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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.
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
The Complexity Of Derivations Of Matrix Identities
, 2001
"... In this thesis we are concerned with building logical foundations for Linear Algebra, from the perspective of proof complexity. As the cornerstone of our logical theories, we use Berkowitz's parallel algorithm for computing the coecients of the characteristic polynomial of a matrix. STandard ..."
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Cited by 1 (0 self)
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In this thesis we are concerned with building logical foundations for Linear Algebra, from the perspective of proof complexity. As the cornerstone of our logical theories, we use Berkowitz's parallel algorithm for computing the coecients of the characteristic polynomial of a matrix. STandard
Many facets of complexity in logic
"... Abstract. There are many ways to define complexity in logic. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among ..."
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Abstract. There are many ways to define complexity in logic. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and their relationship with computational complexity. In particular, we show how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic. For example, the transitive closure function (testing reachability between two given points in a directed graph) is definable using only NLconcepts (where NL is nondeterministic logspace complexity class), and its totality is provable within NLreasoning. 1
Expressing vs. proving: relating forms of complexity in logic
"... Abstract. Complexity in logic comes in many forms. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and ..."
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Abstract. Complexity in logic comes in many forms. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and their relationship with computational complexity. In particular, we show how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic. For example, the transitive closure function (testing reachability between two given points in a directed graph) is definable using only NLconcepts (where NL is the nondeterministic logspace complexity class), and its totality (and, thus, the closure of NL under complementation) is provable within NLreasoning. Lastly, we will touch upon the topic of formalizing complexity theory using logic, and the metaquestion of complexity of logical reasoning about complexitytheoretic statements. This is intended to be a highlevel overview, suitable for readers who are not familiar with complexity theory and complexity in logic. 1