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25
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.
The complexity of propositional proofs
 BULLETIN OF SYMBOLIC LOGIC
"... Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. Thi ..."
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Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.
Short proofs for the determinant identities
 CoRR
, 2011
"... We study arithmetic proof systems Pc(F) and Pf (F) operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field F. We establish a series of structural theorems about these proof systems, the main one stating that Pc(F) proofs can be bala ..."
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We study arithmetic proof systems Pc(F) and Pf (F) operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field F. We establish a series of structural theorems about these proof systems, the main one stating that Pc(F) proofs can be balanced: if a polynomial identity of syntactic degree d and depth k has a Pc(F) proof of size s, then it also has a Pc(F) proof of size poly(s, d) in which every circuit has depth O(k + log2 d + log d ∙ log s). As a corollary, we obtain a quasipolynomial simulation of Pc(F) by Pf (F). Using these results we obtain the following: consider the identities det(XY) = det(X) ∙ det(Y) and det(Z) = z11 ∙ ∙ ∙ znn, where X,Y and Z are n×n square matrices and Z is a triangular matrix with z11,..., znn on the diagonal (and det is the determinant polynomial). Then we can construct a polynomialsize arithmetic circuit det such that the above identities have Pc(F) proofs of polynomialsize using circuits of O(log2 n) depth. Moreover, there exists an arithmetic formula det of size nO(log n) such that the above identities have Pf (F) proofs of size nO(log n). This yields a solution to a basic open problem in propositional proof complexity, namely, whether there are polynomialsize NC2Frege proofs for the determinant identities and the hard matrix identities, as considered, e.g. in Soltys and Cook [SC04] (cf., Beame and Pitassi [BP98]). We show that matrix identities like AB = I → BA = I (for matrices over the two element field) as well as basic properties of the determinant have polynomialsize NC2Frege proofs, and quasipolynomialsize Frege proofs. 1
Weak Pigeonhole Principle, and Randomized Computation
, 2005
"... We study the extension of the theory S1 2 by instances of the dual (onto) weak pigeonhole principle for ptime functions, dWPHP(PV) x x2. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie’s witnessing theo ..."
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We study the extension of the theory S1 2 by instances of the dual (onto) weak pigeonhole principle for ptime functions, dWPHP(PV) x x2. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie’s witnessing theorem for S1 2 + dWPHP(PV). Then we show that dWPHP(PV) is (over S1 2) equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the NisanWigderson construction (conditional derandomization of probabilistic ptime algorithms) in a conservative extension of S1 2 + dWPHP(PV). We also develop in S1 2 the algebraic machinery needed for implicit listdecoding of ReedMuller errorcorrecting codes (including some results on a modification of Soltys ’ theory ∀LAP), and use it to formalize the ImpagliazzoWigderson strengthening of the NisanWigderson theorem. We construct a propositional proof system WF (based on a reformulation
Weak theories of linear algebra
 Archive for Mathematical Logic
"... We investigate the theories LA, LAP, ∀LAP of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment ∃LA of ∀LAP in which we ca ..."
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We investigate the theories LA, LAP, ∀LAP of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment ∃LA of ∀LAP in which we can interpret a weak theory V 1 of bounded arithmetic and carry out polynomial time reasoning about matrices for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, ∃LA proves the commutativity of inverses. 1
Matrix identities and the pigeonhole principle
 ARCH MATH LOGIC
, 2004
"... We show that short boundeddepth Frege proofs of matrix identities, such as PQ I (over the field of two elements), imply short boundeddepth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential size boundeddepth Frege proofs, it follows that t ..."
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We show that short boundeddepth Frege proofs of matrix identities, such as PQ I (over the field of two elements), imply short boundeddepth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential size boundeddepth Frege proofs, it follows that the propositional version of the matrix principle also requires boundeddepth Frege proofs of exponential size.
Formal Theories for Linear Algebra
"... Abstract. We introduce twosorted theories in the style of [CN10] for the complexity classes ⊕L and DET, whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys ’ linear algebra theory LAp over arbitrary integral domains, into each of our ..."
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Abstract. We introduce twosorted theories in the style of [CN10] for the complexity classes ⊕L and DET, whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys ’ linear algebra theory LAp over arbitrary integral domains, into each of our new theories. The result shows equivalences of standard theorems of linear algebra over Z2 and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves can be proved. 1
Effectively polynomial simulations
 IN PROCEEDINGS OF FIRST SYMPOSIUM ON INNOVATIONS IN COMPUTER SCIENCE (ICS
, 2010
"... We introduce a more general notion of efficient simulation between proof systems, which we call effectivelyp simulation. We argue that this notion is more natural from a complexitytheoretic point of view, and by revisiting standard concepts in this light we obtain some surprising new results. Firs ..."
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We introduce a more general notion of efficient simulation between proof systems, which we call effectivelyp simulation. We argue that this notion is more natural from a complexitytheoretic point of view, and by revisiting standard concepts in this light we obtain some surprising new results. First, we give several examples where effectivelyp simulations are possible between different propositional proof systems, but where psimulations are impossible (sometimes under complexity assumptions). Secondly, we prove that the rather weak proof system G0 for quantified propositional logic (QBF) can effectivelyp simulate any proof system for QBF. Thus our definition sheds new light on the comparative power of proof systems. We also give some evidence that with respect to Frege and Extended Frege systems, an effectivelyp simulation may not be possible. Lastly, we prove new relationships between effectivelyp simulations, automatizability, and the P versus NP question.