A. Kojevnikov - Academia.edu (original) (raw)
Papers by A. Kojevnikov
Journal of Mathematical Sciences, 2006
In a paper by , it is shown that any two classical Frege systems polynomially simulate each other... more In a paper by , it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles.
practice, 2007
... 6.1]. Page 12. Acknowledgments The author is very grateful to Dima Grigoriev, Jan Krajıcek an... more ... 6.1]. Page 12. Acknowledgments The author is very grateful to Dima Grigoriev, Jan Krajıcek and Alexander S. Kulikov for helpful comments and is indebted to Edward A. Hirsch for enlight-ening discussions. References 1. Krajıcek ...
Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of inter... more Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of interest. In this work, we show new constructions of cryptosystems based on group invariants and suggest methods to make such cryptosystems secure in practice. Cryptographers still cannot prove security in its cryptographic sense or even reduce it to some statement about regular complexity classes. In this paper we introduce a new notion of cryptographic security, a provable break, and prove that cryptosystems based on matrix group invariants, a variation of the Anshel-Anshel-Goldfeld key agreement protocol, and a non-commutative generalization of the Diffie-Hellman protocol for matrix groups are secure against provable worst-case break unless NP = RP.
St. Petersburg Mathematical Journal, 2009
Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of inter... more Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of interest. In this work, we show new constructions of cryptosystems based on group invariants and suggest methods to make such cryptosystems secure in practice. We do not know any proof of security in its cryptographic sense or even a reduction of it to a sensible statement about regular complexity classes. In this paper we introduce a new notion of cryptographic security, a provable break, and prove that cryptosystems based on matrix group invariants and also a variation of the Anshel-Anshel-Goldfeld key agreement protocol for modular groups are secure against provable worst-case break unless NP ⊆ RP.
The main criticism of known algebraic distributional NP (DistNP) complete problems is based on th... more The main criticism of known algebraic distributional NP (DistNP) complete problems is based on the fact that they contain too many specific relations to simulate a Turing machine. In this paper we present a construction of the semigroup with very few relations and word problem that is DistNP complete. Our construction follows Tseitin ideas . We modify original construction to work with words in standard binary presentation and arbitrary semigroups without any special conditions on its relations.
Lecture Notes in Computer Science, 2001
ABSTRACT In this paper we present a new randomized algorithm for SAT combining unit clause elimin... more ABSTRACT In this paper we present a new randomized algorithm for SAT combining unit clause elimination and local search. The algorithm is inspired by two randomized algorithms having the best current worst- case upper bounds ([9]and [11],[12]). Despite its simplicity, our algorithm performs well on many common benchmarks (we present results of its empirical evaluation). It is also probabilistically approximately complete.
Lecture Notes in Computer Science, 2006
We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin taut... more We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS + proof into Positivstellensatz proof of Grigoriev et al., extracting a "good" expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.
Lecture Notes in Computer Science, 2009
In this paper we report preliminary results of experiments with finding efficient circuits (over ... more In this paper we report preliminary results of experiments with finding efficient circuits (over binary bases) using SAT-solvers. We present upper bounds for functions with constant number of inputs as well as general upper bounds that were found automatically. We focus mainly on MOD-functions. Besides theoretical interest, these functions are also interesting from a practical point of view as they are the core of the residue number system. In particular, we present a circuit of size 3n + c over the full binary basis computing MOD n 3 .
Journal of Mathematical Sciences, 2007
We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin taut... more We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS + proof into Positivstellensatz proof of Grigoriev et al., extracting a "good" expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.
Information Processing Letters, 2010
In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean fun... more In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean functions. In particular, we describe circuits of size 4.5n + o(n) for any symmetric function of n variables, as well as circuits of size 3n for MOD n 3 function.
Annals of Pure and Applied Logic, 2006
We prove that the Cutting Plane proof system based on Gomory-Chvátal cuts polynomially simulates ... more We prove that the Cutting Plane proof system based on Gomory-Chvátal cuts polynomially simulates the lift-and-project system with integer coefficients written in unary. The restriction on the coefficients can be omitted when using Krajíček's cut-free Gentzenstyle extension of both systems. We also prove that Tseitin tautologies have short proofs in this extension (of any of these systems and with any coefficients). (E.A. Hirsch), arist@logic.pdmi.ras.ru (A. Kojevnikov).
Annals of Mathematics and Artificial Intelligence, 2005
... the current formula (which is trivial now) by the input formula, and starts a new period. ...... more ... the current formula (which is trivial now) by the input formula, and starts a new period. ... Method: For t := 1 to MAX_TRIES(F ) do A := random truth assignment for n variables; For p := 1 ... other local search algorithms because at each step it modifies the value of at most one variable. ...
Citeseer
We show that every formula over the basis {∧, ∨, ¬} for a function f :
We describe the basic notions and algorithm of the mixed boolean-algebraic solver being developed... more We describe the basic notions and algorithm of the mixed boolean-algebraic solver being developed in the Laboratory of Mathematical Logic of St.Petersburg Department of Steklov Institute of Mathematics. The solver solves formulas of the propositional logic and checks ...
"Zapiski Nauchnyh Seminarov POMI". VOL. 312 This issue is entitled &... more "Zapiski Nauchnyh Seminarov POMI". VOL. 312 This issue is entitled " Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods. Part 11 "; editor AM Vershik Contents: Preface.....7 (.ps.gz); I. Kantorovich ...
Journal of Mathematical Sciences, 2006
In a paper by , it is shown that any two classical Frege systems polynomially simulate each other... more In a paper by , it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles.
practice, 2007
... 6.1]. Page 12. Acknowledgments The author is very grateful to Dima Grigoriev, Jan Krajıcek an... more ... 6.1]. Page 12. Acknowledgments The author is very grateful to Dima Grigoriev, Jan Krajıcek and Alexander S. Kulikov for helpful comments and is indebted to Edward A. Hirsch for enlight-ening discussions. References 1. Krajıcek ...
Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of inter... more Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of interest. In this work, we show new constructions of cryptosystems based on group invariants and suggest methods to make such cryptosystems secure in practice. Cryptographers still cannot prove security in its cryptographic sense or even reduce it to some statement about regular complexity classes. In this paper we introduce a new notion of cryptographic security, a provable break, and prove that cryptosystems based on matrix group invariants, a variation of the Anshel-Anshel-Goldfeld key agreement protocol, and a non-commutative generalization of the Diffie-Hellman protocol for matrix groups are secure against provable worst-case break unless NP = RP.
St. Petersburg Mathematical Journal, 2009
Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of inter... more Cryptography based on noncommutative algebra still suffers from lack of schemes and lack of interest. In this work, we show new constructions of cryptosystems based on group invariants and suggest methods to make such cryptosystems secure in practice. We do not know any proof of security in its cryptographic sense or even a reduction of it to a sensible statement about regular complexity classes. In this paper we introduce a new notion of cryptographic security, a provable break, and prove that cryptosystems based on matrix group invariants and also a variation of the Anshel-Anshel-Goldfeld key agreement protocol for modular groups are secure against provable worst-case break unless NP ⊆ RP.
The main criticism of known algebraic distributional NP (DistNP) complete problems is based on th... more The main criticism of known algebraic distributional NP (DistNP) complete problems is based on the fact that they contain too many specific relations to simulate a Turing machine. In this paper we present a construction of the semigroup with very few relations and word problem that is DistNP complete. Our construction follows Tseitin ideas . We modify original construction to work with words in standard binary presentation and arbitrary semigroups without any special conditions on its relations.
Lecture Notes in Computer Science, 2001
ABSTRACT In this paper we present a new randomized algorithm for SAT combining unit clause elimin... more ABSTRACT In this paper we present a new randomized algorithm for SAT combining unit clause elimination and local search. The algorithm is inspired by two randomized algorithms having the best current worst- case upper bounds ([9]and [11],[12]). Despite its simplicity, our algorithm performs well on many common benchmarks (we present results of its empirical evaluation). It is also probabilistically approximately complete.
Lecture Notes in Computer Science, 2006
We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin taut... more We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS + proof into Positivstellensatz proof of Grigoriev et al., extracting a "good" expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.
Lecture Notes in Computer Science, 2009
In this paper we report preliminary results of experiments with finding efficient circuits (over ... more In this paper we report preliminary results of experiments with finding efficient circuits (over binary bases) using SAT-solvers. We present upper bounds for functions with constant number of inputs as well as general upper bounds that were found automatically. We focus mainly on MOD-functions. Besides theoretical interest, these functions are also interesting from a practical point of view as they are the core of the residue number system. In particular, we present a circuit of size 3n + c over the full binary basis computing MOD n 3 .
Journal of Mathematical Sciences, 2007
We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin taut... more We prove an exponential lower bound on the size of static Lovász-Schrijver proofs of Tseitin tautologies. We use several techniques, namely, translating static LS + proof into Positivstellensatz proof of Grigoriev et al., extracting a "good" expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.
Information Processing Letters, 2010
In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean fun... more In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean functions. In particular, we describe circuits of size 4.5n + o(n) for any symmetric function of n variables, as well as circuits of size 3n for MOD n 3 function.
Annals of Pure and Applied Logic, 2006
We prove that the Cutting Plane proof system based on Gomory-Chvátal cuts polynomially simulates ... more We prove that the Cutting Plane proof system based on Gomory-Chvátal cuts polynomially simulates the lift-and-project system with integer coefficients written in unary. The restriction on the coefficients can be omitted when using Krajíček's cut-free Gentzenstyle extension of both systems. We also prove that Tseitin tautologies have short proofs in this extension (of any of these systems and with any coefficients). (E.A. Hirsch), arist@logic.pdmi.ras.ru (A. Kojevnikov).
Annals of Mathematics and Artificial Intelligence, 2005
... the current formula (which is trivial now) by the input formula, and starts a new period. ...... more ... the current formula (which is trivial now) by the input formula, and starts a new period. ... Method: For t := 1 to MAX_TRIES(F ) do A := random truth assignment for n variables; For p := 1 ... other local search algorithms because at each step it modifies the value of at most one variable. ...
Citeseer
We show that every formula over the basis {∧, ∨, ¬} for a function f :
We describe the basic notions and algorithm of the mixed boolean-algebraic solver being developed... more We describe the basic notions and algorithm of the mixed boolean-algebraic solver being developed in the Laboratory of Mathematical Logic of St.Petersburg Department of Steklov Institute of Mathematics. The solver solves formulas of the propositional logic and checks ...
"Zapiski Nauchnyh Seminarov POMI". VOL. 312 This issue is entitled &... more "Zapiski Nauchnyh Seminarov POMI". VOL. 312 This issue is entitled " Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods. Part 11 "; editor AM Vershik Contents: Preface.....7 (.ps.gz); I. Kantorovich ...