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34
The Complexity Of Propositional Proofs
 Bulletin of Symbolic Logic
, 1995
"... This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on ..."
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Cited by 105 (2 self)
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This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on
Lower Bounds to the Size of ConstantDepth Propositional Proofs
, 1994
"... 1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp ..."
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Cited by 53 (6 self)
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1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp(O(log 2 n)) but such that every depth d refutation must have the size at least exp(n\Omega\Gamma21 ). The sets T d n express a weaker form of the pigeonhole principle. It is a fundamental problem of mathematical logic and complexity theory whether there exists a proof system for propositional logic in which every tautology has a short proof, where the length (equivalently the size) of a proof is measured essentially by the total number of symbols in it and short means polynomial in the length of the tautology. Equivalently one can ask whether for every theory T there is another theory S (both first order and reasonably axiomatized, e.g. by schemes) having the property that if a statement...
The Taming of the Cut. Classical Refutations with Analytic Cut
 JOURNAL OF LOGIC AND COMPUTATION
, 1994
"... The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of view. F ..."
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Cited by 52 (1 self)
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The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of view. Firstly, they cannot represent the use of auxiliary lemmas in proofs. Secondly, they cannot express the bivalence of classical logic. Thirdly, they are extremely inefficient, as is emphasized by the "computational scandal" that such systems cannot polynomially simulate the truthtables. None of these anomalies occurs if the cut rule is allowed. This raises the problem of formulating a proof system which incorporates a cut rule and yet can provide a suitable model of classical analytic deduction. For this purpose we present an alternative refutation system for classical logic, that we call KE. This system, though being "close" to Smullyan's tableau method, is not cutfree but includes a class...
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.
Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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Cited by 21 (5 self)
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
ON THE NUMBER OF STEPS IN PROOFS
, 1989
"... In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. W ..."
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Cited by 17 (2 self)
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In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length ( = the number of symbols in the proof), depth ( = the maximal depth of a formula in the proof) and number o! steps ( = the number of formulas in the proof). For a particular formaI system and a given formula A we consider the shortest length of a proof of A, the minimal depth ofa proof of A and the minimal number of steps in a proof of A. The main results are the following: (1) a bound on the depth in terms of the number of steps: Theorem 2.2, (2) a bound on the depth in terms of the length: Theorem 2.3, (3) a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: (1) a bound on the number of steps in a cutfree proof, (2) some speedup results, (3) bounds on the number of steps in proofs of ParisHarrington sentences. Some paper
Presenting intuitive deductions via symmetric simplification
 In CADE10: Proceedings of the tenth international conference on Automated deduction
, 1990
"... In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a nontrivial translation procedure to extract humanoriented deductions from machineoriented pro ..."
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Cited by 15 (4 self)
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In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a nontrivial translation procedure to extract humanoriented deductions from machineoriented proofs. Previously known translation procedures, though complete, tend to produce unintuitive deductions. One of the major flaws in such procedures is that too often the rule of indirect proof is used where the introduction of a lemma would result in a shorter and more intuitive proof. We present an algorithm, symmetric simplification, for discovering useful lemmas in deductions of theorems in first and higherorder logic. This algorithm, which has been implemented in the TPS system, has the feature that resulting deductions may no longer have the weak subformula property. It is currently limited, however, in that it only generates lemmas of the form C ∨ ¬C ′ , where C and C ′ have the same negation normal form. 1
Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Cited by 8 (4 self)
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Quasipolynomial normalisation in deep inference via atomic flows and threshold formulae
, 2009
"... ABSTRACT. Jeˇrábek showed that analytic propositionallogic deepinference proofs can be constructed in quasipolynomial time from nonanalytic proofs. In this work, we improve on that as follows: 1) we significantly simplify the technique; 2) our normalisation procedure is direct, i.e., it is interna ..."
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Cited by 8 (4 self)
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ABSTRACT. Jeˇrábek showed that analytic propositionallogic deepinference proofs can be constructed in quasipolynomial time from nonanalytic proofs. In this work, we improve on that as follows: 1) we significantly simplify the technique; 2) our normalisation procedure is direct, i.e., it is internal to deep inference. The paper is selfcontained, and provides a starting point and a good deal of information for tackling the problem of whether a polynomialtime normalisation procedure exists. 1.