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Examining The Fragments of G
"... When restricted to proving \Sigma qi formulas, the quantified propositional proof system G*i is closely related to the \Sigma bitheorems of Buss's theory Si2. Namely, G*i has polynomialsize proofs of the translations of theorems of Si2, and Si2proves that G*i is sound. However, little is known abou ..."
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When restricted to proving \Sigma qi formulas, the quantified propositional proof system G*i is closely related to the \Sigma bitheorems of Buss's theory Si2. Namely, G*i has polynomialsize proofs of the translations of theorems of Si2, and Si2proves that G*i is sound. However, little is known about G*i when proving more complex formulas. In this paper, weprove a witnessing theorem for G*1 similar in style to theKPT witnessing theorem for T i2. This witnessing theorem is then used to show that Si2 proves G*1 is sound with respectto prenex \Sigma q i+1 formulas. Note that unless the polynomial hierarchy collapses Si 2 is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that G*1 is pequivalent to a quantified version of extendedFrege. This is followed by a proof that Gi psimulates G*i+1. We finish by proving that S2 can be axiomatized by S12 plus axioms stating that the cutfree versionof G * is sound. All together this shows that the connection between G*i and Si2 does not extend to more complex formulas.
Proving soundness for the quantified propositional calculus G ⋆ i
, 2010
"... We identify some subsystems of the quantified propositional calculus G that arise in the proof of the Reflection principles for the existing systems G ⋆ i. For each i the new system G ⋆ B,i is sandwiched between G ⋆ i and G ⋆ i+1; it is the extension of G ⋆ i where the cut formulas are allowed to be ..."
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We identify some subsystems of the quantified propositional calculus G that arise in the proof of the Reflection principles for the existing systems G ⋆ i. For each i the new system G ⋆ B,i is sandwiched between G ⋆ i and G ⋆ i+1; it is the extension of G ⋆ i where the cut formulas are allowed to be Boolean combinations of Σ q i and Πqi formulas (instead of being just either Σq i or). We use these systems to correct Steven Perron’s statement of a Π q i kind of Herbrand’s theorem for G ⋆ i. Perron uses this theorem to show that the theory V i proves the Reflection principle for G ⋆ i with respect to Σ q i+1 formulas. Formerly this was known only for Σqi formulas. We give detailed proofs of Perron’s results.
On theories of bounded arithmetic for NC¹
, 2008
"... We develop an arithmetical theory VNC¹∗ and its variant VNC¹∗, corresponding to “slightly nonuniform” NC¹. Our theories sit between VNC¹ and VL, and allow evaluation of logdepth bounded fanin circuits under limited conditions. Propositional translations of ΣB 0 (LVNC 1)formulas provable in VNC¹∗ ..."
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We develop an arithmetical theory VNC¹∗ and its variant VNC¹∗, corresponding to “slightly nonuniform” NC¹. Our theories sit between VNC¹ and VL, and allow evaluation of logdepth bounded fanin circuits under limited conditions. Propositional translations of ΣB 0 (LVNC 1)formulas provable in VNC¹∗ admit Luniform polynomialsize Frege proofs.
Simulation of Gi with prenex cuts
, 2010
"... We show that the quantified propositional proof systems Gi are polynomially equiva(or prenex lent to their restricted versions that require all cut formulas to be prenex Σ q i Π q i). Previously this was known only for the treelike systems G ∗ i. ..."
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We show that the quantified propositional proof systems Gi are polynomially equiva(or prenex lent to their restricted versions that require all cut formulas to be prenex Σ q i Π q i). Previously this was known only for the treelike systems G ∗ i.
Lifting Lower Bounds for TreeLike Proofs
, 2011
"... It is known that constantdepth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider treelike Sequent Calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to b ..."
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It is known that constantdepth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider treelike Sequent Calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning smalldepth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cutfree proofs and “lift ” it so it applies to proofs with constantdepth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constantdepth proofs that use different modular connectives. We show that these treelike proofs with constantdepth cuts cannot polynomially simulate similar daglike proofs, even when the daglike proofs are cutfree. We present a new proof of the nonfinite axiomatizability of the theory of bounded arithmetic I∆0(R). Finally, under a plausible hardness assumption concerning the polynomialtime hierarchy, we show that the hierarchy G ∗ i of quantified propositional proof systems does not collapse.
Quantified Propositional Logspace Reasoning
, 2008
"... In this paper, we develop a quantified propositional proof systems that corresponds to logarithmicspace reasoning. We begin by defining a class ΣCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF(2) formu ..."
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In this paper, we develop a quantified propositional proof systems that corresponds to logarithmicspace reasoning. We begin by defining a class ΣCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF(2) formulas and no cut formula that is not quantifier free contains a free variable that does not appear in the final formula. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of V L into a family of tautologies that have polynomialsize GL ∗ proofs. V L is a theory of bounded arithmetic that is known to correspond to logarithmicspace reasoning. To do the translation, we find an appropriate axiomatization of V L, and put V L proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in V L. This is done by giving a logarithmicspace algorithm that witnesses GL ∗ proofs. 1
Simulating nonprenex cuts in quantified propositional calculus
, 2011
"... We show that the quantified propositional proof systems Gi are polynomially equiva(or prenex lent to their restricted versions that require all cut formulas to be prenex Σ q i Π q i). Previously this was known only for the treelike systems G ∗ i. ..."
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We show that the quantified propositional proof systems Gi are polynomially equiva(or prenex lent to their restricted versions that require all cut formulas to be prenex Σ q i Π q i). Previously this was known only for the treelike systems G ∗ i.
Theories for Subexponentialsize Boundeddepth Frege Proofs ∗
"... This paper is a contribution to our understanding of the relationship between uniform and nonuniform proof complexity. The latter studies the lengths of proofs in various propositional proof systems such as Frege and boundeddepth Frege systems, and the former studies the strength of the correspondi ..."
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This paper is a contribution to our understanding of the relationship between uniform and nonuniform proof complexity. The latter studies the lengths of proofs in various propositional proof systems such as Frege and boundeddepth Frege systems, and the former studies the strength of the corresponding logical theories such as VNC 1 and V 0 in [7]. A superpolynomial lower bound on the length of proofs in a propositional proof system for a family of tautologies expressing a result like the pigeonhole principle implies that the result is not provable in the theory associated with the propositional proof system. We define a new class of bounded arithmetic theories nεioV ∞ for ε < 1 and show that they correspond to complexity classes AltTime(O(1), O(nε)), uniform classes of subexponentialsize boundeddepth circuits DepthSize(O(1), 2O(nε)). To accomplish this we introduce the novel idea of using types to control the amount of composition in our bounded arithmetic theories. This allows our theories to capture complexity classes that have weaker closure properties and are not closed under composition. We show that the proofs of ΣB 0theorems in our theories translate to subexponentialsize boundeddepth Frege proofs.