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Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
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We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
Separating daglike and treelike proof systems
 Accepted in LICS
, 2007
"... We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on ..."
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Cited by 4 (1 self)
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We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on a class of cutformulas that satisfies some plausible hardness assumption. This gives (in some cases, conditional) negative answers to several questions from a recent work of Maciel and Pitassi (LICS 2006). Our technique is inspired by the technique from Maciel and Pitassi. While the sequents used in earlier work are derived from the Pigeonhole principle, here we generalize Statman’s sequents. This gives the desired separation, and at the same time provides stronger results in some cases. 1
Bounded Arithmetic and Constant Depth Frege Proofs
, 2004
"... We discuss the ParisWilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and nonrelativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0depth for PKproofs that makes the translation from boun ..."
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We discuss the ParisWilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and nonrelativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0depth for PKproofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1definable functions of S12are polynomial time computable and that the \Sigma b1definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma
A Propositional Proof System for Log Space
"... Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formulabased proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of ..."
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Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formulabased proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a nonparameter free variable. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of Σ B 0rec into a family of tautologies that have polynomial size GL ∗ proofs. Σ B 0rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ B 0rec, and put Σ B 0rec 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 Σ B 0rec. This is done by giving a log space algorithm that witnesses GL ∗ proofs. 1
EXAMINING FRAGMENTS OF THE QUANTIFIED PROPOSITIONAL CALCULUS
"... Abstract. When restricted to proving Σ q i formulas, the quantified propositional proof system G ∗ i is closely related to the Σbi theorems of Buss’s theory Si 2. Namely, G∗i has polynomialsize proofs of the translations of theorems of S i 2, and Si 2 proves that G∗ i is sound. However, little is k ..."
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Abstract. When restricted to proving Σ q i formulas, the quantified propositional proof system G ∗ i is closely related to the Σbi theorems of Buss’s theory Si 2. Namely, G∗i has polynomialsize proofs of the translations of theorems of S i 2, and Si 2 proves that G∗ i is sound. However, little is known about G ∗ i when proving more complex formulas. In this paper, we prove a witnessing theorem for G ∗ i similar in style to the KPT witnessing theorem for T i 2. This witnessing theorem is then used to show that Si 2 proves G∗ i is sound with respect to Σ q i+1 formulas. Note that unless the polynomialtime hierarchy collapses S i 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 for prenex formulas. This is followed by a proof that Gi psimulates G ∗ i+1. We finish by proving that S2 can be axiomatized by S 1 2 plus axioms stating that the cutfree version of G ∗ 0 is sound. All together this shows that the connection between G∗
Abstract GL\Lambda: A Propositional Proof System For Logspace
"... In recent years, there has been considerable research exploring connections between propositional proof systems, theories of bounded arithmetic, and complexity classes. We know that N C1 corresponds to G\Lambda 0 and that P corresponds to G\Lambda 1, but no proof system corresponding to a complexity ..."
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In recent years, there has been considerable research exploring connections between propositional proof systems, theories of bounded arithmetic, and complexity classes. We know that N C1 corresponds to G\Lambda 0 and that P corresponds to G\Lambda 1, but no proof system corresponding to a complexity class between N C1 and P has been defined. In this work, we construct a proof system GL\Lambda, which corresponds to L. Connections to the theory V L (Zambella's \Sigma p0 \Gamma rec) are also considered. GL\Lambda is defined by restricting cuts in the system G\Lambda 1. The first restriction is syntactic: the cut formulas have to be \Sigma CN F (2), which is a new class of formulas. Unfortunately that is not enough; the free variables in cut formulas must be restricted to parameter variables. We prove that GL\Lambda corresponds to V L by translating theorems of V L into tautologies with small GL\Lambda proof. ii
The Equivalence of Theories that Characterize
, 2007
"... A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory S 1 2 characterizes polytime functions. Among these, ALV ′ (by Clote) is particularly interest ..."
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A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory S 1 2 characterizes polytime functions. Among these, ALV ′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S5 is complete for ALogTime. On the other hand, ALV (by Clote), T 0 NC 0 (by Clote and Takeuti) as well as Arai’s theory AID + Σ B 0CA and its twosorted version VNC 1 (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV ′ has been an open problem. Here we show that ALV ′ is indeed equivalent to the other theories. 1