Urquhart, A. (1995) The complexity of propositional proofs, Bulletin  Home/Search   Document Details and Download   Summary   Related Articles   Check

....tautology has a short proof. More formally, there is a polynomial q such that for every 2 TAUT, there is a string w of length boundedby q(j j) with h(w) Many concrete proof systems for TAUT, like the one given in Example 1. 1, have been shown not to be polynomially bounded (see for example [27]) Besides the interest that concrete proof systems like, for example, resolution or Frege systems have in their own, a main motivation for the study of proof systems comes in fact from the following relation between the NP versus co NP question and the existence of polynomially bounded systems. ....

A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic 1:425-467, 1995.

....when it comes to proof techniques. Given this diversity, as well as quite a substantial amount of material gathered during recent years, it is fairly impossible even to touch all aspects of propositional proof complexity in one lecture. I will not even try that; instead let me refer to the sources [Urq95, Kra95, Raz96, BP98, Pud98] that serve various tastes and, taken together, cover virtually everything which is currently known about the complexity of propositional proofs (see, however, the remark at the end of Section 4) 2. De nitions The framework underlying propositional proof complexity was developed in the seminal ....

A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425-467, 1995.

....The computational pay off may seem doubtful: some time must be spent for scanning the formula replacing the occurrences with ; eliminate boolean constant and so on. So what could be the gain Intuition 2 : The sequent proof of Gamma = Delta is potentially of size O(2 j Gamma [ Delta j ) [39]. If can reduce Gamma = Delta to Gamma so that its size decreases, if only by 1, we would reduce the potential search space at least by half. We trade off some polynomial processing for an exponential gain: scanning the formula and looking for exact copies can be done in polynomial time; ....

....are instances of the local simplification rule (simp) when recursion stops at the main connective of Gamma and Delta formulae and it is only applied to formulae of particular shape. The next restriction that is imposed on clausal tableau proofs or model elimination techniques is regularity [3, 10, 25, 41, 39]: a literal should never appear twice on a branch. In the setting of propositional logic and clausal theorem proving (such as [3, 25] this is equivalent to say that a clause is never selected for the extension of a branch in the tableau if the extension step (an n ary fi rule) will yield a ....

A. Urquhart. The complexity of propositional proofs. The Bulletin of Symbolic Logic, 1(4):425--467, Dec 1995.

....the size of the proof with Pi) but the converse does not hold. The last step is usually proved by exhibiting a family of formulae for which there is an exponential lower bound (in the size of the formula) on every proof in Pi whereas there are short polynomial proofs in . We refer to Urquhart [6] for formal definitions. For instance, the claim that tableaux cannot polynomially simulate tree resolution is based on the fact that the family of formulae Sigma n by Cook Reckhow [1, 2] has only exponential size tableau proofs but a polynomial resolution proof. In the rest of the paper we ....

....node is associated to a different variable; ffl each leaf is associated to a clause whose literals are the atoms of the internal nodes, considered positive if the path (from the root to the leaf) continues at the left of the node and negative if it is at the right. We represent it as follows [6, 3]: Sigma n = Phi SigmaA SigmaA Sigma SigmaA Sigma Sigma : SigmaA Sigmahn Gamma1i Sigma Psi where the string Sigma Delta Delta Delta Sigma is determined by the signs of the previous literals. For instance we have Sigma 1 = fA; Ag Sigma 2 = fA A ; A :A ; A A ....

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A. Urquhart, `The complexity of propositional proofs', Bull. of Symbolic Logic, 1(4), 425--467. Logic Programming / Automated Reasoning 409 F. Massacci

....instructive. Namely, we call a propositional proof system P weak if we (currently) know how to prove super polynomial lower bounds for the accompanying circuit class P and strong otherwise. There is a steady progress in studying the complexity of proofs in weak proof systems surveyed e.g. in [Urq95, Kra95, Raz96, BP98, Pud98, Raz02a] For strong proof systems the current situation is by far more miserable. Although there are no rigorous results along these lines (and, moreover, this feeling is not universal see e.g. Kra02] the empirical evidence strongly suggests that lower bounds for ....

A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425--467, 1995.

....v2V (G) PARITY v; If the maximal degree of G is constant, then the initial size and width of T (G; are small as well. Lemma 3.21. If d is the maximal degree of G, then T (G; is a d CNF with at most n 2 d 1 clauses and at most nd=2 variables. We shall need the following lemma from [Urq95]. Lemma 3.22 (see [Urq95] If G is connected, then T (G; is contradictory i is an odd weight function. Moreover, for any v 2 V (G) there is an assignment satisfying all axioms from fPARITY u; j u 6= v g. The space lower bound on Tseitin formulas will be directly connected to the ....

....the maximal degree of G is constant, then the initial size and width of T (G; are small as well. Lemma 3.21. If d is the maximal degree of G, then T (G; is a d CNF with at most n 2 d 1 clauses and at most nd=2 variables. We shall need the following lemma from [Urq95] Lemma 3. 22 (see [Urq95]) If G is connected, then T (G; is contradictory i is an odd weight function. Moreover, for any v 2 V (G) there is an assignment satisfying all axioms from fPARITY u; j u 6= v g. The space lower bound on Tseitin formulas will be directly connected to the following notion of expansion. ....

A. Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, 1 (1995), pp. 425-467.

....and only if G ; fFg. FACT 3. The Gentzen Cut Elimination Theorem, 18] G ; fFg if and only if G=CUT ; fFg Fact 3 establishes that the CUT rule is not needed in order to derive any provable sequent. Nonetheless, CUT turns out to be an extremely powerful operation: FACT 4. Urquhart[31, 32]) There are (infinite) sequences of formulae hF n i in L for which: a. F n is a propositional tautology of n propositional variables. b. fF n g; G) O(n k ) for k 2 IN) c. fF n g; G=CUT) 2 n (where 0) These constructions by Urquhart are explicit, i.e. a specific ....

....of the Challenger s BACKUP move, then this could be simulated from the initial argument just by repeating the relevant COUNTER and BACKUP moves. Since the size of any dispute tree can be at most the number of arguments within the system itself, a more sophisticated RETRACT semantics 6 In fact, [32], shows G=CUT can be weaker than simple truth tables proving worst case lower bounds of n ) for the former as opposed to upper bounds of n2 n for the latter. csd rep gentzen.tex; 17 10 2001; 12:47; p.25 26 could only shorten the length of a dispute by a polynomial factor not reduce it ....

Urquhart, A.: 1995, `The complexity of propositional proofs'. Bulletin of Symbolic Logic 1(4), 425--467.

.... any unsatisfiable , the length of any TPI dispute over hH ; i is at least the length of the shortest CUT free Gentzen proof of : The exponential lower bound is then simply a matter of noting that there are instances : whose shortest proof in such systems is exponential in the size of , cf [10,11]. While [5] provides a lower bound on how good a propositional proof calculus TPI disputes provide, the question of upper bounds is left open. The present article treats this question. Specifically we show that for any unsatisfiable CNF formula , the number of moves required in a TPI dispute ....

A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1(4):425--467, 1995.

....Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree like versions and the dag like versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known [29, 18, 8]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [25] are extended to monotone real circuits. An exponential separation is also proved between tree like resolution and several refinements of resolution: negative resolution and regular ....

....is also proved between tree like resolution and several refinements of resolution: negative resolution and regular resolution. Actually this last separation also provides a separation between tree like resolution and ordered resolution, thus the corresponding superpolynomial separation of [29] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [13] MSC Classification: 03F20, 68Q17, 68T15 1 ....

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A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425--467, 1995. 21

....of this paper is the homomorphic completeness of MU(1) We pinpoint exactly the e#ciency of #MU(1) by showing that every proof (H, #, F ) # # MU(1) can be transformed into a tree resolution proof of F in polynomial time, and vice versa. Hence, #MU(1) and tree resolution are p equivalent (c.f. [5,24]) Clearly, for fixed k # 1, the set MU(# k) # k i=1 MU(i) is homomorphically complete, since MU(1) # MU(# k) it is conceivable that for k # 2 the proof system # MU(#k) is stronger than # MU(1) We show, however, that # MU(#k) and # MU(1) are p equivalent. Further we show that for ....

....introduced by Monien and Speckenmeyer [20] and have been studied in depth by Kullmann [15,18] 2.2 Proof Systems Cook and Reckhow [5] introduced a general concept of propositional proof systems in terms of functions on sets of strings. We use a more informal concept based on the discussion in [24]. A proof of a formula F is a finite object x which certifies unsatisfiability of F in the sense that, if x is given, then unsatisfiability of F can be verified in polynomial time (proofs of unsatisfiability are also called refutations) A proof system # is a set of proofs such that (i) elements ....

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A. Urquhart. The complexity of propositional proofs. The Bulletin of Symbolic Logic, 1(4):425--467, Dec. 1995.

....at the level of motivations and when it comes to proof techniques. In my talk at the conference I gave a general overview of some important concepts, ideas and proof techniques behind this fascinating theory. A substantial portion of that material was already covered in various surveys (see e.g. [1 5]) For one particular subject from my lecture, however, the situation is very di erent. I am talking about the research on the proof complexity of speci c tautologies that express various forms of the so called Pigeonhole principle. This principle (asserting that there is no injective mapping from ....

Urquhart, A.: The complexity of propositional proofs. Bulletin of Symbolic Logic 1 (1995) 425-467

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Urquhart, A. (1995) The complexity of propositional proofs, Bulletin

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Alasdair Urquhart, The complexity of propositional proofs, Bulletin of Symbolic Logic 1(4) (1995), pp. 425-467.

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A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic 1:425-467, 1995.

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Alasdair Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1(4):425-467, 1995.

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A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic 1, pp. 425467, 1995.

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A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425--467, 1995.

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Alasdair Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425--467, 1995.

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A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425-467, 1995.

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A. Urquhart. The complexity of propositional proofs. the Bulletin of Symbolic Logic, 1:425--467, 1995.

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A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1(4):425-467, 1995. 56

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Alasdair Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1(4):425-467, 1995.

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A. Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, 1 (1995), pp. 425--467.

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Alasdair Urquhart. The complexity of propositional proofs. The Bulletin of Symbolic Logic, 1:425--467, 1995.

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Urquhart, A.: The Complexity of Propositional Proofs. The Bulletin of Symbolic Logic, 1 (

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