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Theory for TC 0 and Other Small Complexity Classes
 Logical Methods in Computer Science
, 2005
"... Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b ..."
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Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b
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.
Thirdorder computation and bounded arithmetic
 University of Wales Swansea
, 2006
"... Abstract. We describe a natural generalization of ordinary computation to a thirdorder setting and give a function calculus with nice properties and recursiontheoretic characterizations of several large complexity classes. We then present a number of thirdorder theories of bounded arithmetic whos ..."
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Abstract. We describe a natural generalization of ordinary computation to a thirdorder setting and give a function calculus with nice properties and recursiontheoretic characterizations of several large complexity classes. We then present a number of thirdorder theories of bounded arithmetic whose definable functions are the classes of the EXPtime hierarchy in the thirdorder setting.
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∗