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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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Cited by 30 (7 self)
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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.
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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Cited by 9 (4 self)
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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
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Cited by 9 (2 self)
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce
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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Cited by 3 (0 self)
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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
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.
Connecting Complexity Classes, Weak Formal Theories, and Propositional Proof Systems
"... This is a survey talk explaining the connection between the three items mentioned in the title. ..."
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This is a survey talk explaining the connection between the three items mentioned in the title.