Proofs LP, which guarantees that the collection of evidence terms t is rich enough to match modal logic. We show that justification logic systems are logically omniscient w.r.t. the usual knowledge and are not logically omniscient w.r.t. the evidencebased knowledge. 1
Extended Clause Learning
"... The past decade has seen clause learning as the most successful algorithm for SAT instances arising from realworld applications. This practical success is accompanied by theoretical results showing clause learning as equivalent in power to resolution. There exist, however, problems that are intract ..."
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Cited by 5 (0 self)
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The past decade has seen clause learning as the most successful algorithm for SAT instances arising from realworld applications. This practical success is accompanied by theoretical results showing clause learning as equivalent in power to resolution. There exist, however, problems that are intractable for resolution, for which clauselearning solvers are hence doomed. In this paper, we present extended clause learning, a practical SAT algorithm that surpasses resolution in power. Indeed, we prove that it is equivalent in power to extended resolution, a proof system strictly more powerful than resolution. Empirical results based on an initial implementation suggest that the additional theoretical power can indeed translate into substantial practical gains. 1.