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We obtain two results about the proof complexity of deep inference: 1) deep-inference 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 deep-inference proof systems that exhibit an exponential speed-up over analytic Gentzen proof systems that they polynomially simulate.

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 speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.

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.

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 clause-set \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 clause-set 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...

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.

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 cut-free 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.

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 worst-case behaviour is very limited. However, many worst-case upper and lower bounds of the form 2 n ( < 1 is a constant) are known for other SAT algorithms, e.g. resolution-like algorithms. In the present paper we prove both upper and lower bounds of this form for local search algorithms. The class of linear-size formulas we consider for the upper bound covers most of the DIMACS benchmarks, the satisability problem for this class of formulas is NP-complete. 1 Introduction Recently there has been an increased interest to local search algorithms for the Boolean satisability problem. Though this problem is NP-complete (see e.g. ...

Abstract. The Hintikka-style modal logic approach to knowledge has a well-known 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 evidence-based knowledge. 1

The past decade has seen clause learning as the most successful algorithm for SAT instances arising from real-world 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 clause-learning 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.